This work is devoted to the analysis of high frequency solutions to theequations of nonlinear elasticity in a half-space. We consider surface waves(or more precisely, Rayleigh waves) arising in the general class of isotropichyperelastic models, which includes in particular the Saint Venant-Kirchhoffsystem. Work has been done by a number of authors since the 1980s on theformulation and well-posedness of a nonlinear evolution equation whose (exact)solution gives the leading term of an emph{approximate} Rayleigh wave solutionto the underlying elasticity equations. This evolution equation, which we referto as "the amplitude equation", is an integrodifferential equation of nonlocalBurgers type. We begin by reviewing and providing some extensions of the theoryof the amplitude equation. The remainder of the paper is devoted to a rigorousproof in 2D that exact, highly oscillatory, Rayleigh wave solutions $u^eps$ tothe nonlinear elasticity equations exist on a fixed time interval independentof the wavelength $eps$, and that the approximate Rayleigh wave solutionprovided by the analysis of the amplitude equation is indeed close in a precisesense to $u^eps$ on a time interval independent of $eps$. The paper focusesmainly on the case of Rayleigh waves that are emph{pulses}, which haveprofiles with continuous Fourier spectrum, but our method applies equally wellto the case of wavetrains, whose Fourier spectrum is discrete.
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机译：这项工作旨在分析半空间中非线性弹性的高频解决方案。我们考虑在Isotropiceperelastic模型的一般类别中产生的表面波（或更精确，瑞利波），其特定地包括圣文尼亚 - 柯彻诺伊系统。自20世纪80年代以来，工作已经完成了许多作者，这是一种关于非线性演化方程的交易和良好的作者，其（精确）解决方案给出了 emph {近似}瑞利波解决方案的前导术语。我们将其引用为“幅度方程”的这种演化方程是非团烧结器类型的积分积分等式。我们首先审查和提供幅度方程的理论的一些延伸。纸张的其余部分在2D中致力于严格，高度振荡的瑞利波解决方案$ U ^ EPS $ u ^ eps $ the nearlyear弹性方程在一个独立于波长$ eps $的固定时间间隔内存在一个固定的时间间隔内，并且默德利近似通过幅度方程分析的波解决方案确实是在$ u ^ eps $上的预ony indy $ eps $的时间间隔收缩。本文侧重于瑞利波的情况下是 emph {脉冲}的情况，其中具有连续傅里叶频谱的预备，但是我们的方法同样适用于波浪的情况，其傅立叶频谱是离散的。
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