Bifurcations in Reaction–Diffusion Problems

Abstract

This chapter discusses bifurcations in reaction-diffusion problems. The general structure related bifurcation theorems is that from some hypotheses about the linearized version of the problem, conclusions are drawn about the full nonlinear one, valid in an appropriate neighborhood. This is analogous to the implicit function theorem, and indeed proofs of such bifurcation theorems typically involve transformations that convert the problem into a form in which the implicit function theorem can be applied. Another consequence of the hypotheses in the theorem stated above relates to the stability properties of the new family of critical points. When this is adjoined to the mere existence of the critical points, the bifurcation theorem becomes genuinely a result about differential equations. Bifurcations are said to exhibit an exchange of stabilities. To obtain a more satisfactory understanding of the shocks and to derive the conditions characterizing their propagation, one should return to the full reaction-diffusion equations and, at least in certain idealized cases, show that they do have solutions with a shock structure, a region of transition between two different plane wave solutions. This can be done for the case of a shock structure joining two plane-wave solutions with nearly equal wave numbers and frequencies by bifurcation methods used for the weak gas dynamic shock structure problem.