On the davey-stewartson and ishimori systems

We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödinger equations IM, + Δu = N(v), (t,x,y) ∈ R³, u(0,x,y) = u₀(x,y), (x,y) ∈ R², (A) where the Laplacian Δ = ∂_x² + ∂_y², the solution u is a complex valued function, the nonlinear term N = N₁ + N₂ consists of the local nonlinear part N₁ (u) which is cubic with respect to the vector v = (u, u_x, u_y, ū, ̄_x, ū_y) in the neighborhood of the origin, and the nonlocal nonlinear part N₂(v) = (v, ∂_x^-1 K_x(v)) + (v, ∂_y^-1 K_y(v)), where (., .) denotes the inner product, ∂_x^-1 = ∫_-∞^x dx′ djc', ∂_y^-1 = ∫_-∞^y dx′ and the vectors K_x ∈ (C⁴(C⁶; C))⁶ and K_y ∈ (C⁴(C⁶; C))⁶ are quadratic with respect to the vector v in the neighborhood of the origin. We assume that the components K_x⁽²⁾ = K_x⁽⁴⁾ ≡ 0, K_y⁽³⁾ = K_y ≡ 0. In particular, Equation (A) includes two physical examples appearing in fluid dynamics. The elliptic-hyperbolic Davey-Stewartson system can be reduced to Equation (A) with N₁ = |u|²u, K_x⁽¹⁾ = ∂_y-(|ul²), K_y⁽¹⁾ = ∂_x(|u|²), and all the rest components of the vectors K_x and K_y are equal to zero. The elliptic-hyperbolic Ishimori system is involved in Equation (A), when N₁ = (1 + |u|²)^-1ū(∇u)², and K_x⁽³⁾ = -K_y⁽²⁾ = (1 + |u|²)^-2(u_xū_y)-(ū _xu_y). Our purpose in this paper is to prove the local existence in time of small solutions to the Cauchy problem (A) in the usual Sobolev space, and the global-in-time existence of small solutions to the Cauchy problem (A) in the weighted Sobolev space under some conditions on the complex conjugate structure of the nonlinear terms, namely if N(e^iθv) = e^iθ N(v) for all θ ∈ R.