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April  2005, 13(3): 811-825. doi: 10.3934/dcds.2005.13.811

Well-posedness for the Benney-Roskes/Zakharov- Rubenchik system

Gustavo Ponce 1, and  Jean-Claude Saut 2,

1. 

Department of Mathematics, University of California, Santa Barbara, CA 93106, United States
2. 

UMR de Mathématiques, Université de Paris-Sud, Bât. 425, 91405 Orsay Cedex, France

Received  October 2004 Revised  February 2005 Published  May 2005

We consider the system obtained by D. J. Benney-G. J. Roskes and V. E. Zakharov-A. M. Rubenchik to model the interaction of low amplitude high frequency waves with low frequency, acoustic type waves. We reduce it to a nonlinear Schrödinger equation with nonlinear terms involving nonlocal terms and derivatives of the unknown. Using various smoothing effects associated to the Schrödinger group and the structure of the nonlinearity we prove that the Cauchy problem is locally well-posed in $H^s(\mathbb R^n), s>n/2$ where $n=2,3$.
Keywords: local well-posedness., Wave propagation.
Mathematics Subject Classification: Primary: 35Q35; Secondary: 35Q5.
Citation: Gustavo Ponce, Jean-Claude Saut. Well-posedness for the Benney-Roskes/Zakharov- Rubenchik system. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 811-825. doi: 10.3934/dcds.2005.13.811
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