Pulse solutions in Gierer–Meinhardt equation with slowly degenerate nonlinearity

Abstract

Based on geometric singular perturbation theory (GSPT) and nonlocal eigenvalue problem (NLEP) method, this article studies the existence and stability of algebraically delaying pulses in Gierer–Meinhardt equation with slowly degenerate nonlinearity. By utilizing the fact that the critical manifold is both normally hyperbolic and invariant, we rigorously establish the existence of algebraically decaying pulses by combining GSPT with the Melnikov method. It is proven that the model has a unique algebraically decaying pulse. On the other hand, the slowly degenerate nonlinearity results in that the linearized matrix associated with the eigenvalue problem no longer approaches the constant matrix exponentially. Hence, we must solve the resulting linear “time-varying” problem. By classifying the power of the slowly degenerate nonlinearity, we introduce different special functions including the Whittaker function and the Bessel function to solve this linear problem explicitly. Thus the spectral (in)stability criteria on the algebraically delaying pulse can be set up by matching the slow and fast segments of the eigenfunctions. An example is also provided to illustrate the theoretical framework.