Large stable pulse solutions in reaction-diffusion equations

By Arjen Doelman, R. Gardner, T. Kaper. 2001

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Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations. (2005, methodology): “This is a relatively straightforward procedure that leads, at leading order in ε, to the following ‘normal form’ [3],”
Asymmetric Spotty Patterns for the Gray–Scott Model in R2 (2003, background): “When Ω = R(1), Doelman, Gardner and Kaper [2] studied the stability of single and multiple pulses for any τ > 0.”
Chapter 18 - Stability of Travelling Waves (2002, methodology): “We also refer to [45, 46, 47] for applications, and extensions, of the analytic extension of the Evans function across the essential spectrum to the stability of pulses in singularly perturbed reaction-diffusion equations with a strong coupling between the fast and slow subsystems.”
An Explicit Theory for Pulses in Two Component, Singularly Perturbed, Reaction–Diffusion Equations (2015, unknown): “It can be shown by a straightforward Melnikov calculation that the two-dimensional first intersections I+1 = Wu(M) ∩ {q = 0} and I−1 = Ws(M) ∩ {q = 0} must intersect in a one-dimensional manifold I ∩ I−1 = {(u0, 0, vf,h(0;u0) +O( √ ε), 0);u0 > 0} ⊂ {p = q = 0}, parameterized by u0 (see [3]).”