Radiating dispersive shock waves in non-local optical media

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 Schrödinger equation type, the undular bore generated from this initial condition has major differences from the standard undular bore solution of the defocusing nonlinear Schrödinger equation. In particular, it is found that the bore has positive polarity and generates resonant radiation which propagates ahead of it. Remarkably, the velocity of the lead soliton of the bore 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 undular bore is asymptotically governed by a Korteweg-de Vries equation with fifth order dispersion, which explicitly shows the resonance generating the radiation ahead of the bore. The constructed asymptotic theory is shown to be in good agreement with the results of direct numerical simulations.