## Abstract

We study the initial value problem for the two-dimensional nonlinear nonlocal Schrödinger equations IM, + Δu = N(v), (t,x,y) ∈ R^{3}, u(0,x,y) = u_{0}(x,y), (x,y) ∈ R^{2}, (A) where the Laplacian Δ = ∂_{x}^{2} + ∂_{y}^{2}, the solution u is a complex valued function, the nonlinear term N = N_{1} + N_{2} consists of the local nonlinear part N_{1} (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 N_{2}(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} ∈ (C^{4}(C^{6}; C))^{6} and K_{y} ∈ (C^{4}(C^{6}; 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 N_{1} = |u|^{2}u, K_{x}^{(1)} = ∂_{y}-(|ul^{2}), 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 N_{1} = (1 + |u|^{2})^{-1}ū(∇u)^{2}, and K_{x}^{(3)} = -K_{y}^{(2)} = (1 + |u|^{2})^{-2}(u_{x}ū_{y})-(ū _{x}u_{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^{iθ}v) = e^{iθ} N(v) for all θ ∈ R.

Original language | English |
---|---|

Pages (from-to) | 53-81 |

Number of pages | 29 |

Journal | Mathematical Physics Analysis and Geometry |

Volume | 2 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1999 |

## Keywords

- Davey-stewartson system
- Elliptic-hyperbolic case
- Ishimori system
- Nonlocal nonlinear schrödinger equation