While solitary waves and other coherent phenomena dominate the long-time behavior of solutions of nonlinear, dispersive wave equations, real phenomena usually features dissipation as well. In the absence of continual forcing, the coherent structures no longer exist and finite energy disturbances decay back to the rest state.

Motivated by some water wave experiments, we pose some delicate questions about the long-time asymptotics of this decay. Theory pertaining to this question has been worked out in a number of contexts, one or two of which will be discussed in a bit more depth.