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, ux, uy, ū, ̄x, ūy) in the neighborhood of the origin, and the nonlocal nonlinear part N2(v) = (v, ∂x-1 Kx(v)) + (v, ∂y-1 Ky(v)), where (., .) denotes the inner product, ∂x-1 = ∫-∞x dx′ djc', ∂y-1 = ∫-∞y dx′ and the vectors Kx ∈ (C4(C6; C))6 and Ky ∈ (C4(C6; C))6 are quadratic with respect to the vector v in the neighborhood of the origin. We assume that the components Kx(2) = Kx(4) ≡ 0, Ky(3) = Ky ≡ 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, Kx(1) = ∂y-(|ul2), Ky(1) = ∂x(|u|2), and all the rest components of the vectors Kx and Ky are equal to zero. The elliptic-hyperbolic Ishimori system is involved in Equation (A), when N1 = (1 + |u|2)-1ū(∇u)2, and Kx(3) = -Ky(2) = (1 + |u|2)-2(uxūy)-(ū xuy). 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(eiθv) = eiθ N(v) for all θ ∈ R.
- Davey-stewartson system
- Elliptic-hyperbolic case
- Ishimori system
- Nonlocal nonlinear schrödinger equation