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 nonlocal generalization of Burgers’ 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 ‘ideal’ subsonic phase boundaries, which were shown by the first author to be associated with linear surface waves. The amplitude equation for corresponding weakly nonlinear surface waves is calculated explicitly and the stability condition is investigated analytically and numerically. 2000 Mathematics Subject Classification: 35C20; 35L50; 35L67; 35R35; 76T10.