Large stable pulse solutions in reaction-diffusion equations

In this paper we study the existence and stability of asymptotically large stationary multi-pulse solutions in a family of singularly perturbed reacti on-diffusion equations. This family includes the generalized Gierer-Meinhar dt equation. The existence of N-pulse homoclinic orbits (N greater than or equal to 1) is established by the methods of geometric singular perturbatio n theory. A theory, called the NLEP (=NonLocal Eigenvalue Problem) approach , is developed, by which the stability of these patterns can be studied exp licitly. This theory is based on the ideas developed in our earlier work on the Gray-Scott model. It is known that the Evans function of the linear ei genvalue problem associated to the stability of the pattern can be decompos ed into the product of a slow and a fast transmission function. The NLEP ap proach determines explicit leading order approximations of these transmissi on functions. It is shown that the zero/pole cancellation in the decomposit ion of the Evans function, called the NLEP paradox, is a phenomenon that oc curs naturally in singularly perturbed eigenvalue problems. It follows that the zeroes of the Evans function, and thus the spectrum of the stability p roblem, can be studied by the slow transmission function. The key ingredien t of the analysis of this expression is a transformation of the associated nonlocal eigenvalue problem into an inhomogeneous hypergeometric differenti al equation. By this transformation it is possible to determine both the nu mber and the position of all elements in the discrete, spectrum of the line ar eigenvalue problem. The method is applied to a special case that corresp onds to the classical model proposed by Gierer and Meinhardt. It is shown t hat the one-pulse pattern can gain (or lose) stability through a Hopf bifur cation at a certain value mu (Hopf) of the main parameter mu. The NLEP appr oach not only yields a leading order approximation of mu (Hopf), but it als o shows that there is another bifurcation value, wedge, at which a new (sta ble) eigenvalue bifurcates from the edge of the essential spectrum. Finally , it is shown that the N-pulse patterns are always unstable when N greater than or equal to 2.