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Bäcklund transformation, superposition formulae and \(N\)-soliton solutions for the perturbed Korteweg-de Vries equation. (English) Zbl 1252.35243
Summary: With symbolic computation, under investigation in this paper is the perturbed Korteweg-de Vries equation for the nonlocal solitary waves and arrays of wave crests. Via the Hirota method, the bilinear form, Bäcklund transformation and superposition formulae are obtained. \(N\)-soliton solutions in terms of the Wronskian are constructed. Asymptotic analysis is used to analyze the collision dynamics, and figures are plotted to illustrate the influence of the perturbation. We find that the perturbation affects the propagation velocities of the solitons, but does not affect the amplitudes and widths of the solitons. Besides, the solitonic collisions turn out to be elastic.

35Q53 KdV equations (Korteweg-de Vries equations)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
35C08 Soliton solutions
68W30 Symbolic computation and algebraic computation
Full Text: DOI
[1] Wazwaz, A.M., A study on an integrable system of coupled KdV equations, Commun nonlinear sci numer simulat, 15, 2846-2850, (2010) · Zbl 1222.37070
[2] Gai, X.L.; Gao, Y.T.; Wang, L.; Meng, D.X.; Lü, X.; Sun, Z.Y., Lax pair and Darboux transformation of the variable-coefficient modified kortweg – de Vries model in fluid-filled elastic tubes, Commun nonlinear sci numer simulat, 16, 1776-1782, (2011) · Zbl 1221.35334
[3] Sakovich, S. Yu., On integrability of a (2+1)-dimensional perturbed KdV equation, J nonlinear math phys, 5, 230-233, (1998) · Zbl 0946.35092
[4] Akylas, T.R., On the excitation of long nonlinear water waves by a moving pressure distribution, J fluid mech, 141, 455-466, (1984) · Zbl 0551.76018
[5] Grimshaw, R.H.J.; Smyth, N., Resonant flow of a stratified fluid over topography, J fluid mech, 169, 429-464, (1984) · Zbl 0614.76108
[6] Veksler, A.; Zarmi, Y., Freedom in the expansion and obstacles to integrability in multiple-soliton solutions of the perturbed KdV equation, Physica D, 217, 77-87, (2006) · Zbl 1099.35124
[7] Antonova, M.; Biswas, A., Adiabatic parameter dynamics of perturbed solitary waves, Commun nonlinear sci numer simulat, 14, 734-748, (2009) · Zbl 1221.35321
[8] Girgis, L.; Biswas, A., Soliton perturbation theory for nonlinear wave equations, Appl math comput, 216, 2226-2231, (2010) · Zbl 1192.35150
[9] Hai, W.; Xiao, Y., Soliton solution of a singularly perturbed KdV equation, Phys lett A, 208, 79-83, (1995) · Zbl 1020.35520
[10] Fan, E., A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation, Chaos soliton fract, 15, 567-574, (2003) · Zbl 1037.76049
[11] Ablowitz, M.J.; Clarkson, P.A., Solitons, nonlinear evolution equations and inverse scattering, (1991), Cambridge Univ. Press New York · Zbl 0762.35001
[12] Zabusky, N.J.; Kruskal, M.D.; Gardner, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M., Method for solving the korteweg – de Vries equation, Phys rev lett, Phys rev lett, 19, 1095-1097, (1967) · Zbl 1061.35520
[13] Triki, H.; Taha, T.R.; Wazwaz, A.M., Solitary wave solutions for a generalized kdv – mkdv equation with variable coefficients, Math comput simulat, 80, 1867-1873, (2010) · Zbl 1346.35184
[14] Wazwaz, A.M., Multiple-soliton solutions of the perturbed KdV equation, Commun nonlinear sci numer simulat, 15, 3270, (2010) · Zbl 1222.65134
[15] Hirota, R.; Ito, M., Resonance of solitons in one dimension, J phys soc jpn, 52, 744-748, (1983)
[16] Kraenkel, R.A., First-order perturbed korteweg – de Vries solitons, Phys rev E, 57, 4775-4777, (1998)
[17] Lin, G.D.; Gao, Y.T.; Wang, L.; Meng, D.X.; Yu, X., Elastic-inelastic-interaction coexistence and double Wronskian solutions for the whitham – broer – kaup shallow-water-wave model, Commun nonlinear sci numer simulat, 16, 3090-3096, (2011) · Zbl 1419.76089
[18] Tian, B.; Shan, W.R.; Zhang, C.Y.; Wei, G.M.; Gao, Y.T., Transformations for a generalized variable-coefficient nonlinear schrö dinger model from plasma physics, arterial mechanics and optical fibers with symbolic computation, Eur phys J B (rapid not), 47, 329-332, (2005)
[19] Gao, Y.T.; Tian, B., Reply to: “comment on: ‘spherical kadomtsev – petviashvili equation and nebulons for dust ion-acoustic waves with symbolic computation”, Phys lett A, 361, 523-528, (2007) · Zbl 1325.35192
[20] Tian, B.; Gao, Y.T., Comment on: ‘exact solutions of cylindrical and spherical dust ion acoustic waves’, Phys plasmas, 12, 054701, (2005)
[21] Hirota, R.; Satsumam, J., N-soliton solutions of model equations for shallow water waves, Phys soc jpn, 40, 611-612, (1976) · Zbl 1334.76016
[22] Hirota, R., The direct method in soliton theory, (2004), Cambridge Univ. Press Cambridge
[23] Hu, X.B., The higher-order KdV equation with a source and nonlinear superposition formula, Chaos soliton fract, 7, 211-215, (1996) · Zbl 1080.35535
[24] Xu, T.; Tian, B.; Li, L.L.; Lü, X.; Zhang, C., Dynamics of alfvn solitons in inhomogeneous plasmas, Phys plasmas, 15, 102307, (2008)
[25] Chen, D.Y., Bäcklund transformation and N soliton solutions, J math res exposition, 25, 479-488, (2005)
[26] Yu, X.; Gao, Y.T.; Sun, Z.Y.; Liu, Y., Solitonic propagation and interaction for a generalized variable-coefficient forced Korteweg-de Vries equation in fluids, Phys rev E, 83, 056601, (2011)
[27] Yu, X.; Gao, Y.T.; Sun, Z.Y.; Liu, Y., N-soliton solutions, backlund transformation and Lax pair for a generalized variable-coefficient fifth-order Korteweg-de Vries equation, Phys scr, 81, 045402, (2010) · Zbl 1191.35242
[28] Sun, Z.Y.; Gao, Y.T.; Yu, X.; Meng, X.H.; Liu, Y., Inelastic interactions of the multiple-front waves for the modified Kadomtsev-Petviashvili equation in fluid dynamics, plasma physics and electrodynamics, Wave motion, 46, 511, (2009) · Zbl 1231.37045
[29] Sun, Z.Y.; Gao, Y.T.; Yu, X.; Liu, Y., Formation of vortices in a combined pressure-driven electro-osmotic flow through the insulated sharp tips under finite Debye length effects, Colloid surface A, 366, 1, (2010)
[30] Wang, L.; Gao, Y.T.; Gai, X.L.; Sun, Z.Y., Inelastic interactions and double Wronskian solutions for the whitham – broer – kaup model in shallow water, Phys scr, 80, 065017, (2009) · Zbl 1423.35321
[31] Wang, L.; Gao, Y.T.; Gai, X.L., Odd-soliton-like solutions for the variable-coefficient variant Boussinesq model in the long gravity waves, Z naturforsch A, 65, 1, (2010)
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