On localized solutions in nonlinear Faraday resonance

The dynamics of a nonlinear modulated cross-wave of resonant frequency ω1 and carrier frequency ω ≈ ω1 is considered. The wave is excited in a long channel of width 6 that contains water of depth d, which is subjected to a vertical oscillation of frequency 2ω. As has been shown by Miles (1984b), the complex amplitude satisfies a cubic Schrödinger equation with weak damping and parametric driving. The stability of its solitary wave solution is considered here in various parameter regions. We find that in a certain regime the solitary wave is stable. Completely new is the result of instability outside this parameter regime. The instability has also been verified numerically. It is shown that the final stage of solitary wave instability is a cnoidal-wave-type solution.