Schrödinger equations with nonlinearity of integral type

Nakao Hayashi; Tohru Ozawa

doi:10.3934/dcds.1995.1.475

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1995, Volume 1, Issue 4: 475-484. Doi: 10.3934/dcds.1995.1.475

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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
2.
Department of Mathematics, Hokkaido University, Sapporo 060-0810

Received: October 31, 1994

Published: September 1995

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Abstract

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.

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Mathematics Subject Classification: 35D05, 35E15, 35Q20.

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