On the polynomial swallowtail and cusp singularities of stochastic Burgers equation

• Main Author: Reynolds, Christopher
• Published: Swansea University 2002
• Subjects: 519
• Online Access: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.638648

This thesis is concerned with singularities of the stochastic heat and Burgers equations. We study the classification of caustics (shockwaves) for Burgers equation and the level surfaces of the corresponding heat equation. Particular attention is paid to two examples of a two dimensional caustic, namely the semicubical parabolic cusp and the polynomial swallowtail. These examples, whose names have been adopted in recognition of Thom's list of seven elementary catastrophes, may be viewed as special cases of the larger class of initial functions <i>S₀</i>(<i>x₀</i>) = <i>f </i>(<i>x₀</i>) + <i>g</i>(<i>x</i>₀)<i>y₀</i> where <i>f </i>and <i>g </i>are polynomials in <i>x₀</i>. The thesis is structured as follows: Chapter 1 introduces many of the concepts required throughout the thesis. In particular the stochastic heat and Burgers equations are introduced and the notion of shockwaves discussed. In Chapter 2 we restrict ourselves to the deterministic free case and set about deriving a polynomial initial condition that produces a swallowtail type caustic. This is considered for both the two and three dimensional cases. In Chapter 3 the examples of the cusp and polynomial swallowtail are considered under the presence of white noise. The stochastic heat kernel is derived by a direct approach and used to obtain explicit formulae for the stochastic caustic and corresponding level surfaces. Chapter 4 is dedicated to the study of hot and cool parts of the caustic. Building upon the work of Truman, Davies and Zhao we develop a new method for determining whether one side of the caustic is hot or cool and show, that under a certain type of noise, only the deterministic case need be considered. In Chapter 5 we consider touching points of the pre-curves and show how this leads to the concept of turbulent times in the stochastic case. We derive a stochastic process whose zeros are the turbulent times for a particular class of examples and study the properties of this process. In Chapter 6 we repeat much of our earlier analysis in the presence of a harmonic oscillator potential.