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Representations of the multisoliton solutions of the Korteweg-de Vries equation. (English) Zbl 0805.35111

From the introduction: One of the interesting problems in the theory of soliton equations is that of finding the simplest representation of their multisoliton solutions. By simplest we mean a computationally efficient algorithm which contains the maximum amount of explicit information. A different approach to the problem is to look for the simplest homotopy between the asymptotic states. Apart from its usefulness in phenomenological applications of soliton theory, such representations have generic implications for the class of soliton equations.
In a previous paper, one of us presented such a representation for the sine-Gordon equation [A. C. Bryan, Nonlinear Anal., Theory Methods Appl. 12, No. 10, 1047-1052 (1988; Zbl 0701.35131)] and its general properties enabled us to extend the method to the modified Korteweg-de Vries equation. In this paper we present a further extension of the method to the Korteweg-de Vries equation itself. We also show that this representation can be reduced to a simpler structural form, i.e. a linear superposition, but at the expense of a loss of explicit information.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)

Citations:

Zbl 0701.35131
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References:

[1] Bryan, A. C., On representations of the multisoliton solutions of the sine-Gordon equation, Nonlinear Analysis, 12, 1047-1052 (1988) · Zbl 0701.35131
[2] Bryan, A. C.; Miller, J.; Stuart, A. E.G., Superposition formula for multisolition II.—The modifoed Korteweg-de Vries equation, Il Nuovo Cimento B, 101, 715-720 (1988)
[3] Hirota, R., Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett., 27, 1192-1194 (1971) · Zbl 1168.35423
[4] Wadati, M.; Toda, M., The exact \(N\)-soliton solution of the Korteweg-de Vries equation, J. phys. Soc. Jpn, 32, 1403-1411 (1972)
[5] Wahlquist, H. D.; Estabrook, F. B., Bäcklund transformations for solutions of the Korteweg-de Vries equation, Phys. Rev. Lett., 31, 1386-1390 (1973)
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[7] Freeman, N. C.; Nimmo, J. J.C., Soliton solutions of the Korteweg-de Vries and the Kadomtsev-Petviashvili equations: the Wronskian technique, Proc. Roy. Soc. A, 389, 319-329 (1983) · Zbl 0588.35077
[8] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Korteweg-de Vries equation and generalizations. VI. Methods for exact solution, Communs pure appl. Math., 27, 97-133 (1974) · Zbl 0291.35012
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