Radiating dispersive shock waves in nonlocal optical media

TL;DR

This paper investigates the unique properties of dispersive shock waves in nematic liquid crystals, revealing their positive polarity, resonant radiation, and connection to KdV equations with fifth order dispersion, supported by numerical simulations.

Contribution

It introduces an asymptotic theory for radiating dispersive shock waves in nematic media, highlighting their distinct features and deriving their behavior using WKB and KdV approximations.

Findings

DSW has positive polarity and generates resonant radiation.

Lead soliton velocity matches classical shock velocity.

Asymptotic behavior governed by a fifth-order dispersive KdV equation.

Abstract

We consider the step Riemann problem for the system of equations describing the propagation of a coherent light beam in nematic liquid crystals, which is a general system describing nonlinear wave propagation in a number of different physical applications. While the equation governing the light beam is of defocusing nonlinear Schr\"odinger equation type, the dispersive shock wave (DSW) generated from this initial condition has major differences from the standard DSW solution of the defocusing nonlinear Schr\"odinger equation. In particular, it is found that the DSW has positive polarity and generates resonant radiation which propagates ahead of it. Remarkably, the velocity of the lead soliton of the DSW is determined by the classical shock velocity. The solution for the radiative wavetrain is obtained using the WKB approximation. It is shown that for sufficiently small initial jumps the nematic DSW is asymptotically governed by a Korteweg-de Vries equation with fifth order dispersion, which explicitly shows the resonance generating the radiation ahead of the DSW. The constructed asymptotic theory is shown to be in good agreement with the results of direct numerical simulations.