American Institute of Mathematical Sciences

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October  1995, 1(4): 475-484. doi: 10.3934/dcds.1995.1.475

Schrödinger equations with nonlinearity of integral type

Nakao Hayashi 1, and  Tohru Ozawa 2,

1. 

Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162, Japan
2. 

Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan

Received  November 1994 Published  August 1995

We consider the Cauchy problem for the nonlinear Schrödinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. Concerning the problem with finite initial time, we prove the global well-posedness in the largest space $L^2(\mathbb R^2)$. Concerning the problem with infinite initial time, we prove the existence of modified wave operators on a dense set of small and sufficiently regular asymptotic states.
Keywords: propagation of laser beams., Cauchy problem, Schrödinger equation.
Mathematics Subject Classification: 35D05, 35E15, 35Q2.
Citation: Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475
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