论文标题
WHITHAM调制理论，用于广义Whitham方程和模块不稳定性的一般标准
Whitham modulation theory for generalized Whitham equations and a general criterion for modulational instability
论文作者
论文摘要
提出了将WHITHAM方程作为地表水波的模型，该模型结合了二次通量非线性$ f（u）= \ tfrac {1} {1} {2} {2} u^2 $ korteweg-de vries方程和全线性分散$ω（k）= \ sqr的whate wray wray wray wraws wraws whate \ k}适当缩放变量。本文提出并分析了Whitham模型对由一般非线性通量函数$ f（u）$和一般线性分散关系$ω（k）$组成的单向非线性波方程的概括。假设存在该广义WHITHAM方程的周期性波动解决方案，则在WHITHAM调制理论的背景下研究了它们的缓慢调制。多个量表计算得出调制方程，这是一个三种保护定律的系统，这些系统描述了周期性波动波数，振幅和平均值的缓慢演变。在弱的非线性限制中，确定了建立严格的高光质性和调制方程的真实非线性的一般$ f（u）$和$ω（k）$方面的明确，简单的标准。该结果被解释为用于模量不稳定的广义灯塔 - Whitham标准。
The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity $f(u) = \tfrac{1}{2}u^2$ of the Korteweg-de Vries equation and the full linear dispersion relation $Ω(k) = \sqrt{k\tanh k}$ of uni-directional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux function $f(u)$ and a general linear dispersion relation $Ω(k)$. Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit, simple criteria in terms of general $f(u)$ and $Ω(k)$ establishing the strict hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill-Whitham criterion for modulational instability.
