Asymmetric travelling wave solutions of the capillary-gravity Whitham Equation

Abstract: By a bifurcation argument we prove that the capillary-gravity Whitham equation features asymmetrical periodic travelling wave solution of arbitrarily small amplitude. Such waves exist only in the weak surface tension regime $0<T<\frac{1}{3}$ and are necessarily bimodal; they are located at double bifurcation points satisfying a certain symmetry breaking condition. Our bifurcation argument is an extension of the one applied by Ehrnstr\"om et al to find symmetric waves: Here, two additional scalar equations arise. Combining the variational structure of our problem with its translation symmetry, we show that these two additional equations are in fact linearly dependent, and can (at 'symmetry breaking' bifurcation points) be solved by incorporating the surface tension as a bifurcation parameter. Contrary to the symmetric case, only very specific modal pairs $(k_1,k_2)$ give rise to (small) asymmetrical periodic waves and we here provide a partial characterization of such pairs.