Weakly nonlinear surface waves and subsonic phase boundaries

The aim of this work is twofold. In a first, abstract part, it is shown how to derive an asymptotic equation for the amplitude of weakly nonlinear surface waves associated with neutrally stable undercompressive shocks. The amplitude equation obtained is a non-local generalization of Burgersʹ equation, for which an explicit stability condition is exhibited. This is an extension of earlier results by J. Hunter. The second part is devoted to `idealʹ subsonic phase boundaries, which were shown by the first author to be associated with linear surface waves. The amplitude equation for corresponding weakly non-linear surface waves is calculated explicitly and the stability condition is investigated analytically and numerically.