Full dispersion models in coastal oceanography

This thesis is on the rigorous justification of full dispersion models in coastal oceanography. The first full dispersion model was introduced by G. Whitham in 1967 to study the Stokes waves of maximal amplitude and the wavebreaking phenomenon. It is a full dispersion modification of the Korteweg-de Vries equations which have the same dispersion relation as the general water-waves model. Afterwards, numerous unidirectional and bidirectional full dispersion models were introduced in the litterature. In the first part of the thesis, we use classical techniques on free surface elliptic equations to derive rigorously some Whitham-Boussinesq and Whitham-Green-Naghdi models. In the second part, we justify rigorously a class of non-local quasi-linear Whitham-Boussinesq systems. In the next part, we justify rigorously Whitham's model using two different methods. One is adapted to the propagation of unidirectional waves and use pseudo-differential calculus. The other is adapted to the propagation of bidirectionnal waves. It is based on a generalisation of Birkhoff's normal form algorithm.