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

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 sech2 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.