×

zbMATH — the first resource for mathematics

The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method. (English) Zbl 1067.37106
Summary: A Bäcklund transformation both in bilinear form and in ordinary form for the transformed Vakhnenko equation is derived. An inverse scattering problem is formulated. The inverse scattering method has a third-order eigenvalue problem. A procedure for finding the exact \(N\)-soliton solution of the Vakhnenko equation via the inverse scattering method is described. The procedure is illustrated by considering the cases \(N=1\) and \(N=2\).

MSC:
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Vakhnenko, V.A., Solitons in a nonlinear model medium, J. phys. A math. nucl. gen., 25, 4181-4187, (1992) · Zbl 0754.35132
[2] Vakhnenko, V.O., High-frequency soliton-like waves in a relaxing medium, J. math. phys., 40, 2011-2020, (1999) · Zbl 0946.35094
[3] Parkes, E.J., The stability of solutions of Vakhnenko’s equation, J. phys. A math. nucl. gen., 26, 6469-6475, (1993) · Zbl 0809.35086
[4] Boyko, V.M., The symmetrical properties of some equations of the hydrodynamic type, (), 32-37
[5] Vakhnenko, V.O.; Parkes, E.J., The two loop soliton solution of the Vakhnenko equation, Nonlinearity, 11, 1457-1464, (1998) · Zbl 0914.35115
[6] Vakhnenko, V.O.; Parkes, E.J.; Michtchenko, A.V., The Vakhnenko equation from the viewpoint of the inverse scattering method for the KdV equation, Int. J. differential equations applic., 1, 429-449, (2000)
[7] Morrison, A.J.; Parkes, E.J.; Vakhnenko, V.O., The N loop soliton solution of the Vakhnenko equation, Nonlinearity, 12, 1427-1437, (1999) · Zbl 0935.35129
[8] Hirota, R., Direct methods in soliton theory, (), 157-176
[9] Hirota, R., A new form of Bäcklund transformations and its relation to the inverse scattering problem, Prog. theor. phys., 52, 1498-1512, (1974) · Zbl 1168.37322
[10] ()
[11] Kaup, D.J., On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx+6Qψx+6Rψ=λψ, Stud. appl. math., 62, 189-216, (1980) · Zbl 0431.35073
[12] Caudrey, P.J., The inverse problem for a general N×N spectral equation, Physica D, 6, 51-66, (1982) · Zbl 1194.35524
[13] Caudrey, P.J., The inverse problem for the third order equation uxxx+q(x)ux+r(x)u=−iζ3u, Phys. lett. A, 79, 264-268, (1980)
[14] Hirota, R.; Satsuma, J., N-soliton solutions of model equations for shallow water waves, J. phys. soc. jpn., 40, 611-612, (1976) · Zbl 1334.76016
[15] Hirota, R.; Satsuma, J., A variety of nonlinear network equations generated from the Bäcklund transformation for the Toda lattice, Prog. theor. phys. suppl., 59, 64-100, (1976)
[16] Satsuma, J.; Kaup, D.J., A Bäcklund transformation for a higher order korteweg – de Vries equation, J. phys. soc. jpn., 43, 692-697, (1977) · Zbl 1334.81041
[17] Zakharov, V.E., On stochastization of one-dimensional chains of nonlinear oscillators, Sov. phys. JETP, 38, 108-110, (1974)
[18] Deift, P.; Tomei, C.; Trubowitz, E., Inverse scattering and the Boussinesq equation, Commun. pure appl. math., 35, 567-628, (1982) · Zbl 0479.35074
[19] Musette, M.; Conte, R., Algorithmic method for deriving Lax pairs from the invariant Painlevé analysis of nonlinear partial differential equations, J. math. phys., 32, 1450-1457, (1991) · Zbl 0734.35086
[20] Clarkson, P.; Mansfield, E.L., Symmetry reductions and exact solutions of shallow water wave equations, Acta appl. math., 39, 245-276, (1995) · Zbl 0835.35006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.