TY - JOUR

T1 - Radiating dispersive shock waves in non-local optical media

AU - El, Gennady A.

AU - Smyth, Noel F.

PY - 2016/3

Y1 - 2016/3

N2 - 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 (NLS) equation type, the dispersive shock wave (DSW) generated from this initial condition has major differences from the standard DSW solution of the defocusing NLS 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 Wentzel-Kramers-Brillouin approximation. It is shown that for sufficiently small initial jumps the nematic DSW is asymptotically governed by a Korteweg-de Vries equation with the 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.

AB - 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 (NLS) equation type, the dispersive shock wave (DSW) generated from this initial condition has major differences from the standard DSW solution of the defocusing NLS 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 Wentzel-Kramers-Brillouin approximation. It is shown that for sufficiently small initial jumps the nematic DSW is asymptotically governed by a Korteweg-de Vries equation with the 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.

KW - Dispersive shock wave

KW - Nematic liquid crystals

KW - Resonant radiation

KW - Undular bore

UR - http://www.scopus.com/inward/record.url?scp=84962909722&partnerID=8YFLogxK

U2 - 10.1098/rspa.2015.0633

DO - 10.1098/rspa.2015.0633

M3 - Article

AN - SCOPUS:84962909722

SN - 1364-5021

VL - 472

JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

IS - 2187

M1 - 20150633

ER -