arXiv subjects

Ikkei Shimizu

Publications and source records attributed to Ikkei Shimizu.

9 recordsLinked to original sources

arXiv2026

Global perturbation of isolated equivariant chiral skyrmions from the Bogomol'nyi case

Isolated skyrmion solutions to the two-dimensional Landau-Lifshitz equation with Dzyaloshinskii-Moriya interaction, Zeeman term, and easy-plane anisotropy of various strengths are studied. In the full range of parameter values for which the energy is a positive variation of the Bogomol'nyi case, we construct solutions to the corresponding Euler-Lagrange equation and analyze their qualitative properties, including monotonicity, exponential decay, and stability. Our analysis is global and non-perturbative. Moreover, we derive precise estimates qualifying the difference between these solutions and those in the Bogomol'nyi regime. A key ingredient of our approach is a novel resolvent estimate for the linearized operator, which remains uniform with respect to additional implicit potentials arising in the problem.

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arXiv2026

Time decay estimates for localized perturbations around a helical state for the Landau-Lifshitz-Gilbert equation

We study the dynamics of the Landau--Lifshitz--Gilbert equation with the Dzyaloshinskii--Moriya interaction. The equation admits a family of exact stationary solutions, referred to as helical states, which are periodic in one spatial variable and constant in the others. We investigate the dynamical stability of a helical state with respect to perturbations belonging to suitable Lebesgue and Sobolev spaces. Under a smallness assumption on the initial perturbation, we prove global existence and time decay estimates for solutions, demonstrating that the above helical state is stable. The analysis of the relevant linear operator is carried out via the Bloch--Fourier-wave decomposition, where the eigenvalue problem for the reduced operator is characterized by certain Mathieu equations.

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arXiv2025

Phase transition thresholds and chiral magnetic fields of general degree

We study a variational problem for the Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions arising in 2D micromagnetics, focusing on the Bogomol'nyi regime. We first determine the minimal energy for arbitrary topological degree, thereby revealing two types of phase transitions consistent with physical observations. In addition, we prove the uniqueness of the energy minimizer in degrees $0$ and $-1$, and nonexistence of minimizers for all other degrees. Finally, we show that the homogeneous state remains stable even beyond the threshold at which the skyrmion loses stability, and we uncover a new stability transition driven by the Zeeman energy.

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arXiv2023

Multi-solitons for the nonlinear Schrödinger equation with repulsive Dirac delta potential

We prove the existence of multi-soliton solutions for the nonlinear Schrödinger equation with repulsive Dirac delta potential and $L^2$-supercritical focusing nonlinear term. Our main contribution is to treat the unmoving part of the multi-solitons, which is the ground state of the equation. The linearized operator around it has two unstable eigenvalues. This is the main difference from NLS without potential, whose existence of multi-solitons is investigated by Côte, Martel, and Merle (2011).

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arXiv2022

Global dynamics below a threshold for the nonlinear Schrödinger equations with the Kirchhoff boundary and the repulsive Dirac delta boundary on a star graph

We consider the nonlinear Schrödinger equations on the star graph with the Kirchhoff boundary and the repulsive Dirac delta boundary at the origin. In the present paper, we show the scattering-blowup dichotomy result below the mass-energy of the ground state on the real line. The proof of the scattering part is based on a concentration compactness and rigidity argument. Our main contribution is to give a linear profile decomposition on the star graph by using a symmetrical decomposition.

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arXiv2022

Phase transition threshold and stability of magnetic skyrmions

We examine the stability of vortex-like configuration of magnetization in magnetic materials, so-called the magnetic skyrmion. These correspond to critical points of the Landau-Lifshitz energy with the Dzyaloshinskii-Moriya (DM) interactions. In an earlier work of the Döring and Melcher, it is known that the skyrmion is a ground state when the coefficient of the DM term is small. In this paper, we prove that there is an explicit critical value of the coefficient above which the skyrmion is unstable, while stable below this threshold. Moreover, we show that in the unstable regime, the infimum of energy is not bounded below, by giving an explicit counterexample with a sort of helical configuration. This mathematically explains the occurrence of phase transition observed in some experiments.

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arXiv2020

Local well-posedness for the Landau-Lifshitz equation with helicity term

We consider the initial value problem for the Landau-Lifshitz equation with helicity term (chiral interaction term), which arises from the Dzyaloshinskii-Moriya interaction. We prove that it is well-posed locally-in-time in the space $\bar{k} +H^s$ for $s\ge 3$ with $s\in \mathbb{Z}$ and $\bar{k}={}^t(0,0,1)$. We also show that if we further assume that the solution is homotopic to constant maps, then local well-posedness holds in the space $\bar{k} + H^s$ for $s>2$ with $s\in \mathbb{R}$. Our proof is base on the analysis via the modified Schrödinger map equation.

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arXiv2020

On Uniqueness for Schrödinger maps with low regularity large data

We prove that the solutions to the initial-value problem for 2-dimensional Schrödinger maps are unique in $C_tL_x^{\infty} \cap L_t^{\infty} (\dot{H}^1_x\cap \dot{H}^2_x)$. For the proof, we follow McGahagan's argument with improving its technical part, combining Yudovich's argument.

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arXiv2018

Remarks on local theory for Schrödinger maps near harmonic maps

We consider the initial-value problem for the equivariant Schrödinger maps near a family of harmonic maps. We provide some supplemental arguments for the proof of local well-posedness result by Gustafson, Kang and Tsai in [Duke Math. J. 145(3) 537--583, 2008]. We also prove that the solution near harmonic maps is unique in $C(I;\dot{H}^1(\mathbb{R}^2)\cap\dot{H}^2(\mathbb{R}^2))$ for time interval $I$. In the proof, we give a justification of the derivation of the modified Schrödinger map equation in low regularity settings without smallness of energy.

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