Forced Waves Near Resonance at a Phase-Speed Minimum

When a dispersive wave system is subject to forcing by a moving external disturbance, a maximum or minimum of the phase speed is associated with a critical forcing speed at which the linear response is resonant. Nonlinear effects can play an important part near such resonances, and the salient characteristics of the nonlinear response depend on whether the maximum or minimum of the phase speed is realized in the long-wave limit (zero wavenumber) or at a finite wavenumber. The focus here is on the latter case that, among other physical systems, applies to gravity-capillary waves on water of finite or infinite depth. The analysis, for simplicity, is based on a forced-damped fifth-order Korteweg-de Vries equation, a model problem that features a phase-speed minimum at a finite wavenumber. When damping is not too strong compared with forcing, multiple subcritical finite-amplitude steady-solution branches coexist with the small-amplitude response predicted by linear theory. For forcing speed well below critical, the transient response from rest approaches the small-amplitude state, but at speeds close to critical, jump phenomena can occur, and reaching a time-periodic state that involves shedding of wavepacket solitary waves is also possible.