Abstract
Interfacial water waves of permanent profile between two fluids of different densities are considered. We will show that interfacial waves are generalizations of surface waves, which have been studied extensively in both mathematical and physical papers. The purpose of the present paper is to give a mathematical explanation for numerical results on bifurcations of surface waves by Shoji and ourselves. A hypothesis of degeneracy plays a key role in the present analysis. In fact, we showed in an early paper that a certain degenerate bifurcation point, if it is assumed to be present, can elucidate the complicated bifurcation structure of the surface waves by Shoji. However, in previous papers, we proved unexpectedly that any degenerate bifurcation point does not appear if we vary the depth of the flow. So, the idea of degeneracy has not been physically substantiated in the category of surface waves. In this paper we prove that such a degenerate bifurcation point actually exists when we vary the ratio of the propagation speeds between the upper and lower fluids. Consequently the complicated structure of the surface waves can be explained by regarding the surface waves as special cases of the interfacial waves.
