Local well-posedness for hyperbolic–elliptic Ishimori equation

In this paper we consider the hyperbolic–elliptic Ishimori initial-value problem with the form:{∂ts=s×□xs+b(ϕx1sx2+ϕx2sx1)on R2×[−1,1];Δϕ=2s⋅(sx1×sx2);s(0)=s0 where s(⋅,t):R2→S2⊂R3s(⋅,t):R2→S2⊂R3, × denotes the wedge product in R3R3, □x=∂x12−∂x22, b∈Rb∈R. We prove that such system is locally well-posed for small data s0∈HQσ0(R2;S2), σ0>3/2σ0>3/2, Q∈S2Q∈S2. The new ingredient is that we develop the methods of Ionescu and Kenig (2006) [6] and (2007) [7] to approach the problem in a perturbative way.