Nonlinear and Localized Modes in Hydrodynamics and Vortex Dynamics

This dissertation, "Nonlinear and Localized Modes in Hydrodynamics and Vortex Dynamics" by Lai-pan, Yip, 葉禮彬, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of the thesis entitled Nonlinear and Localized Modes in Hydrodynamics and Vortex dynamics Submitted by Lai-Pan YIP For the degree of Master of Philosophy at the University of Hong Kong In July 2007 Soliton refers to the wave travels with permanent shape which results from the balance between dispersive and nonlinear effects. Since its discovery, it has received considerable research attention. Extensive studies on the application of soliton solution and nonlinear evolution equations in different physical models have been carried out. The remarkable particle behaviour of soliton has also been demonstrated both theoretically and experimentally in various studies. The main focus of the present work is to study the nonlinear evolution equations in some physical systems and obtain the corresponding soliton solution. To deal with the nonlinear evolution equations, the Hirota's bilinear method is adopted to obtain the exact soliton. Previous studies have shown that this method is applicable in various types of nonlinear evolution equations, and it does not involve sophisticated mathematical techniques, but simple differentiation. In hydrodynamics, the evolution of water packets on water surface of finite depth can be described by the nonlinear Schrodinger (NLS) equation. For situations where higher order nonlinear effects need to be restored, a family of higher order NLS equations iis obtained. Such equations are important in the study of the modulational stability of water wave and optical systems involve additional nonlinearity. In particular, the solitary solution of the derivative nonlinear Schrodinger (DNLS) equation of Chen-Lee-Liu (CLL) type is investigated under different dispersion coefficients. Special exact solutions in the form of solitary pulse are obtained by employing special chirp factors and wavenumbers. The 1-soliton and 2-soliton solutions are obtained and they show special features of pulse broadening or compression. The soliton solution is also found to be applicable in the capillarity model. A nonlinear system of equations governing the potential flow of a fluid in the presence of capillarity effects is shown to be reducible to a resonant Davey-Stewartson (RDS) type system, with a class of three-parameter free energy functions. The equation exhibits the usual cubic nonlinearity present in the classical NLS system, together with an additional nonlinear term involving the modulus of the wave envelope. The system is transformed into bilinear equations and the 1-soliton and 2-soliton solutions are obtained. The dynamic of vortex soliton is investigated numerically. More precisely, the dynamic evolution of two dimensional vortex in a periodic channel with slip boundary condition is studied using a semi-Lagrangian code. The small-scale initial vorticity distribution is driven to large-scale quasi-stationary dipole or multi-pole structure by the inverse cascade. The geometrical effect of the channel, namely aspect ratio a (channel width over its stream-wise period) is examined specifically. As the ratio increases, the relaxed flow undergoes a sequence of bifurcations, including dipole number, shape of vortex cores etc. Vortex structures of "dipole array," "distorted dipole" and "2-jet structure" are also identified. For specific values of aspect ratio, the stream-vorticity iirelation of the relaxed flow is found to obey the sinh-Poisson equation. The approximate analytical des...