Amplitude Equations for Weakly Nonlinear Surface Waves in Variational Problems

Abstract

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 236(2):409–446, 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 Eng Sci 21(11):1331–1342, 1983), Parker and co-workers (J Elast 15(4):389–426, 1985) in the framework of elasticity, and by Hunter (Nonlinear surface waves. In: Current progress in hyberbolic systems: Riemann problems and computations (Brunswick, 1988). Contemporary mathematics, vol 100. American Mathematical Society, pp 185–202, 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 (Benzoni-Gavage and Coulombel Arch Ration Mech Anal 205(3):871–925, 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 (Dynamic instability of the liquid crystal director. In: Current progress in hyperbolic systems: Riemann problems and computations (Brunswick, 1988). Contemporary mathematics, vol 100. American Mathematical Society, Providence, pp 325–330, 1989) and studied by Austria and Hunter (Commun Inf Syst 13(1):3–43, 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 57(3–4):1463–1484, 2009).