Pattern formations in nonlinear reaction-diffusion systems with strong localized impurities. (English) Zbl 07893990
Summary: In this manuscript, a general geometric singular perturbation framework to study pattern formations in nonlinear reaction-diffusion (R-D) systems with single or multiple strong localized impurities is developed. In the method, we divide the spatial domain into fast and slow regions respectively near and far away from the positions of impurities. In each region, the limiting fast or slow system is defined separately in terms of different scales. By so doing, the R-D system with strong localized impurities can be transformed into a singularly perturbed problem with suitable slow-fast structure. We then solve the limiting fast and slow flow explicitly and matching them approximately. Accordingly, the existence and stability criterion on the pinned 1-pulse and multiple-pulse solutions can be set up analytically. Numerical simulations are also performed to verify the theoretical predictions.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35K40 | Second-order parabolic systems |
| 35K57 | Reaction-diffusion equations |
Keywords:
pattern formation; geometric singular perturbation theory; pinned pulse; impurity; spectral stability; one space dimensionSoftware:
DLMFReferences:
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