Lie symmetry analysis, conservation laws and analytical solutions for chiral nonlinear Schrödinger equation in (2 + 1)-dimensions

In this work, we consider the chiral nonlinear Schrödinger equation in (2 + 1)-dimensions, which describes the envelope of amplitude in many physical media. We employ the Lie symmetry analysis method to study the vector field and the optimal system of the equation. The similarity reductions are analyzed by considering the optimal system. Furthermore, we find the power series solution of the equation with convergence analysis. Based on a new conservation law, we construct the conservation laws of the equation by using the resulting symmetries.&nbsp

Simulating lake dynamics: the effects of bathymetry and bottom drag

This work seeks, through numerical simulations as well as analysis, to derive from relatively simple models an intuitive understanding of the dynamics and behaviour of flow in lakes near the bottom boundary. The main body is divided into two equally important sections, the analysis and simulation of the effects of nonlinear (quadratic) bottom drag on the flow, and the simulation of the effects of topography on lake dynamics as it relates to the redistribution of sediment from the lakebed. The simulations all follow a structured scheme, beginning with relatively simple one-dimensional models to build intuition and proceeding to full two-dimensional simulations using the weakly nonhydrostatic shallow water equations. Thus this work seeks to build an understanding of the behaviour of the modified shallow water equations (a good representation of lake behaviour) and to analyze the effects of nonlinear drag and bottom topography on these systems. The nonlinear drag chapters demonstrate that the addition of a nonlinear friction term, while very efficient at removing energy from the system, also causes interesting new behaviour. In the pendulum (a good one dimensional analogy to the shallow water equations) the presence of nonlinear drag alters the parameter space enough to induce or destroy chaotic behaviour. A phenomenon worth considering in relation the shallow water equations. Additionally, the presence of drag causes as a cascade in spectral space, similar to the classical turbulent cascade. This work considers this effect and seeks to differentiate it from the turbulent cascade wherever and whenever possible. The final section of the thesis deals with the presence and effects of bottom topography (namely protrusions from the lake bed) on wave velocities due to a basin-scale seiche. This section examines both the dynamics of the system, through deflection about topography and the modification of the wave due to nonlinearity and bathymetry, as well as the relationship between lake dynamics and sediment redistribution. Finally, possible future directions are suggested as natural extensions to the work already done, as well as more sophisticated numerical models which could provide further insight into the problems discussed herein

Integrability, Recursion Operators and Soliton Interactions

This volume contains selected papers based on the talks,presented at the Conference "Integrability, Recursion Operators and Soliton Interactions", held in Sofia, Bulgaria (29 - 31 August 2012) at the Institute for Nuclear Research and Nuclear Energy of the Bulgarian Academy of Sciences. Included are also invited papers presenting new research developments in the thematic area. The Conference was dedicated to the 65-th birthday of our esteemed colleague and friend Vladimir Gerdjikov. The event brought together more than 30 scientists, from 6 European countries to celebrate Vladimir's scientific achievements. All participants enjoyed a variety of excellent talks in a friendly and stimulating atmosphere. The main topics of the conference were those where Vladimir has contributed enormously during his career: integrable nonlinear partial differential equations, underlying algebraic and geometric structures of the integrable systems, soliton solutions, soliton interactions, quantum integrable systems, discrete integrable systems and applications of the nonlinear models. The papers, included in this volume will be useful to researchers with interests in these areas