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STABILITY AND SYMMETRY OF SOLITARY-WAVE SOLUTIONS TO SYSTEMS MODELING INTERACTIONS OF LONG WAVES
 

Summary: STABILITY AND SYMMETRY OF SOLITARY-WAVE SOLUTIONS
TO SYSTEMS MODELING INTERACTIONS OF LONG WAVES
John Albert and Felipe Linares
Abstract. We consider systems of equations which arise in modelling strong interactions
of weakly nonlinear long waves in dispersive media. For a certain class of such systems, we
prove the existence and stability of localized solutions representing coupled solitary waves
travelling at a common speed. Our results apply in particular to the systems derived by
Gear and Grimshaw and by Liu, Kubota, and Ko as models for interacting gravity waves in
a density-stratified fluid. For the latter system, we also prove that any coupled solitary-wave
solution must have components which are all symmetric about a common vertical axis.
1. Introduction.
Model equations for long, weakly nonlinear waves in fluids are typically derived by
expanding the full equations of motion to first order in a small parameter determining
the size of the wave amplitude and inverse wavelength. (The use of one small parameter
to describe these two small quantities implicitly assumes a certain balance between them.)
The solutions of the model equations describe the slow evolution, due to weak dispersive
and nonlinear effects, of a wave which in the linear, non-dispersive limit corresponds to a
mode of a linear eigenvalue problem.
The well-known Korteweg-de Vries equation, for example, was derived in this way by
Benney [7] as a model for internal waves in a vertically stratified fluid. To zeroth order in ,

  

Source: Albert, John - Department of Mathematics, University of Oklahoma

 

Collections: Mathematics