In this work, the dispersive shock wave (DSW) solution of a Boussinesq Benjamin–Ono (BBO) equation, the standard Boussinesq equation with dispersion replaced by nonlocal Benjamin–Ono dispersion, is derived. This DSW solution is derived using two methods, DSW fitting and from a simple wave solution of the Whitham modulation equations for the BBO equation. The first of these yields the two edges of the DSW, while the second yields the complete DSW solution. As the Whitham modulation equations could not be set in Riemann invariant form, the ordinary differential equations governing the simple wave are solved using a hybrid numerical method coupled to the dispersive shock fitting which provides a suitable boundary condition. The full DSW solution is then determined, which is found to be in excellent agreement with numerical solutions of the BBO equation. This hybrid method is a suitable and relatively simple method to fully determine the DSW solution of a nonlinear dispersive wave equation for which the (hyperbolic) Whitham modulation equations are known, but their Riemann invariant form is not.

中文翻译：

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Boussinesq Benjamin-Ono 方程的色散冲击波

在这项工作中，Boussinesq Benjamin-Ono (BBO) 方程的色散冲击波 (DSW) 解被推导出来，标准 Boussinesq 方程用非局部 Benjamin-Ono 色散代替色散。该 DSW 解是使用两种方法推导出来的，DSW 拟合和 BBO 方程的 Whitham 调制方程的简单波解。第一个产生 DSW 的两条边，而第二个产生完整的 DSW 解。由于惠瑟姆调制方程不能以黎曼不变形式设置，控制简单波的常微分方程使用混合数值方法与色散激波拟合相结合，提供了合适的边界条件。然后确定完整的 DSW 解决方案，发现它与 BBO 方程的数值解非常一致。这种混合方法是一种合适且相对简单的方法，可以完全确定非线性色散波动方程的 DSW 解，其中（双曲线）惠瑟姆调制方程已知，但其黎曼不变量形式未知。