Absolute instabilities of standing pulses

Abstract

We analyse instabilities of standing pulses in reaction-diffusion systems that are caused by an absolute instability of the homogeneous background state. Specifically, we investigate the impact of pitchfork, Turing and oscillatory bifurcations of the rest state on the standing pulse. At a pitchfork bifurcation, the standing pulse continues through the bifurcation point where it selects precisely one of the two bifurcating equilibria. At a Turing instability, symmetric pulses emerge that are spatially asymptotic to the birfurcating spatially-periodic Turing patterns. These pulses exist for any wavenumber inside the Eckhaus stability band. Oscillatory instabilities of the background state lead to genuinely time-periodic pulses that emit small wave trains with a unique selected wavenumber. We analyse these three birfurcations by studying the standing-wave and modulated-wave equations: in this setup, pulses correspond to homoclinic orbits to equilibria that undergo reversible bifurcations. We use blow-up techniques to show that the relevant stable and unstable manifolds can be continued across the bifurcation point and to investigate both existence and stability of the bifurcating waves.

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