Finding Lie Symmetries of PDEs with MATHEMATICA: Applications to Nonlinear Fiber Optics Vladimir Pulov Department of Physics, Technical University-Varna, Bulgaria Ivan Uzunov Department of Applied Physics, Technical University-Sofia, Bulgaria Eddy Chacarov Department of Informatics and Mathematics, Varna Free University, Bulgaria Geometry, Geometry, Integrability Integrability and and Quatization Quatization − − June 8 June 8 - - 13, 2007 13, 2007
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Finding Lie Symmetries of PDEs with MATHEMATICA:Applications to Nonlinear Fiber Optics
Vladimir PulovDepartment of Physics, Technical University-Varna, Bulgaria
Ivan UzunovDepartment of Applied Physics, Technical University-Sofia, Bulgaria
Eddy ChacarovDepartment of Informatics and Mathematics, Varna Free University, Bulgaria
Geometry, Geometry, IntegrabilityIntegrability and and QuatizationQuatization −− June 8June 8--13, 200713, 2007
Plan of Presentation
1. MATHEMATICA package for finding Lie symmetries of PDE1.1. Block-scheme and algorithm1.2. Input and output 1.3. Tracing the evaluation1.4. Trial run
2. Applications to nonlinear fiber optics2.1. Physical model2.2. Results obtained
3. Conclusion
Symmetry Group of Δ
{ } 0 , Ω∈⊂Ω∈= ra
r RaTG
System of PDE( ) ( )( ) lkuuuxF n
k ,,2,1,0,...,,, 1 K== ( )Δ
Creating Defining System( ) ( )( )[ ] 0zF pr n =Xn for
Solving Defining System
( ) ( )uxuxii , , , αα ηηξξ ==
( )F
nz Δ∈
MA
THE
MA
TIC
A ( )
( ) ufdad
xffdadf
a
a
==
==
=
=
0
0
,,
,,
ϕϕηϕ
ϕξ
Solving the Lie Equation
( ) ( )∑∑== ∂
∂+
∂∂
=qp
ii
i
uux
xuxX
11,,
αα
αννν ηξ
Basic Infinitesimal Generators
Each solution of after transformation of the group remains a solution of .G ΔΔ
Lie Group of Symmetry Transformations
( ) ( )( ) lkuuuxF nk ,,2,1,0,...,,, 1 K==
{ } 0 , δδ ∈⊂∈= RaTG a
( )Δ
u
x
( )xfu =u′
x′
( )xfu ′′=′aT
If is a solution of then is also a solution of .fTf a ⋅=′ Δf Δ
two mode fiberstwo mode fibersstrong birefringent fibersstrong birefringent fibers
( )( ) 0BBB
2B
021
222
2
222
2
=++∂∂
+∂∂
=++∂∂
+∂∂
Atx
i
ABAt
AxAi
γν
γ
Invariants tJ =1 zJ =2 ς=3J xJ δα −=3 xJ εβ −=4
New variables ( )xpz =
( ) ( )βςα iBizA expexp ==
( )xq=ς ( ) xtf δα += ( ) xtg εβ +=
Reduced system ( )( ) 0222
0222
0202
232
232
=−++′−′′
=−++′−′′
=′′+′′=′′+′′
qqpqgqq
ppqpfpp
gqgqfpfp
ενγνν
δγ
Exact solution for Case C(two-mode fibers and strongly birefringent fibers)
( )⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+Π+
= xmjnbh
CiUA ε
λ| ;
12 exp
1
1
( )⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+Π+
±= xmjn
bhC
iUB ελ
| ; 12
exp 1
1
( ) ( ) 2122
21 21 , 12 ,| cn U bbthjbmjbb −=+=+−= λλ
( ) ( )[ ] dwmwnmjnj
∫−
−=Π0
12 | sn 1| ;
1 0, , ,1
21
31
21 ±=−
=−−
= εb
bbn
bbbb
m
are the roots of the polynomial321 bbb >> ( ) ( ) 11412 2
1223
++
+−
+−=
hC
hC
hQ θθεθθ
and are the Jacobean sine and cosine elliptic functions ( )mj |sn ( )mj |cn
Approximate vector solitary wavesApproximate vector solitary wavesStrong birefringent fibers with Raman scattering Strong birefringent fibers with Raman scattering A generalized version of previously obtained scalar solitary-wave solutionA generalized version of previously obtained scalar solitary-wave solution
( ) ( ) zzazazazF tanh sech ln5
8158
1516 2 ⎟
⎠⎞
⎜⎝⎛ −+−=
( ) zzzG 21 sech sinh =
( ) zzG 22 sech=
( ) ( ) βα ∂−+∂−+∂+∂= bctactcxX tx
( ) ( ) sech zF sech2 zzaA θ+−=
( )zGB θ=Galilean-like symmetry reduced system
( ) ( )
( ) ( ) 0 2 21 )(
0 2 21)(
2223
22
2223
21
=+−++−+−
=+−++−+−
yyyy
yyyy
qppqqqhpqqCqqcyb
pqqppphqppCppcya
θ
θ
1. L. Gagnon and P. A. Bélanger, Soliton self-frequency shift versus Galilean-like symmetry, Opt. Lett., Vol. 15, No. 9 (1990), pp. 466-468.
1<<θ − Raman parameter
-3 -2 -1 1 2 3z
0.2
0.4
0.6
0.8
1
τ02»A»2
-3 -2 -1 1 2 3z
5×10-7
1×10-6
1.5×10-6
2×10-6
τ02»B»2
-3 -2 -1 1 2 3z
2×10-6
4×10-6
6×10-6
8×10-6
τ02»B»2
( ) ( ) zzazazazF tanh sech ln5
8158
1516 2 ⎟
⎠⎞
⎜⎝⎛ −+−=
( ) zzzG 21 sech sinh =
( ) zzG 22 sech=
( ) ( ) sech zF sech2 zzaA θ+−=
( )zGB θ=
( )zF ( )zG1 ( )zG2
1. L. Gagnon and P. A. Bélanger, Soliton self-frequency shift versus Galilean-like symmetry, Opt. Lett., Vol. 15, No. 9 (1990), pp. 466-468.
2. N. Akhmediev and A. Ankiewicz, Novel soliton states and bifurcation phenomena in nonlinear fiber couplers, Phys. Rev. Lett., Vol. 70, No. 16 (1993), pp. 2395-2398.
LAWSLAWS OFOF CONSERVATIONCONSERVATION
TwoTwo--modemode fibers and strongfibers and strong birefringentbirefringent fibersfibers
( )
( ) ( )
( ) 122
5
24
23
2244222
**1
J
J
21
21-HJ
J
ixJdtBAtJ
dtB
dtA
dtBAhBABA
dtBBAA
x
x
x
x
x
x
x
xtt
x
xtt
++=
=
=
⎥⎦⎤
⎢⎣⎡ ++++=≡
+=
∫
∫
∫
∫
∫
−
−
−
−
−
ν
ν
TIME TRANSLATION
SYMMETRY LAWS OF CONSERVATION
SPACE TRANSLATION
TRANSLATION OF THE PHASE α
TRANSLATION OF THE PHASE β
GALILEAN-LIKESYMMETRY
References
[1] Christodoulides, D.N. and R.I. Joseph, Optics Lett., 13(1), 53-55 (1988).[2] Tratnik, M. V. and J. E. Sipe, Phys. Rev. A, 38(4), 2011-2017 (1988).[3] Christodoulides, D.N., Phys. Lett. A, 132(8, 9), 451-452 (1988).[4] Florjanczyk, M. and R. Tremblay, Phys. Lett. A, 141(1,2), 34-36 (1989).[5] Kostov, N. A. and I. M. Uzunov, Opt. Commun., 89, 389-392 (1992).[6] Florjanczyk, M. and R. Tremblay, Opt. Commun., 109, 405-409 (1994).[7] Pulov V., I. Uzunov, and E. Chacarov, Phys. Rev E, 57 (3), 3468-3477 (1998).
Conclusion
• The symbolic computational tools of MATHEMATICA have been applied to determining the Lie symmetries of PDE.
• An algorithm for creating and solving the defining system of the symmetry transformations has been developed and implemented in MATHEMATICA package.
• The package has been successfully applied to basic physical equations from nonlinear fiber optics.
• Future work: The package capabilities can be extended byadding new programming modules for transforming andsolving other wider classes of differential equations.