Geodesic
Explicit Solutions of Boundary-Value Problems for $(2+1)$-Dimensional Integrable Systems
Matematičeskie zametki, Tome 93 (2013) no. 3, pp. 333-346

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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, dressing method
Mots-clés : Gelfand–Levitan–Marchenko equation, soliton solution. 
@article{MZM_2013_93_3_a1,
     author = {V. L. Vereshchagin},
     title = {Explicit {Solutions} of {Boundary-Value} {Problems} for $(2+1)${-Dimensional} {Integrable} {Systems}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {333--346},
     publisher = {mathdoc},
     volume = {93},
     number = {3},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_93_3_a1/}
}
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V. L. Vereshchagin. Explicit Solutions of Boundary-Value Problems for $(2+1)$-Dimensional Integrable Systems. Matematičeskie zametki, Tome 93 (2013) no. 3, pp. 333-346. http://geodesic.mathdoc.fr/item/MZM_2013_93_3_a1/