Generalized solitary waves and fronts in coupled Korteweg–de Vries systems

Abstract

A variety of problems in nonlinear science can be modelled by a system of two coupled long wave equations. In such systems, a resonance between a solitary wave of one of the two equations and a co-propagating periodic wave of the other equation can occur. The resulting wave is a generalized solitary wave, with non-vanishing oscillatory tails. It is shown that in the case of a ‘table-top’ solitary wave, which is solution to an extended Korteweg–de Vries equation with a cubic nonlinearity, the generalized solitary waves do not behave like standard sech 2 generalized solitary waves. In particular, it is shown that the oscillations can vanish in the tails or in the central core, but not in both simultaneously. A simplified model is introduced, which allows a better understanding of these stationary long wave solutions and the occurrence of embedded solitons.