Stability in the energy space for chains of solitons of the Landau–Lifshitz equation

Laire, André de; Gravejat, Philippe

doi:10.1016/j.jde.2014.09.003

Cited by 16 publications

(29 citation statements)

References 29 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: 54%

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: 54%

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

Laire¹,

Mennuni²

2020

Discrete &Amp; Continuous Dynamical Systems - A

Self Cite

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“…In the second section, we recall the orbital stability result for the multi-solitons, stated by de Laire and Gravejat in [8], which is an important tool to prove our results.…”

Section: Plan Of the Papermentioning

confidence: 91%

“…Remark 2.1. The second orthogonality condition in (2.8) is not the same as the one used by de Laire and Gravejat in [8]. However, the result remains true by the same argument used in [1] (see Section 3 in [1] for more details).…”

Section: )mentioning

confidence: 99%

“…In this section, we first recall the orbital stability result proved by de Laire and Gravejat in [8]. In order to quantify it precisely, we set…”

Section: Orbital Stability In the Hydrodynamical Frameworkmentioning

confidence: 99%

“…. , N } (see [8] for more details). For α and L given by Proposition 2.1, we also infer from the results in [8] that…”

Section: )mentioning

confidence: 99%

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On the asymptotic stability in the energy space for multi-solitons of the Landau-Lifshitz equation

Bahri

2017

Trans. Amer. Math. Soc.

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We establish the asymptotic stability of multi-solitons for the one-dimensional LandauLifshitz equation with an easy-plane anisotropy. The solitons have non-zero speed, are ordered according to their speeds and have sufficiently separated initial positions. We provide the asymptotic stability around solitons and between solitons. More precisely, we show that for an initial datum close to a sum of N dark solitons, the corresponding solution converges weakly to one of the solitons in the sum, when it is translated to the centre of this soliton, and converges weakly to zero when it is translated between solitons.

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Some explicit solutions of the Landau‐Lifshitz equation for anisotropic ferromagnets

Han

Yang

2018

Math Methods in App Sciences

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The aim of this paper is to construct explicit solutions of the Landau‐Lifshitz equation for anisotropic ferromagnets. Firstly, we present some explicit solutions for n‐dimensional Landau‐Lifshitz equation with a generalized uniaxial anisotropy field (depending only on time) under a suitable transformation, where the generalized uniaxial anisotropy field contains the classical easy‐axis case and the important higher‐order approximations. Some relevant examples are indicated that contains the cubic system proposed by Landau and Lifshitz in 1960. Secondly, an explicit plane‐wave solution of Landau‐Lifshitz equation for the completely anisotropic case is obtained. Finally, some properties of magnetization for ferromagnets are provided by analysis of the solutions.

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Stability in the energy space for chains of solitons of the Landau–Lifshitz equation

Cited by 16 publications

References 29 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 in the energy space for multi-solitons of the Landau-Lifshitz equation

Some explicit solutions of the Landau‐Lifshitz equation for anisotropic ferromagnets

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