Large stable pulse solutions in reaction-diffusion equations (Q2758109)

The authors study the existence and stability of asymptotically large stationary multi-pulse solutions in a family of singularly perturbed reaction-diffusion equations. This family includes the generalized Gierer-Meinhardt equation. The existence of \(N\)-pulse homoclinic orbits is established by the methods of geometric singular perturbation theory. The stability of these solutions is studied explicitely by the NLEP (nonlocal eigenvalue problem) approach, which allows to determine the number and position of all elements in the discrete spectrum of the linear eigenvalue problem. The method is applied to the Gierer-Meinhardt equation. It is shown that the one-pulse solution can gain or lose stability through a Hopf bifurcation. The NLEP approach not only yields a leading order approximation of the Hopf bifurcation point, but it also shows that there is also another bifurcation value, at which a new stable eigenvalue bifurcates from the edge of the essential spectrum. Finally it is shown that the \(N\)-pulse solutions are always unstable when \(N\geq 2\).