  The travelling wave problem for a particular bidirectional Whitham system
modelling surface water waves is under consideration. This system firstly
appeared in [Dinvay, Dutykh, Kalisch 2018], where it was numerically shown to
be stable and a good approximation to the incompressible Euler equations. In
subsequent papers [Dinvay 2018], [Dinvay, Selberg, Tesfahun 2019] the
initial-value problem was studied and well-posedness in classical Sobolev
spaces was proved. Here we prove existence of solitary wave solutions and
provide their asymptotic description. Our proof relies on a variational
approach and a concentration-compactness argument. The main difficulties stem
from the fact that in the considered Euler-Lagrange equation we have a
non-local operator of positive order appearing both in the linear and
non-linear parts. Our approach allows us to obtain solitary waves for a
particular Boussinesq system as well.
