Solitary wave solutions to a class of Whitham–Boussinesq systems

In this note, we study solitary wave solutions of a class of Whitham–Boussinesq systems which include the bidirectional Whitham system as a special example. The travelling wave version of the evolution system can be reduced to a single evolution equation, similar to a class of equations studied by Ehrnstrom et al. (Nonlinearity 25:2903–2936, 2012). In that paper, the authors prove the existence of solitary wave solutions using a constrained minimization argument adapted to noncoercive functionals, developed by Buffoni (Arch Ration Mech Anal 173:25–68, 2004), Groves and Wahlen (J Math Fluid Mech 13:593–627, 2011), together with the concentration–compactness principle.