2001 Fiscal Year Annual Research Report

薄膜領域におけるギンツブルグ・ランダウ方程式の渦糸解

Research Project

Project/Area Number	13640142
Research Institution	Ryukoku University
Principal Investigator	森田善久龍谷大学, 理工学部, 教授 (10192783)
Co-Investigator(Kenkyū-buntansha)	岡宏枝龍谷大学, 理工学部, 教授 (20215221) 池田勉龍谷大学, 理工学部, 教授 (50151296) 四ツ谷晶二龍谷大学, 理工学部, 教授 (60128361) 神保秀一北海道大学, 大学院・理学研究科, 教授 (80201565) 松本和一郎龍谷大学, 理工学部, 教授 (40093314)
Keywords	ギンツブルグ・ランダウ方程式 / 渦糸解 / 薄膜領域 / 解の安定性 / 超伝導 / 非線形偏微分方程式 / 力学系
Research Abstract	ギンツブルグ・ランダウ方程式は超伝導の巨視的な現象を記述するモデルである.方程式は秩序変数と磁場のポテンシャルを結合した形で書き表される.この方程式において渦糸を表現する解は数学的な研究対象として,大変注目されているが,方程式は変数が多く数学的な取り扱いが難しい.そこで,誤定を単純化して秩序変数だけの方程式を考え,2次元領域での渦点のダイナミクスを考えると,あるパラメータの特異極限でそのダイナミクスはグリーン関数やグリーン関数の正則な項を使って表現できることが,神保と森田によって示された. 次に3次元の薄膜領域で磁場の効果がある場合を考える.神保と森田は薄膜の薄さを制御パラメータにとり,その極限を考えると,方程式の磁場の効果はゼロに近づき,2次元の秩序変数だけの問題に帰着できることを示した.これにより,最初の場合のように,磁場の効果を無視することについて数学的な正当性が示された. 2次元の渦点のダイナミクスの特徴的な性質は,有限次元の力学系として表せる.岡は力学系理論の応用について成果をあげている. ギンツブルグ・ランダウ方程式は偏微分方程式系になっているので,偏微分方程式の基礎的な研究は重要な関連テーマである.四ツ谷は楕円型方程式の対称解の研究で統一的なアプローチの方法を確立し,松本は初期値問題の基礎的定理を発展させた. また,ギンツブルグ・ランダウ方程式において数値シミュレーションは数学的理論の補完や,新しい問題の発展のために欠かせない.池田や中根は数値シミュレーションや数値解析の技術を発展させている.

Research Products
(12 results)

All Other

All Publications (12 results)

[Publications] Jimbo, S., Morita, Y.: "Ginzburg-Landau equation with magnetic effect in a thin domain"Calculus of Variations and Partial Differential Equations. (掲載予定).
[Publications] Jimbo, S., Morita, Y.: "Notes on the limit equation of vortex motion for the Ginzburg-Landau equation with Neumann condition"Japan Journal of Industrial and Applied Mathematics. Vol.18,No.2. 483-501 (2001)
[Publications] Morita, Y., Dockery, J., Pernarowski, M.: "Symmetry breaking homoclinic bifurcations in diffusively coupled equations"Journal of Dynamics and Differential Equations. Vol.13,No.3. 613-649 (2001)
[Publications] Yanagida, E., Yotsutani, S.: "A unified approach to the structure of radial solutions for semilinear elliptic problems"Japan Journal of Industrial and Applied Mathematics. Vol.18,No.2. 503-519 (2001)
[Publications] Kabeya, Y., Yanagida, E., Yotsutani, S.: "Global structure of solutions for equations of Brezis-Nirenburg type on the uniti ball"Proceedings of the Royal Society of Edinburgh Sect. A. Vol.131,No.3. 647-665 (2001)
[Publications] Yotsutani, S.: "Canonical form related with radial solutions of semilinear elliptic eouations and its applications"Taiwanese Journal of Mathematics. Vol.5,No.3. 507-517 (2001)
[Publications] Nagayama, M., Ikeda, T., Ishiwata, T., Tamura, N., Ohyanagi, M.: "Three-dimensional numerical simulation of helically propagating combustion waves"Journal of Materials Synthesis and Processing. Vol.9,No.3. 153-163 (2001)
[Publications] Gedeon, T., Kokubu, H., Mischaikow, K., Oka, H.: "Chaotic solutions in slowly varying perturbations of Hamiltonian systems with applications to shallow water sloshing"Journal of Dynamics and Differential Equations. Vol.14,No.1. 63-84 (2002)
[Publications] Matsumoto, W., Murai, M., Nagase, T.: "On the Cauchy-Kowalevskaya theorem of Nagumo type for systems"Hyperbolic differential operators and related problems, Ed V. Ancina et al., Marcel Dekker. (掲載予定).
[Publications] Jimbo, S., Zhai, J.: "Domain perturbation method and local minimizers to Ginzburg-Landau functional with magnetic effect"Abstract and Applied Analysis. (掲載予定).
[Publications] Nakane, K.: "Global existence of solutions to one-dimensional free boundary problem of hyperbolic type"Proceedings of the Japan Academy. Vol.77,No.7. 103-107 (2001)
[Publications] Nakane, K.: "Numerical calculation of singularities for Ginzburg-Landau functionals with a variable coefficient"Nonlinear Analysis : TMA. Vol.47. 3803-3811 (2001)

2001 Fiscal Year Annual Research Report

薄膜領域におけるギンツブルグ・ランダウ方程式の渦糸解

Principal Investigator

森田 善久 龍谷大学, 理工学部, 教授 (10192783)

Research Products

[Publications] Jimbo, S., Morita, Y.: "Ginzburg-Landau equation with magnetic effect in a thin domain"Calculus of Variations and Partial Differential Equations. (掲載予定).

[Publications] Jimbo, S., Morita, Y.: "Notes on the limit equation of vortex motion for the Ginzburg-Landau equation with Neumann condition"Japan Journal of Industrial and Applied Mathematics. Vol.18,No.2. 483-501 (2001)

[Publications] Morita, Y., Dockery, J., Pernarowski, M.: "Symmetry breaking homoclinic bifurcations in diffusively coupled equations"Journal of Dynamics and Differential Equations. Vol.13,No.3. 613-649 (2001)

[Publications] Yanagida, E., Yotsutani, S.: "A unified approach to the structure of radial solutions for semilinear elliptic problems"Japan Journal of Industrial and Applied Mathematics. Vol.18,No.2. 503-519 (2001)

[Publications] Kabeya, Y., Yanagida, E., Yotsutani, S.: "Global structure of solutions for equations of Brezis-Nirenburg type on the uniti ball"Proceedings of the Royal Society of Edinburgh Sect. A. Vol.131,No.3. 647-665 (2001)

[Publications] Yotsutani, S.: "Canonical form related with radial solutions of semilinear elliptic eouations and its applications"Taiwanese Journal of Mathematics. Vol.5,No.3. 507-517 (2001)

[Publications] Nagayama, M., Ikeda, T., Ishiwata, T., Tamura, N., Ohyanagi, M.: "Three-dimensional numerical simulation of helically propagating combustion waves"Journal of Materials Synthesis and Processing. Vol.9,No.3. 153-163 (2001)

[Publications] Gedeon, T., Kokubu, H., Mischaikow, K., Oka, H.: "Chaotic solutions in slowly varying perturbations of Hamiltonian systems with applications to shallow water sloshing"Journal of Dynamics and Differential Equations. Vol.14,No.1. 63-84 (2002)

[Publications] Matsumoto, W., Murai, M., Nagase, T.: "On the Cauchy-Kowalevskaya theorem of Nagumo type for systems"Hyperbolic differential operators and related problems, Ed V. Ancina et al., Marcel Dekker. (掲載予定).

[Publications] Jimbo, S., Zhai, J.: "Domain perturbation method and local minimizers to Ginzburg-Landau functional with magnetic effect"Abstract and Applied Analysis. (掲載予定).

[Publications] Nakane, K.: "Global existence of solutions to one-dimensional free boundary problem of hyperbolic type"Proceedings of the Japan Academy. Vol.77,No.7. 103-107 (2001)

[Publications] Nakane, K.: "Numerical calculation of singularities for Ginzburg-Landau functionals with a variable coefficient"Nonlinear Analysis : TMA. Vol.47. 3803-3811 (2001)

森田善久龍谷大学, 理工学部, 教授 (10192783)