Abstract

Modulation instability of the Stokes wave solution of Whitham-Boussinesq systems derived from the water wave equations is analyzed in an attempt to examine Benjamin-Feir Instability for these systems and its connection with modulational instability of undular bores previously identified for these systems.  These Whitham-Boussinesq systems are in the weakly nonlinear limit, but with the full linear dispersion of the water wave equations.  This instability is related to previous work on the modulational instability of dispersive shock wave solutions of the Whitham-Boussinesq equations and of the full water wave equations.  The modulation instability on Stokes waves is analyzed for the previously identified wavenumbers associated with the water wave dispersive shock wave. We also examine the existence of solitary wave solutions of the Whitham-Boussinesq systems with details given of their shape and amplitude-velocity relation. Finally, the implications of these results for the Whitham-Boussinesq systems with the addition of surface tension for gravity-capillary RDSW and CDSW solutions are discussed, which is work in progress. This is joint work with Prof. Noel Smyth from the University of Edinburgh, Prof. Tim Marchant from the University of Melbourne, and Dr. Marcelo Flamarion from the Universidade Federal Rural de Pernambu.