Zhao, Huaizhong (1994) The stochastic elementary formula method and approximate travelling waves for semi-linear reaction diffusion equations. PhD thesis, University of Warwick.

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Abstract

In this thesis we consider approximate travelling wave solutions for stochastic and
generalised KPP equations and systems by using the stochastic elementary formula
method of Elworthy and Truman. We begin with the semi-classical analysis for generalised
KPP equations. With a so-called "late caustic" assumption we prove that
the global wave front is given by the Hamilton Jacobi function. We prove a Huygens
principle on complete Riemannian manifolds without cut locus, with some bounds on
their volume elements, in particular Cartan-Hadamard manifolds. Based on the semiclassical
analysis we then consider the propagation of approximate travelling waves
for stochastic generalised KPP equations. Three regimes of perturbation are considered:
weak, mild, and strong. We show that weak perturbations have little effect
on the wave like solutions of the unperturbed equations while strong perturbations
essentially destroy the wave and force the solutions to decay rapidly. In the more difficult
mild case we show the existence of a 'wave front', in front of which the solution
is close to zero (of order exp(-c1μ-2) as μ~0 for c1 random) and behind which it
has at least order exp(-c2μ-1) for some random c2 depending on the increment of
the noise. For an alternative stochastic equation we classify the effect of the noise
by the Lyapunov exponent of a corresponding stochastic ODE. Finally we study the
asymptotic behaviour of reaction-diffusion systems with a small parameter by using
the n-dimensional Feynman-Kac formula and Freidlin's large deviation theory. We
obtain the travelling wave with nonlinear ergodic interactions and a special case with
nonlinear reducible interactions.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Reaction-diffusion equations, Waves, Stochastic processes
Official Date:	July 1994
Dates:	Date Event July 1994 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Elworthy, K. D. ; Truman, A. (Aubrey)
Extent:	119 leaves
Language:	eng

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