Travelling Waves for a Certain First-Order Coupled PDE System
Owen D. Lyne, University of Nottingham
Abstract
This paper concentrates on a particular first-order coupled PDE
system. It provides both a detailed treatment of the existence and
uniqueness of monotone travelling waves to various equilibria, by
differential-equation theory and by probability theory and a treatment
of the corresponding hyperbolic initial-value problem, by analytic
methods.
The initial-value problem is studied using characteristics to show
existence and uniqueness of a bounded solution for bounded initial
data (subject to certain smoothness conditions). The concept of
weak solutions to partial differential equations is used to
rigorously examine bounded initial data with jump discontinuities.
For the travelling wave problem the differential-equation treatment
makes use of a shooting argument and explicit calculations of the
eigenvectors of stability matrices.
The probabilistic treatment is careful in its proofs of
martingale (as opposed to merely local-martingale) properties. A
modern change-of-measure technique is used to obtain the best
lower bound on the speed of the monotone travelling wave --- with
Heaviside initial conditions the solution converges to an approximate
travelling wave of that speed (the solution tends to one ahead of the
wave-front and to zero behind it). Waves to different equilibria are
shown to be related by Doob h-transforms. Large-deviation
theory provides heuristic links between alternative descriptions of
minimum wave speeds, rigorous algebraic proofs of which are provided.
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