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Biswas, Anjan; Konar, Swapan

Soliton perturbation theory for the compound KdV equation. (English) Zbl 1147.35346

Int. J. Theor. Phys. 46, No. 2, 237-243 (2007).
Summary: The soliton perturbation theory is used to study the solitons that are governed by the compound Korteweg de-Vries equation
\[ q_t+(aq^p+bq^{2p})q_x+cq_{xxx}=\varepsilon R \] where \(\varepsilon\) is the perturbation parameter and \(0<\varepsilon\ll 1\), in presence of the perturbation terms \(R=\alpha q+\beta q_{xx}+\gamma q^mq_x+\delta q_x^3\) with \(\alpha,\beta\) small dissipative coefficients, \(\gamma\) the coefficient of higher order nonlinear dispersive term, and \(m\) a positive integer \(1\leq m\leq 4\). The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained.

Cited in 10 Documents

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems

Keywords:

soliton perturbation theory; adiabatic parameter dynamics; KdV equation

Software:

RATH
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\textit{A. Biswas} and \textit{S. Konar}, Int. J. Theor. Phys. 46, No. 2, 237--243 (2007; Zbl 1147.35346)
Full Text: DOI

References:

[1] Feng, B. F. and Kawahara, T. (2000a). Stationary travelling wave solutions of an unstable KdV-Burgers equation. Physica D 137(3–4), 228–236. · Zbl 0943.35081
[2] Feng, B. F. and Kawahara, T. (2000b). Multi-hump stationary waves for a Korteweg-de Vries equation with nonlocal perturbations. Physica D 137(3–4), 237–246. · Zbl 0945.35078
[3] Feng, Z. (2002). Qualitative analysis and exact solutions to the Burgers-Korteweg-de Vries equation. Dynamics of Continous, Discrete and Impulsive Systems, Series A 9(4), 563–580. · Zbl 1017.35100
[4] Feng, Z. (2003). Exact solution in terms of elliptic functions for the Burgers-Korteweg-de Vries equation. Wave Motion 38(2), 109–115. · Zbl 1163.74349
[5] Johnson, R. S. (1996). A two-dimensional Boussinesq equation for water waves and some of its solutions. Journal of Fluid Mechanics 323, 65–78. · Zbl 0896.76007
[6] Kaya, D. (2004). Solitary-wave solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order. Applied Mathematics and Computation 152(3), 709–720. · Zbl 1077.35110
[7] Kichenassamy, S. (1997). Existence of solitary waves for water-wave models. Nonlinearity 10(1), 133–151. · Zbl 0906.76012
[8] Kivshar, Y. and Malomed, B. A. (1989). Dynamics of solitons in nearly integrable systems. Reviews of Modern Physics 61(4), 763–915.
[9] Kodama, Y. and Ablowitz, M. J. (1981). Perturbations of solitons and solitary waves. Studies in Applied Mathematics 64, 225–245. · Zbl 0486.76029
[10] Li, Z. and Liu, Y. (2002). RATH: A Maple package for finding travelling solitary wave solutions to nonlinear evolution equations. Computer Physics Communications 148(2), 256–266. · Zbl 1196.35008
[11] Mann, E. (1997). The perturbed Korteweg de-Vries equation considered anew. Journal of Mathematical Physics 38(7), 3772–3785. · Zbl 0880.35107
[12] Marchant, T. R. and Smyth, N. F. (1996). Soliton interaction for the extended Korteweg-de Vries equation. IMA Journal of Applied Mathematics 56(2), 157–176. · Zbl 0857.35113
[13] Osborne, A. R. (1997). Approximate asymptotic integration of a higher order water-wave equation using the inverse scattering transform. Nonlinear Processes in Geophysics 4(1), 29–53.
[14] Ostrovsky, L. A. and Stepanyants, Y. A. (1989). Do internal solitons exist in the ocean? Reviews of Geophysics 27, 293–310.
[15] Parkes, E. J. and Duffy, B. R. (1996). An automated tanh function method for finding solitary wave solutions to non-linear evolution equations. Computer Physics Communications 98(3), 288–300. · Zbl 0948.76595
[16] Wazwaz, A. M. (2003). Compactons and solitary patterns structures for variants of the KdV and the KP equations. Applied Mathematics and Computations 139(1), 37–54. · Zbl 1029.35200
[17] Wazwaz, A. M. (2006). Analytic study on the generalized fifth-order KdV equation: New solitons and periodic solutions. Communications in Nonlinear Science and Numerical Simulation, to appear. · Zbl 1179.35296
[18] Wazwaz, A. M. (2006). Abundant solitons solutions for several forms of the fifth-orde KdV equation by using the tanh method. Applied Mathematics and Computation, to appear. · Zbl 1107.65092
[19] Zhang, W., Chang, Q., and Jiang, B. (2002). Explicit exact solitary-wave solutions for compound KdV-type and compound KdV-Burgerstype equations with nonlinear terms of any order. Chaos, Solitons & Fractals 13(2), 311–319. · Zbl 1028.35133
[20] Zhidkov, P. E. (2001). Korteweg-de Vries and Nonlinear Schrödinger’s Equations: Qualitative Theory. Springer Verlag. New York, NY. · Zbl 0987.35001
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