Linear and nonlinear waves on a finite interval

We illustrate the behaviour of a system of nonlinear dispersive PDEs modelling the propagation of water waves in a finite space interval, by comparing it with the behaviour of the corresponding linearized PDE system. We review how in general the interaction with the boundary destroys the nonlinear effects, reducing the evolution to an essentially linear dispersive phenomenon. In one particular case of homogeneous boundary conditions, it has been observed numerically that the evolution and in particular the solitary waves generated from the initial waveform persist beyond the interaction with the boundary, which is in this sense transparent. We show how in this case the solution of the linearized problem has a specific structure and propose that this can be used as a characterizing property of those boundary conditions that lead to “transparent” boundaries.