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Fronts in Non-Linear Wave Equations With Spatial Inhomogeneity.

Knight, Christopher J. K. (2011) Fronts in Non-Linear Wave Equations With Spatial Inhomogeneity. Doctoral thesis, University of Surrey (United Kingdom)..

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In this thesis fronts of an inhomogeneous non-linear wave equation are considered. The spatial domain is split up into n + 2 intervals where the middle n intervals have length Li. The potentials for the nonlinear wave equation are (in general) different in each interval. The thesis is split up into two parts, in the first part we consider stationary solutions and in particular their stability. Within each interval the homogeneous non-linear wave equation is considered. By constructing solutions on each interval and patching them together at the n + 1 boundaries between intervals, continuously differentiable solutions to the inhomogeneous equation can be found. A new stability criterion for such stationary fronts is suggested. The criterion gives a necessary and sufficient condition for the linearisation of the inhomogeneous non-linear wave equation about a stationary front to have an eigenvalue zero. This is a necessary condition for a change of stability. Examples are given to illustrate that this criterion, Sturm-Liouville theory and continuation arguments together lead to stability conclusions. In particular, it is shown that, in the context of a 0-π Joseph-son junction, a defect can stabilise a non-monotonic stationary front. A rigorous mathematical justification of why this is the case is given for the first time. The second part of this thesis focuses on the evolution of fronts in a specific example of the inhomogeneous non-linear wave equation. We consider an inhomogeneous sine-Gordon type equation with the domain split in to 3 intervals. We derive a system of two first order ODEs to approximate the behaviour of fronts in this PDE. We show that for a wide range of parameter values the system of ODEs mimics the qualitative behaviour of the PDE and gives quantitative agreement when the parameters are sufficiently small. The behaviour which is mimicked includes fronts turning round (before and after passing through the middle interval), fronts passing through the middle interval, fronts oscillating round the middle interval and fronts which become pinned (that is, are made stationary) because of the inhomogeneity. Behaviour of the PDE can be predicted from the system of ODEs, including choosing parameter regimes where specific behaviour will occur.

Item Type: Thesis (Doctoral)
Divisions : Theses
Authors : Knight, Christopher J. K.
Date : 2011
Additional Information : Thesis (Ph.D.)--University of Surrey (United Kingdom), 2011.
Depositing User : EPrints Services
Date Deposited : 06 May 2020 12:07
Last Modified : 06 May 2020 12:13

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