On the Amplitude Equations for Weakly Nonlinear Surface Waves

Abstract

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.

Geometric Optics for Surface Waves in Nonlinear Elasticity

“…In this Section, we show formally that high frequency weakly nonlinear solutions to the nonlinear equations (2.1), (2.2) are governed by a nonlocal Burgers type equation that is similar to the ones derived in [BGC12] or [BGC15] for second-order equations, or in [Hun89,Mar10] for first-order hyperbolic systems. In the case of elastodynamics, the derivation of such amplitude equations dates back at least to [Lar83] for two-dimensional elasticity.…”

“…The first-order 'Hamiltonian' formulation (2.8) was already introduced in [Ser06] and it was used in [BGC12] in order to show structural properties for the amplitude equation which we shall derive here in a slightly more general context. The second-order formulation (2.7) was adopted in [BGC15] and we mainly follow this approach here.…”

“…The second-order formulation (2.7) was adopted in [BGC15] and we mainly follow this approach here. However, the equivalent formulation (2.8) will be used at some point, which is why we recall it here with notation similar to those in [BGC12]. The aim of the normal mode analysis is to determine the frequency pairs (τ − i γ, η) with γ ≥ 0 for which (2.7) admits a nontrivial solution.…”

“…This well-posedness problem has been solved in [Hun06,BG09], so constructing the leading profile of u app ε will not be our main concern here. However, in order to have a functional framework in which we can show that approximate solutions are close to exact solutions, we shall need to extend somewhat the constructions in [Lar83,Lar86] and recent generalizations in [BGC12,BGC15].…”

Geometric Optics for Rayleigh Wavetrains in d-Dimensional

“…The problem of constructing oscillatory, approximate solutions to the traction problem in nonlinear elasticity (0.1) has been considered by a number of authors including [Lar83,Hun06,BGC12,BGC16,CW16]. These papers construct 2-scale, WKB-type, approximate solutions consisting of a leading term and, in the case of [CW16], a first corrector as well.…”