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 Electronic Journal of Probability > Vol. 5 (2000) > Paper 14 open journal systems 


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.


Full text: PDF

Pages: 1-40

Published on: August 17, 2000


Bibliography
  1. Beale, J. T. (1986) Large-Time Behavior of Discrete Velocity Boltzmann Equations. Commun. Math. Phys., 106, 659-678. Math. Review 87j:82056
  2. Champneys, A., Harris, S., Toland, J., Warren, J. & Williams, D. (1995) Algebra, analysis and probability for a coupled system of reaction-diffusion equations. Phil. Trans. R. Soc. Lond., A 350, 69--112. Math. Review 96e:35080
  3. Coddington, E. A. & Levinson, N. (1955) Theory of Ordinary Differential Equations. New York: McGraw-Hill. Math. Review 16,1022b
  4. Cohen, J.E. (1981) Convexity of the dominant eigenvalue of an essentially non-negative matrix. Proc. Amer. Math. Soc., 81, 657--658. Math. Review 82a:15016
  5. Courant, R. & Hilbert, D. (1962) Methods of Mathematical Physics, Volume II. New York: Interscience. Math. Review 25 #4216
  6. Crooks, E. C. M. (1996) On the Vol'pert theory of travelling-wave solutions for parabolic systems. Nonlinear Analysis, Theory, Methods and Applications. 26, 1621--1642. Math. Review 97a:35094
  7. Deuschel, J-D. & Stroock, D.W. (1989) Large Deviations. Boston: Academic Press. Math. Review 90h:60026
  8. Dunbar, S.R. (1988) A branching random evolution and a nonlinear hyperbolic equation. SIAM J. Appl. Math., 48, 1510--1526. Math. Review 90a:60183
  9. Hadeler, K. P. (1995) Travelling Fronts in Random Walk Systems. Forma, 10, 223--233. Math. Review 98m:35098
  10. Holmes, E. E. (1993) Are diffusion models too simple? A comparison with telegraph models of invasion. Amer. Naturalist, 142, 779--795.
  11. McKean, H. P. (1975) Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Comm. Pure Appl. Math., 28, 323--331. Math. Review 53 #4262
  12. McKean, H. P. (1976) Correction to the above. Comm. Pure Appl. Math., 29, 553--554. Math. Review 54 #11534
  13. Neveu, J. (1987) Multiplicative martingales for spatial branching processes. Seminar on Stochastic Processes (ed. E. Cinlar, K. L Chung and R. K. Getoor), Progress in Probability and Statistics 15. pp. 223--241. Boston: Birkhauser. Math. Review 91f:60144
  14. Rogers, L. C. G. & Williams, D. (1994) Diffusions, Markov Processes and Martingales, Volume 1: Foundations, 2nd edition. Chichester: Wiley. Math. Review 96h:60116
  15. Seneta, E. (1981) Non-negative Matrices and Markov Chains. Heidelberg, New York: Springer-Verlag. Math. Review 85i:60058
















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Electronic Journal of Probability. ISSN: 1083-6489