Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Solitary waves in nonlinear dispersive systems

Pages: 313 - 378, Volume 2, Issue 3, August 2002      doi:10.3934/dcdsb.2002.2.313

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Jerry Bona - Department of Mathematics, Statistics and Computer Science, The University of Illinois at Chicago , 851 S. Morgan Street MC 249, Chicago, Illinois 60607-7045, United States (email)
Hongqiu Chen - The Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, The Ph.D Program in Mathematics, The Graduate Center of the City University of New York, New York, NY 10016, United States (email)

Abstract: Evolution equations that feature both nonlinear and dispersive effects often possess solitary-wave solutions. Exact theory for such waves has been developed and applied to single equations of Korteweg-de Vries type, Schrödinger-type and regularized long-wave-type for example. Much less common has been the analysis of solitary-wave solutions for systems of equations. The present paper is concerned with solitary travelling-wave solutions to systems of equations arising in fluid mechanics and other areas of science and engineering. The aim is to show that appropriate modification of the methods coming to the fore for single equations may be effectively applied to systems as well. This contention is demonstrated explicitly for the Gear- Grimshaw system modeling the interaction of internal waves and for the Boussinesq systems that arise in describing the two-way propagation of long-crested surface water waves.

Keywords:  Concentration compactness, dispersion, Fréchet space, internal waves, positive operator, solitary wave, travelling waves.
Mathematics Subject Classification:  35A15, 35B38, 35B40, 35B65, 35Q35, 35Q51, 35Q53, 35Q72, 45G10, 45G15, 45M20, 45P05, 46A04, 46A40, 46A50, 46M20, 46N20, 46T20, 47G10,47G30, 47H07, 47H10, 47H11, 47J20, 47J25, 47J30, 55M25, 76B55, 76B70, 76B15, 76B25, 76B45, 76M30.

Received: October 2001;      Revised: January 2002;      Available Online: May 2002.