Explicit solutions of boundary-value problems for \((2+1)\)-dimensional integrable systems. (English) Zbl 1307.35269
Math. Notes 93, No. 3, 360-372 (2013); translation from Mat. Zametki 93, No. 3, 333-346 (2013).
Summary: 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.
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35C05 | Solutions to PDEs in closed form |
| 35C08 | Soliton solutions |
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 solutionReferences:
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