Nonlinear elastic surface gravity waves: Continuous and discontinuous solutions

An asymptotic method for the periodic solutions to forced nonlinear surface gravity waves is presented. The phenomena studied are governed by the equation utt − uxx + eF(u, ux, uxxt, ut, …) = 0. The forcing is through the boundaries of the finite region and the forcing frequency is in resonance with the free waves in the linearized system. A perturbation scheme valid at resonance is developed. it is shown that the first order perturbation beyond the linear solution brings no contribution and the second order perturbation leads to a nonlinear integro-differential equation. In the steady nondissipative case, it becomes possible to integrate this equation completely to obtain a cubic algebraic equation. The study of this equation reveals the existence of the discontiuous solutions along with the continuous ones. Furthermore, the exact solution to the integro-differential equation helps to explain the meaning and to assess the range of validity of the commonly used modal (i.e. Fourier) decomposition. The effect of dissipation is also studied and a method of multiple time scales is used for the study of the transient behavior including the evolution of the catastrophes and the stability of various solution branches.