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:
| [1] | Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Phys. Rev. Lett., 19, 1095 (1967), (GGKM) · Zbl 1061.35520 |
| [2] | M.D. Kruskal, this meeting.; M.D. Kruskal, this meeting. |
| [3] | Zakharov, V. E.; Faddeev, L. D., Funct. Anal. Appl., 5, 280 (1971) |
| [4] | Ablowitz, M. J.; Bar Yacov, D.; Fokas, A. S., Stud. Appl. Math., 69, 135 (1983) · Zbl 0527.35080 |
| [5] | Dodd, R. K.; Bullough, R. K., Physica Scripta, 20, 514 (1979) · Zbl 1063.37560 |
| [6] | Manakov, S. V., Phys., D3, 420 (1981) · Zbl 1194.35507 |
| [7] | Fokas, A. S.; Ablowitz, M. J., Stud. Appl. Math., 69, 211 (1983) · Zbl 0528.35079 |
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