Douglas Svensson Seth: Non-symmetric Solutions to the Capillary-Gravity Whitham Equation

Abstract:

Over the last two decades the Whitham equation has been proven to be a particularly rich shallow water wave model for gravity waves. In the presence of surface tension, the natural generalisation of this water wave model is the capillary-gravity Whitham equation. Here we focus on periodic travelling waves that solve the capillary-gravity Whitham equation. These have been fully characterized in the case of small and even waves. It is complemented by the work presented in this talk characterizing small non-symmetric periodic travelling waves. By non-symmetric we mean waves that they are not even under any spacial translation. Due to the translational symmetry of the problem this is the counterpart to even waves. Such non-symmetric solutions are far more scarce than symmetric ones and small non-symmetric waves do not exist for strong surface tensions nor for the Whitham equation without surface tension. However, for weak surface tension it is possible to construct arbitrarily small non-symmetric solutions.