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Zbl 0827.35120
Hayashi, Nakao; Saut, Jean-Claude
Global existence of small solutions to the Davey-Stewartson and the Ishimori systems. (English)
[J] Differ. Integral Equ. 8, No.7, 1657-1675 (1995). ISSN 0893-4983

Summary: We study the initial-value problems for the Davey-Stewartson systems and the Ishimori equations. Elliptic-hyperbolic and hyperbolic-elliptic cases were treated by the inverse scattering techniques. Elliptic-elliptic and hyperbolic-elliptic cases were studied without the use of the inverse scattering techniques. Existence of a weak solution to the Davey- Stewartson systems for the elliptic-hyperbolic case was also obtained in [{\it J. M. Ghidaglia} and {\it J. C. Saut}, Nonlinearity 3, No. 2, 475- 506 (1990; Zbl 0727.35111)] with a smallness condition on the data in $L^2$ and a blow-up result was also obtained for the elliptic-elliptic case. By using the sharp smoothing property of solutions to the linear Schrödinger equations the local existence of a unique solution to the Davey-Stewartson systems for the elliptic-hyperbolic and hyperbolic- hyperbolic cases was established in [{\it F. Linares} and {\it G. Ponce}, Ann. Inst. Henri Poincaré, Anal. nonlinéaire 10, No. 5, 523-548 (1993; Zbl 0807.35136)] in the usual Sobolev spaces with a smallness condition on the data.\par We prove the local existence of a unique solution to the Davey-Stewartson systems for the elliptic-hyperbolic and hyperbolic-hyperbolic cases in some analytic function spaces without a smallness condition on the data. Furthermore we prove existence of global small solutions of these equations for the elliptic-hyperbolic and hyperbolic-hyperbolic cases in some analytic function spaces.

MSC 2000:: *35Q55 NLS-like (nonlinear Schroedinger) equations
35D05 Existence of generalized solutions of PDE
35E15 Initial value problems of PDE with constant coefficients
76B15 Wave motions (fluid mechanics)

Keywords: local existence; unique solution; global small solutions

Citations: Zbl 0727.35111; Zbl 0807.35136

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