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The cubic Schrödinger regime of the Landau–Lifshitz equation with a strong easy-axis anisotropy
Revista Matemática Iberoamericana ( IF 1.2 ) Pub Date : 2020-08-05 , DOI: 10.4171/rmi/1202
André de Laire 1 , Philippe Gravejat 2
Affiliation  

We pursue our work on the asymptotic regimes of the Landau–Lifshitz equation. We put the focus on the cubic Schrödinger equation, which is known to describe the dynamics of uniaxial ferromagnets in a regime of strong easy-axis anisotropy. In any dimension, we rigorously prove this claim for solutions with sufficient regularity. In this regime, we additionally classify the one-dimensional solitons of the Landau–Lifshitz equation and quantify their convergence towards the solitons of the one-dimensional cubic Schrödinger equation.

Our proof is based on a nonlinear rewriting of the Landau–Lifshitz equation for uniaxial ferromagnets as a nonlinear Schrödinger equation. This latter equation turns out to be consistent with the cubic Schrödinger equation in a regime of strong easy-axis anisotropy. We rely on this consistency to perform suitable energy estimates and then control the difference between the solutions of the two equations.



中文翻译:

具有强易轴各向异性的Landau-Lifshitz方程的三次Schrödinger态

我们继续研究Landau-Lifshitz方程的渐近状态。我们将重点放在三次Schrödinger方程上,该方程描述了在强易轴各向异性状态下的单轴铁磁体的动力学。在任何维度上,我们都严格证明了对具有足够规律性的解决方案的要求。在这种情况下,我们还对Landau–Lifshitz方程的一维孤子进行分类,并量化它们对一维三次Schrödinger方程的孤子的收敛性。

我们的证明基于将单轴铁磁体的Landau-Lifshitz方程非线性改写为非线性Schrödinger方程。事实证明,在强易轴各向异性的情况下,后一个方程与三次Schrödinger方程一致。我们依靠这种一致性来执行合适的能量估计,然后控制两个方程的解之间的差异。

更新日期:2020-08-05
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