Abstract

Certain nonlinear approximations to the Euler equations of hydrodynamics in the weakly nonlinear and weakly dispersive waves regime will be discussed in the context of their associated initial-boundary value problems on the half-line and the finite interval. In particular, explicit solution formulae will be derived via the unified transform of Fokas for the linearizations of these Boussinesq-type systems in the case of nonzero boundary conditions. Furthermore, well-posedness will be established for a certain regularization of these systems in the context of an initial-boundary value problem that describes the dynamics of water waves in a basin with wall- boundary conditions. Numerical experiments illustrating the effect of the boundary conditions on the reflection of solitary waves by a vertical wall will also be discussed. This is joint work with Dimitrios Mitsotakis.