On the amplitude equations for weakly nonlinear surface waves

Nonlocal generalizations of Burgers’ equation were derived in earlier work by Hunter [Contemp. Math. 1989], and more recently by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009], as weakly nonlinear amplitude equations for hyperbolic boundary value problems admitting linear surface waves. The local-in-time well-posedness of such equations in Sobolev spaces was proved by Benzoni-Gavage [Diff. Int. Eq. 2009] under an appropriate stability condition originally pointed out by Hunter. The latter stability condition has also been shown to be necessary for well-posedness in Sobolev spaces in a previous work of the authors in collaboration with Tzvetkov [Adv. Math. 2011]. In this article, we show how the verification of Hunter’s stability condition follows from natural stability assumptions on the original hyperbolic boundary value problem, thus avoiding lengthy computations in each particular situation. When the original boundary value problem has a variational origin, we also show that the resulting amplitude equation has a Hamiltonian structure. Our analysis encompasses previous equations derived for nonlinear Rayleigh waves in elasticity. AMS subject classification: 35L53, 35L50, 74B20, 35L20.