Solitary wave solutions of a Whitham–Boussinesq system

The travelling wave problem for a particular bidirectional Whitham system modelling surface water waves is under consideration. This firstly appeared in Dinvay et al. (2019), where it was numerically shown to be stable and good approximation the incompressible Euler equations. In subsequent papers (Dinvay, 2019; al., 2019) initial-value studied well-posedness classical Sobolev spaces proved. Here we prove existence of solitary solutions provide their asymptotic description. Our proof relies on variational approach concentration-compactness argument. main difficulties stem from fact that considered Euler–Lagrange equation have non-local operator positive order appearing both linear non-linear parts. allows us obtain Boussinesq as well.