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 bifurcating 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 bifurcations by studying the standingwave 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.

Defects in Oscillatory Media: Toward a Classification

“…Analogous bifurcations can also occur at fronts when the rest state ahead of the front destabilizes [48]. Last, we proved in [51] that both flip-flops and one-dimensional target patterns (see Figure 1.2(i) and (ii)) will bifurcate from standing pulses whose background states undergo a Hopf instability. This appears to be the mechanism that creates the chemical flip-flops observed in [38].…”

Dark solitons in external potentials

“…Information about small eigenvalues is drawn from the derivatives of the Evans function with respect to λ and ǫ near λ = 0 and ǫ = 0. However, computational formulas become more and more involved when higher-order derivatives of the Evans function are needed and this has caused some miscalculations in the past (see Appendix A in [44]). …”

Nonlinear stability of source defects in the complex Ginzburg–Landau equation

“…This has been investigated in a variety of previous studies, including [Leg01,PSAK95,CM92,KR00,SS05,LF97]. Partial analytical results can be found in [KR00,SS05]. Numerical and asymptotic evidence in [CM92,PSAK95] suggests that the sources are stable in an open region of parameter space near the NLS limit of (1.1), which corresponds to the limit |α|, |β| → ∞ and γ 1 , γ 2 → 0.…”

“…To find a spectrally stable source, one needs to find parameter values for which both the unstable eigenvalue (from the rGL limit) and the perturbed zero eigenvalue (from the cCGL limit) become stable. This has been investigated in a variety of previous studies, including [Leg01,PSAK95,CM92,KR00,SS05,LF97]. Partial analytical results can be found in [KR00,SS05].…”