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Pitchfork bifurcation along a slow parameter ramp: coherent structures in the critical scaling

Ryan Goh, Tasso J. Kaper, Arnd Scheel

Abstract

We investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed $c\sim \varepsilon^{1/3}$, where $\varepsilon$ is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of $c$, that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings-McLeod solution of the Painlevé-II equation to Painlevé-II equations with a drift term.

Pitchfork bifurcation along a slow parameter ramp: coherent structures in the critical scaling

Abstract

We investigate the slow passage through a pitchfork bifurcation in a spatially extended system, when the onset of instability is slowly varying in space. We focus here on the critical parameter scaling, when the instability locus propagates with speed , where is a small parameter that measures the gradient of the parameter ramp. Our results establish how the instability is mediated by a front traveling with the speed of the parameter ramp, and demonstrate scalings for a delay or advance of the instability relative to the bifurcation locus depending on the sign of , that is on the direction of propagation of the parameter ramp through the pitchfork bifurcation. The results also include a generalization of the classical Hastings-McLeod solution of the Painlevé-II equation to Painlevé-II equations with a drift term.
Paper Structure (11 sections, 12 theorems, 20 equations, 2 figures)

This paper contains 11 sections, 12 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline of the proof.
  3. Asymptotics at Lg
  4. The linearization at monotone solutions
  5. Existence and qualitative properties for Lg
  6. Existence and qualitative properties for Lg
  7. Existence and monotonicity for all c: continuation analysis
  8. Results from numerical simulations
  9. Conclusions
  10. Summary of the main results
  11. Discussion and future directions

Key Result

Theorem 1.1

For any $c\in\mathbb{R}$, the equation e:aclin has a unique monotonically decreasing stationary solution $u_*(x;c)$ with the properties that

Figures (2)

  • Figure 1: Top: Sample profiles for $c=-200,-10,1,0,1,10$ of admissible solutions $u(x;c)$ of \ref{['e:aclin']} computed using AUTO07p. This illustrates the transition from fronts with diffusive spill-over (for $c\le 0$) to fronts which exhibit significant delays in the onset of the instability past the pitchfork bifurcation point (for $c>0$). Bottom left: The value $u(0;c)$ plotted as a function of $(-c)^{1/4}$ for $c<0$ (blue) along with linear fit (orange), of data for large $c$ values. Here, the slope of the fit line was found to be 0.7527, within 0.0016 of the predicted $\pi^{-1/4}$; see \ref{['e:u(0)-cnegative']}. Bottom center: Value of the invasion point $x_\delta(c)$ defined by $u(x_\delta(c);c)=\delta$ with $\delta = 0.1$. Bottom right: numerically measured crossover point $x_0<0$ from Lemma \ref{['l:cross']} for a range of positive $c$ values. For $c$ larger, the measured value was within machine precision, and we found that $x u+ u^3<0$ for all grid points with $x<0.$
  • Figure 2: Left: Comparison of front solutions of \ref{['e:actanh']}, $u(x;c)$ (solid colored lines) with scaled inner solutions given by $\varepsilon^{1/3}\tilde{u}(\varepsilon^{-1/3} x)$ (dashed black lines) for a range of un-scaled $c$ values (see legend). Right: Comparison of the front interface location $x_\delta(c)$ (blue solid) in \ref{['e:actanh']} with the scaled front interface, $\tilde{x}_\delta(\varepsilon^{-1/3} c)$ (black dashed), for \ref{['e:aclin']}. Here, $\delta = 0.1$ and $\varepsilon = 0.001$ throughout.

Theorems & Definitions (17)

  • Theorem 1.1: Existence and Uniqueness of Quenched Fronts
  • Definition 1.2
  • Proposition 1.3: Qualitative Properties of Quenched Fronts
  • Lemma 2.1: Stable manifold at $+\infty$
  • Remark 2.2
  • Lemma 2.3: $c=0$: Unstable manifold at $-\infty$
  • Remark 2.4
  • Lemma 2.5: $c \ne 0$: Unstable manifold at $-\infty$
  • Lemma 3.1
  • proof
  • ...and 7 more