Near-critical reflection of internal waves

We consider a two-dimensional weakly nonlinear Boussinesq system for internal waves in a domain with a sloping boundary.
Starting from an incident wave packet hitting the boundary with a near-critical angle with respect to the angle of the slope, we construct an approximate solution given by: the incident wave packet, some boundary layer terms, a second harmonic wave packet, a mean flow and other nonlinear terms of lower order.
We prove the stability of the approximation and the convergence to the weak solutions of the original system.
Our main reference is the paper ‘’Near-critical reflection of internal waves’’ by T. Dauxois and W. R. Young.
This is a joint work with Anne-Laure Dalibard and Laure Saint-Raymond.