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

## TL;DR

This paper introduces a versatile method for analyzing dispersive shock waves in nonlinear systems with nonlocal Benjamin-Ono type dispersion, extending previous techniques beyond integrable equations and applying it to Calogero-Sutherland hydrodynamics.

## Contribution

A new approach to describe dispersive shock waves in nonlocal dispersive systems, applicable without integrability, and demonstrated on Calogero-Sutherland hydrodynamics.

## Key findings

- Classification of solution types from Riemann step problem
- Explicit formulas for DSW edge speeds and soliton amplitudes
- Excellent agreement between analytical results and numerical simulations

## Abstract

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.