Dispersive shock waves in systems with nonlocal dispersion of Benjamin–Ono type

We develop a general approach to the description of dispersive shock waves (DSWs) for a class of nonlinear wave equations with a nonlocal Benjamin–Ono type dispersion term involving the Hilbert transform. Integrability of the governing equation is not a pre-requisite for the application of this method which represents a modification of the DSW fitting method previously developed for dispersive-hydrodynamic systems of Korteweg-de Vries (KdV) type (i.e. reducible to the KdV equation in the weakly nonlinear, long wave, unidirectional approximation). The developed method is applied to the Calogero–Sutherland dispersive hydrodynamics for which the classification of all solution types arising from the Riemann step problem is constructed and the key physical parameters (DSW edge speeds, lead soliton amplitude, intermediate shelf level) of all but one solution type are obtained in terms of the initial step data. The analytical results are shown to be in excellent agreement with results of direct numerical simulations.

我们针对一类具有涉及希尔伯特变换的非局部本杰明 - 昂诺型色散项的非线性波动方程，开发了一种描述色散激波（DSW）的通用方法。控制方程的可积性并非应用此方法的先决条件，该方法是对先前为科特韦格 - 德弗里斯（KdV）型色散 - 流体动力学系统（即在弱非线性、长波、单向近似下可简化为KdV方程）所开发的DSW拟合方法的一种改进。所开发的方法应用于卡洛杰罗 - 萨瑟兰色散流体动力学，针对该理论构建了由黎曼阶跃问题产生的所有解类型的分类，并根据初始阶跃数据得到了除一种解类型之外的所有解类型的关键物理参数（DSW边缘速度、先导孤子振幅、中间平台高度）。分析结果表明与直接数值模拟的结果高度吻合。