Water waves as a spatial dynamical system; infinite depth case

Abstract

We review the mathematical results on traveling waves in one or several superposed layers of potential flow, subject to gravity, with or without surface and interfacial tension, where the bottom layer is infinitely deep. The problem is formulated as a "spatial dynamical system," and it is shown that the linearized operator of the resulting reversible system has an essential spectrum filling the real line. We consider three cases where bifurcation occurs. (i) The first case is when, in moving a parameter, two pairs of imaginary eigenvalues merge into one pair of double eigenvalues, and then split into four symmetric complex conjugate eigenvalues. (ii) The second case is when one pair of imaginary eigenvalues meet in 0, and disappear; (iii) the third case is when the phenomenon described in (ii) is superposed to the presence of another pair of imaginary eigenvalues sitting at finite distance from 0. We give a physical example for each case and more specially study the solitary waves and generalized solitary waves, emphasizing the differences, in the methods and in the results, between these cases and the finite depth case.