Nonlinear geometric optics for reflecting uniformly stable pulses

J. Hyper. Differential Equations

Williams

2014

Self Cite

In this companion paper to our study of amplification of wavetrains [CGW13], we study weakly stable semilinear hyperbolic boundary value problems with pulse data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency in the hyperbolic region. As a consequence of this degeneracy there is again an amplification phenomenon: outgoing pulses of amplitude O(ε 2 ) and wavelength ε give rise to reflected pulses of amplitude O(ε), so the overall solution has amplitude O(ε). Moreover, the reflecting pulses emanate from a radiating pulse that propagates in the boundary along a characteristic of the Lopatinskii determinant.In the case of N ×N systems considered here, a single outgoing pulse produces on reflection a family of incoming pulses traveling at different group velocities. Unlike wavetrains, pulses do not interact to produce resonances that affect the leading order profiles. However, pulse interactions do affect lower order profiles and so these interactions have to be estimated carefully in the error analysis. Whereas the error analysis in the wavetrain case dealt with small divisor problems by approximating periodic profiles by trigonometric polynomials (which amounts to using a high frequency cutoff), in the pulse case we approximate decaying profiles with nonzero moments by profiles with zero moments (a low frequency cutoff). Unlike the wavetrain case, we are now able to obtain a rate of convergence in the limit describing convergence of approximate to exact solutions.

Section: Error Analysismentioning

confidence: 82%

Section: Error Analysismentioning

confidence: 98%

See 1 more Smart Citation

Amplification of pulses in nonlinear geometric optics

J. Hyper. Differential Equations

Williams

2014

Self Cite

“…In the pulse setting the presence of Fourier spectrum arbitrarily close to k = 0 makes it impossible to construct such "strongly evanescent" profiles; indeed, now δ = δ(k) → 0 as k → 0. In the case of surface waves given by pulses, as in other problems involving pulses [CW13,CW14], there is no hope of constructing high order approximate solutions. Indeed, in the pulse setting it is usually impossible to construct high order correctors even in problems where the uniform Lopatinski condition is satisfied [CW13].…”

Section: Chapter 1 General Introductionmentioning

confidence: 99%

Geometric Optics for Surface Waves in Nonlinear Elasticity

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.

“…(c)) on the F i (resp. H i ).8 Elements of VF with terms involving no triple products were called functions of type F in[CW13,CW14].…”

mentioning

confidence: 99%

The Mach stem equation and amplification in strongly nonlinear geometric optics

American Journal of Mathematics

Williams²

2017

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We study highly oscillating solutions to a class of weakly well-posed hyperbolic initial boundary value problems. Weak well-posedness is associated with an amplification phenomenon of oscillating waves on the boundary. In the previous works [CGW14, CW14], we have rigorously justified a weakly nonlinear regime for semilinear problems. In that case, the forcing term on the boundary has amplitude O(ε 2 ) and oscillates at a frequency O(1/ε). The corresponding exact solution, which has been shown to exist on a time interval that is independent of ε ∈ (0, 1], has amplitude O(ε). In this paper, we deal with the exact same scaling, namely O(ε 2 ) forcing term on the boundary and O(ε) solution, for quasilinear problems. In analogy with [CGM03], this corresponds to a strongly nonlinear regime, and our main result proves solvability for the corresponding WKB cascade of equations, which yields existence of approximate solutions on a time interval that is independent of ε ∈ (0, 1]. Existence of exact solutions close to approximate ones is a stability issue which, as shown in [CGM03], highly depends on the hyperbolic system and on the boundary conditions; we do not address that question here.This work encompasses previous formal expansions in the case of weakly stable shock waves [MR83] and two-dimensional compressible vortex sheets [AM87]. In particular, we prove well-posedness for the leading amplitude equation (the "Mach stem equation") of [MR83] and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms. The latter case is solved under a crucial nonresonant assumption and a small divisor condition.