Comparison between Boussinesq‐ and Whitham–Boussinesq‐type systems

摘要

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 O1/ϵ$$ mathcal{O}left(1/sqrt{epsilon}right) $$, where ϵ>0$$ epsilon amp;amp;0 $$ 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 O1/ϵ$$ mathcal{O}left(1/sqrt{epsilon}right) $$.