Dynamics of Pinned Pulses in a Class of Nonlinear Reaction-Diffusion Equations with Strong Localized Impurities

For linear reaction-diffusion equations, a general geometric singular perturbation framework was developed, to study the impact of strong, spatially localized, smooth nonlinear impurities on the existence, stability, and bifurcation of localized structure, in the paper [Doelman et al., 2018]. The multiscale nature enables deriving algebraic conditions determining the existence of pinned single- and multi-pulses. Moreover, linearity enables treating the spectral stability issue for pinned pulses similarly to the problem of existence. In this paper, we move one step further to treat a special type of nonlinear reaction-diffusion equation with the same type of impurity. The additional nonlinear term generates richer and more complex dynamics. We derive algebraic conditions for determining the existence and stability of pinned pulses in terms of Legendre functions.