Nonlinear stability of fast invading fronts in a Ginzburg–Landau equation with an additional conservation law

Hilder, Bastian

doi:10.1088/1361-6544/abd612

Cited by 8 publications

(4 citation statements)

References 19 publications

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“…In the special case q = 0, when (1.6) degenerates to a homogeneous solution of the modified Ginzburg-Landau system (1.5), the nonlinear stability of (1.6) against localized perturbations has been shown in [11] without the restriction to mean-zero perturbations. The proof exploits the special structure of the linearity for q = 0, see also Lemma 3.11 and Theorem 3.4. In this paper we consider the behavior of the periodic solutions (1.6) in the modified Ginzburg-Landau system (1.5) under C m ub -perturbations.…”

Section: Additional Conservation Lawmentioning

confidence: 99%

Nonlinear Stability of Periodic Roll Solutions in the Real Ginzburg–Landau Equation Against $$C_{\textrm{ub}}^m$$-Perturbations

Hilder

Rijk

Schneider

2023

Commun. Math. Phys.

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The real Ginzburg–Landau equation arises as a universal amplitude equation for the description of pattern-forming systems exhibiting a Turing bifurcation. It possesses spatially periodic roll solutions which are known to be stable against localized perturbations. It is the purpose of this paper to prove their stability against bounded perturbations, which are not necessarily localized. Since all state-of-the-art techniques rely on localization or periodicity properties of perturbations, we develop a new method, which employs pure $$L^\infty $$ L ∞ -estimates only. By fully exploiting the smoothing properties of the semigroup generated by the linearization, we are able to close the nonlinear iteration despite the slower decay rates. To show the wider relevance of our method, we also apply it to the amplitude equation as it appears for pattern-forming systems with an additional conservation law.

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Section: Additional Conservation Lawmentioning

confidence: 99%

Nonlinear Stability of Periodic Roll Solutions in the Real Ginzburg–Landau Equation Against $$C_{\textrm{ub}}^m$$-Perturbations

Hilder

Rijk

Schneider

2023

Commun. Math. Phys.

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“…We also mention related work establishing stability of supercritical fronts -moving faster than the linear spreading speed -in the Swift-Hohenberg equation [9] and in the Ginzburg-Landau equation coupled to an additional conservation law [16], where the main difficulty is again to characterize diffusive decay in the presence of outward transport. The methods there are specifically adapted to supercritical fronts, relying crucially on the fact that one can obtain exponential in time linear stability of the unstable rest state in a suitable exponentially weighted norm.…”

Section: The Linearization In Thementioning

confidence: 99%

Sharp decay rates for localized perturbations to the critical front in the Ginzburg-Landau equation

Avery¹,

Scheel²

2021

Preprint

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We revisit the nonlinear stability of the critical invasion front in the Ginzburg-Landau equation. Our main result shows that the amplitude of localized perturbations decays with rate t −3/2 , while the phase decays diffusively. We thereby refine earlier work of Bricmont and Kupiainen as well as Eckmann and Wayne, who separately established nonlinear stability but with slower decay rates. On a technical level, we rely on sharp linear estimates obtained through analysis of the resolvent near the essential spectrum via a far-field/core decomposition which is well suited to accurately describing the dynamics of separate neutrally stable modes arising from far-field behavior on the left and right.

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“…For general perturbations, we impose the spatial exponential decay on the initial data in order to gain the asymptotic decay of perturbations in time. This technique is definitely not novel, see [9,15,31] for earlier studies in similar contexts. Further improvements of the asymptotic stability results in less restrictive function spaces are left for future work.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic stability of viscous shocks in the modular Burgers equation

Pelinovsky

Poullet

2021

Nonlinearity

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Dynamics of viscous shocks is considered in the modular Burgers equation, where the time evolution becomes complicated due to singularities produced by the modular nonlinearity. We prove that the viscous shocks are asymptotically stable under odd and general perturbations. For the odd perturbations, the proof relies on the reduction of the modular Burgers equation to a linear diffusion equation on a half-line. For the general perturbations, the proof is developed by converting the time-evolution problem to a system of linear equations coupled with a nonlinear equation for the interface position. Exponential weights in space are imposed on the initial data of general perturbations in order to gain the asymptotic decay of perturbations in time. We give numerical illustrations of asymptotic stability of the viscous shocks under general perturbations.

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Nonlinear stability of fast invading fronts in a Ginzburg–Landau equation with an additional conservation law

Cited by 8 publications

References 19 publications

Nonlinear Stability of Periodic Roll Solutions in the Real Ginzburg–Landau Equation Against $$C_{\textrm{ub}}^m$$-Perturbations

Nonlinear Stability of Periodic Roll Solutions in the Real Ginzburg–Landau Equation Against $$C_{\textrm{ub}}^m$$-Perturbations

Sharp decay rates for localized perturbations to the critical front in the Ginzburg-Landau equation

Asymptotic stability of viscous shocks in the modular Burgers equation

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