The Cauchy problem for the hyperbolic–elliptic Ishimori system and Schrödinger maps**Any opinions, findings and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Abstract

We show an improved local in time existence and uniqueness result for Schrödinger maps and for the hyperbolic–elliptic nonlinear system proposed by Ishimori in analogy with the two-dimensional classical continuous isotropic Heisenberg spin (2d-CCIHS) chain. The proof uses fairly standard gauge geometric tools and energy estimates in combination with Kenig's version of the Koch–Tzvetkov method, to obtain a priori estimates for classical solutions to certain dispersive equations.