On the Davey–Stewartson and Ishimori Systems

Abstract

We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödinger equations i u_t + Δ u = N(v), (t, x, y) ∈ R³, u(0, x, y) = u₀(x, y), (x, y) ∈ R² (A), where the Laplacian Δ = ∂² _x + ∂² _y, the solution u is a complex valued function, the nonlinear term N = N₁ + N₂ consists of the local nonlinear part N₁(v) which is cubic with respect to the vector v=(u,u_x,u_y,\overline{u},\overline{u}_{x},\overline{u}_{y}) in the neighborhood of the origin, and the nonlocal nonlinear part N₂(v) =(v, ∂ ^{− 1} _x K_x(v)) + (v, ∂^{− 1} _y K_y(v)), where (⋅, ⋅) denotes the inner product, $\partial _x^{ - 1} = \int_{ - \infty }^x {{\text{d}}x',} \partial _y^{ - 1} = \int_{ - \infty }^x {{\text{d}}x',}$ 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⁽²⁾ _x = K⁽⁴⁾ _x ≡ 0, K⁽³⁾ _y = K⁽⁶⁾ _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 $\mathcal{N}_1 = \left| u \right|^2 u,\mathcal{K}_x^{\left( 1 \right)} = \partial _y \left( {\left| u \right|^2 } \right),\mathcal{K}_y^{\left( 1 \right)} = \partial _x \left( {\left| u \right|^2 } \right)$, 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 $\mathcal{N}_1 = \left( {1 + \left| u \right|^2 } \right)^{ - 1} \bar u\left({\nabla u} \right)^2$, and $\mathcal{K}_y^{\left( 3 \right)} = - \mathcal{K}_y^{\left( 2 \right)} \left( {1 + \left| \right|^2 } \right)^{ - 2} \left( {u_x \bar u_y - \bar u_x u_y } \right)$. 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^{i θ} v) = e^{i θ} N(v) for all θ ∈ R.