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

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

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