Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-16T03:26:13.305Z Has data issue: false hasContentIssue false
  • English
  • Français

Adiabatic Evolution of Coupled Waves for aSchrödinger-Korteweg-de Vries System

Published online by Cambridge University Press:  29 February 2012

W. Abou Salem
W. Abou Salem*
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, SK, S7N 5E6, Canada
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

The effective dynamics of interacting waves for coupled Schrödinger-Korteweg-de Vriesequations over a slowly varying random bottom is rigorously studied. One motivation forstudying such a system is better understanding the unidirectional motion of interactingsurface and internal waves for a fluid system that is formed of two immiscible layers. Itwas shown recently by Craig-Guyenne-Sulem [1] thatin the regime where the internal wave has a large amplitude and a long wavelength, thedynamics of the surface of the fluid is described by the Schrödinger equation, while thatof the internal wave is described by the Korteweg-de Vries equation. The purpose of thisletter is to show that in the presence of a slowly varying random bottom, the coupledwaves evolve adiabatically over a long time scale. The analysis covers the cases when thesurface wave is a stable bound state or a long-lived metastable state.

Keywords

Korteweg-de Vries equationSchrödinger equationcoupled waveseffective dynamicsadiabatic theorem
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 2: Solitary waves , 2012 , pp. 1 - 12
Copyright
© EDP Sciences, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

W. Craig, P. Guyenne, C. Sulem. Coupling between internal and surface waves, Natural Hazards, Special Issue on “Internal waves in the oceans and estuaries : modeling and observations”, (2010), doi :10.1007/s11069-010-9535-4, 26pp.
Craig, W., Guyenne, P., Sulem, C.. A Hamiltonian approach to nonlinear modulation analysis. Wave Motion 47 (2010), 552–563. CrossRefGoogle Scholar
van Groesen, E., Pudjaprasetya, S. R.. Uni-directional waves over slowly varying bottom. I. Derivation of a KdV-type of equation. Wave Motion 18 (1993), 345–370. CrossRefGoogle Scholar
Yoon, S. B., Liu, Philip L.-F.. A note on Hamiltonian for long water waves in varying depth. Wave Motion 20 (1994), 359–370. CrossRefGoogle Scholar
Dejak, S.I, Segal, I.M.. Long time dynamics of KdV solitary waves over a variable bottom. Comm. Pure Appl. Math. 59 (2006), 869–905. CrossRefGoogle Scholar
S.I. Dejak, B.L.G Jonsson. Long time dynamics of variable coefficient mKdV solitary waves. J. Math. Phys. 47 (2006), 072703, 16pp.
J. Holmer. Dynamics of KdV solitons in the presence of a slowly varying potential. IMRN (2011), doi :10.1093/imrn/rnq284, 31pp.
Holmer, J., Perelman, G., Zworski, M.. Effective dynamics of double solitons for perturbed mKdV. Commun. Math. Phys. 305 (2011), 363–425. CrossRefGoogle Scholar
C. Munoz. On the soliton dynamics under a slowly varying medium for generalized KdV equations. arxiv.org arXiv :0912.4725 [math.AP] (2009). To appear in Analysis and PDE.
Abou Salem, W., Fröhlich, J.. Adiabatic theorems for quantum resonances. Commun. Math. Phys. 273 (2007), 651–675. CrossRefGoogle Scholar
Kato, T.. On the adiabatic theorem of quantum mechanics. Phys. Soc. Jap. 5 (1958), 435–439. CrossRefGoogle Scholar
Avron, J.E., Elgart, A.. Adiabatic theorem without a gap condition. Commun. Math. Phys. 203 (1999), 445–463. CrossRefGoogle Scholar
Teufel, S.. A note on the adiabatic theorem without a gap condition. Lett. Math. Phys. 58 (2002), 261–266. CrossRefGoogle Scholar
Joye, A.. General adiabatic evolution with a gap condition. Commun. Math. Phys. 275 (2007), 139-162. CrossRefGoogle Scholar
Buslaev, V. S., Sulem, C.. Linear adiabatic dynamics generated by operators with continuous spectrum. Asymptotic Anal. 58 (2008), 17–45. Google Scholar
Elgart, A., Hagedorn, G. A.. An adiabatic theorem for resonances. Comm. Pure Appl. Math. 64 (2011), 1029–1058. CrossRefGoogle Scholar
Bona, J.L., Souganidis, P.E., Strauss, W.A.. Stability and instability of solitary waves of Korteweg de Vries type. Proc. Roy. Soc. London Ser. A 411 (1987), 395–412. CrossRefGoogle Scholar
Guillopé, L., Zworski, M.. Upper bounds on the number of resonances on noncompact Riemann e surfaces. J. Func. Anal. 129 (1995), 364–389. CrossRefGoogle Scholar
T. Kato. Perturbation Theory for Linear Operators. Springer-Verlag New York, 1991.
Hunziker, W.. Resonances, metastable states and exponential decay laws in perturbation theory. Commun. Math. Phys. 132 (1990), 177. CrossRefGoogle Scholar
Kenig, C.E., Ponce, G., Vega, L.. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math. 46 (1993), 527–620. CrossRefGoogle Scholar
J. Holmer, M. Zworski. Soliton interaction with slowly varying potentials. IMRN (2008), doi : 10.1093/imrn/rnn026, 36 pp.
Martel, Y., Merle, F.. Asymptotic stability of solitons for subcritical gKdV equations revisited. Nonlinearity 18 (2005), 55–80. CrossRefGoogle Scholar