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Soliton asymptotics of nondecreasing solutions of nonlinear completely integrable evolution equations. (English) Zbl 0819.58015

Marchenko, V. A. (ed.), Spectral operator theory and related topics. Collection of papers. Providence, RI: American Mathematical Society. Adv. Sov. Math. 19, 129-180 (1994).
This survey gives an account of studies done by the Kharkov mathematicians on soliton asymptotics of the Cauchy problem for a number of nonlinear integrable differential equations with nondecreasing initial data whose typical representative is a step-like function. The authors express the initial data of the nonlinear integrable equations in terms of spectra of the Lax pair operators. A special spectral structure corresponding to these initial data leads to a special type of asymptotic. Besides of traditional solitons generated by a discrete spectrum the authors investigate mainly the asymptotic solitons generated by a continuous spectrum. For the first time these asymptotic solitons were discovered by the first author in 1975 and investigated intensively further by the second author. According to their results the nondecreasing initial data of the Cauchy problem split themselves into superposition of traditional and asymptotic solitons when time goes to infinity. The authors give a rigorous proof of this fact by means of the inverse scattering method for the KdV, MKdV, NS, sine-Gordon and KP equations.
For the entire collection see [Zbl 0802.00007].

MSC:

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.)
35Q51 Soliton equations
35B40 Asymptotic behavior of solutions to PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
58J37 Perturbations of PDEs on manifolds; asymptotics
81U40 Inverse scattering problems in quantum theory
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