Dispersive hydrodynamics in a non-local non-linear medium
Baqer, Saleh Ahmad
Dispersive shock wave (DSW), sometimes referred to as an undular bore in fluid mechanics, is a non-linear dispersive wave phenomenon which arises in non-linear dispersive media for which viscosity effects are negligible or non-existent. It is generated when physical quantities, such as fluid pressure, density, temperature and electromagnetic wave intensity, undergo rapid variations as time evolves. Its structure is a non-stationary modulated wavetrain which links two distinct physical states. DSW's occurrence in nature is quite omnipresent in classical/quantum fluids and non-linear optics. The main purpose of this thesis is to fully analyse all regimes for DSW propagation in the non-linear optical medium of a nematic liquid crystal in the defocusing regime. These DSWs are generated from step initial conditions for the intensity of the optical field and are resonant in that linear diffractive waves (termed dispersive waves in the context of fluid mechanics) are in resonance with the DSW, leading to a resonant wavetrain propagating ahead of it. It is found that there are six hydrodynamic regimes, which are distinct and require different solution methods. In previous studies, a reductive nematic Korteweg-de Vries equation and gas dynamic shock wave theory were used to understand all nematic dispersive hydrodynamics, which do not yield solutions in full agreement with numerical solutions. Indeed, the standard DSW structure disappears and a ``Whitham shock'' emerges for sufficiently large initial jumps. Asymptotic theory, approximate methods or Whitham's modulation theory are used to find solutions for these resonant DSWs in a given regime. It is found that for small initial intensity jumps, the resonant wavetrain is unstable, but that it stabilises above a critical jump height. It is additionally found that the DSW is unstable, except for small jump heights for which there is no resonance and large jump heights for which there is no standard DSW structure. The theoretical solutions are found to be in excellent agreement with numerical solutions of the nematic equations in all hydrodynamic regimes.