Existence of Large Amplitude Periodic Waves in Two-Fluid Flows of Infinite Depth

Abstract

Two-dimensional periodic traveling gravity waves in a two-fluid flow are considered, where the flow has no rigid boundaries. Each fluid is inviscid, incompressible, and irrotational and the density ratio of the upper fluid to the lower fluid is between zero and one. The governing equations are first transformed into a single nonlinear integral equation using the Hilbert transform and the corresponding integral operator is compact in certain Banach spaces after a cut-off function is introduced. By a global bifurcation theorem, it is shown that there exist periodic waves of large amplitude on the interface until either the bifurcation parameter goes to infinity or the function of the wave profile and its first-order derivative are not in the classical Holder space. It is also noted that the nonlinear integral equation is very general and can be used to study the waves of large amplitude numerically.