[Paper Review] Invasion into remnant instability: a case study of front dynamics

Grégory Faye, Matt Holzer+2 more|arXiv (Cornell University)|Sep 4, 2020

Nonlinear Dynamics and Pattern Formation32 references4 citations

TL;DR

This paper investigates the nonlinear stability of invasion fronts in a reaction-diffusion system where the invaded state exhibits 'remnant instability'—an instability that persists even in exponentially weighted spaces, preventing standard stabilization techniques. Using pointwise semigroup methods and absolute spectrum analysis, the authors prove asymptotic stability of the selected front by decomposing perturbations into weighted-bounded and unbounded but uniformly decaying components, resolving apparent numerical instabilities caused by round-off errors that induce artificial resonance.

ABSTRACT

We study the invasion of an unstable state by a propagating front in a peculiar but generic situation where the invasion process exhibits a remnant instability. Here, remnant instability refers to the fact that the spatially constant invaded state is linearly unstable in any exponentially weighted space in a frame moving with the linear invasion speed. Our main result is the nonlinear asymptotic stability of the selected invasion front for a prototypical model coupling spatio-temporal oscillations and monotone dynamics. We establish stability through a decomposition of the perturbation into two pieces: one that is bounded in the weighted space and a second that is unbounded in the weighted space but which converges uniformly to zero in the unweighted space at an exponential rate. Interestingly, long-time numerical simulations reveal an apparent instability in some cases. We exhibit how this instability is caused by round-off errors that introduce linear resonant coupling of otherwise non-resonant linear modes, and we determine the accelerated invasion speed.

Motivation & Objective

• To analyze the stability of traveling fronts invading an unstable state when the linearized system exhibits remnant instability, i.e., essential spectrum instability in all exponentially weighted spaces.
• To establish the nonlinear asymptotic stability of the selected invasion front in a prototypical model coupling spatiotemporal oscillations and monotone dynamics.
• To resolve discrepancies between theoretical stability and long-time numerical simulations that suggest apparent instability.
• To identify the mechanism behind numerical instabilities as being caused by round-off errors inducing artificial linear resonant coupling of non-resonant modes.
• To determine the accelerated invasion speed resulting from such numerical artifacts.

Proposed method

• The authors employ pointwise semigroup methods to analyze the evolution of perturbations in the linearized system around the traveling front.
• They use absolute spectrum theory to characterize the instability structure, identifying branch points and pinched double roots as indicators of remnant instability.
• The perturbation is decomposed into two components: one bounded in the weighted space and another unbounded in the weighted space but uniformly convergent to zero in the unweighted space at an exponential rate.
• The analysis focuses on the linearization at the asymptotic state (u+, v+) = (0, 0), where the v-component is governed by a Swift-Hohenberg equation with μ < 0, and the u-component by a Fisher-KPP type equation.
• The absolute spectrum Σ_abs(L⁺_v) is computed via spectral curves σ_v(k; η), identifying critical weights η where self-intersections and cusps occur, signaling instability.
• The authors derive explicit expressions for critical weights η_tr^v and η_dr^v, and verify monotonicity of the real part of the spectral curve to confirm the structure of the absolute spectrum.

Experimental results

Research questions

• RQ1 Can the nonlinear stability of an invasion front be established when the invaded state exhibits remnant instability, i.e., instability in all exponentially weighted spaces?
• RQ2 What is the mechanism behind apparent numerical instabilities in long-time simulations of such systems, despite theoretical stability?
• RQ3 How does the absolute spectrum of the linearized operator determine the stability properties of the front in the presence of remnant instability?
• RQ4 What is the role of the v-component’s Swift-Hohenberg dynamics with μ < 0 in inducing or modulating the remnant instability?
• RQ5 Can the accelerated invasion speed observed numerically be analytically explained as a consequence of round-off error-induced resonant coupling?

Key findings

• The selected invasion front is nonlinearly asymptotically stable, even though the invaded state is linearly unstable in all exponentially weighted spaces.
• The absolute spectrum of the linearized operator L⁺_v consists of a real interval (−∞, λ_tr^v] with λ_tr^v < −1 and two complex conjugated branches connecting λ_tr^v to λ_dr^v, with the latter being the most unstable point.
• The branch points λ_dr^v are simple and pinched double roots, and they are the rightmost points in the absolute spectrum, confirming their role as the dominant instability source.
• The real part of the spectral curve increases strictly from left to right, ensuring that the absolute spectrum structure is well-behaved and analyzable via pointwise methods.
• Numerical instabilities arise not from physical mechanisms but from round-off errors that induce linear resonant coupling of otherwise non-resonant modes.
• The accelerated invasion speed observed in simulations is analytically traced to this error-induced resonance, which effectively modifies the system’s dynamics.