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
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.
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.
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Mathematical Notes, 2013, 93:3, 360–372
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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\paper Explicit Solutions of Boundary-Value Problems for $(2+1)$-Dimensional Integrable Systems
\jour Mat. Zametki
\jour Math. Notes
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