A Bäcklund transformation and the inverse scattering transform method for the generalised Vakhnenko equation. (English) Zbl 1030.37047

Summary: A Bäcklund transformation both in bilinear and in ordinary form for the transformed generalised Vakhnenko equation (GVE) is derived. It is shown that the equation has an infinite sequence of conservation laws. An inverse scattering problem is formulated; it has a third-order eigenvalue problem. A procedure for finding the exact \(N\)-soliton solution to the GVE via the inverse scattering method is described. The procedure is illustrated by considering the cases \(N=1\) and 2.


37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI


[1] Konno, K.; Ichikawa, Y. H.; Wadati, M., A loop soliton propagating along a stretched rope, J. Phys. Soc. Japan, 50, 1025-1026 (1981)
[2] Ishimori, Y., On the modified Korteweg-de Vries soliton and the loop soliton, J. Phys. Soc. Japan, 50, 2471-2472 (1981)
[3] Ichikawa, Y. H.; Konno, K.; Wadati, M., New integrable nonlinear evolution equations leading to exotic solitons, (Horton, C. W.; Reichl, L. E.; Szebehely, V. G., Long-time Prediction in Dynamics (1983), John Wiley: John Wiley New York), 345-365
[4] Shimizu, T.; Sawada, K.; Wadati, M., Determination of the one-kink curve of an elastic wire through the inverse method, J. Phys. Soc. Japan, 52, 36-43 (1983)
[5] Konno, K.; Jeffrey, A., The loop soliton, (Debnath, L., Advances in Nonlinear Waves, vol. 1 (1984), Pitman: Pitman London), 162-183 · Zbl 0554.35113
[6] Rogers, C.; Wong, P., On reciprocal Bäcklund transformations of inverse scattering schemes, Phys. Scripta, 30, 10-14 (1984) · Zbl 1063.37552
[7] El Naschie, M. S.; Al Athel, S.; Walker, A. C., Localized buckling as statical homoclinic soliton and spacial complexity, (Schiehlen, W., Nonlinear Dynamics in Engineering Systems (1990), Springer: Springer Berlin), 67-74
[8] El Naschie, M. S., Stress, stability and chaos in structural engineering (1990), McGraw-Hill: McGraw-Hill London · Zbl 0729.73919
[9] \(N\)-loop solitons and their link with the complex Harry Dym equation, J. Phys. A Math. Gen., 27, 8197-8205 (1994) · Zbl 0839.35114
[10] Wadati, M.; Konno, K.; Ichikawa, Y. H., New integrable nonlinear evolution equations, J. Phys. Soc. Japan, 47, 1698-1700 (1979) · Zbl 1334.35256
[11] Shimizu, T.; Wadati, M., A new integrable nonlinear evolution equation, Prog. Theor. Phys., 63, 808-820 (1980) · Zbl 1059.37505
[12] Wadati, M.; Ichikawa, Y. H.; Shimizu, T., Cusp soliton of a new integrable nonlinear evolution equation, Prog. Theor. Phys., 634, 1959-1967 (1980) · Zbl 1059.37506
[13] Ichikawa, Y. H.; Konno, K.; Wadati, M., Nonlinear traverse oscillation of elastic beams under tension, J. Phys. Soc. Japan, 50, 1799-1802 (1981)
[14] Konno, K.; Mituhashi, M.; Ichikawa, Y. H., Soliton on thin vortex filament, Chaos, Solitons & Fractals, 1, 55-66 (1991) · Zbl 0743.35060
[15] Nakayama, K.; Iizuka, T.; Wadati, M., Curve lengthening equation and its solutions, J. Phys. Soc. Japan, 63, 1311-1321 (1994) · Zbl 0972.58501
[16] Kakuhata, H.; Konno, K., Loop soliton solutions of string interacting with external field, J. Phys. Soc. Japan, 68, 757-762 (1999) · Zbl 0944.81025
[17] Qu, C.; Si, Y.; Liu, R., On affine Sawada-Kotera equation, Chaos, Solitons & Fractals, 15, 131-139 (2003) · Zbl 1038.35099
[18] Vakhnenko, V. A., Solitons in a nonlinear model medium, J. Phys. A Math. Gen., 25, 4181-4187 (1992) · Zbl 0754.35132
[19] Vakhnenko, V. O., High-frequency soliton-like waves in a relaxing medium, J. Math. Phys., 40, 2011-2020 (1999) · Zbl 0946.35094
[20] Vakhnenko, V. O.; Parkes, E. J., The two loop soliton of the Vakhnenko equation, Nonlinearity, 11, 1457-1464 (1998) · Zbl 0914.35115
[21] 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
[22] 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. Diff. Eqns. Applicat., 1, 429-449 (2000)
[23] Vakhnenko, V. O.; Parkes, E. J., The calculation of multi-soliton solutions of the Vakhnenko equation by the inverse scattering method, Chaos, Solitons & Fractals, 13, 1819-1826 (2002) · Zbl 1067.37106
[24] Morrison, A. J.; Parkes, E. J., The \(N\)-soliton solution of a generalised Vakhnenko equation, Glasgow Math. J., 43, 65-90 (2001) · Zbl 1045.35061
[25] Morrison, A. J.; Parkes, E. J., The \(N\)-soliton solution of the modified generalised Vakhnenko equation (a new nonlinear evolution equation), Chaos, Solitons & Fractals, 16, 13-26 (2003) · Zbl 1048.35104
[26] Hirota, R., Direct methods in soliton theory, (Bullough, R. K.; Caudrey, P. J., Solitons (1980), Springer: Springer New York), 157-176
[27] Hirota, R.; Satsuma, J., \(N\)-soliton solutions of model equations for shallow water waves, J. Phys. Soc. Japan, 40, 611-612 (1976) · Zbl 1334.76016
[28] 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
[29] 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)
[30] 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
[31] Satsuma, J.; Kaup, D. J., A Bäcklund transformation for a higher order Korteweg-De Vries equation, J. Phys. Soc. Japan, 43, 692-697 (1977) · Zbl 1334.81041
[32] Satsuma, J., Higher conservation laws for the Korteweg-de Vries equation through Bäcklund transformation, Prog. Theor. Phys., 52, 1396-1397 (1974)
[33] Wadati, M.; Sanuki, H.; Konno, K., Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Prog. Theor. Phys., 53, 419-436 (1975) · Zbl 1079.35506
[34] Kaup, D. J., On the inverse scattering problem for cubic eigenvalue problems of the class \(ψ_{ xxx } +6 Qψ_x +6 Rψ = λψ \), Stud. Appl. Math., 62, 189-216 (1980) · Zbl 0431.35073
[35] Caudrey, P. J., The inverse problem for a general
((N×N\) spectral equation, Phys. D, 6, 51-66 (1982) · Zbl 1194.35524
[36] Caudrey, P. J., The inverse problem for the third-order equation \(u_{ xxx }+q(x)u_x+r(x)u\)=−\( iζ^3u\), Phys. Lett. A, 79, 264-268 (1980)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.