The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearity of the Korteweg–de Vries equation and the full linear dispersion relation of unidirectional 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 and a general linear dispersion relation . 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 and 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.

中文翻译：

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广义 Whitham 方程的 Whitham 调制理论和调制不稳定性的一般准则

Whitham 方程被提议作为地表水波模型，它结合了Korteweg-de Vries 方程的二次通量非线性和适当缩放的变量中单向重力水波的完整线性色散关系。本文提出并分析了 Whitham 模型对由一般非线性通量函数和一般线性色散关系组成的单向非线性波动方程的推广。. 假设这个广义惠瑟姆方程存在周期性行波解，在惠瑟姆调制理论的背景下研究它们的慢调制。多尺度计算产生调制方程，这是一个由三个守恒定律组成的系统，描述了周期性行波的波数、幅度和平均值的缓慢演变。在弱非线性极限中，确定了广义和建立调制方程的严格双曲性和真正非线性方面的明确的、简单的标准。该结果被解释为调制不稳定性的广义 Lighthill-Whitham 标准。