Rigorous weakly nonlinear geometric optics for surface waves

Marcou, Alice

doi:10.3233/asy-2010-0996

Cited by 25 publications

(56 citation statements)

References 14 publications

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“…This result is formulated in Theorem 12.6, which is a more precise version of Theorem 0.1. The writing of part 3 was strongly influenced by [Mar10], which treats surface waves for first-order conservation laws with linear, homogeneous boundary conditions Cu = 0, and the second Chapter of [Mar11], which constructs approximate solutions consisting of a leading term U 2 and (part of a) first corrector U 3 for a simplified version of the SVK model. We give more detail later about our debt to these works; here we just note that the main novelty in our construction of approximate solutions lies in our construction of arbitrarily high order profiles.…”

Section: Part 1 General Introduction and Main Resultsmentioning

confidence: 99%

“…18 We considered (v, Dv) instead of Dxv here in order to include a proof of Lemma 4.4(b). 19 A similar use of the product estimate is made in section 1.10 of [Mar10] in her study of first-order hyperbolic conservation laws.…”

Section: Uniform Estimates For the Coupled Nonlinear Systemsmentioning

confidence: 86%

“…When proving properties of the | · | s,k,ε norms the following simple observation (used already for the | · | s,0,ε norm in [Mar10]) is quite useful.…”

Section: Norms and Basic Estimatesmentioning

confidence: 99%

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Geometric Optics for Rayleigh Wavetrains in d-Dimensional

Wheeler

Williams²

2018

SIAM J. Math. Anal.

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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.

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Section: Part 1 General Introduction and Main Resultsmentioning

confidence: 99%

Section: Uniform Estimates For the Coupled Nonlinear Systemsmentioning

confidence: 86%

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Geometric Optics for Rayleigh Wavetrains in d-Dimensional

Wheeler

Williams²

2018

SIAM J. Math. Anal.

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“…The degeneracy of this condition may occur in different ways, and we consider here the case where surface waves of finite energy occur, see [BGS07,Chapter 7]. Let us mention right away the work of Marcou [Mar10] for first order nonlinear systems in the surface wavetrain case. In [Mar10] Marcou provided a complete justification of weakly nonlinear geometric optics expansions for surface wavetrains arising in first order conservation laws with linear, homogeneous boundary conditions.…”

Section: Chapter 1 General Introductionmentioning

confidence: 99%

“…Let us mention right away the work of Marcou [Mar10] for first order nonlinear systems in the surface wavetrain case. In [Mar10] Marcou provided a complete justification of weakly nonlinear geometric optics expansions for surface wavetrains arising in first order conservation laws with linear, homogeneous boundary conditions. In contrast our main focus will be on the second order hyperbolic systems with fully nonlinear, nonhomogeneous boundary conditions arising in elasticity theory.…”

Section: Chapter 1 General Introductionmentioning

confidence: 99%

Geometric Optics for Surface Waves in Nonlinear Elasticity

Coulombel

Williams

2020

Memoirs of the AMS

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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 integrodifferential 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 ε to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength ε, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to u ε on a time interval independent of ε. 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.

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Geometric Optics for Surface Waves on the Plasma–Vacuum Interface: Higher Order Expansion

Secchi,

Yuan

2024

CIM Series in Mathematical Sciences

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Rigorous weakly nonlinear geometric optics for surface waves

Cited by 25 publications

References 14 publications

Geometric Optics for Rayleigh Wavetrains in d-Dimensional

Geometric Optics for Rayleigh Wavetrains in d-Dimensional

Geometric Optics for Surface Waves in Nonlinear Elasticity

Geometric Optics for Surface Waves on the Plasma–Vacuum Interface: Higher Order Expansion

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