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On gradients of approximate travelling waves for generalised KPP equations

Published online by Cambridge University Press:  14 November 2011

H. Z. Zhao
H. Z. Zhao
Affiliation:
Department of Mathematics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.
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Abstract

In this paper we use stochastic semiclassical analysis and the logarithmic transformation to study the gradients of the approximate travelling wave solutions for the generalised KPP equations with Gaussian and Dirac delta initial distributions. We apply the logarithmic transformation to the nonlinear reaction diffusion equations and obtain a Maruyama–Girsanov–Cameron–Martin formula for the drift μ2 ∇ log uμ, uμ being a solution of a generalised KPP equation. We obtain that μ2|∇ log uμ(t,x)| is bounded and the trough is flat. The difficult problem in this paper is to prove that the corresponding crest is flat. A probabilistic approach is used in this paper to treat this problem successfully.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 2 , 1997 , pp. 423 - 439
Copyright
Copyright © Royal Society of Edinburgh 1997

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