  Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a
variational principle 'generically' admit linear surface waves, as was shown by
Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of
surface waves is expected to be governed by an amplitude equation that can be
derived by means of a formal asymptotic expansion. Amplitude equations for
weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci.
1983], Parker and co-workers [J. Elasticity 1985] in the framework of
elasticity, and by Hunter [Contemp. Math. 1989] for abstract hyperbolic
problems. They consist of nonlocal evolution equations involving a complicated,
bilinear Fourier multiplier in the direction of propagation along the boundary.
It was shown by the authors in an earlier work [Arch. Ration. Mech. Anal. 2012]
that this multiplier, or kernel, inherits some algebraic properties from the
original IBVP. These properties are crucial for the (local) well-posedness of
the amplitude equation, as shown together with Tzvetkov [Adv. Math., 2011].
Properties of amplitude equations are revisited here in a somehow simpler way,
for surface waves in a variational setting. Applications include various
physical models, from elasticity of course to the director-field system for
liquid crystals introduced by Saxton [Contemp. Math. 1989] and studied by
Austria and Hunter [Commun. Inf. Syst. 2013]. Similar properties are eventually
shown for the amplitude equation associated with surface waves at reversible
phase boundaries in compressible fluids, thus completing a work initiated by
Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009].
