Local well-posedness for hyperbolic-elliptic Ishimori equation

In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem with the form: [GRAPHICS] where s(center dot, t) : R-2 -> S-2 subset of R-3, x denotes the wedge product in R-3, rectangle(x)= partial derivative(2)(x1) - partial derivative(2)(x2) 42, b is an element of R. We prove that such system is locally well-posed for small data s(0) is an element of H-Q(sigma 0)(R-2;S-2), sigma(0) > 3/2, Q is an element of S-2. The new ingredient is that we develop the methods of lonescu and Kenig (2006) [6] and (2007) [7] to approach the problem in a perturbative way. (C) 2012 Elsevier Inc. All rights reserved.