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Mat. Zametki, 2013, Volume 93, Issue 3, Pages 333–346 (Mi mz9060)  

Explicit Solutions of Boundary-Value Problems for $(2+1)$-Dimensional Integrable Systems

V. L. Vereshchagin

Institute of Mathematics with Computing Centre, Ufa Science Centre, Russian Academy of Sciences

Abstract: Two nonlinear integrable models with two space variables and one time variable, the Kadomtsev–Petviashvili equation and the two-dimensional Toda chain, are studied as well-posed boundary-value problems that can be solved by the inverse scattering method. It is shown that there exists a multitude of integrable boundary-value problems and, for these problems, various curves can be chosen as boundary contours; besides, the problems in question become problems with moving boundaries. A method for deriving explicit solutions of integrable boundary-value problems is described and its efficiency is illustrated by several examples. This allows us to interpret the integrability phenomenon of the boundary condition in the traditional sense, namely as a condition for the availability of wide classes of solutions that can be written in terms of well-known functions.

Keywords: Kadomtsev–Petviashvili equation, Toda chain, boundary-value problem, inverse scattering method, $(2+1)$-dimensional integrable systems, Lax representation, Gelfand–Levitan–Marchenko equation, dressing method, soliton solution.

DOI: https://doi.org/10.4213/mzm9060

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English version:
Mathematical Notes, 2013, 93:3, 360–372

Bibliographic databases:

UDC: 517.953
Received: 17.01.2011

Citation: V. L. Vereshchagin, “Explicit Solutions of Boundary-Value Problems for $(2+1)$-Dimensional Integrable Systems”, Mat. Zametki, 93:3 (2013), 333–346; Math. Notes, 93:3 (2013), 360–372

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