Gravity and Capillary-Gravity Periodic Travelling Waves for Two Superposed Fluid Layers, One Being of Infinite Depth

The mathematical study of 2D travelling waves in the potential flow of two superposed layers of perfect fluid, with free surface and interfaces (with or without surface tensions) and with the bottom layer of infinite depth, is set as an ill-posed reversible evolution problem, where the horizontal space variable plays the role of a "time". We give the structure of the spectrum of the linearized operator near equilibrium. This spectrum contains a set of isolated eigenvalues of finite multiplicities, a small number of which lie near or on the imaginary axis, and the entire real axis constitutes the essential spectrum, where there is no eigenvalue, except 0 in some cases. We give a general constructive proof of bifurcating periodic waves, adapting the Lyapunov-Schmidt method to the present (reversible) case where 0 (which is "resonant") belongs to the continuous spectrum. In particular we give the results for the generic case and for the 1 : 1 resonance case.