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Explicit solutions of boundary-value problems for (2 + 1)-dimensional integrable systems

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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.

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  1. Institute of Mathematics and Computer Center, Ural Science Center, Russian Academy of Sciences, Ufa, Russia

    V. L. Vereshchagin

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  1. V. L. Vereshchagin
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Correspondence to V. L. Vereshchagin.

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Original Russian Text © V. L. Vereshchagin, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 3, pp. 333–346.

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Vereshchagin, V.L. Explicit solutions of boundary-value problems for (2 + 1)-dimensional integrable systems. Math Notes 93, 360–372 (2013). https://doi.org/10.1134/S0001434613030024

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