A normal form for nonlinear resonance of embedded solitons

A normal form for nonlinear resonance of embedded solitons is derived for a coupled two–wave system that generalizes the second–harmonic–generating model. This wave system is non–Hamiltonian in general. An embedded soliton is a localized mode of the nonlinear system that coexists with the linear wave spectrum. It occurs as a result of a codimension–one bifurcation of non–local wave solutions. Nonlinearity couples the embedded soliton and the linear wave spectrum and induces a one–sided radiation–driven decay of embedded solitons. The normal form shows that the embedded soliton is semi–stable, i.e. it survives under perturbations of one sign, but is destroyed by perturbations of the opposite sign. When a perturbed embedded soliton sheds continuous wave radiation, the radiation amplitude is generally not minimal, even if the wave system is Hamiltonian. The results of the analytical theory are confirmed by numerical computations.