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Global existence of small solutions to the Davey-Stewartson and the Ishimori systems. (English) Zbl 0827.35120

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 [J. M. Ghidaglia and 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 [F. Linares and 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.

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:
 35Q55 NLS-like (nonlinear Schrödinger) equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35E15 Initial value problems of PDE with constant coefficients 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction