Comparison between Boussinesq- and Whitham–Boussinesq-type systems

In this paper, we consider a Whitham–Boussinesq-type system modeling surface water waves of an inviscid incompressible fluid layer. The system describes the evolution with time of surface waves of a liquid layer in the two-dimensional physical space. Using fixed point argument, we prove that the system is locally well-posed on the time scale of order (Formula presented.), where (Formula presented.) is the shallowness parameter measuring the ratio of amplitude of the wave to mean depth of fluid. We also show that the solution to the Whitham–Boussinesq system approximates the solution of a Boussinesq system on the time scale of order (Formula presented.).