Universal unfoldings of group invariant equations which model second and third harmonic resonant capillary-gravity waves

Abstract

An investigation is made into the free surface capillary-gravity waves caused by the interaction of the fundamental mode and a harmonic of either one-half or one-third its wavelength. The problem is studied by casting it as an integro-differential equation in a suitable function space. This equation is found to be invariant under certain group actions. The infinite dimensional problem is then reduced by means of the classical Lyapunov-Schmidt procedure to a set of finite equations known as the bifurcation equations. Although it is not possible to calculate the bifurcation equations exactly, the symmetries inherent in the problem enable us to make quite strong statements about the structure of the bifurcation equations. In particular, it allows us to replace them by a simpler set of equations, the solution set of which is qualitatively the same as that of the original equations. We then discuss unfoldings of the bifurcation equations. This is a procedure whereby a finite set of new parameters is adjoined to the equations. These possess the property that any perturbation of the equations can be represented by a certain choice of these parameters. Their physical significance is discussed and the bifurcaion diagrams are interpreted in the context of the original hydrodynamical problem.