The stochastic elementary formula method and approximate travelling waves for semilinear reaction diffusion equations
Doctoral thesis
English
OPEN
Zhao, Huaizhong
In this thesis we consider approximate travelling wave solutions for stochastic and\ud generalised KPP equations and systems by using the stochastic elementary formula\ud method of Elworthy and Truman. We begin with the semiclassical analysis for generalised\ud KPP equations. With a socalled "late caustic" assumption we prove that\ud the global wave front is given by the Hamilton Jacobi function. We prove a Huygens\ud principle on complete Riemannian manifolds without cut locus, with some bounds on\ud their volume elements, in particular CartanHadamard manifolds. Based on the semiclassical\ud analysis we then consider the propagation of approximate travelling waves\ud for stochastic generalised KPP equations. Three regimes of perturbation are considered:\ud weak, mild, and strong. We show that weak perturbations have little effect\ud on the wave like solutions of the unperturbed equations while strong perturbations\ud essentially destroy the wave and force the solutions to decay rapidly. In the more difficult\ud mild case we show the existence of a 'wave front', in front of which the solution\ud is close to zero (of order exp(c1μ2) as μ~0 for c1 random) and behind which it\ud has at least order exp(c2μ1) for some random c2 depending on the increment of\ud the noise. For an alternative stochastic equation we classify the effect of the noise\ud by the Lyapunov exponent of a corresponding stochastic ODE. Finally we study the\ud asymptotic behaviour of reactiondiffusion systems with a small parameter by using\ud the ndimensional FeynmanKac formula and Freidlin's large deviation theory. We\ud obtain the travelling wave with nonlinear ergodic interactions and a special case with\ud nonlinear reducible interactions.

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