Travelling waves in heterogeneous media

Adam Boden

Research output: ThesisDoctoral Thesis

Abstract

In this thesis we study the existence of travelling wave type solutions for a reaction diffusion equation in R2 with a nonlinearity which depends periodically on the spatial variable. Specifically we will consider a particular class of nonlinearities where we treat the coefficient of the linear term as a parameter. For this class of nonlinearities we formulate the problem as a spatial dynamical system and use a centre manifold reduction to find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. We then consider what happens if the parameter and the wave speed vary close to zero; by analysing the bifurcations in this case we are able to find travelling wave solutions with periodic and homoclinic structures. Finally we examine what happens to the travelling wave solutions as the period of the periodic dependence in the nonlinearity tends to zero.
Original languageEnglish
QualificationPh.D.
Awarding Institution
  • University of Bath
Supervisors/Advisors
  • Matthies, Karsten, Supervisor
Date of Award30 Apr 2013
StateUnpublished - 2013

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Nonlinearity
Traveling wave
Wave speed
Traveling wave solutions
Center manifold reduction
Heterogeneous media
Homoclinic
Zero
Reaction-diffusion equations
Bifurcation
Dynamical system
Vary
Tend
Coefficient
Term

Cite this

Travelling waves in heterogeneous media. / Boden, Adam.

2013.

Research output: ThesisDoctoral Thesis

Boden, A 2013, 'Travelling waves in heterogeneous media', Ph.D., University of Bath.
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AB - In this thesis we study the existence of travelling wave type solutions for a reaction diffusion equation in R2 with a nonlinearity which depends periodically on the spatial variable. Specifically we will consider a particular class of nonlinearities where we treat the coefficient of the linear term as a parameter. For this class of nonlinearities we formulate the problem as a spatial dynamical system and use a centre manifold reduction to find conditions on the parameter and nonlinearity for the existence of travelling wave type solutions with particular wave speeds. We then consider what happens if the parameter and the wave speed vary close to zero; by analysing the bifurcations in this case we are able to find travelling wave solutions with periodic and homoclinic structures. Finally we examine what happens to the travelling wave solutions as the period of the periodic dependence in the nonlinearity tends to zero.

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