Geometric Optics for Rayleigh Wavetrains in d-Dimensional

Abstract

A Rayleigh wave is a type of surface wave that propagates in the boundary of an elastic solid with traction (or Neumann) boundary conditions. Since the 1980s much work has been done on the problem of constructing a leading term in an approximate solution to the rather complicated second-order quasilinear hyperbolic boundary value problem with fully nonlinear Neumann boundary conditions that governs the propagation of Rayleigh waves. The question has remained open whether or not this leading term approximate solution is really close in a precise sense to the exact solution of the governing equations. We prove a positive answer to this question for the case of Rayleigh wavetrains in any space dimension d ≥ 2. The case of Rayleigh pulses in dimension d = 2 has already been treated by Coulombel and Williams. For highly oscillatory Rayleigh wavetrains we are able to construct high-order approximate solutions consisting of the leading term plus an arbitrary number of correctors. Using those high-order solutions we then perform an error analysis which shows (among other things) that for small wavelengths the leading term is close to the exact solution in L ∞ on a fixed time interval independent of the wavelength. The error analysis is carried out in a somewhat general setting and is applicable to other types of waves for which high order approximate solutions can be constructed.

Weakly nonlinear surface waves in magnetohydrodynamics

“…The transport operator in (4.32) is governed by the group velocity associated with the manifold along which the Lopatinskii determinant vanishes. This observation is a general fact that we can directly check for our particular problem, see [Mar10,CW18,WW17]. Let us now define a bilinear symmetric operator B that acts on C ∞ , 2π−periodic functions in θ as follows:…”

“…Let us observe that in [Mar10], functions in S ± are chosen not to depend on the fast tangential variable θ (the same in [WW17]). It does not seem possible to use this framework here due to the form of the source terms in the WKB cascade below.…”

“…It does not seem possible to use this framework here due to the form of the source terms in the WKB cascade below. Our source terms differ from those in [Mar10,WW17] because we deal here with a free boundary problem and we consider additional fixed top and bottom boundaries. We believe that our extension of the functional framework might be useful in other contexts which also give rise to surface waves on free discontinuities.…”

“…The well-posedness of nonlocal equations of the type (4.33) has been systematically studied in [Hun06, BG09, Mar10, CW18, WW17] in either the pulse or wavetrain framework (k ∈ R or k ∈ Z). The most convenient references for our purpose here are [Hun06,WW17] where the following result is proved 5 . For notational convenience, we introduce H s , s ≥ 0, as the Sobolev space of functions g ∈ H s (T 2 y × T θ ) with zero mean with respect to θ, and H ∞ := ∩ s H s .…”

Weakly nonlinear surface waves on the plasma-vacuum interface

“…In [25,Theorem 4.2], the constant K depends on s, but it is shown to be independent of s in [52]. The global existence for (4.20) is an open question.…”