Ratliff D J
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK.
Proc Math Phys Eng Sci. 2019 Mar;475(2223):20180784. doi: 10.1098/rspa.2018.0784. Epub 2019 Mar 13.
A mechanism for dispersion to automatically arise from the dispersionless Whitham Modulation equations (WMEs) is presented, relying on the use of a moving frame. The speed of this is chosen to be one of the characteristics which emerge from the linearization of the Whitham system, and assuming these are real (and thus the WMEs are hyperbolic) morphs the WMEs into the Korteweg-de Vries (KdV) equation in the boosted coordinate. Strikingly, the coefficients of the KdV equation are universal, in the sense that they are determined by abstract properties of the original Lagrangian density. Two illustrative examples of the theory are given to illustrate how the KdV may be constructed in practice. The first being a revisitation of the derivation of the KdV equation from shallow water flows, to highlight how the theory of this paper fits into the existing literature. The second is a complex Klein-Gordon system, providing a case where the KdV equation may only arise with the use of a moving frame.
提出了一种依赖于移动坐标系从无色散惠特姆调制方程（WMEs）自动产生色散的机制。其速度被选为从惠特姆系统线性化中出现的特征之一，并且假设这些特征是实数（因此WMEs是双曲型的），这会在加速坐标系中将WMEs转变为科特韦格 - 德弗里斯（KdV）方程。引人注目的是，KdV方程的系数是通用的，从某种意义上说，它们由原始拉格朗日密度的抽象性质决定。给出了该理论的两个示例，以说明在实践中如何构建KdV方程。第一个是重新审视从浅水流推导KdV方程，以突出本文理论如何融入现有文献。第二个是一个复克莱因 - 戈登系统，提供了一个只有使用移动坐标系才会出现KdV方程的情况。
