THE SOLITON SOLUTIONS OF SOME NONLINEAR DYNAMICAL MODEL EQUATIONS SUMMARY OF THE THESIS SUBMITTED TO THE FACULTY OF SCIENCE KURUKSHETRA UNIVERSITY, KURUKSHETRA FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN PHYSICS BY HITENDER KUMAR DEPARTMENT OF PHYSICS KURUKSHETRA UNIVERSITY KURUKSHETRA ‒ 136 119, INDIA July ‒ 2013
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THE SOLITON SOLUTIONS OF SOME
NONLINEAR DYNAMICAL MODEL
EQUATIONS
SUMMARY
OF THE
THESIS
SUBMITTED TO
THE FACULTY OF SCIENCE
KURUKSHETRA UNIVERSITY, KURUKSHETRA
FOR THE DEGREE
OF
DOCTOR OF PHILOSOPHY IN
PHYSICS
BY
HITENDER KUMAR
DEPARTMENT OF PHYSICS
KURUKSHETRA UNIVERSITY
KURUKSHETRA ‒ 136 119, INDIA
July ‒ 2013
1 Introduction
The origin of nonlinear partial differential equations (PDEs) is very old. They had undergone ma-
jor new developments during the second half of the twentieth century. One of the main needs for
developing nonlinear PDEs has been the study of nonlinear wave propagation problems. These
problems arise in diverse areas of applied mathematics, physics and engineering, including fluid
dynamics, nonlinear optics, solid mechanics, plasma physics, quantum field theory and condensed-
matter physics. Nonlinear wave equations, in particular, have provided several examples of new
solutions that are remarkably different from those obtained for linear wave problems. The best
known examples of these are the corresponding shock waves, water waves, solitons and solitary
waves. Indeed, the theory of nonlinear waves and soliions has experienced a revolution over the
past few decades. During this revolution, many remarkable and unexpected phenomena have also
been observed in physical, chemical, and biological systems. Other major achievements include
the discovery of soliton interactions, the inverse scattering transform method for finding the ex-
plicit exact solution for several PDEs and asymptotic perturbation analysis for the investigation of
nonlinear evolution equations.
Historically, the famous 1965 paper of Zabusky and Kruskal marked the birth of the new concept
of the soliton, a name intended to signify particle like quantities [1]. Subsequently, Zabusky (1967)
[2] confirmed, numerically, the actual physical interaction of two solitons, and Lax (1968) [3] gave
a rigorous analytical proof that the identities of two distinct solitons are preserved through the
nonlinear interaction governed by the KdV equation. Physically, when two solitons of different
amplitudes (and hence, of different speeds) are placed far apart on the real line, the taller (faster)
wave to the left of the shorter (slower), the taller one eventually catches up to the shorter one and,
then, overtakes it. When this happens, they undergo a nonlinear interaction according to the KdV
equation and emerge from the interaction completely preserved in form and speed with only a phase
shift. These discoveries have, in turn, led to extensive theoretical, experimental, and computational
studies over the last few years. Many nonlinear model equations have now been found in nonlinear
science to explain many of the novel features of dynamical systems.
Keeping in view many interesting applications of nonlinear PDEs in nonlinear dynamics, the sub-
ject matter of present thesis is to investigate analytic soliton solutions of some nonlinear PDEs
having vital applications in different fields. So to achieve this goal, newly developed analytic tech-
niques have been utilized and a variety of soliton solutions are obtained.
1
2 Methodology
Nonlinear evolution equations are widely used as models to describe complex physical phenomena
in various fields of sciences, especially in fluid mechanics, solid state physics, plasma physics and
chemical physics. Given a nonlinear PDE, there is no general way of knowing whether it has soliton
solutions or not, or how the soliton solutions can be found. In order to get a better understanding
of the underlying phenomena as well as their further applications in practical life, it is important
to seek their exact solutions. Analytical solutions to nonlinear PDEs play an important role in
nonlinear science, especially in nonlinear physical science since they can provide much physical
information and more insight into the physical aspects of the problem and thus lead to further ap-
plications. Moreover, new exact solutions may help researchers to find new phenomena. The exact
solutions, if available, of nonlinear PDEs facilitate the verification of results for numerical solvers
and aid in the stability analysis of solutions.
A literature survey reveals that, researchers usually employed a variety of distinct methods to ana-
lyze nonlinear evolution equations. These methods range from reasonable to difficult that require
a huge size of work. In fact there is no unified method that can be used for all types of nonlinear
evolution equations. For single soliton solutions, several methods, such as the inverse scatter-
ing transformation method [4], Hirota’s bilinear method [5], the truncated Painleve expansion [6],
wave ansatz method [15], He’s variational method [16] and so on.
3 Work carried out
Here we present a brief account of the studies carried out on the analytic soliton solutions of some
important nonlinear physical models.
(1). The higher order nonlinear Schrodinger (HNLS) equationThe HNLS equation describes the propagation of femtosecond optical pulses in optical fibers and
Here, the complex valued function E represents the wave profile where the independent variables
are the spatial x and time t and a1, a2, a3, a4, a5 are distributed parameters which are all time de-
pendent.
The intensity of topological soliton takes the form
|E(x, t)|2 = A20 tanh
2
[A0
√− a2(t) + a4(t)κ0
2a1(t) + 6a3(t)κ0
(x− v(t)t)
]. (27)
8
The time varying soliton parameters, the soliton frequency ω(t) and velocity v(t) have been cal-
culated during the course of derivation of solitons. Note that this solution exists provided that
constraint equation between the model coefficients a1(t), a2(t), a3(t), a4(t) and a5(t) is satisfied.
(3). Inhomogeneous NLSE with time-dependent coefficientsIt is well known that NLSE describes numerous nonlinear physical phenomena in the field of non-
linear science such as optical solitons in optical fibres, solitons in the mean-field theory of Bose-
Einstein condensates and the rogue waves (RWs) in the nonlinear oceanography etc. The oceanic
RWs can be, under the nonlinear theories of ocean waves, modeled by the dimensionless NLSE
iut +1
2uxx + |u|2u = 0, (28)
which describes the two-dimensional quasi-periodic deep-water trains in the lowest order in wave
steepness and spectral width. In the present work, we extend the NLSE (28) to the inhomogeneous
NLSE with variable coefficients, including group velocity dispersion β(t), linear potential V (x, t),
nonlinearity g(t) and the gain/loss term γ(t), in the form [19]
iut +β(t)
2uxx + V (x, t)u+ g(t)|u|2u = iγ(t)u, (29)
and found the bright and dark 1-soliton solutions using solitary wave ansatz under some parametric
restrictions.
(4). Variable coefficient NLSEThe varying dispersion and Kerr nonlinearity are of practical importance in a real optical-fiber
transmission system with the consideration of the inhomogeneities resulting from such factors as
the variation in the lattice parameters of the fiber media and fluctuation of the fiber’s diameters
[20]. Therefore, investigations on the variable-coefficient NLSE-type models for optical fibers
have become desirable.
Here, our emphasis is on the following variable-coefficient NLSE model [21]:
i[Ψz +
α(z)
2Ψ + σ(z)Ψt
]− 1
2β(z)Ψtt + γ(z)|Ψ|2Ψ = 0, (30)
where Ψ is a complex function of z and t. The function α(z) is the linear attenuation coefficient,
σ(z), β(z), and γ(z) are the inhomogeneous functions, respectively, related to the intermodal dis-
Table 1: Some single-JEF soliton solutionsSoliton type Soliton intensity ξ expression Existence condition
Case 1:Unchirped BS |Ψ′21|
2 = f220 exp
(−
∫ z0 α(z)dz
)sech2(ξ) ξ = p0t +
∫ z0 p0[βφ0 − σ]dz + q0
c4β(z)γ(z)
< 0
Case 2:Chirped BS |Ψ′22|
2 = f220 exp
[ ∫ z0
(κzκ
− α)dz
]sech2(ξ) ξ = p0κt + 1
2p0
∫ z0 κ[(φ0 − 2
∫ z0 σdz)κz − 2σ]dz + q0
c42γ(z)
dkdz
< 0
Case 3:Unchirped DS |Ψ′11|
2 = f220 exp
(−
∫ z0 α(z)dz
)tanh2(ξ) ξ = p0t +
∫ z0 p0[βφ0 − σ]dz + q0
c4β(z)γ(z)
> 0
Case 4:Chirped DS |Ψ′13|
2 = f220 exp
[ ∫ z0
(κzκ
− α)dz
]tanh2(ξ) ξ = p0κt + 1
2p0
∫ z0 κ[(φ0 − 2
∫ z0 σdz)κz − 2σ]dz + q0
c42γ(z)
dkdz
> 0
persion, GVD and nonlinear loss or gain. In practical applications, Eq.(30) and their various forms
9
are of considerable importance for the description of amplification, absorption, compression and
broadening of optical solitons in inhomogeneous optical fiber systems and also for the study of
stable transmission of solitons [22].
By using the F-expansion method, we construct exact solutions of the generalized NLSE with
varying intermodal dispersion and nonlinear gain or loss. Here we describe the dynamics of a few
analytic solutions which may be vital to improve the soliton transmission features in some actual
physical situations.
Table 1 depicts some interesting single-JEF soliton solutions. To display unique behavior of these
exact soliton solutions, one can choose the dispersion coefficient β(z) and the phase chirp parame-
ter k(z) in terms of trigonometric, hyperbolic functions, some constants and linear functions. Here,
we present some examples using the linear, trigonometric and hyperbolic distributed control sys-
tems.
From Table 1, note that the speed of chirpless bright soliton (BS) or dark soliton (DS) is related
to p0[β(z)φ0 − σ(z)] and phase shift is determined by∫ z
0[β(z)(φ2
0 − p20)− 2σ(z)φ0]dz. The wave
amplitude of BS is given by f20 exp(−12
∫ z
0α(z)dz) =
√−β(z)p20c4/γ(z), where c4β(z)
γ(z)< 0. So by
choosing suitable initial conditions and the intermodal dispersion and GVD parameters, the speed
and phase shift of unchirped bright and dark solitons in optical fiber communication systems can
be controlled. Similarly, in Table 1 we obtained different existence conditions for chirped and chirp
free soliton solutions. In unchirped case, the amplitude and solitary wave characteristics can exclu-
sively be controlled by the dispersion coefficients β(z) and the gain/loss coefficients γ(z). But for
solitary waves with chirp function κ(z) and the gain/loss coefficient γ(z) will determine the wave
propagation characteristics. To visualize the propagation characteristics of chirp and unchirped
dark-bright soliton solutions, few numerical simulations are given in the thesis.
(5). The (3+1)-dimensional NLSEThe pulse propagation in a cubic-quintic (CQ) isotropic dispersive medium is governed by the
where Ψ(x, y, t; z) is the spatiotemporal field envelope propagating along the longitudinal distance
z, t is the retarded time, i.e., the time in the frame of reference moving with the wave packet
and ∆⊥ stands for the transverse Laplacian operator acting on spatial coordinates (x, y). The dis-
tributed parameters β, g3, g5 and γ are, respectively, the diffraction/dispersion coefficient, the cubic
and quintic nonlinearities and gain/loss coefficients.
We obtain exact spatiotemporal periodic traveling wave solutions to the generalized (3+1)D
CQNLSE by using F-expansion technique. For restrictive parameters, these periodic wave solutions
acquire the form of localized spatial solitons. Such solutions exist under certain conditions, and im-
pose constraints on the functions describing dispersion, nonlinearity, and gain (or loss). We present
10
some cases of the periodic wave and light bullet soliton solutions, taking the diffraction (disper-
sion) coefficient β(z) of the form β(z) = β0 cos(kbz) and the gain (loss) coefficient γ(z) as a small
constant. This choice leads to alternating regions of positive and negative values of β(z), g3(z)
and g5(z), which is required for an eventual stability of soliton solutions. We then demonstrate the
nonlinear tunneling effects and controllable compression technique of 3-dimensional bright and
dark solitons when they pass unchanged through the potential barriers and wells affected by special
choices of the diffraction and/or the nonlinearity parameters. We show that when the bright soliton
(BS) passes through the diffraction barrier, its intensity grows and forms a peak near barrier, af-
terwards it again recovers its original shape while the intensity of dark soliton (DS) first decreases
and then increases in corresponding two asymptotic states near barrier and then it again attains its
original shape. Conversely, when the BS crosses the diffraction well, the intensity of the soliton
vanishes and forms a valley near well, while the intensity of DS first increases and then decreasing
in two asymptotic states near well respectively and after the tunneling, the solitons are restored
to their original shapes. For compression of soliton pulses, we consider a system with decaying
diffraction barrier and nonlinearity on an exponential background. It can be seen that after passing
the well the soliton is compressed about the position of well, which indicates that the pulse can be
compressed to a desired width and amplitude in a controllable (lossless, increasing or decreasing)
manner by the choice of the well (barrier) parameters. This indicates that a new pulse compression
technique might be developed for 3D soliton pulses. Direct numerical simulation has been per-
formed to show the stable propagation of bright soliton with 5% white noise perturbation.
(6). The (2+1)-dimensional NLSEIn this study, we consider the coupled (2+1)D nonlinear system of Schrodinger equations as
iut − uxx + uyy + |u|2u− 2uv = 0, (32)
vxx − vyy − (|u|2)xx = 0, (33)
where u(x, y, t) and v(x, y, t) are complex-valued functions. The (2+1)D NLSE play an vital role
in atomic physics, and the functions u and v have different physical meanings in different disci-
plines of physics [24, 25]. Main applications are, for instance, in fluid dynamics [24] and plasma
physics [25]. In the context of water waves, u is the amplitude of a surface wave packet while v
is the velocity potential of the mean flow interacting with the surface waves [25]. However, in the
hydrodynamic context, u is the envelope of the wave packet and v is the induced mean flow. In
addition, Eqs. (32) and (33) are relevant in a number of different physical contexts, describing slow
modulation effects of the complex amplitude v, due to a small nonlinearity, or a monochromatic
wave in a dispersive medium.
In the present work, we have reduced (2+1)D NLSE to a nonlinear ODE by using a simple trans-
formation then various solutions of the nonlinear ODE are obtained by using extended F-expansion
11
and projective Riccati equation methods. With the aid of solutions of the nonlinear ODE more ex-
plicit traveling wave solutions expressed by the hyperbolic functions, trigonometric functions and
rational functions are found out.
(7). Density-independent diffusion reaction (DR) equationsWhile a variety of simplified versions of nonlinear DR equation are studied [26, 27, 28] in liter-
ature, but a DR equation with quadratic and cubic nonlinearities has been found useful in several
interesting applications recently [26, 27]. In fact, the presence of velocity-term in DR equations,
which is also attributed to the turbulent or anomalous diffusion in this later version is found to play
an important role in several applications, particularly in biological studies [26].
To obtain the exact solutions of the DR equations with quadratic and cubic nonlinearities, namely
Ct + vCx = DCxx + [a+ U(x)]C − β|C|C, (34)
Ct + vCx = DCxx + [a+ U(x)]C − β|C|2C, (35)
some simplifying assumptions are made, mainly to obtain the closed form solutions of Eqs.(34)
and (35) for complex concentration function C(x, t). Here, we relax this assumption of complex
C(x, t) and demonstrate that Eqs.(34) and (35) for real C(x, t) do admit a variety of exact solutions
including the solitonic ones in terms of hyperbolic functions. In particular, the existence of soliton
solutions of Eq.(35) are explicitly demonstrated in a certain parametric domain. We shall, however,
restrict ourselves to the case of constant value of the random potential U(x), say U0, in Eqs.(34)
and (35). In some sense Eqs.(34) and (35) can be considered as the generalization of the Fisher and
Nagumo equations respectively [26, 29]. For the solution of Eqs.(34) and (35), we introduce the
variable ζ = x− ωt and recast them for the real C(x, t) in the form
DC ′′ − (v − ω)C ′ + αC − βC2 = 0, (36)
DC ′′ − (v − ω)C ′ + αC − βC3 = 0, (37)
where U(x) is assumed to be constant (say U0) and α = (a+ U0).
Then, we apply the extended tanh-method for constructing a variety of exact traveling wave so-
lutions of Eqs.(34) and (35). The kink and antikink and some other traveling wave solutions are
expressed in terms of hyperbolic tangent functions. The kink and antikink soliton solutions of
Eqs.(34) and (35) are given by
C1(ζ) =α
4β
[1± tanh
(±√
α
24Dζ)]2
, (38)
C2(ζ) = ±√
α
4β
[1 + tanh
(±√
α
8Dζ)]
, (39)
respectively. Eqs.(38) and (39) have four traveling wave solutions corresponding to the two values
of µ and we label these solutions as C++(ζ), C+−(ζ), C−+(ζ) and C−−(ζ) corresponding to the
12
(upper, upper), (upper, lower), (lower, upper), (lower, lower) signs on the right side of these equa-
tions.
(8). Density-dependent DR equationsThere are various real physical phenomena in which, the diffusion coefficient D itself becomes
concentration or density dependent [26, 28]. For example, in certain insect dispersal model, D
depends on the population concentration C [26]. Thus, if concentration of a species increases then
its diffusion coefficient will also increase. In this work, we look for the exact solutions of the DR
equation with quadratic and quartic type nonlinearities along with a nonlinear convective flux term.
In particular, we investigate the solutions of following equations
Ct + kCCx = DCxx + αC − βC2, (40)
Ct + kC2Cx = DCxx + αC − βC4, (41)
where C = C(x, t) has the varying meaning depending on the phenomenon under study, D is
the diffusion coefficient and k, α and β are real constants. Eqs.(40) and (41) describe a trans-
port phenomenon in which both diffusion and convection processes are of equal importance i.e.,
the nonlinear diffusion could be thought of as equivalent to the nonlinear convection effects. In
particular, the second term on left hand side, kCCx (or kC2Cx) is the replacement of the conven-
tional vCx-term. It may be mentioned that the presence of vCx-type convective flux term in the DR
Eq.(35) makes the system nonconservative, whereas a nonlinear convection term in Eq.(40) or in
Eq.(41) arises as a natural extension of a conservation law. The soliton solutions of Eqs.(40) and
(41) are given by
C1(ζ) =α
2β
[1∓ tanh
(± αk
4βDζ)]
, (42)
C2(ζ) =βD
k2
[1± tanh
(± β2D
k3ζ)]
. (43)
For the positive value of k in Eq.(42), we represent these solutions as C−+(ζ) and C+−(ζ), which
in fact turn out to be kink soliton solutions and on the other hand negative value of k in Eq.(42),
represented by C−−(ζ) and C++(ζ) gives antikink soliton solutions.
(9). (2+1)-dimensional Maccari systemHere, we are interested to investigate new exact traveling wave solutions for the following (2 + 1)D
soliton equation
iut + uxx + uv = 0, (44a)
vt + vy + (uu∗)x = 0, (44b)
where u(x, y, t) is complex function and v(x, y, t) is real function. Eqs.(44a) and (44b) are similar
to integrable Zakharov equation in plasma physics to describe the behavior of sonic Langmuir soli-
tons, which are Langmuir oscillations trapped in regions of reduced plasma density caused by the
13
ponderomotive force due to a high-frequency field (when x = y in Eq. (44b). Recently Maccari
[30] obtained Eqs. (44a) and (44b) by an asymptotically exact reduction method based on Fourier
expansion and spatiotemporal rescaling from the Kadomtsev-Petviashvili equation. Maccari’s sys-
tem is a kind of nonlinear evolution equation that are often presented to describe the motion of the
isolated waves, localized in a small part of space, in many fields such as hydrodynamic, plasma
physics, nonlinear optic, etc.
Similarly, like (2+1)D NLSE, we construct large number of traveling wave solutions of this system
using extended F-expansion and projective Ricatti equation methods [9].
(10). Modified KdV (mKdV) equation with time-dependent coefficientsThe mKdV equation appears in many fields such as acoustic waves in certain anharmonic lattices,
Alfven waves in a collisionless plasma, transmission lines in Schottky barrier, models of traffic con-
gestion, ion acoustic solitons, elastic media, etc.[6] Here we studied the following form of mKdV
equation with variable coefficients by using solitary wave ansatz:
ut + α(t)ux − β(t)u2ux + γ(t)uxxx = 0, (45)
where α(t), β(t) and γ(t) are all analytic functions of t. The bright soliton solution is given by
u(x, t) =
√−6γ(t)
β(t)B0 sech[B0(x− vt)], (46)
where the velocity v(t) is determined during integration of Eq.(45) and it should be noted that the
soliton solution Eq.(46) exists under the condition γ(t)β(t) < 0. Finally, the dark soliton solution
for the time-dependent mKdV Eq.(45) is written as
u(x, t) =
√6γ(t)
β(t)B0 tanh[B0(x− vt)], (47)
which shows that it is necessary to have β(t)γ(t) > 0 for soliton to exist.
(11). Generalized Gardner equation (GE)The generalized GE, with full nonlinearity is given by
9. “Optical solitary wave solutions for the higher order nonlinear Schrodinger equation with
self-steepening and self-frequency shift effects ”, Hitender Kumar and Fakir Chand, Opt.
Laser Tech. 54, (2013) 265.
10. “1-soliton solutions of complex modified KdV equation with time-dependent coefficients”,
Hitender Kumar and Fakir Chand, Indian J. Phys. (2013), DOI 10.1007/s12648-013-0310-
8.
11. “Chirped and chirpfree soliton solutions of generalized nonlinear Schrodinger equation with
variable coefficients ”, Hitender Kumar and Fakir Chand, (communicated).
20
Research Papers presented in Conferences/Workshops:
1. “ Exact soliton solutions of nonlinear diffusion reaction equation with quadratic and cubic
nonlinearities”, Hitender Kumar and Fakir Chand, “PNLD 2010”, held at IISc. Bangalore
from July 26-29, 2010.
2. “Exact traveling wave solutions of time-delayed and diffusion reaction nonlinear evolution
equations”, Hitender Kumar, Anand Malik and Fakir Chand, “Sixth NCNSD”, held at
Bharathidasan University, Tiruchirapalli from January, 27-30, 2011.
3. “Exact traveling wave solutions of the Bogoyavlenskii equation”, Anand Malik, HitenderKumar, Fakir Chand and S.C. Mishra, “19th IEEE workshop on Nonlinear Dynamics in
Electronic Systems”, held at IICB and SINP Kolkata from March, 8-11, 2011.
4. “Dark and Bright Solitary Wave Solutions of Higher Order Nonlinear Schrodinger Equation”,
Hitender Kumar, Anand Malik and Fakir Chand, “19th IEEE workshop on Nonlinear Dy-
namics in Electronic Systems”, held at IICB and SINP, Kolkata from March, 8-11, 2011.
5. “Analytic study of some Nonlinear Diffusion Reaction Equations”, Hitender Kumar, Anand
Malik and Fakir Chand, “19th IEEE workshop on Nonlinear Dynamics in Electronic Sys-
tems”, held at IICB and SINP, Kolkata from March, 8-11, 2011.
6. “Chirped and unchirped soliton solution of generalized nonlinear Schrodinger equation with
distributed coefficients”, Hitender Kumar and Fakir Chand, National Conference on Ad-
vances in Physics held at Department of Physics, IIT Roorkee from February 25-26, 2012.