Double three-wave interaction of four waves: Lax representations and exact solutions

Abstract

The nonlinear resonant interaction of coherent waves is a fundamental process in the study of wave phenomena which has received a great deal of attention in its many aspects. In the present article a system of four interacting waves which constitute two resonant triplets is considered. The system is described, in a simplified model, by a Hamiltonian system of eight autonomous ordinary differential equations, with time as the independent variable; both positive and negative energy waves are allowed in the interaction. Two distinct Lax representations for this system, two new classes of exact solutions in terms of elliptic functions, a solution in the form of a convergent generic Laurent series expansion around a movable pole in the independent variable, and a stabilization criterion for the explosive instability that may occur when waves of different energy sign interact are obtained herein.