The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy

Laire, André de; Gravejat, Philippe

doi:10.1016/j.anihpc.2018.03.005

Cited by 13 publications

(14 citation statements)

References 38 publications

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“…We also notice that other statements for Cauchy problem for the Gross-Pitaevskii equation have been established in different topologies when W = δ 0 (see e.g. [61,35,33,10,31,30] and the reference there in), and these results can probably be adapted to our nonlocal framework.…”

Section: Stabilitymentioning

confidence: 53%

Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity

Laire¹,

Mennuni²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum.As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.

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Section: Stabilitymentioning

confidence: 53%

Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity

Laire¹,

Mennuni²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…Real-valued solutions of (1.1) that initially are in H 1 sin × L 2 are preserved for all time, see e.g [17] and [52]. Additionally, they are globally well-defined thanks to the fact that sin(•) is a smooth bounded function.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Note that in Theorem 6.1 we do not specify the space where (φ, φ t ) are posed, this because (Q, 0)(t) in (1.5) does not belong to H 1 × L 2 . However, it is possible to show local and global well-posedness (LWP), such that H 1 ×L 2 perturbations are naturally allowed [17]. Remark 6.2 (Explicit examples).…”

Section: Now Recalling That H Satisfies the Equation Hmentioning

confidence: 99%

On the asymptotic stability of the sine-Gordon kink in the energy space

Alejo¹,

Muñoz²,

Palacios³

2020

Preprint

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We consider the sine-Gordon (SG) equation in 1+1 dimensions. The kink is a static, non symmetric exact solution to SG, stable in the energy space H 1 × L 2 . It is well-known that the linearized operator around the kink has a simple kernel and no internal modes. However, it possesses an odd resonance at the bottom of the continuum spectrum, deeply related to the existence of the (in)famous wobbling kink, an explicit periodic-in-time solution of SG around the kink that contradicts the asymptotic stability of the kink in the energy space.In this paper we further investigate the influence of resonances in the asymptotic stability question. We also discuss the relationship between breathers, wobbling kinks and resonances in the SG setting. By gathering Bäcklund transformations (BT) as in [24,52] and Virial estimates around odd perturbations of the vacuum solution, in the spirit of [32], we first identify the manifold of initial data around zero under which BTs are related to the wobbling kink solution. It turns out that (even) small breathers are deeply related to odd perturbations around the kink, including the wobbling kink itself. As a consequence of this result and [32], using BTs we can construct a smooth manifold of initial data close to the kink, for which there is asymptotic stability in the energy space. The initial data has spatial symmetry of the form (kink + odd, even), non resonant in principle, and not preserved by the flow. This asymptotic stability property holds despite the existence of wobbling kinks in SG. We also show that wobbling kinks are orbitally stable under odd data, and clarify some interesting connections between SG and φ 4 at the level of linear Bäcklund transformations. Date: March 23, 2020. * M.A. Alejo was partially supported by CNPq grant no. 305205/2016-1. * * C. M. work was funded in part by Chilean research grants FONDECYT 1191412, project France-Chile ECOS-Sud C18E06 and CMM Conicyt PIA AFB170001. * * * J. M. P. was partially supported by Chilean research grants FONDECYT 1191412 and project France-Chile ECOS-Sud C18E06.

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“…When 0 < α < 1, (LLG) is of parabolic type. We refer the reader to [40,29,32,33,12,14,15,13] and the references therein for more details and surveys on these equations.…”

Section: The Landau-lifshitz-gilbert Equation: Self-similar Solutionsmentioning

confidence: 99%

Self-similar shrinkers of the one-dimensional Landau–Lifshitz–Gilbert equation

Gutiérrez

Laire

2020

J. Evol. Equ.

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The main purpose of this paper is the analytical study of self-shrinker solutions of the one-dimensional Landau-Lifshitz-Gilbert equation (LLG), a model describing the dynamics for the spin in ferromagnetic materials. We show that there is a unique smooth family of backward self-similar solutions to the LLG equation, up to symmetries, and we establish their asymptotics. Moreover, we obtain that in the presence of damping, the trajectories of the self-similar profiles converge to great circles on the sphere S 2 , at an exponential rate.In particular, the results presented in this paper provide examples of blow-up in finite time, where the singularity develops due to rapid oscillations forming limit circles.

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The Sine-Gordon regime of the Landau–Lifshitz equation with a strong easy-plane anisotropy

Cited by 13 publications

References 38 publications

Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity

Traveling waves for some nonlocal 1D Gross–Pitaevskii equations with nonzero conditions at infinity

On the asymptotic stability of the sine-Gordon kink in the energy space

Self-similar shrinkers of the one-dimensional Landau–Lifshitz–Gilbert equation

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