Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation

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January 1999, 5(1): 93-106. doi: 10.3934/dcds.1999.5.93

Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation

Nakao Hayashi ^1, , Elena I. Kaikina ^2, and Pavel I. Naumkin ^3,

1.	Department of Applied Mathematics, Science University of Tokyo, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162
2.	Departamento de Ciencias Básicas, Instituto Tecnologico de Morelia, CP 58080, Morelia, Michoacan, Mexico
3.	Instituto de Física y Matemáticas, Universidad Michoacana, AP 2-82, CP 58040, Morelia, Michoacana, Mexico

Received September 1997 Revised May 1998 Published October 1998

Abstract
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We study the Cauchy problem for a nonlinear Schrödinger equation which is the generalization of a one arising in plasma physics. We focus on the so called subcritical case and prove that when the initial datum is "small", the solution exists globally in time and decays in time just like in the linear case. For a certain range of the exponent in the nonlinear term, we prove that the solution is asymptotic to a "final state" and the nonexistence of asymptotically free solutions. The method used in this paper is based on some gauge transformation and on a certain phase function.

Keywords: large time asymptotics., Derivative nonlinear Schrödinger equation.

Mathematics Subject Classification: 35Q5.

Citation: Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93

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