Rigorous derivation of the Whitham equations from the water waves equations in the shallow water regime

Abstract We derive the Whitham equations from water waves in shallow regime using two different methods, thus obtaining a direct and rigorous link between these models. The first one is based on construction of approximate Riemann invariants for Whitham–Boussinesq system adapted to unidirectional waves. second generalisation Birkhoff’s normal form algorithm almost smooth Hamiltonians bidirectional propagation. In both cases we clarify improved accuracy fully dispersive model with respect long wave Korteweg–de Vries approximation.