Doctor of Philosophy
School of Mathematics and Applied Statistics
Dispersive shock waves (DSWs), also termed undular bores in fluid mechanics, are generated from a jump discontinuity. DSWs form due to the balance between nonlinearity and dispersion and have an oscillating wave structure where the leading and trailing edges have different velocities. DSW has been investigated intensively over the past few decades, ever since Whitham’s pioneering invention of modulation theory  and Gurevich and Pitaevsky’s construction of the DSW solution for the Korteweg-de Vries equation . The theory was subsequently used to study DSWs governed by integrable equations. Then, based on Whitham’s and Gurevich and Pitaevsky’s research, El proposed the framework of a DSW fitting method which enables the analysis of DSWs governed by non-integrable equations [37, 41].
In this thesis, we consider the analysis of DSW in three different applications, all governed by non-integrable equations. The first is the analysis of the propagation of an optical DSW in a defocussing colloidal medium. The equations governing nonlinear light propagation in a colloidal medium consist of an NLS-type equation for the beam and an algebraic equation for the medium response. Solutions for the leading and trailing edges of the colloidal DSW are found using El’s theory.
The second is an investigation of the DSWs governed by the nonlocal Whitham equation. This equation allows the study of short wavelength effects, that led to peaked cusped waves within the DSW. The equation combines the weak nonlinearity of the KdV equation with full linear dispersion. Various dispersion relations are considered, for surface gravity waves, the intermediate long wave equation and a model dispersion relation introduced by Whitham to investigate the 120o peaked Stokes wave of highest amplitude. El’s DSW fitting method is used to find the leading (solitary wave) and trailing (linear wave) edges. This method is found to produce results in excellent agreement with numerical solutions up until the lead solitary wave of the DSW reaches its highest amplitude. Numerical solutions show that the DSWs for the water wave and Whitham peaking kernels become modulationally unstable and evolve into multi-phase wavetrains after a critical amplitude, which is just below the DSW of maximum amplitude.
The third is the investigation of DSWs in quadratic media. A quadratic medium gives rise to a second harmonic generation, which is a nonlinear process that induces two photons with same frequency to interact with each other and generates a photon with twice the energy as before. As a second harmonic is considered, a phase locking assumption is required for the analytic solution of DSW in the quadratic system. The DSW fitting technique is again used to determine the leading and trailing edges. Excellent agreements between theory and numerical solutions are found for all three problems considered.
An, Xin, Dispersive shock waves in optical and fluid media, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2019. https://ro.uow.edu.au/theses1/458