Dispersive shock waves for the Boussinesq Benjamin–Ono equation

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.