Complete integrability of the Kadomtsev-Petviashvili equations in \(2+1\) dimensions. (English) Zbl 0615.35072
The authors discuss the complete integrability of the equations
\[
(u_ t+6uu_ x+u_{xxx})_ x=-3\alpha^ 2 u_{yy},
\]
where \(\alpha =i\) or \(\alpha =-1\), which appear to be an analogue in \(2+1\) dimensions of the Korteweg-de Vries equation. It is stated that the case \(\alpha =-1\) is solved by the ”D-bar” method, and the case \(\alpha =i\) by means of a nonlocal Riemann-Hilbert problem.
Reviewer: C.Constanda
MSC:
35Q99 | Partial differential equations of mathematical physics and other areas of application |
35A30 | Geometric theory, characteristics, transformations in context of PDEs |
35G20 | Nonlinear higher-order PDEs |
Keywords:
Kadomtsev-Petviashvili equations; complete integrability; Korteweg-de Vries equation; nonlocal Riemann-Hilbert problemReferences:
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