On a two dimensional nonlocal shallow-water model

Abstract

In the present study we describe the asymptotic perturbation method to derive a two-dimensional nonlocal shallow-water model equation in the context of full water waves. Starting from the incompressible and irrotational governing equations in the three-dimensional water waves, we show that such a equation arises in the modeling of the propagation of shallow water waves over a flat bed. The resulting equation is a two dimensional Camassa-Holm equation-type with weakly transverse effect for the horizontal velocity component. The equation captures stronger nonlinear effects than the classical dispersive integrable equations like the Korteweg-de Vries and Kadomtsev-Petviashvili equations. We also address some properties of this model equation and how it relates to the surface wave. Analytically, we establish the local well-posedness of this model in a suitable Sobolev space. We then investigate formation of singularities and existence of traveling-wave solutions to this quasi-linear model equation with an emphasis on the understanding of weak transverse effect. Finally, we provide a Liouville-type property to obtain unique continuation result for the strong solution.