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Geometric Optics for Surface Waves in Nonlinear Elasticity
About this Title
Jean-François Coulombel, CNRS and Université de Nantes, Laboratoire de mathématiques Jean Leray (UMR CNRS 6629), 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France and Mark Williams, University of North Carolina, Mathematics Department, CB 3250, Phillips Hall, Chapel Hill, NC 27599
Publication: Memoirs of the American Mathematical Society
Publication Year:
2020; Volume 263, Number 1271
ISBNs: 978-1-4704-4037-4 (print); 978-1-4704-5650-4 (online)
DOI: https://doi.org/10.1090/memo/1271
Published electronically: March 2, 2020
MSC: Primary 35L70, 74B20, 78A05
Table of Contents
Chapters
- 1. General introduction
- 2. Derivation of the weakly nonlinear amplitude equation
- 3. Existence of exact solutions
- 4. Approximate solutions
- 5. Error Analysis and proof of Theorem 3.8
- 6. Some extensions
- A. Singular pseudodifferential calculus for pulses
Abstract
This work is devoted to the analysis of high frequency solutions to the equations of nonlinear elasticity in a half-space. We consider surface waves (or more precisely, Rayleigh waves) arising in the general class of isotropic hyperelastic models, which includes in particular the Saint Venant-Kirchhoff system. Work has been done by a number of authors since the 1980s on the formulation and well-posedness of a nonlinear evolution equation whose (exact) solution gives the leading term of an approximate Rayleigh wave solution to the underlying elasticity equations. This evolution equation, which we refer to as “the amplitude equation”, is an integro-differential equation of nonlocal Burgers type. We begin by reviewing and providing some extensions of the theory of the amplitude equation. The remainder of the paper is devoted to a rigorous proof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^\varepsilon$ to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength $\varepsilon$, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to $u^\varepsilon$ on a time interval independent of $\varepsilon$. The paper focuses mainly on the case of Rayleigh waves that are pulses, which have profiles with continuous Fourier spectrum, but our method applies equally well to the case of wavetrains, whose Fourier spectrum is discrete.- Giuseppe Alì and John K. Hunter, Nonlinear surface waves on a tangential discontinuity in magnetohydrodynamics, Quart. Appl. Math. 61 (2003), no. 3, 451–474. MR 1999831, DOI 10.1090/qam/1999831
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