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Research ArticleThe Johnson Equation Fredholm and WronskianRepresentations of Solutions and the Case of Order Three
Pierre Gaillard
Universite de Bourgogne Institut de Mathematiques de Bourgogne 9 avenue Alain Savary BP 47870 21078 Dijon Cedex France
Correspondence should be addressed to Pierre Gaillard pierregaillardu-bourgognefr
Received 10 November 2017 Accepted 8 May 2018 Published 1 August 2018
Academic Editor Giampaolo Cristadoro
Copyright copy 2018 PierreGaillardThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians oforder 2119873 giving solutions of order 119873 depending on 2119873 minus 1 parameters We obtain 119873 order rational solutions that can be writtenas a quotient of two polynomials of degree 2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2 parameters This methodgives an infinite hierarchy of solutions to the Johnson equation In particular rational solutions are obtainedThe solutions of order3 with 4 parameters are constructed and studied in detail by means of their modulus in the (119909 119910) plane in function of time 119905 andparameters 1198861 1198862 1198871 and 1198872
1 Introduction
The Johnson equation was introduced in 1980 by Johnson [1]to describe waves surfaces in shallow incompressible fluids[2 3] This equation was derived for internal waves in astratified medium [4] The Johnson equation is dissipativeit is well known that there is no solution with a linear frontlocalized along straight lines in the (119909 119910) planeThis Johnsonequation is for example able to explain the existence ofthe horseshoe-like solitons and multisoliton solutions quitenaturally
We consider the Johnson equation (J) in the followingnormalization
where as usual subscripts 119909 119910 and 119905mean partial derivativesThe first solutions were constructed in 1980 by Johnson
[1] Other types of solutions were found in [5] A newapproach to solve this equation was given in 1986 [6] bygiving a link between solutions of the Kadomtsev-Petviashvili(KP) [7] and solutions of the Johnson equation In 2007other types of solutions were obtained by using the Darbouxtransformation [8] More recently in 2013 other extensionshave been considered as the elliptic case [9]
Here we consider the famous Kadomtsev-Petviashvili(KPI) which can be written in the following form
(4119906119905 minus 6119906119906119909 + 119906119909119909119909)119909 minus 3119906119910119910 = 0 (2)
The KPI equation first appeared in 1970 [7] in a paper writtenby Kadomtsev and Petviashvili This equation is consideredas a model for surface and internal water waves [10] and innonlinear optics [11]
In the following wewill use theKPI equation to constructsolutions to the Johnson equation but in anotherway differentfrom this used in [6] Indeed these last authors consideranother representation of KPI equation given by
(119906119905 + 6119906119906119909 + 119906119909119909119909)119909 minus 3119906119910119910 = 0 (3)
and so the transformations between solutions of (3) and (1)are different from those we use to transform solutions to (2)in solutions to (1)
In fact to obtain solutions to (1) from solutions to (2) weuse the following transformation
In this paper we give solutions by means of Fredholmdeterminants of order 2119873 depending on 2119873 minus 1 parameters
HindawiAdvances in Mathematical PhysicsVolume 2018 Article ID 1642139 18 pageshttpsdoiorg10115520181642139
2 Advances in Mathematical Physics
and then by means of Wronskians of order 2119873 with 2119873 minus 1parameters So we construct an infinite hierarchy of solutionsto the Johnson equation depending on 2119873 minus 1 real parame-ters
New rational solutions depending a priori on 2119873 minus 2parameters at order 119873 are constructed when one parametertends to 0
We obtain families depending on 2119873 minus 2 parameters forthe119873th order as a ratio of two polynomials of degree 2119873(119873+1) in 119909 119905 and of degree 4119873(119873 + 1) in 119910
In this paper we construct only rational solutions oforder 3 depending on 4 real parameters we constructthe representations of their modulus in the plane of thecoordinates (119909 119910) according to the four real parameters 119886119894 and119887119894 for 1 le 119894 le 2 and time 1199052 Solutions to Johnson Equation Expressed byMeans of Fredholm Determinants
Some notations are given We define first real numbers 120582119895such that minus1 lt 120582] lt 1 ] = 1 2119873 they depend on aparameter 120598 and can be written as
120582119895 = 1 minus 212059821198952120582119873+119895 = minus120582119895
1 le 119895 le 119873(5)
Then we define 120581] 120575] 120574] and 119909119903] they are functions of120582] 1 le ] le 2119873 and are defined by the following formulas
120581119895 = 2radic1 minus 1205822119895120575119895 = 120581119895120582119895120574119895 = radic 1 minus 1205821198951 + 120582119895
119909119903119895 = (119903 minus 1) ln 120574119895 minus 119894120574119895 + 119894 119903 = 1 3120591119895 = minus121198941205822119895radic1 minus 1205822119895 minus 4119894 (1 minus 1205822119895)radic1 minus 1205822119895
(120574120578 + 120574]120574120578 minus 120574120583)sdot exp (119894120581]119909 minus 2120575]119910 + 120591]119905 + 119909119903] + 119890])
(17)
where 120581] 120575] 119909119903] 120574] 120591] 119890] and 120598] are defined in (6) (5) (7)and (8)
The connection between the solutions to the Johnsonequation and these to the KPI equation was already explainedin [6] but with another expression of the KPI equation (3)
Here the knowledge of a solution 119906 to the KPI equation(2) gives a solution to the Johnson equation (1) Let usconsider 119906(119909 119910 119905) a solution of the KPI equation (2) then thefunction
(1199091 1199101 1199051) (18)
for
1199091 = minus119894119909 minus 119894119910211990512 1199101 = 1199101199051199051 = 4119894119905
(19)
is a solution to the KPI equation (2) Using this crucialtransformation the solution to the Johnson equation takesthe form
119903 = det (120572119895119896119890119895 + 120573119895119896) (28)
where 120572119895119896 = (minus1)119896(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 1 le 119895 le 119873 1 le 119896 le 2119873120572119895119896 = (minus1)119896minus1(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 119873 + 1 le 119895 le 2119873 1 le 119896 le 2119873We have to calculate 119903 So we use the following lemma
Lemma 3 Let 119860 = (119886119894119895)119894119895isin[1119873] let 119861 = (119887119894119895)119894119895isin[1119873] andlet (119867119894119895)119894119895isin[1119873] be the matrix formed by replacing in 119860 thejth row of 119860 by the ith row of 119861 Then
det (119886119894119895119909119894 + 119887119894119895) = det (119886119894119895)times det(120575119894119895119909119894 + det (119867119894119895)
det (119886119894119895) ) (29)
Proof Let 119860 = (119886119895119894)119894119895isin[1119873] be the transposed matrix incofactors of 119860 Then 119860 times 119860 = det119860 times 119868
So det(119860) = (det(119860))119873minus1Then the general term of the product (119888119894119895)119894119895isin[1119873] =(119886119894119895119909119894 + 119887119894119895)119894119895isin[1119873] times (119886119895119894)119894119895isin[1119873] can be expressed by
119888119894119895 = 119873sum119904=1
(119886119894119904119909119894 + 119887119894119904) times 119886119895119904 = 119909119894 119899sum119904=1
119887119894119904119886119895119904= 120575119894119895 det (119860) 119909119894 + det (119867119894119895)
(30)
We obtain
det (119888119894119895) = det (119886119894119895119909119894 + 119887119894119895) times (det (119860))119873minus1= (det (119860))119873 times det(120575119894119895119909119894 + det (119867119894119895)
det (119860) ) (31)
So det(119886119894119895119909119894+119887119894119895) = det(119860)timesdet(120575119894119895119909119894+det(119867119894119895) det(119860))We use the notations 119880 = (120572119894119895)119894119895isin[12119873] and 119881 =(120573119894119895)119894119895isin[12119873]Using the preceding lemma we get
119903 = det (120572119894119895119890119894 + 120573119894119895)= det (120572119894119895) times det(120575119894119895119890119894 + det (119867119894119895)
det (120572119894119895) )= det (119880) times det(120575119894119895119890119894 + det (119867119894119895)
det (119880) ) (32)
where (119867119894119895)119894119895isin[1119873] is the matrix obtained by replacing in 119880the jth row of 119880 by the ith row of 119881 defined previously119880 is the classical Vandermonde determinant that is equalto
det (119880) = 119894119873(2119873minus1) prod2119873ge119897gt119898ge1
(120574119897 minus 120574119898) (33)
We have to compute det(119867119894119895) to evaluate the determinant 119903To do that we study two cases
(1) For 1 le 119895 le 119873 the matrix 119867119894119895 is a Vandermondematrix where the 119895th row of 119880 in 119880 is replaced by the 119894throw of 119881 Then we have
det (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (34)
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Thus we get
det (119867119894119895) = minus (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= minus (119894)119873(2119873minus1) times prod
times prod2119873ge119897gt119898ge1119897 =119895119898 =119895
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894)
(35)
To compute 119903 we have to simplify the quotient 119902119894119895 fldet(119867119894119895) det(119880)119902119894119895= (minus1)119895 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898)= (minus1)119895prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895) = (minus1)119895prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895)= minus prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(36)
119902119894119895 is equal to 119903119894119895 defined byminusprod119897 =119895(120574119897+120574119894)prod119897 =119894(120574119897minus120574119894) becausedet(120575119894119895119909119894 + det(119902119894119895) det(119860)) = det(120575119894119895119909119894 + det(119903119894119895) det(119860))
120574119897 + 120574119894120574119897 minus 120574119895) = 119888119894119895119890minus2119894Θ119903119894(0) (37)
with the notations given in (17)(2) We can do the same estimations for 119873 + 1 le 119895 le 2119873
are made det119867119894119895 is first as followsdet (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (38)
Advances in Mathematical Physics 5
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Then we get
det (119867119894119895) = (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= (119894)119873(2119873minus1) times prod
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894) (39)
Then 119902119894119895 fl det(119867119894119895) det(119880) can be expressed as
119902119894119895 = (minus1)119895minus1 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898) = (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895)= (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895) = prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(40)
119902119894119895 is replaced by 119903119894119895 defined byprod119897 =119895(120574119897 + 120574119894)prod119897 =119894(120574119897 minus 120574119894) forthe same reason as previously exposed
With (24) the following link between Fredholm determi-nants and Wronskians is obtained
det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)
6 Advances in Mathematical Physics
and
det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)
As Θ3119895(0) contains 119873 terms 1199093119895 1 le 119895 le 119873 and 119873 termsminus1199093119895 1 le 119895 le 119873 we have the relation 1198963(0) = 1198961(0) and weget the following theorem
is a solution of the Johnson equation which depends on 2119873 minus 1real parameters 119886119896 119887119896 and 120598 with 120601119903] defined in (21)
120601119903] = sin(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 1 le ] le 119873
120601119903] = cos(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 119873 + 1 le ] le 2119873 119903 = 1 3
(50)
where 120581] 120575] 119909119903] 120574] and 119890] are defined in (6) (5) and (7)
4 Study of the Limit Case When 120598 Tends to 041 Rational Solutions of Order 119873 Depending on 2119873 minus 2Parameters An infinite hierarchy of rational solutions tothe Johnson equation depending on 2119873 minus 2 parameters isobtained For this we take the limit when the parameter 120598tends to 0
We get the following statement
Theorem 5 The function V
V (119909 119910 119905) = lim120598997888rarr0
minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)
is a rational solution to the Johnson equation It is a quotient oftwo polynomials 119899(119909 119910 119905) and 119889(119909 119910 119905) depending on 2119873 minus 2real parameters 119886119895 and 119895 1 le 119895 le 119873minus1 of degrees 2119873(119873+1)in 119909 119905 and 4119873(119873 + 1) in 11991042 Families of Rational Solutions of Order 3 Depending on 4Parameters Here we construct families of rational solutionsto the Johnson equation of order 3 explicitly they depend on4 parameters
We only give the expression without parameters and wegive it in the appendix because of the length of the solutions
We construct the patterns of the modulus of the solutionsin the plane (119909 119910) of coordinates in functions of parameters119886119894 119887 1 le 119894 le 2 and time 119905
The role of the parameters 119886119894 and 119887119894 for the same integer119894 is the same one one will be interested primarily only inparameters 119886119894
The study of these configurationsmakes it possible to givethe following conclusions The variation of the configurationof the module of the solutions is very fast according to time119905 When time 119905 grows from 0 to 0 01 one passes froma rectilinear structure with a height of 98 to a horseshoestructure with a maximum height equal to 4 The role playedby the parameters 119886119894 and 119887119894 is the same for same index 119894Whenvariables 119909 119910 and time tend towards infinity the modulus ofthe solutions tends towards 2 in accordancewith the structureof the polynomials which will be studied in a forthcomingarticle
5 Conclusion
We have constructed solutions to the Johnson equationstarting from the solutions of the KPI equation what makesit possible to obtain rational solutions These solutions areexpressed bymeans of quotients of two polynomials of degree2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2parameters
Here we have given a new method to construct solutionsto the Johnson equation related to previous results [12ndash14]
We have given two types of representations of the solu-tions to the Johnson equation An expression by means ofFredholm determinants of order 2119873 depending on 2119873 minus 1real parameters is given Another expression by means ofWronskians of order 2119873 depending on 2119873minus1 real parametersis also constructed Also rational solutions to the Johnsonequation depending on 2119873 minus 2 real parameters are obtainedwhen one of parameters (120598) tends to zero
The patterns of the modulus of the solutions in the plane(119909 119910) and their evolution according to time and parametershave been studied in Figures 1 2 3 4 and 5
In another study we will give a more general representa-tion of rational solutions to the Johnson equation It can bewritten without limit at order 119873 depending on 2119873 minus 2 realparameters We will prove that these solutions can be writtenas a quotient of polynomials of degree 2119873(119873 + 1) in 119909 119905 and4119873(119873 + 1) in 119910Appendix
The solutions to the Johnson equation can be written as
Figure 1 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and1198872 = 0 and on the right for 119905 = 0 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0
Figure 2 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and1198872 = 106 and on the right for 119905 = 0 01 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0a12 = 8916100448256a11 = 89161004482561199051199102 + 2567836929097728119905a10
Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
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2 Advances in Mathematical Physics
and then by means of Wronskians of order 2119873 with 2119873 minus 1parameters So we construct an infinite hierarchy of solutionsto the Johnson equation depending on 2119873 minus 1 real parame-ters
New rational solutions depending a priori on 2119873 minus 2parameters at order 119873 are constructed when one parametertends to 0
We obtain families depending on 2119873 minus 2 parameters forthe119873th order as a ratio of two polynomials of degree 2119873(119873+1) in 119909 119905 and of degree 4119873(119873 + 1) in 119910
In this paper we construct only rational solutions oforder 3 depending on 4 real parameters we constructthe representations of their modulus in the plane of thecoordinates (119909 119910) according to the four real parameters 119886119894 and119887119894 for 1 le 119894 le 2 and time 1199052 Solutions to Johnson Equation Expressed byMeans of Fredholm Determinants
Some notations are given We define first real numbers 120582119895such that minus1 lt 120582] lt 1 ] = 1 2119873 they depend on aparameter 120598 and can be written as
120582119895 = 1 minus 212059821198952120582119873+119895 = minus120582119895
1 le 119895 le 119873(5)
Then we define 120581] 120575] 120574] and 119909119903] they are functions of120582] 1 le ] le 2119873 and are defined by the following formulas
120581119895 = 2radic1 minus 1205822119895120575119895 = 120581119895120582119895120574119895 = radic 1 minus 1205821198951 + 120582119895
119909119903119895 = (119903 minus 1) ln 120574119895 minus 119894120574119895 + 119894 119903 = 1 3120591119895 = minus121198941205822119895radic1 minus 1205822119895 minus 4119894 (1 minus 1205822119895)radic1 minus 1205822119895
(120574120578 + 120574]120574120578 minus 120574120583)sdot exp (119894120581]119909 minus 2120575]119910 + 120591]119905 + 119909119903] + 119890])
(17)
where 120581] 120575] 119909119903] 120574] 120591] 119890] and 120598] are defined in (6) (5) (7)and (8)
The connection between the solutions to the Johnsonequation and these to the KPI equation was already explainedin [6] but with another expression of the KPI equation (3)
Here the knowledge of a solution 119906 to the KPI equation(2) gives a solution to the Johnson equation (1) Let usconsider 119906(119909 119910 119905) a solution of the KPI equation (2) then thefunction
(1199091 1199101 1199051) (18)
for
1199091 = minus119894119909 minus 119894119910211990512 1199101 = 1199101199051199051 = 4119894119905
(19)
is a solution to the KPI equation (2) Using this crucialtransformation the solution to the Johnson equation takesthe form
119903 = det (120572119895119896119890119895 + 120573119895119896) (28)
where 120572119895119896 = (minus1)119896(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 1 le 119895 le 119873 1 le 119896 le 2119873120572119895119896 = (minus1)119896minus1(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 119873 + 1 le 119895 le 2119873 1 le 119896 le 2119873We have to calculate 119903 So we use the following lemma
Lemma 3 Let 119860 = (119886119894119895)119894119895isin[1119873] let 119861 = (119887119894119895)119894119895isin[1119873] andlet (119867119894119895)119894119895isin[1119873] be the matrix formed by replacing in 119860 thejth row of 119860 by the ith row of 119861 Then
det (119886119894119895119909119894 + 119887119894119895) = det (119886119894119895)times det(120575119894119895119909119894 + det (119867119894119895)
det (119886119894119895) ) (29)
Proof Let 119860 = (119886119895119894)119894119895isin[1119873] be the transposed matrix incofactors of 119860 Then 119860 times 119860 = det119860 times 119868
So det(119860) = (det(119860))119873minus1Then the general term of the product (119888119894119895)119894119895isin[1119873] =(119886119894119895119909119894 + 119887119894119895)119894119895isin[1119873] times (119886119895119894)119894119895isin[1119873] can be expressed by
119888119894119895 = 119873sum119904=1
(119886119894119904119909119894 + 119887119894119904) times 119886119895119904 = 119909119894 119899sum119904=1
119887119894119904119886119895119904= 120575119894119895 det (119860) 119909119894 + det (119867119894119895)
(30)
We obtain
det (119888119894119895) = det (119886119894119895119909119894 + 119887119894119895) times (det (119860))119873minus1= (det (119860))119873 times det(120575119894119895119909119894 + det (119867119894119895)
det (119860) ) (31)
So det(119886119894119895119909119894+119887119894119895) = det(119860)timesdet(120575119894119895119909119894+det(119867119894119895) det(119860))We use the notations 119880 = (120572119894119895)119894119895isin[12119873] and 119881 =(120573119894119895)119894119895isin[12119873]Using the preceding lemma we get
119903 = det (120572119894119895119890119894 + 120573119894119895)= det (120572119894119895) times det(120575119894119895119890119894 + det (119867119894119895)
det (120572119894119895) )= det (119880) times det(120575119894119895119890119894 + det (119867119894119895)
det (119880) ) (32)
where (119867119894119895)119894119895isin[1119873] is the matrix obtained by replacing in 119880the jth row of 119880 by the ith row of 119881 defined previously119880 is the classical Vandermonde determinant that is equalto
det (119880) = 119894119873(2119873minus1) prod2119873ge119897gt119898ge1
(120574119897 minus 120574119898) (33)
We have to compute det(119867119894119895) to evaluate the determinant 119903To do that we study two cases
(1) For 1 le 119895 le 119873 the matrix 119867119894119895 is a Vandermondematrix where the 119895th row of 119880 in 119880 is replaced by the 119894throw of 119881 Then we have
det (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (34)
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Thus we get
det (119867119894119895) = minus (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= minus (119894)119873(2119873minus1) times prod
times prod2119873ge119897gt119898ge1119897 =119895119898 =119895
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894)
(35)
To compute 119903 we have to simplify the quotient 119902119894119895 fldet(119867119894119895) det(119880)119902119894119895= (minus1)119895 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898)= (minus1)119895prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895) = (minus1)119895prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895)= minus prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(36)
119902119894119895 is equal to 119903119894119895 defined byminusprod119897 =119895(120574119897+120574119894)prod119897 =119894(120574119897minus120574119894) becausedet(120575119894119895119909119894 + det(119902119894119895) det(119860)) = det(120575119894119895119909119894 + det(119903119894119895) det(119860))
120574119897 + 120574119894120574119897 minus 120574119895) = 119888119894119895119890minus2119894Θ119903119894(0) (37)
with the notations given in (17)(2) We can do the same estimations for 119873 + 1 le 119895 le 2119873
are made det119867119894119895 is first as followsdet (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (38)
Advances in Mathematical Physics 5
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Then we get
det (119867119894119895) = (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= (119894)119873(2119873minus1) times prod
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894) (39)
Then 119902119894119895 fl det(119867119894119895) det(119880) can be expressed as
119902119894119895 = (minus1)119895minus1 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898) = (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895)= (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895) = prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(40)
119902119894119895 is replaced by 119903119894119895 defined byprod119897 =119895(120574119897 + 120574119894)prod119897 =119894(120574119897 minus 120574119894) forthe same reason as previously exposed
With (24) the following link between Fredholm determi-nants and Wronskians is obtained
det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)
6 Advances in Mathematical Physics
and
det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)
As Θ3119895(0) contains 119873 terms 1199093119895 1 le 119895 le 119873 and 119873 termsminus1199093119895 1 le 119895 le 119873 we have the relation 1198963(0) = 1198961(0) and weget the following theorem
is a solution of the Johnson equation which depends on 2119873 minus 1real parameters 119886119896 119887119896 and 120598 with 120601119903] defined in (21)
120601119903] = sin(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 1 le ] le 119873
120601119903] = cos(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 119873 + 1 le ] le 2119873 119903 = 1 3
(50)
where 120581] 120575] 119909119903] 120574] and 119890] are defined in (6) (5) and (7)
4 Study of the Limit Case When 120598 Tends to 041 Rational Solutions of Order 119873 Depending on 2119873 minus 2Parameters An infinite hierarchy of rational solutions tothe Johnson equation depending on 2119873 minus 2 parameters isobtained For this we take the limit when the parameter 120598tends to 0
We get the following statement
Theorem 5 The function V
V (119909 119910 119905) = lim120598997888rarr0
minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)
is a rational solution to the Johnson equation It is a quotient oftwo polynomials 119899(119909 119910 119905) and 119889(119909 119910 119905) depending on 2119873 minus 2real parameters 119886119895 and 119895 1 le 119895 le 119873minus1 of degrees 2119873(119873+1)in 119909 119905 and 4119873(119873 + 1) in 11991042 Families of Rational Solutions of Order 3 Depending on 4Parameters Here we construct families of rational solutionsto the Johnson equation of order 3 explicitly they depend on4 parameters
We only give the expression without parameters and wegive it in the appendix because of the length of the solutions
We construct the patterns of the modulus of the solutionsin the plane (119909 119910) of coordinates in functions of parameters119886119894 119887 1 le 119894 le 2 and time 119905
The role of the parameters 119886119894 and 119887119894 for the same integer119894 is the same one one will be interested primarily only inparameters 119886119894
The study of these configurationsmakes it possible to givethe following conclusions The variation of the configurationof the module of the solutions is very fast according to time119905 When time 119905 grows from 0 to 0 01 one passes froma rectilinear structure with a height of 98 to a horseshoestructure with a maximum height equal to 4 The role playedby the parameters 119886119894 and 119887119894 is the same for same index 119894Whenvariables 119909 119910 and time tend towards infinity the modulus ofthe solutions tends towards 2 in accordancewith the structureof the polynomials which will be studied in a forthcomingarticle
5 Conclusion
We have constructed solutions to the Johnson equationstarting from the solutions of the KPI equation what makesit possible to obtain rational solutions These solutions areexpressed bymeans of quotients of two polynomials of degree2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2parameters
Here we have given a new method to construct solutionsto the Johnson equation related to previous results [12ndash14]
We have given two types of representations of the solu-tions to the Johnson equation An expression by means ofFredholm determinants of order 2119873 depending on 2119873 minus 1real parameters is given Another expression by means ofWronskians of order 2119873 depending on 2119873minus1 real parametersis also constructed Also rational solutions to the Johnsonequation depending on 2119873 minus 2 real parameters are obtainedwhen one of parameters (120598) tends to zero
The patterns of the modulus of the solutions in the plane(119909 119910) and their evolution according to time and parametershave been studied in Figures 1 2 3 4 and 5
In another study we will give a more general representa-tion of rational solutions to the Johnson equation It can bewritten without limit at order 119873 depending on 2119873 minus 2 realparameters We will prove that these solutions can be writtenas a quotient of polynomials of degree 2119873(119873 + 1) in 119909 119905 and4119873(119873 + 1) in 119910Appendix
The solutions to the Johnson equation can be written as
Figure 1 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and1198872 = 0 and on the right for 119905 = 0 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0
Figure 2 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and1198872 = 106 and on the right for 119905 = 0 01 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0a12 = 8916100448256a11 = 89161004482561199051199102 + 2567836929097728119905a10
Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
(120574120578 + 120574]120574120578 minus 120574120583)sdot exp (119894120581]119909 minus 2120575]119910 + 120591]119905 + 119909119903] + 119890])
(17)
where 120581] 120575] 119909119903] 120574] 120591] 119890] and 120598] are defined in (6) (5) (7)and (8)
The connection between the solutions to the Johnsonequation and these to the KPI equation was already explainedin [6] but with another expression of the KPI equation (3)
Here the knowledge of a solution 119906 to the KPI equation(2) gives a solution to the Johnson equation (1) Let usconsider 119906(119909 119910 119905) a solution of the KPI equation (2) then thefunction
(1199091 1199101 1199051) (18)
for
1199091 = minus119894119909 minus 119894119910211990512 1199101 = 1199101199051199051 = 4119894119905
(19)
is a solution to the KPI equation (2) Using this crucialtransformation the solution to the Johnson equation takesthe form
119903 = det (120572119895119896119890119895 + 120573119895119896) (28)
where 120572119895119896 = (minus1)119896(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 1 le 119895 le 119873 1 le 119896 le 2119873120572119895119896 = (minus1)119896minus1(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 119873 + 1 le 119895 le 2119873 1 le 119896 le 2119873We have to calculate 119903 So we use the following lemma
Lemma 3 Let 119860 = (119886119894119895)119894119895isin[1119873] let 119861 = (119887119894119895)119894119895isin[1119873] andlet (119867119894119895)119894119895isin[1119873] be the matrix formed by replacing in 119860 thejth row of 119860 by the ith row of 119861 Then
det (119886119894119895119909119894 + 119887119894119895) = det (119886119894119895)times det(120575119894119895119909119894 + det (119867119894119895)
det (119886119894119895) ) (29)
Proof Let 119860 = (119886119895119894)119894119895isin[1119873] be the transposed matrix incofactors of 119860 Then 119860 times 119860 = det119860 times 119868
So det(119860) = (det(119860))119873minus1Then the general term of the product (119888119894119895)119894119895isin[1119873] =(119886119894119895119909119894 + 119887119894119895)119894119895isin[1119873] times (119886119895119894)119894119895isin[1119873] can be expressed by
119888119894119895 = 119873sum119904=1
(119886119894119904119909119894 + 119887119894119904) times 119886119895119904 = 119909119894 119899sum119904=1
119887119894119904119886119895119904= 120575119894119895 det (119860) 119909119894 + det (119867119894119895)
(30)
We obtain
det (119888119894119895) = det (119886119894119895119909119894 + 119887119894119895) times (det (119860))119873minus1= (det (119860))119873 times det(120575119894119895119909119894 + det (119867119894119895)
det (119860) ) (31)
So det(119886119894119895119909119894+119887119894119895) = det(119860)timesdet(120575119894119895119909119894+det(119867119894119895) det(119860))We use the notations 119880 = (120572119894119895)119894119895isin[12119873] and 119881 =(120573119894119895)119894119895isin[12119873]Using the preceding lemma we get
119903 = det (120572119894119895119890119894 + 120573119894119895)= det (120572119894119895) times det(120575119894119895119890119894 + det (119867119894119895)
det (120572119894119895) )= det (119880) times det(120575119894119895119890119894 + det (119867119894119895)
det (119880) ) (32)
where (119867119894119895)119894119895isin[1119873] is the matrix obtained by replacing in 119880the jth row of 119880 by the ith row of 119881 defined previously119880 is the classical Vandermonde determinant that is equalto
det (119880) = 119894119873(2119873minus1) prod2119873ge119897gt119898ge1
(120574119897 minus 120574119898) (33)
We have to compute det(119867119894119895) to evaluate the determinant 119903To do that we study two cases
(1) For 1 le 119895 le 119873 the matrix 119867119894119895 is a Vandermondematrix where the 119895th row of 119880 in 119880 is replaced by the 119894throw of 119881 Then we have
det (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (34)
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Thus we get
det (119867119894119895) = minus (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= minus (119894)119873(2119873minus1) times prod
times prod2119873ge119897gt119898ge1119897 =119895119898 =119895
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894)
(35)
To compute 119903 we have to simplify the quotient 119902119894119895 fldet(119867119894119895) det(119880)119902119894119895= (minus1)119895 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898)= (minus1)119895prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895) = (minus1)119895prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895)= minus prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(36)
119902119894119895 is equal to 119903119894119895 defined byminusprod119897 =119895(120574119897+120574119894)prod119897 =119894(120574119897minus120574119894) becausedet(120575119894119895119909119894 + det(119902119894119895) det(119860)) = det(120575119894119895119909119894 + det(119903119894119895) det(119860))
120574119897 + 120574119894120574119897 minus 120574119895) = 119888119894119895119890minus2119894Θ119903119894(0) (37)
with the notations given in (17)(2) We can do the same estimations for 119873 + 1 le 119895 le 2119873
are made det119867119894119895 is first as followsdet (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (38)
Advances in Mathematical Physics 5
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Then we get
det (119867119894119895) = (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= (119894)119873(2119873minus1) times prod
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894) (39)
Then 119902119894119895 fl det(119867119894119895) det(119880) can be expressed as
119902119894119895 = (minus1)119895minus1 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898) = (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895)= (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895) = prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(40)
119902119894119895 is replaced by 119903119894119895 defined byprod119897 =119895(120574119897 + 120574119894)prod119897 =119894(120574119897 minus 120574119894) forthe same reason as previously exposed
With (24) the following link between Fredholm determi-nants and Wronskians is obtained
det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)
6 Advances in Mathematical Physics
and
det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)
As Θ3119895(0) contains 119873 terms 1199093119895 1 le 119895 le 119873 and 119873 termsminus1199093119895 1 le 119895 le 119873 we have the relation 1198963(0) = 1198961(0) and weget the following theorem
is a solution of the Johnson equation which depends on 2119873 minus 1real parameters 119886119896 119887119896 and 120598 with 120601119903] defined in (21)
120601119903] = sin(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 1 le ] le 119873
120601119903] = cos(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 119873 + 1 le ] le 2119873 119903 = 1 3
(50)
where 120581] 120575] 119909119903] 120574] and 119890] are defined in (6) (5) and (7)
4 Study of the Limit Case When 120598 Tends to 041 Rational Solutions of Order 119873 Depending on 2119873 minus 2Parameters An infinite hierarchy of rational solutions tothe Johnson equation depending on 2119873 minus 2 parameters isobtained For this we take the limit when the parameter 120598tends to 0
We get the following statement
Theorem 5 The function V
V (119909 119910 119905) = lim120598997888rarr0
minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)
is a rational solution to the Johnson equation It is a quotient oftwo polynomials 119899(119909 119910 119905) and 119889(119909 119910 119905) depending on 2119873 minus 2real parameters 119886119895 and 119895 1 le 119895 le 119873minus1 of degrees 2119873(119873+1)in 119909 119905 and 4119873(119873 + 1) in 11991042 Families of Rational Solutions of Order 3 Depending on 4Parameters Here we construct families of rational solutionsto the Johnson equation of order 3 explicitly they depend on4 parameters
We only give the expression without parameters and wegive it in the appendix because of the length of the solutions
We construct the patterns of the modulus of the solutionsin the plane (119909 119910) of coordinates in functions of parameters119886119894 119887 1 le 119894 le 2 and time 119905
The role of the parameters 119886119894 and 119887119894 for the same integer119894 is the same one one will be interested primarily only inparameters 119886119894
The study of these configurationsmakes it possible to givethe following conclusions The variation of the configurationof the module of the solutions is very fast according to time119905 When time 119905 grows from 0 to 0 01 one passes froma rectilinear structure with a height of 98 to a horseshoestructure with a maximum height equal to 4 The role playedby the parameters 119886119894 and 119887119894 is the same for same index 119894Whenvariables 119909 119910 and time tend towards infinity the modulus ofthe solutions tends towards 2 in accordancewith the structureof the polynomials which will be studied in a forthcomingarticle
5 Conclusion
We have constructed solutions to the Johnson equationstarting from the solutions of the KPI equation what makesit possible to obtain rational solutions These solutions areexpressed bymeans of quotients of two polynomials of degree2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2parameters
Here we have given a new method to construct solutionsto the Johnson equation related to previous results [12ndash14]
We have given two types of representations of the solu-tions to the Johnson equation An expression by means ofFredholm determinants of order 2119873 depending on 2119873 minus 1real parameters is given Another expression by means ofWronskians of order 2119873 depending on 2119873minus1 real parametersis also constructed Also rational solutions to the Johnsonequation depending on 2119873 minus 2 real parameters are obtainedwhen one of parameters (120598) tends to zero
The patterns of the modulus of the solutions in the plane(119909 119910) and their evolution according to time and parametershave been studied in Figures 1 2 3 4 and 5
In another study we will give a more general representa-tion of rational solutions to the Johnson equation It can bewritten without limit at order 119873 depending on 2119873 minus 2 realparameters We will prove that these solutions can be writtenas a quotient of polynomials of degree 2119873(119873 + 1) in 119909 119905 and4119873(119873 + 1) in 119910Appendix
The solutions to the Johnson equation can be written as
Figure 1 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and1198872 = 0 and on the right for 119905 = 0 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0
Figure 2 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and1198872 = 106 and on the right for 119905 = 0 01 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0a12 = 8916100448256a11 = 89161004482561199051199102 + 2567836929097728119905a10
Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
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4 Advances in Mathematical Physics
The determinant 119903 can be written as
119903 = det (120572119895119896119890119895 + 120573119895119896) (28)
where 120572119895119896 = (minus1)119896(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 1 le 119895 le 119873 1 le 119896 le 2119873120572119895119896 = (minus1)119896minus1(119894120574119895)119896minus1 119890119895 = 119890minus2119894Θ119903119895 and 120573119895119896 =(119894120574119895)119896minus1 119873 + 1 le 119895 le 2119873 1 le 119896 le 2119873We have to calculate 119903 So we use the following lemma
Lemma 3 Let 119860 = (119886119894119895)119894119895isin[1119873] let 119861 = (119887119894119895)119894119895isin[1119873] andlet (119867119894119895)119894119895isin[1119873] be the matrix formed by replacing in 119860 thejth row of 119860 by the ith row of 119861 Then
det (119886119894119895119909119894 + 119887119894119895) = det (119886119894119895)times det(120575119894119895119909119894 + det (119867119894119895)
det (119886119894119895) ) (29)
Proof Let 119860 = (119886119895119894)119894119895isin[1119873] be the transposed matrix incofactors of 119860 Then 119860 times 119860 = det119860 times 119868
So det(119860) = (det(119860))119873minus1Then the general term of the product (119888119894119895)119894119895isin[1119873] =(119886119894119895119909119894 + 119887119894119895)119894119895isin[1119873] times (119886119895119894)119894119895isin[1119873] can be expressed by
119888119894119895 = 119873sum119904=1
(119886119894119904119909119894 + 119887119894119904) times 119886119895119904 = 119909119894 119899sum119904=1
119887119894119904119886119895119904= 120575119894119895 det (119860) 119909119894 + det (119867119894119895)
(30)
We obtain
det (119888119894119895) = det (119886119894119895119909119894 + 119887119894119895) times (det (119860))119873minus1= (det (119860))119873 times det(120575119894119895119909119894 + det (119867119894119895)
det (119860) ) (31)
So det(119886119894119895119909119894+119887119894119895) = det(119860)timesdet(120575119894119895119909119894+det(119867119894119895) det(119860))We use the notations 119880 = (120572119894119895)119894119895isin[12119873] and 119881 =(120573119894119895)119894119895isin[12119873]Using the preceding lemma we get
119903 = det (120572119894119895119890119894 + 120573119894119895)= det (120572119894119895) times det(120575119894119895119890119894 + det (119867119894119895)
det (120572119894119895) )= det (119880) times det(120575119894119895119890119894 + det (119867119894119895)
det (119880) ) (32)
where (119867119894119895)119894119895isin[1119873] is the matrix obtained by replacing in 119880the jth row of 119880 by the ith row of 119881 defined previously119880 is the classical Vandermonde determinant that is equalto
det (119880) = 119894119873(2119873minus1) prod2119873ge119897gt119898ge1
(120574119897 minus 120574119898) (33)
We have to compute det(119867119894119895) to evaluate the determinant 119903To do that we study two cases
(1) For 1 le 119895 le 119873 the matrix 119867119894119895 is a Vandermondematrix where the 119895th row of 119880 in 119880 is replaced by the 119894throw of 119881 Then we have
det (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (34)
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Thus we get
det (119867119894119895) = minus (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= minus (119894)119873(2119873minus1) times prod
times prod2119873ge119897gt119898ge1119897 =119895119898 =119895
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894)
(35)
To compute 119903 we have to simplify the quotient 119902119894119895 fldet(119867119894119895) det(119880)119902119894119895= (minus1)119895 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898)= (minus1)119895prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895) = (minus1)119895prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895)= minus prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(36)
119902119894119895 is equal to 119903119894119895 defined byminusprod119897 =119895(120574119897+120574119894)prod119897 =119894(120574119897minus120574119894) becausedet(120575119894119895119909119894 + det(119902119894119895) det(119860)) = det(120575119894119895119909119894 + det(119903119894119895) det(119860))
120574119897 + 120574119894120574119897 minus 120574119895) = 119888119894119895119890minus2119894Θ119903119894(0) (37)
with the notations given in (17)(2) We can do the same estimations for 119873 + 1 le 119895 le 2119873
are made det119867119894119895 is first as followsdet (119867119894119895) = (minus1)119873(2119873+1)+119873minus1 (119894)119873(2119873minus1) times 119872 (38)
Advances in Mathematical Physics 5
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Then we get
det (119867119894119895) = (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= (119894)119873(2119873minus1) times prod
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894) (39)
Then 119902119894119895 fl det(119867119894119895) det(119880) can be expressed as
119902119894119895 = (minus1)119895minus1 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898) = (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895)= (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895) = prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(40)
119902119894119895 is replaced by 119903119894119895 defined byprod119897 =119895(120574119897 + 120574119894)prod119897 =119894(120574119897 minus 120574119894) forthe same reason as previously exposed
With (24) the following link between Fredholm determi-nants and Wronskians is obtained
det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)
6 Advances in Mathematical Physics
and
det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)
As Θ3119895(0) contains 119873 terms 1199093119895 1 le 119895 le 119873 and 119873 termsminus1199093119895 1 le 119895 le 119873 we have the relation 1198963(0) = 1198961(0) and weget the following theorem
is a solution of the Johnson equation which depends on 2119873 minus 1real parameters 119886119896 119887119896 and 120598 with 120601119903] defined in (21)
120601119903] = sin(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 1 le ] le 119873
120601119903] = cos(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 119873 + 1 le ] le 2119873 119903 = 1 3
(50)
where 120581] 120575] 119909119903] 120574] and 119890] are defined in (6) (5) and (7)
4 Study of the Limit Case When 120598 Tends to 041 Rational Solutions of Order 119873 Depending on 2119873 minus 2Parameters An infinite hierarchy of rational solutions tothe Johnson equation depending on 2119873 minus 2 parameters isobtained For this we take the limit when the parameter 120598tends to 0
We get the following statement
Theorem 5 The function V
V (119909 119910 119905) = lim120598997888rarr0
minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)
is a rational solution to the Johnson equation It is a quotient oftwo polynomials 119899(119909 119910 119905) and 119889(119909 119910 119905) depending on 2119873 minus 2real parameters 119886119895 and 119895 1 le 119895 le 119873minus1 of degrees 2119873(119873+1)in 119909 119905 and 4119873(119873 + 1) in 11991042 Families of Rational Solutions of Order 3 Depending on 4Parameters Here we construct families of rational solutionsto the Johnson equation of order 3 explicitly they depend on4 parameters
We only give the expression without parameters and wegive it in the appendix because of the length of the solutions
We construct the patterns of the modulus of the solutionsin the plane (119909 119910) of coordinates in functions of parameters119886119894 119887 1 le 119894 le 2 and time 119905
The role of the parameters 119886119894 and 119887119894 for the same integer119894 is the same one one will be interested primarily only inparameters 119886119894
The study of these configurationsmakes it possible to givethe following conclusions The variation of the configurationof the module of the solutions is very fast according to time119905 When time 119905 grows from 0 to 0 01 one passes froma rectilinear structure with a height of 98 to a horseshoestructure with a maximum height equal to 4 The role playedby the parameters 119886119894 and 119887119894 is the same for same index 119894Whenvariables 119909 119910 and time tend towards infinity the modulus ofthe solutions tends towards 2 in accordancewith the structureof the polynomials which will be studied in a forthcomingarticle
5 Conclusion
We have constructed solutions to the Johnson equationstarting from the solutions of the KPI equation what makesit possible to obtain rational solutions These solutions areexpressed bymeans of quotients of two polynomials of degree2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2parameters
Here we have given a new method to construct solutionsto the Johnson equation related to previous results [12ndash14]
We have given two types of representations of the solu-tions to the Johnson equation An expression by means ofFredholm determinants of order 2119873 depending on 2119873 minus 1real parameters is given Another expression by means ofWronskians of order 2119873 depending on 2119873minus1 real parametersis also constructed Also rational solutions to the Johnsonequation depending on 2119873 minus 2 real parameters are obtainedwhen one of parameters (120598) tends to zero
The patterns of the modulus of the solutions in the plane(119909 119910) and their evolution according to time and parametershave been studied in Figures 1 2 3 4 and 5
In another study we will give a more general representa-tion of rational solutions to the Johnson equation It can bewritten without limit at order 119873 depending on 2119873 minus 2 realparameters We will prove that these solutions can be writtenas a quotient of polynomials of degree 2119873(119873 + 1) in 119909 119905 and4119873(119873 + 1) in 119910Appendix
The solutions to the Johnson equation can be written as
Figure 1 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and1198872 = 0 and on the right for 119905 = 0 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0
Figure 2 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and1198872 = 106 and on the right for 119905 = 0 01 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0a12 = 8916100448256a11 = 89161004482561199051199102 + 2567836929097728119905a10
Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Advances in Mathematical Physics 5
with119872 = 119872(1198981 1198982119873) being the determinant defined by119898119896 = 120574119896 for 119896 = 119895 and119898119895 = minus120574119894 Then we get
det (119867119894119895) = (119894)119873(2119873minus1) times prod2119873ge119897gt119896ge1
(119898119897 minus 119898119896)= (119894)119873(2119873minus1) times prod
(120574119897 minus 120574119898)times prod119897 =119895
(120574119897 + 120574119894) (39)
Then 119902119894119895 fl det(119867119894119895) det(119880) can be expressed as
119902119894119895 = (minus1)119895minus1 (119894)119873(2119873minus1) times prod2119873ge119897gt119898ge1119897 =119895119898 =119895 (120574119897 minus 120574119898) times prod119897 =119895 (120574119897 + 120574119894)119894119873(2119873minus1)prod2119873ge119897gt119898ge1 (120574119897 minus 120574119898) = (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)prod119897lt119895 (120574119895 minus 120574119897)prod119897gt119895 (120574119897 minus 120574119895)= (minus1)119895minus1prod119897 =119895 (120574119897 + 120574119894)(minus1)119895minus1prod119897 =119895 (120574119897 minus 120574119895) = prod119897 =119895 (120574119897 + 120574119894)prod119897 =119895 (120574119897 minus 120574119895)
(40)
119902119894119895 is replaced by 119903119894119895 defined byprod119897 =119895(120574119897 + 120574119894)prod119897 =119894(120574119897 minus 120574119894) forthe same reason as previously exposed
With (24) the following link between Fredholm determi-nants and Wronskians is obtained
det (119868 + 1198633) = 1198963 (0) times 1198823 (1206011199031 1206011199032119873) (0) (47)
6 Advances in Mathematical Physics
and
det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)
As Θ3119895(0) contains 119873 terms 1199093119895 1 le 119895 le 119873 and 119873 termsminus1199093119895 1 le 119895 le 119873 we have the relation 1198963(0) = 1198961(0) and weget the following theorem
is a solution of the Johnson equation which depends on 2119873 minus 1real parameters 119886119896 119887119896 and 120598 with 120601119903] defined in (21)
120601119903] = sin(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 1 le ] le 119873
120601119903] = cos(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 119873 + 1 le ] le 2119873 119903 = 1 3
(50)
where 120581] 120575] 119909119903] 120574] and 119890] are defined in (6) (5) and (7)
4 Study of the Limit Case When 120598 Tends to 041 Rational Solutions of Order 119873 Depending on 2119873 minus 2Parameters An infinite hierarchy of rational solutions tothe Johnson equation depending on 2119873 minus 2 parameters isobtained For this we take the limit when the parameter 120598tends to 0
We get the following statement
Theorem 5 The function V
V (119909 119910 119905) = lim120598997888rarr0
minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)
is a rational solution to the Johnson equation It is a quotient oftwo polynomials 119899(119909 119910 119905) and 119889(119909 119910 119905) depending on 2119873 minus 2real parameters 119886119895 and 119895 1 le 119895 le 119873minus1 of degrees 2119873(119873+1)in 119909 119905 and 4119873(119873 + 1) in 11991042 Families of Rational Solutions of Order 3 Depending on 4Parameters Here we construct families of rational solutionsto the Johnson equation of order 3 explicitly they depend on4 parameters
We only give the expression without parameters and wegive it in the appendix because of the length of the solutions
We construct the patterns of the modulus of the solutionsin the plane (119909 119910) of coordinates in functions of parameters119886119894 119887 1 le 119894 le 2 and time 119905
The role of the parameters 119886119894 and 119887119894 for the same integer119894 is the same one one will be interested primarily only inparameters 119886119894
The study of these configurationsmakes it possible to givethe following conclusions The variation of the configurationof the module of the solutions is very fast according to time119905 When time 119905 grows from 0 to 0 01 one passes froma rectilinear structure with a height of 98 to a horseshoestructure with a maximum height equal to 4 The role playedby the parameters 119886119894 and 119887119894 is the same for same index 119894Whenvariables 119909 119910 and time tend towards infinity the modulus ofthe solutions tends towards 2 in accordancewith the structureof the polynomials which will be studied in a forthcomingarticle
5 Conclusion
We have constructed solutions to the Johnson equationstarting from the solutions of the KPI equation what makesit possible to obtain rational solutions These solutions areexpressed bymeans of quotients of two polynomials of degree2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2parameters
Here we have given a new method to construct solutionsto the Johnson equation related to previous results [12ndash14]
We have given two types of representations of the solu-tions to the Johnson equation An expression by means ofFredholm determinants of order 2119873 depending on 2119873 minus 1real parameters is given Another expression by means ofWronskians of order 2119873 depending on 2119873minus1 real parametersis also constructed Also rational solutions to the Johnsonequation depending on 2119873 minus 2 real parameters are obtainedwhen one of parameters (120598) tends to zero
The patterns of the modulus of the solutions in the plane(119909 119910) and their evolution according to time and parametershave been studied in Figures 1 2 3 4 and 5
In another study we will give a more general representa-tion of rational solutions to the Johnson equation It can bewritten without limit at order 119873 depending on 2119873 minus 2 realparameters We will prove that these solutions can be writtenas a quotient of polynomials of degree 2119873(119873 + 1) in 119909 119905 and4119873(119873 + 1) in 119910Appendix
The solutions to the Johnson equation can be written as
Figure 1 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and1198872 = 0 and on the right for 119905 = 0 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0
Figure 2 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and1198872 = 106 and on the right for 119905 = 0 01 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0a12 = 8916100448256a11 = 89161004482561199051199102 + 2567836929097728119905a10
Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
6 Advances in Mathematical Physics
and
det (119868 + 1198631) = 1198961 (0) times 1198821 (1206011199031 1206011199032119873) (0) (48)
As Θ3119895(0) contains 119873 terms 1199093119895 1 le 119895 le 119873 and 119873 termsminus1199093119895 1 le 119895 le 119873 we have the relation 1198963(0) = 1198961(0) and weget the following theorem
is a solution of the Johnson equation which depends on 2119873 minus 1real parameters 119886119896 119887119896 and 120598 with 120601119903] defined in (21)
120601119903] = sin(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 1 le ] le 119873
120601119903] = cos(minus119894120581]1199092 + 119894 (minus120581]11991024 + 120575])119910119905 minus 119894119909119903]2 + 2120591]119905+ 120574]119908 minus 119894119890]2 ) 119873 + 1 le ] le 2119873 119903 = 1 3
(50)
where 120581] 120575] 119909119903] 120574] and 119890] are defined in (6) (5) and (7)
4 Study of the Limit Case When 120598 Tends to 041 Rational Solutions of Order 119873 Depending on 2119873 minus 2Parameters An infinite hierarchy of rational solutions tothe Johnson equation depending on 2119873 minus 2 parameters isobtained For this we take the limit when the parameter 120598tends to 0
We get the following statement
Theorem 5 The function V
V (119909 119910 119905) = lim120598997888rarr0
minus 2 10038161003816100381610038161198823 (119909 119910 119905)10038161003816100381610038162(1198821 (119909 119910 119905))2 (51)
is a rational solution to the Johnson equation It is a quotient oftwo polynomials 119899(119909 119910 119905) and 119889(119909 119910 119905) depending on 2119873 minus 2real parameters 119886119895 and 119895 1 le 119895 le 119873minus1 of degrees 2119873(119873+1)in 119909 119905 and 4119873(119873 + 1) in 11991042 Families of Rational Solutions of Order 3 Depending on 4Parameters Here we construct families of rational solutionsto the Johnson equation of order 3 explicitly they depend on4 parameters
We only give the expression without parameters and wegive it in the appendix because of the length of the solutions
We construct the patterns of the modulus of the solutionsin the plane (119909 119910) of coordinates in functions of parameters119886119894 119887 1 le 119894 le 2 and time 119905
The role of the parameters 119886119894 and 119887119894 for the same integer119894 is the same one one will be interested primarily only inparameters 119886119894
The study of these configurationsmakes it possible to givethe following conclusions The variation of the configurationof the module of the solutions is very fast according to time119905 When time 119905 grows from 0 to 0 01 one passes froma rectilinear structure with a height of 98 to a horseshoestructure with a maximum height equal to 4 The role playedby the parameters 119886119894 and 119887119894 is the same for same index 119894Whenvariables 119909 119910 and time tend towards infinity the modulus ofthe solutions tends towards 2 in accordancewith the structureof the polynomials which will be studied in a forthcomingarticle
5 Conclusion
We have constructed solutions to the Johnson equationstarting from the solutions of the KPI equation what makesit possible to obtain rational solutions These solutions areexpressed bymeans of quotients of two polynomials of degree2119873(119873 + 1) in 119909 119905 and 4119873(119873 + 1) in 119910 depending on 2119873 minus 2parameters
Here we have given a new method to construct solutionsto the Johnson equation related to previous results [12ndash14]
We have given two types of representations of the solu-tions to the Johnson equation An expression by means ofFredholm determinants of order 2119873 depending on 2119873 minus 1real parameters is given Another expression by means ofWronskians of order 2119873 depending on 2119873minus1 real parametersis also constructed Also rational solutions to the Johnsonequation depending on 2119873 minus 2 real parameters are obtainedwhen one of parameters (120598) tends to zero
The patterns of the modulus of the solutions in the plane(119909 119910) and their evolution according to time and parametershave been studied in Figures 1 2 3 4 and 5
In another study we will give a more general representa-tion of rational solutions to the Johnson equation It can bewritten without limit at order 119873 depending on 2119873 minus 2 realparameters We will prove that these solutions can be writtenas a quotient of polynomials of degree 2119873(119873 + 1) in 119909 119905 and4119873(119873 + 1) in 119910Appendix
The solutions to the Johnson equation can be written as
Figure 1 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and1198872 = 0 and on the right for 119905 = 0 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0
Figure 2 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and1198872 = 106 and on the right for 119905 = 0 01 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0a12 = 8916100448256a11 = 89161004482561199051199102 + 2567836929097728119905a10
Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Advances in Mathematical Physics 7
Figure 1 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and1198872 = 0 and on the right for 119905 = 0 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0
Figure 2 Solution of order 3 to (1) on the left for 119905 = 0 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0 in the center for 119905 = 0 1198861 = 0 1198871 = 0 1198862 = 0 and1198872 = 106 and on the right for 119905 = 0 01 1198861 = 0 1198871 = 103 1198862 = 0 and 1198872 = 0a12 = 8916100448256a11 = 89161004482561199051199102 + 2567836929097728119905a10
Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
8 Advances in Mathematical Physics
Figure 3 Solution of order 3 to (1) on the left for 119905 = 0 01 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 0 1 1198861 = 103 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 1 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 4 Solution of order 3 to (1) on the left for 119905 = 0 1 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 1 1198861 = 0 1198871 = 0 1198862 = 106and 1198872 = 0 and on the right for 119905 = 10 1198861 = 103 1198871 = 0 1198862 = 0 and 1198872 = 0
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Advances in Mathematical Physics 9
Figure 5 Solution of order 3 to (1) on the left for 119905 = 10 1198861 = 0 1198871 = 0 1198862 = 106 and 1198872 = 0 in the center for 119905 = 100 1198861 = 106 1198871 = 01198862 = 0 and 1198872 = 0 and on the right for 119905 = 103 1198861 = 105 1198871 = 103 1198862 = 0 and 1198872 = 0
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
[4] V D Lipovskii ldquoOn the nonlinear internal wave theory in fluidof finite depthrdquo Izv Akad Nauka Phys of Atmosphere andOcean vol 21 no 8 pp 864ndash871 1985
[5] V I Golinko V S Dryuma and Y A Stepanyants ldquoNonlin-ear quasicylindrical waves exact solutions of the cylindricalKadomtsev-Petviashvili equationrdquo in Nonlinear and TurbulentProcesses in Physics Proceedings of the Second InternationalWorkshop on Nonlinear and Turbulent Processes in Physics KievUSSR 10ndash25 October 1983 pp 1353ndash1360 Harwood AcademicPublishers Gordon and Breach 1984
[6] V D Lipovskii V B Matveev and A O Smirnov ldquoOn aconnection between the Kadomtsev-Petviashvili equation andthe Johnson equationrdquo Zapiski Nauchnykh Seminarov LOMIvol 150 pp 70ndash75 1986
[7] B B Kadomtsev and W I Petviashvili ldquoOn the stability ofsolitary waves in weakly dispersing mediardquo Soviet PhysicsDoklady vol 15 no 6 pp 539ndash541 1970
[8] K Klein V B Matveev and A O Smirnov ldquoCylindricalKadomtsev-Petviashvili equation old and new resultsrdquo Theo-retical and Mathematical Physics vol 152 no 2 pp 1132ndash11452007
[9] K R Khusnutdinova C Klein V B Matveev and A OSmirnov ldquoOn the integrable elliptic cylindrical Kadomtsev-Petviashvili equationrdquo Chaos vol 23 no 1 Article ID 01312615 pages 2013
[10] M J Ablowitz and H Segur ldquoOn the evolution of packets ofwater wavesrdquo Journal of Fluid Mechanics vol 92 no 4 pp 691ndash715 1979
[11] D E Pelinovsky Y A Stepanyants and Y S Kivshar ldquoSelf-focusing of plane dark solitons in nonlinear defocusing mediardquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 51 no 5 pp 5016ndash5026 1995
[12] P Gaillard ldquoFamilies of rational solutions of order 5 to theKPI equation depending on 8 parametersrdquo New Horizons inMathematical Physics vol 1 no 1 pp 26ndash31 2017
[13] P Gaillard ldquoFamilies of quasi-rational solutions of the NLSequation and multi-rogue wavesrdquo Journal of Physics A Math-ematical and Theoretical vol 44 pp 1ndash15 2010
[14] P Gaillard ldquoDegenerate determinant representation of solutionof the NLS equation higher Peregrine breathers and multi-rogue wavesrdquo Journal of Mathematical Physics vol 54 ArticleID 013504 32 pages 2013
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences