论文标题
非线性Schrödinger近似之外的极化高频波传播
Polarized high-frequency wave propagation beyond the nonlinear Schrödinger approximation
论文作者
论文摘要
本文在包括Klein-Gordon方程和Maxwell-Lorentz系统的设置中研究了具有小参数的一类半线性双曲程方程系统的高度振荡解决方案。这里的兴趣在于，从某种意义上说，解决方案的溶液是最大的误差，解决方案中的振荡仅取决于一种满足分散性关系与给定波矢量出现在初始波浪包中的频率。这种极化溶液的构建和分析是使用调制的傅立叶扩展进行的。这种方法包括较高的谐波，并在小参数中对偏振溶液的近似值产生近似值，远远超出了通过非线性schrödinger方程的已知一阶近似值。极化溶液的给定构造是显式的，此外，对于每个进一步的近似顺序，也可以使用线性schrödinger方程，并且可以访问直接数值近似。
This paper studies highly oscillatory solutions to a class of systems of semilinear hyperbolic equations with a small parameter, in a setting that includes Klein--Gordon equations and the Maxwell--Lorentz system. The interest here is in solutions that are polarized in the sense that up to a small error, the oscillations in the solution depend on only one of the frequencies that satisfy the dispersion relation with a given wave vector appearing in the initial wave packet. The construction and analysis of such polarized solutions is done using modulated Fourier expansions. This approach includes higher harmonics and yields approximations to polarized solutions that are of arbitrary order in the small parameter, going well beyond the known first-order approximation via a nonlinear Schrödinger equation. The given construction of polarized solutions is explicit, uses in addition a linear Schrödinger equation for each further order of approximation, and is accessible to direct numerical approximation.
