Water waves as a spatial dynamical system; infinite depth case

International audienceWe review the mathematical results on travelling waves in one or several superposed layers of potential flow, subject to gravity, with or without surface and interfacial tension, where the bottom layer in 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 3 examples where bifurcation occurs. i) The first example is when in moving a parameter, two pairs of imaginary eigenvalues merge into one pair of double eigenvalues, and then split into 4 symmetric complex conjugate eigenvalues. ii) The second example is when one pair of imaginary eigenvalues meet in 0, and disappear; iii) the third example is when the phenomenon described at 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