Global existence of small solutions to the Davey-Stewartson and the Ishimori systems

Abstract

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 ([2--4, 10, 13--15, 32] for the Davey-Stewartson systems and [28, 29, 33] for the Ishimori equations). Elliptic-elliptic and hyperbolic-elliptic cases were studied (in [16, 17] for the Davey-Stewartson systems and [31] for the Ishimori equations) without the use of the inverse scattering techniques. Existence of a weak solution to the Davey-Stewartson systems for the elliptic-hyperbolic case is also obtained in [16] 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 [30] 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.