Fullwood, Timothy Brent (1995) Pattern formation and travelling waves in reaction-diffusion equations. PhD thesis, University of Warwick.

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Abstract

This thesis is about pattern formation in reaction - diffusion equations, particularly
Turing patterns and travelling waves. In chapter one we concentrate
on Turing patterns. We give the classical approach to proving the existence of
these patterns, and then our own, which uses the reversibility of the associated
travelling wave equations when the wave speed is zero. We use a Lyapunov -
Schmidt reduction to prove the existence of periodic solutions when there is
a purely imaginary eigenvalue. We pay particular attention to the bifurcation
point where these patterns arise, the 1: 1 resonance. We prove the existence of
steady patterns near a Hopf bifurcation and then include a similar result for
dynamics close to a Takens - Bogdanov point.
Chapter two concentrates on travelling waves and looks for the existence of
such in three different ways. Firstly we prove the conditions that are needed for
the travelling wave equations to go through a Hopf bifurcation. Secondly, we
look for the existence of travelling waves as the wave speed is perturbed from
zero and prove when this occurs, again, using a Lyapunov - Schmidt reduction.
Thirdly we describe a result proving the existence of periodic travelling waves
when the wave speed is perturbed from infinity. In the last part of chapter
two we prove the stability of such waves for A-w systems.
In chapter three we discuss computer simulations of the work done in the
earlier chapters. We present the mappings used and prove that their behaviour
is similar to the original partial differential equations. The two specific examples
we give are a predator prey model and the complex Ginzburg - Landau
equations.

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Pattern formation (Physical sciences), Reaction-diffusion equations, Waves
Official Date:	March 1995
Dates:	Date Event March 1995 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Roberts, Mark
Sponsors:	Science and Engineering Research Council (Great Britain) (SERC)
Extent:	vii, 70 leaves
Language:	eng
URI:	http://wrap.warwick.ac.uk/id/eprint/4251

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