Skip to main content
Article thumbnail
Location of Repository

The stochastic elementary formula method and approximate travelling waves for semi-linear reaction diffusion equations

By Huaizhong Zhao

Abstract

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 semi-classical analysis for generalised\ud KPP equations. With a so-called "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 Cartan-Hadamard 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(-c1μ-2) as μ~0 for c1 random) and behind which it\ud has at least order exp(-c2μ-1) 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 reaction-diffusion systems with a small parameter by using\ud the n-dimensional Feynman-Kac 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

Topics: QA
OAI identifier: oai:wrap.warwick.ac.uk:4236

Suggested articles

Citations

  1. (1975). A Correction to: Application of Brownian Motion to tile Equation of Kolmogorov-Petrovskii-Piskiinov,
  2. (1983). A Numerical Study of the Belousov-Zhabotinskii Reaction Using Galerkin Finite Elements Methods, doi
  3. (1993). Algebra, Analysis and Probability for a Coupled System of Reaction Diffusion
  4. (1984). An Introduction to tile Theory of Large Deviations,
  5. (1975). Application of Brownian Motion to the Equation of Kolmogorov-PetrovskiiPiskunov,
  6. (1990). Applications of the Rotation Theory to Vector Fields to the Study of the Wave Solutions of Parabolic Equations,
  7. (1993). Approximate Týravelling Waves for the Generalised KPP Equations and Classical Mechanics with an Appendix by
  8. (1990). Approximation for Diffiision in Random Fields,
  9. (1982). Asymptotic Methods in Nonlinear Wave Theory,
  10. (1991). Coupled React! on-Diffusion Equations, The Ann. doi
  11. (1969). Differential and Integral Inequalities, Theory and Applications, doi
  12. (1967). Diffusion Processes in a Small Time Interval, doi
  13. (1937). Etude de 1'equation de la Diffusion Avec Croissance de la Matiere et soil Application a un Probleme Biologiqtie,
  14. (1992). Formulae for Solutions to (Possibly Degenerate) Diffusion Equations Exhibiting Semi-classical Asymptotics. In Stochastics and Quantum Mechanics
  15. (1985). Functional Integration and Partial Differential equations, doi
  16. (1979). Functional Integration and Quantum Physics, doi
  17. (1994). Generalised KPP Equations with Random Noise
  18. (1989). Large Deviation,
  19. (1982). Location of Wave fronts for the Multi- Dimensional K-P-P Equation and Brownian First Exist Densities, doi
  20. (1980). Mathematical Models in Molecular and Celler Biology,
  21. (1970). On Certain Systems of Parabolic Equations, doi
  22. (1970). On Small Random Perturbations of Dynamical Systems, doi
  23. (1990). Parabolic Problems for the Anderson Model,
  24. (1979). Propagation of a Concentration Wave in the Presence of Random A/lotion Association with the Growth of a Substance,
  25. (1986). Reaction- Diffusion Equations and Their Applications to Biology, doi
  26. (1992). Semi-linear PDE's and Limit Theorem for Large Deviations. Ecole d'Ete de doi
  27. (1983). Shock Waves and Reaction- Diffusion Equations, doi
  28. (1966). Some Asymptotic Formula for Wiener Integrals, 'R-ans.
  29. (1992). Some Remarks on Regularity Theory arid Stochastic Flows for Parabolic SPDE'S,
  30. (1977). Spatial Contact Models for Ecological and Epidemic Spread,
  31. (1982). Stochastic Differential Equations on Manifold, London Mathematical Society Lecture Notes Series 70, Cambridge
  32. (1972). Stochastic Integral Representations of Multiplicative Operator Functionals of a Weiner Process,
  33. (1990). Stochastic Parabolic Equations in Bounded Domains: Random Evolution Operator arid Lyapunov Exponents, doi
  34. (1977). The Approach of Solutions of Nonlinear Diffusion Equations to 'Ravelling Wave Solutions,
  35. (1977). The Asymptotic Speed of Propagation of a Simple Epidemic, in: Nonlinear Diffusion,
  36. (1994). The Convergence of the Solutions of Reaction-diffusion Equations with Small Parameters
  37. (1982). The Diffusion Equation and Classical Mechanics: an Elementary Formula, in: doi
  38. (1994). The Nonlinear Stability of Front Solutions for Parabolic Partial Differential Equations, doi
  39. (1993). The Propagation of Travelling Waves for Stochastic Generalised KPP Eqllations with an Appendix by
  40. (1994). The Ravelling Wave Ronts of Nonlinear Reaction-Diffusion Systems via Freidlin's Stochastic Approaches,
  41. (1994). The Stochastic Hamilton Jacobi Equation, Stochastic Heat Equations and Schr6dinger Equations, submitted to Stochastic Processes and Their Applications.
  42. (1993). Týavelling Waves for the KPP Equations with Noise, Talk at AMS Summer Institute on Stochastic Analysis, Cornell,
  43. (1909). Uber Matrizen aus Positiven Elementen,
  44. (1912). Uber Matrizen nicht negative Elementen,
  45. (1970). Wiener Integral Representations for Certain Semigroups Which Have infinitesimal Generators with Matrix Coefficients, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.