Selection of pushed pattern-forming fronts in the FitzHugh-Nagumo system

Abstract

We establish nonlinear stability of fronts that describe the creation of a periodic pattern through the invasion of an unstable state. Our results concern pushed fronts, that is, fronts whose propagation is driven by a localized mode at the front interface. We prove that these pushed pattern-forming fronts attract initial data supported on a half-line, and therefore determine both propagation speeds and selected wave numbers for invasion from localized initial conditions. This provides to our knowledge the first proof of the marginal stability conjecture for pattern-forming fronts, thereby confirming universal wave number selection laws widely used in the physics literature. We present our analysis in the specific setting of the FitzHugh-Nagumo system, but our methods can be applied to general dissipative PDE models which exhibit pattern formation. The main technical challenge is to control the interaction between the localized mode driving the propagation and outgoing diffusive modes in the wake of the front. Through a subtle far-field/core decomposition of the linearized evolution, we resolve this interaction and describe the nonlinear response of the front to perturbations as a dynamically driven phase mixing problem for the pattern in the wake. The methods we develop are generally useful in any setting involving the interaction of localized modes and outward diffusive transport, such as in the nonlinear stability of undercompressive viscous shock waves or source defects.

Figures (3)

• Figure 1: Left three panels: spacetime diagrams of numerical solutions to \ref{['e: FHN']}; insets show the graph of $u(x,t)$ against $x$ at 100 time units before the final time. Far left: simulation with initial condition consisting of the unstable equilibrium $\mathbf{u} \equiv (0,0)^\top$ perturbed with low amplitude white noise. Second from left: simulation with initial condition consisting of a small compactly supported perturbation of $\mathbf{u} \equiv (0,0)^\top$ near the left boundary. Third from left: initially the same simulation as the previous panel, but at $t = 600$ we pause the simulation and add a perturbation ahead of the front; the front adjusts its position and continues to propagate, leaving behind a phase defect. Right: sketch of the dynamics of the phase defect, in the frame $\xi = x - c_\mathrm{ps} t$ in which $\mathbf{u}_\mathrm{ps}(\xi)$ is stationary, with time increasing from bottom to top. Bottom right: front profile $\mathbf{u}_\mathrm{ps}(\xi)$ (red) and initial perturbation $\mathbf{w}_0(\xi)$ (tan). The plots at subsequent times show the response to the perturbation. The front adjusts its position, forming a phase defect $\psi(x,t)$ (blue) which propagates to the left (in this frame) with the group velocity.
• Figure 2: Left: schematic of a pushed pattern-forming front $\mathbf{u}_\mathrm{ps}(\xi)$. Pattern formation is driven by the front interface (blue) which creates wave trains in its wake with outward pointing group velocity $c_g < 0$. Right: spectrum of the linearization about the pushed pattern-forming front $\mathbf{u}_\mathrm{ps}(\xi)$, after stabilization with an exponential weight. Red and purple curves denote the spectra of $u \equiv 0$ and $\mathbf{u}_\mathrm{wt}$, which form the boundaries of the essential spectrum of $\mathcal{L}_\mathrm{ps}$ (shaded in grey). The blue cross at $\lambda = 0$ denotes the simple translational eigenvalue embedded in the essential spectrum.
• Figure 3: Spectrum of $\mathcal{L}_\mathrm{ps}$ together with integration contours used in the inverse Laplace transform. Left: the initial contour $\mathrm{Re} \, \lambda = \Lambda > 0$ (tan) used in Proposition \ref{['p: C0 semigroup']}. Right: the shifted contour $\Gamma_\delta$ (tan) used in Proposition \ref{['p: contour shifting']}. The contour $\Gamma_\delta$ lies in the open left-half plane, except for a portion lying in the disk $B(0, \delta)$, whose boundary is denoted by the dashed curve.

Theorems & Definitions (61)

• Theorem 1.1: Existence of pushed pattern-forming fronts CarterScheel
• Remark 2.1
• Remark 2.2
• Theorem 2.3: SpectralFronts
• Theorem 2.4: Nonlinear stability of pushed pattern-forming fronts
• Corollary 2.5: Selection of pushed pattern-forming fronts
• Remark 2.6
• Proposition 3.1: Inverse Laplace representation of semigroup
• Theorem 3.2: Semigroup decomposition and linear estimates
• Proposition 4.1