Pulse replication and accumulation of eigenvalues

Carter, Paul; Rademacher, Jens D. M.; Sandstede, Björn

doi:10.48550/arxiv.2005.11683

Cited by 3 publications

(5 citation statements)

References 33 publications

(69 reference statements)

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“…Here of course the heterogeneity at ξ = 0 precludes such a mode, but as ε 1 0, we found that the associated eigenfunction is localised near the front interface at ξ = ξ tf , which moves farther and farther away ξ = 0 as c c lin , and approximately resembles the spatial derivative of the front. We note that similar behaviour can be observed for the standard Fitzhugh-Nagume pulse with a small 'critical' eigenvalue resembling an approximate spatial derivative of the Nagumo pulse along the back, see [11] for numerical computations.…”

Section: Numerical Results and Evidencesupporting

confidence: 75%

“…At the phenomenological level, our result shows the somewhat subtle and unexpected phenomena of a (spectrally) stable pattern-forming front with a long plateau state lying near an absolutely unstable base state. More generally, it contributes a novel and explanatory example to the recently growing set of works where absolute spectrum plays a role in governing the stability and bifurcation of coherent structures [11,14,21,57]. We note that our case is novel as we exhibit a situation where the absolute spectrum of the plateau state is unstable while the spectrum of the linearisation about the front is stable.…”

Section: Overview Of Our Approachmentioning

confidence: 75%

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Spectral stability of pattern-forming fronts in the complex Ginzburg–Landau equation with a quenching mechanism

Goh

Rijk

2021

Nonlinearity

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We consider pattern-forming fronts in the complex Ginzburg–Landau equation with a traveling spatial heterogeneity which destabilises, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed c just below the linear invasion speed of the pattern-forming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearisation about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed c increases towards the linear invasion speed, the absolute spectrum stabilises with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectral stability of the front in L 2 ( R ) . The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivise the linear flow, and use Riemann surface unfolding in combination with a superposition principle to study the evolution of subspaces as solutions to the associated matrix Riccati differential equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati–Evans function, and can be located using winding number and parity arguments.

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Section: Numerical Results and Evidencesupporting

confidence: 75%

Section: Overview Of Our Approachmentioning

confidence: 75%

Spectral stability of pattern-forming fronts in the complex Ginzburg–Landau equation with a quenching mechanism

Goh

Rijk

2021

Nonlinearity

View full text Add to dashboard Cite

show abstract

“…Here of course the heterogeneity at ξ = 0 precludes such a mode, but as ε 1 0, we found that the associated eigenfunction, being localized near the front interface at ξ = ξ tf , which moves farther and farther away ξ = 0 as c c lin , approximately resembles the spatial derivative of the front, and hence an approximate translational mode. We note that similar behavior can be observed for the standard Fitzhugh-Nagume pulse with a small "critical" eigenvalue resembling an approximate spatial derivative of the Nagumo pulse along the back, see [9] for numerical computations.…”

Section: Restriction To the Real Linesupporting

confidence: 74%

“…At the phenomenological level, our result shows the somewhat subtle and unexpected phenomena of a (spectrally) stable pattern-forming front with a long plateau state lying near an absolutely unstable base state. More generally, it contributes a novel and explanatory example to the recently growing set of works where absolute spectrum plays a role in governing the stability and bifurcation of coherent structures [9,12,51]. We note that our case is novel as we exhibit a situation where the absolute spectrum of the plateau state is unstable while the spectrum of the linearization about the front is stable.…”

Section: Overview Of Our Approachmentioning

confidence: 76%

Spectral stability of pattern-forming fronts in the complex Ginzburg-Landau equation with a quenching mechanism

Goh,

de Rijk

2020

Preprint

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We consider pattern-forming fronts in the complex Ginzburg-Landau equation with a traveling spatial heterogeneity which destabilizes, or quenches, the trivial ground state while progressing through the domain. We consider the regime where the heterogeneity propagates with speed c just below the linear invasion speed of the pattern-forming front in the associated homogeneous system. In this situation, the front locks to the interface of the heterogeneity leaving a long intermediate state lying near the unstable ground state, possibly allowing for growth of perturbations. This manifests itself in the spectrum of the linearization about the front through the accumulation of eigenvalues onto the absolute spectrum associated with the unstable ground state. As the quench speed c increases towards the linear invasion speed, the absolute spectrum stabilizes with the same rate at which eigenvalues accumulate onto it allowing us to rigorously establish spectrally stability of the front in L 2 (R).The presence of unstable absolute spectrum poses a technical challenge as spatial eigenvalues along the intermediate state no longer admit a hyperbolic splitting and standard tools such as exponential dichotomies are unavailable. Instead, we projectivize the linear flow, and use Riemann surface unfolding in combination with a superposition principle to study the evolution of subspaces as solutions to the associated matrix Riccati differential equation on the Grassmannian manifold. Eigenvalues can then be identified as the roots of the meromorphic Riccati-Evans function, and can be located using winding number and parity arguments.

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“…An interesting extension of this work would be to calculate the Maslov index of a wave which passes through a fold point, e.g. [7].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation to instability through the lens of the Maslov index

Cornwell¹,

Jones²,

Kiers³

2021

Preprint

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The Maslov index is a powerful tool for assessing the stability of solitary waves. Although it is difficult to calculate in general, a framework for doing so was recently established for singularly perturbed systems [14]. In this paper, we apply this framework to standing wave solutions of a three-component activator-inhibitor model. These standing waves are known to become unstable as parameters vary. Our goal is to see how this established stability criterion manifests itself in the Maslov index calculation. In so doing, we obtain new insight into the mechanism for instability. We further suggest how this mechanism might be used to reveal new instabilities in singularly perturbed models.

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Pulse replication and accumulation of eigenvalues

Cited by 3 publications

References 33 publications

Spectral stability of pattern-forming fronts in the complex Ginzburg–Landau equation with a quenching mechanism

Spectral stability of pattern-forming fronts in the complex Ginzburg–Landau equation with a quenching mechanism

Spectral stability of pattern-forming fronts in the complex Ginzburg-Landau equation with a quenching mechanism

Bifurcation to instability through the lens of the Maslov index

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