Resonant generation and refraction of dispersive shock waves in one-dimensional nonlinear Schrödinger flows

In the Thesis, two important theoretical problems arising in the theory of one-dimensional defocusing nonlinear Schrödinger (NLS) flows are investigated analytically and numerically: (i) the resonant generation of dispersive shock waves (DSWs) in one-dimensional NLS flow past a broad repulsive penetrable barrier; and (ii) the interaction of counter-propagating DSW and a simple rarefaction wave (RW), which is referred to as the DSW refraction problem. The first problem is motivated by the recent experimental observations of dark soliton radiation in a cigar-shaped BEC by sweeping through it a localised repulsive potential; the second problem represents a dispersive-hydrodynamic counterpart of the classical gas-dynamics problem of the shock wave refraction on a RW, and, apart from its theoretical significance could also find applications in superfluid dynamics. Both problems also naturally arise in nonlinear optics, where the NLS equation is a standard mathematical model and the `superfluid dynamics of light' can be used for an all-optical modelling of BEC flows. The main results of the Thesis are as follows: (i) In the problem of the transcritical flow of a BEC through a wide repulsive penetrable barrier an asymptotic analytical description of the arising wave pattern is developed using the combination of the localised ``hydraulic'' solution of the 1D Gross-Pitaevskii (GP) equation with repulsion (the defocusing NLS equation with an added external potential) and the appropriate exact solutions of the Whitham-NLS modulation equations describing the resolution of the upstream and downstream discontinuities through DSWs. We show that the downstream DSW effectively represents the train of dark solitons, which can be associated with the excitations observed experimentally by Engels and Atherton (2008). (ii) The refraction of a DSW due to its head-on collision with the centred RW is considered in the frameworks of two one-dimensional defocusing NLS models: the standard cubic NLS equation and the NLS equation with saturable nonlinearity, the latter being a standard model for the light propagation through photorefractive optical crystals. For the cubic nonlinearity case we present a full asymptotic description of the DSW refraction by constructing appropriate exact solutions of the Whitham modulation equations in Riemann invariants. For the NLS equation with saturable nonlinearity, whose modulation system does not possess Riemann invariants, we take advantage of the recently developed method for the DSW description in non-integrable dispersive systems to obtain key parameters of the DSW refraction. In both problems, we undertake a detailed analysis of the flow structure for different parametric regimes and calculate physical quantities characterising the output flows in terms of relevant input parameters. Our modulation theory analytical results are supported by direct numerical simulations of the corresponding full dispersive initial value problems (IVP).