It is well-known that the existence of traveling wave solutions for reaction-diüsion partial diérential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar diérential equations. We introduce a method for finding explicit upper and lower bounds of these heteroclinic orbits. In particular, for the classical Fisher- Kolmogorov equation we give rational upper and lower bounds which allow to locate these solutions analytically and with very high accuracy. These results allow one to construct analytical approximate expressions for the traveling wave solutions with a rigorous control of the errors for arbitrary values of the independent variables. These explicit expressions are very simple and tractable for practical purposes. They are constructed with exponential and rational functions.
|Journal||Discrete and Continuous Dynamical Systems|
|Publication status||Published - 1 Aug 2013|
- Fisher-Kolmogorov equation
- Heteroclinic orbit
- Invariant manifold
- Reaction-diüsion partial diérential equation
- Traveling wave