Math. Model. Nat. Phenom.
Volume 7, Number 2, 2012Solitary waves
|Page(s)||1 - 12|
|Published online||29 February 2012|
Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System
Department of Mathematics and Statistics, University of
Saskatchewan, SK, S7N
⋆ Corresponding author. E-mail: email@example.com
The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vries equations over a slowly varying random bottom is rigorously studied. One motivation for studying such a system is better understanding the unidirectional motion of interacting surface and internal waves for a fluid system that is formed of two immiscible layers. It was shown recently by Craig-Guyenne-Sulem  that in the regime where the internal wave has a large amplitude and a long wavelength, the dynamics of the surface of the fluid is described by the Schrödinger equation, while that of the internal wave is described by the Korteweg-de Vries equation. The purpose of this letter is to show that in the presence of a slowly varying random bottom, the coupled waves evolve adiabatically over a long time scale. The analysis covers the cases when the surface wave is a stable bound state or a long-lived metastable state.
Mathematics Subject Classification: 35Q35 / 35Q40 / 35Q51 / 81Q99
Key words: Korteweg-de Vries equation / Schrödinger equation / coupled waves / effective dynamics / adiabatic theorem
© EDP Sciences, 2012
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