Forced nonlinear gravity-capillary waves near the minimum phase speed

For Bond number less than 1/3, the minimum gravity-capillary phase speed, Cmin, is known to be the bifurcation point of wavepacket-type plane as well as fully localized solitary waves (lumps) on water of finite or infinite depth. Also, according to linear inviscid theory, the steady-state forced response to a moving 2-D or 3-D disturbance (Rayleigh's solution) becomes singular when the speed of the forcing approaches Cmin. We examine the role that plane solitary waves and lumps play in the nonlinear forced response near this critical speed, using as a model a forced fifth-order KP equation with an additional term representing the effect of dissipation. For sub-critical forcing speed, there are multiple steady-state finite-amplitude response branches. Moreover, for a range of transcritical forcing speeds, it is possible for the response, rather than reaching steady state, to remain transient owing to periodic shedding of solitary waves. The results are discussed in connection with experiments on forced deep-water gravity- capillary waves conducted by Prof J. Duncan's group.