Geometric Optics for Rayleigh Pulses in Nonlinear Elasticity

An amplitude equation for weakly nonlinear surface pulses in nonlinear elasticity was introduced by Lardner (Int J Eng Sci 21:1331–1342, 1983). This was a nonlocal, Burgers-type equation with a very complicated integral kernel. Hunter (Contemp Math 100:185–202, 1989) derived a similar equation for first-order conservation laws and proposed a solvability condition for this equation. Benzoni-Gavage (Differ Integral Equ 22(3–4):303–320, 2009) proved the sufficiency of this condition for local well-posedness in Sobolev spaces. More recently, in a general variational setting that includes elasticity, Benzoni-Gavage and Coulombel (Amplitude equations for weakly nonlinear surface waves in variational problems. arXiv:1510.01119 and this proceedings, 2015) derived the amplitude equation for surface pulses (and proved its well-posedness) directly from the second-order formulation of the equations. The solution of the amplitude equation for surface pulses in nonlinear elasticity gives the leading term of an approximate geometric optics solution to the underlying elasticity equations. Previous work has left open the question of whether exact surface pulse solutions to these equations exist on a time interval independent of the wavelength ɛ, and also the question of whether the approximate solutions constructed in earlier work converge in some sense to the exact solutions as ɛ → 0. Here we describe joint work with Coulombel and Williams (Geometric optics for surface waves in nonlinear elasticity. Memoirs of the AMS, to appear) that resolves both questions. This work is dedicated to my friend and collaborator Guy Métivier on the occasion of his 65th birthday.