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Comments on inverse scattering for the Kadomtsev-Petviashvili equation. (English) Zbl 0544.76026

Mathematical methods in hydrodynamics and integrability in dynamical systems, La Jolla Inst. 1981, AIP Conf. Proc. 88, 211-228 (1982).
Summary: [For the entire collection see Zbl 0521.00026.]
S. V. Manakov [Physica 3D, 420 ff. (1981)] has given an inverse scattering formalism to solve the Kadomtsev-Petviashvili equation [B. B. Kadomtsev and V. I. Petviashvili, Sov. Phys., Dokl. 15, 539- 541 (1970; Zbl 0217.250)] with positive dispersion as an initial-value problem. A consequence of this formulation is that neither plane-wave solitons nor algebraically decaying ”lumps” evolve from spatially confined initial data. The present paper shows that an inverse scattering formulation similar to Manakov’s can be justified for initial data that are small enough in a certain norm, and that lumps are excluded by this requirement of smallness.

MSC:

76B25 Solitary waves for incompressible inviscid fluids