Kawahara dynamics in dispersive media

In a previous paper it was shown that in the long wave limit the water wave problem without surface tension can be described approximately by two decoupled KdV equations. Here, we study a model problem and prove that in a degenerate case which occurs for the water wave problem with surface tension and near other codimension two points at which the coefficient in front of the leading order dispersive term in the equation of motion vanishes, the long wave limit can be rigorously approximated by two decoupled Kawahara equations.