# Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System

Mathematical Modelling of Natural Phenomena (2012)

- Volume: 7, Issue: 2, page 1-12
- ISSN: 0973-5348

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topAbou Salem, W.. "Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System." Mathematical Modelling of Natural Phenomena 7.2 (2012): 1-12. <http://eudml.org/doc/222294>.

@article{AbouSalem2012,

abstract = {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 [1] 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.},

author = {Abou Salem, W.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {Korteweg-de Vries equation; Schrödinger equation; coupled waves; effective dynamics; adiabatic theorem},

language = {eng},

month = {2},

number = {2},

pages = {1-12},

publisher = {EDP Sciences},

title = {Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System},

url = {http://eudml.org/doc/222294},

volume = {7},

year = {2012},

}

TY - JOUR

AU - Abou Salem, W.

TI - Adiabatic Evolution of Coupled Waves for a Schrödinger-Korteweg-de Vries System

JO - Mathematical Modelling of Natural Phenomena

DA - 2012/2//

PB - EDP Sciences

VL - 7

IS - 2

SP - 1

EP - 12

AB - 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 [1] 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.

LA - eng

KW - Korteweg-de Vries equation; Schrödinger equation; coupled waves; effective dynamics; adiabatic theorem

UR - http://eudml.org/doc/222294

ER -

## References

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