Remarks on the Bifurcation of Two Dimensional Capillary-Gravity Waves of Finite Depth

Abstract

We consider the problem to determine the two dimensional water waves of permanent configuration. The problem is a bifurcation problem ([16]). In this paper, we define a mapping G between two Hilbert spaces and prove that the solutions of the above problem are in one-to-one correspondence to the zeros of the mapping G. Our most important contribution in this paper is to clarify the role of the aspect ratio, i.e., the ratio between the mean depth of the flow and the wave length. In particular, we prove that there is no degenerate bifurcation point, whatever the aspect ratio may be.