Abstract
We apply the theory of invariant manifolds for singularly perturbed ordinary differential equations and results about the persistence of homoclinic orbits in autonomous differential systems with several parameters in order to establish the existence of pulses in reaction-diffusion systems. Essential assumptions for the existence of pulses are the following: (i) Existence of a homoclinic orbit to a hyperbolic equilibrium in the corresponding reaction system. (ii) The quotient of some measure for the diffusivities and the square of the puls speed is sufficiently small. (iii) Validity of some transversality condition. The last assumption requires the occurence of parameters in the reaction term.
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Dedicated to Professor H. W. Knobloch on the occasion of his 65th birthday.
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Schneider, K.R. On the Existence of Pulses in Reaction- Diffusion- Equations. Results. Math. 21, 200–210 (1992). https://doi.org/10.1007/BF03323079
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DOI: https://doi.org/10.1007/BF03323079