非线性弹性表面波的几何光学
Geometric optics for surface waves in nonlinear elasticity
摘要：
本文研究半空间非线性弹性力学方程的高频解。我们考虑表面波（或更准确地说，瑞利波）产生的一般类各向同性超弹性模型，其中特别包括圣维南基尔霍夫系统。自20世纪80年代以来，许多作者已经完成了关于非线性发展方程的制定和适定性的工作，该方程的（精确）解给出了基础弹性方程的\emph{approximate}瑞利波解的首项。这个发展方程，我们称之为“振幅方程”，是一个非局部Burgers型的积分微分方程。我们首先回顾并提供振幅方程理论的一些扩展。本文的其余部分是致力于一个严格的证明，在2D的精确，高振荡，瑞利波解决方案$u^\eps$的非线性弹性方程存在于一个固定的时间间隔独立的波长$\eps$，和近似瑞利波解的振幅方程的分析提供的确实是在一个精确的意义上接近$u^\eps$的时间间隔独立的$\eps$。本文主要集中在瑞利波的情况下，是\emph{pulses}，它具有连续傅立叶频谱的轮廓，但我们的方法同样适用于波列的情况下，其傅立叶频谱是离散的。
英文摘要：
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 \emph{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 $u^\eps$ to the nonlinear elasticity equations exist on a fixed time interval independent of the wavelength $\eps$, and that the approximate Rayleigh wave solution provided by the analysis of the amplitude equation is indeed close in a precise sense to $u^\eps$ on a time interval independent of $\eps$. The paper focuses mainly on the case of Rayleigh waves that are \emph{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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