The stability of capillary waves of finite amplitude

Abstract

Stability (in the sense of a relaxed definition of Lyapunov stability) of Crapper's exact solution for capillary waves is proven by Lyapunov's direct method. The wave surface is described using coefficients of the Laurent series of the conformal mapping of one period of the wave onto the unit circle interior (the Stokes coefficients). The Stokes coefficients are treated as generalized wave coordinates. The dynamical equations for a capillary wave are represented in the form of an infinite chain of Euler–Lagrange equations for the Stokes coefficients. A steady solution is found for these equations, and it is found to be the Crapper solution for capillary waves. The Lyapunov function is constructed basing on the energy and momentum conservation laws, and it is shown that it is positive definite with respect to arbitrary perturbations of the wave surface with period equal to the wavelength.