Global existence of small solutions to the Ishimori system without exponential decay on the data

Abstract

We study the initial value problem for the Ishimori system. Elliptic-hyperbolic and hyperbolic-elliptic cases were treated by inverse scattering techniques [6, 7], [9]. Elliptic-elliptic and hyperbolic-elliptic cases were studied in [8] without using inverse scattering techniques. In the previous paper [4] we proved the global existence of small solutions to the Ishimori and the Davey-Stewartson systems in the elliptic-hyperbolic and hyperbolic-hyperbolic cases under exponential decay on the data. Our purpose in this paper is to remove that condition in the case of the Ishimori system.