Contrast structures: Theory and applications (Q1128393)

A number of important applied problems in chemical kinetics, synergetics, biology, astrophysics, and semiconductor physics lead to reaction-diffusion-advection equations. In many cases (fast reactions, small diffusion, etc.) these equations are singularly perturbed, and their solutions therefore have zones characterized by fast spatial-temporal variation (boundary and interior layers). The most difficult case to study is when the solution undergoes fast changes in the interior of the domain. Such solutions will be called contrast structures. We qualitatively classify these contrast structures into two groups: bursts and steps. Numerical solution of such problems is quite difficult, and it must allow for the qualitative behavior of the solution, information about which is supplied by the asymptotic solution of the problem. In this paper, we present the main directions and results for research into the asymptotic theory of contrast structures carried out at the Department of Mathematics of the Faculty of Physics at the Moscow State University. The main topics include: existence and evolution of contrast structures, stability analysis of contrast structures, and contrast structures in applied problems.