Deformations of discrete curves and their applications

• Project/Area Number: 15K04862
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kurume Institute of Technology (2018); Fukuoka University (2015-2017)
• Principal Investigator: Matsuura Nozomu 久留米工業大学, 工学部, 准教授 (00389339)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Project Status: Completed (Fiscal Year 2018)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: 離散曲線 / 離散曲面 / 局所誘導方程式 / アフィン球面 / 弾性曲線 / 曲線短縮流 / 曲線短縮方程式 / 離散微分幾何 / 差分幾何 / 離散非固有アフィン球面 / 離散modified KdV方程式 / 離散リウヴィル方程式 / 離散非線形シュレディンガー方程式 / 渦糸

Outline of Final Research Achievements

We constructed discrete models for some deformations of curves in plane and in space. In particular, from the viewpoint of integrable systems, an integrable discrete model of the local induction equation, which is known as a mathematical model of vortex filaments, is formulated to construct a family of characteristic exact solutions. From the viewpoint of non-integrable systems, a simple discrete model of the curve shortening flow is proposed.

Academic Significance and Societal Importance of the Research Achievements

離散的な図形の性質と構成方法を研究した。本研究では例えば曲率に相当するような離散的概念を見出すことを通して離散的図形をコントロールしたため、構成した図形は汎用的な離散化アルゴリズムで生成したものに比べてよい特性を持っている。具体的には特に渦糸の運動に対する可積分離散モデルを作り、特徴的な厳密解の族を構成した。このモデルを雛形のひとつとして今後さらに離散的図形の幾何学が進展することが期待できる。

Research Products

• [Journal Article] Discrete local induction equation (2019)
  • Author(s): Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura and Yasuhiro Ohta
  • Journal Title: Journal of Integrable Systems Volume: 印刷中 Issue: 1 Pages: 1-43
  • DOI: 10.1093/integr/xyz003
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Isoperimetric deformations of curves on the Minkowski plane (2019)
  • Author(s): Hyeongki Park, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Nozomu Matsuura and Yasuhiro Ohta
  • Journal Title: International Journal of Geometric Methods in Modern Physics Volume: 印刷中 Issue: 07 Pages: 1950100-1950100
  • DOI: 10.1142/s0219887819501007
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Defocusing mKdV flow on centroaffine plane curves (2018)
  • Author(s): Hyeongki Park, Kenji Kajiwara, Takashi Kurose and Nozomu Matsuura
  • Journal Title: JSIAM Letters Volume: 10
  • NAID: 130007420864
  • Peer Reviewed
• [Journal Article] dNLS Flow on Discrete Space Curves (2016)
  • Author(s): Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara,Nozomu Matsuura, Yasuhiro Ohta
  • Journal Title: Mathematics for Industry Volume: 24 Pages: 137-150
  • DOI: 10.1007/978-981-10-1076-7_14
  • ISBN: 9789811010750, 9789811010767
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Isogonal deformation of discrete plane curves and discrete Burgers hierarchy (2016)
  • Author(s): Kenji Kajiwara, Toshinobu Kuroda and Nozomu Matsuura
  • Journal Title: Pacific Journal of Mathematics for Industry Volume: 8 Issue: 1
  • DOI: 10.1186/s40736-016-0022-z
  • NAID: 40021103648
  • Peer Reviewed / Open Access / Acknowledgement Compliant
• [Journal Article] 曲線と曲面の差分幾何 (2016)
  • Author(s): 松浦望
  • Journal Title: 日本応用数理学会誌「応用数理」
  • NAID: 130007043096
  • Peer Reviewed
• [Presentation] 曲線短縮方程式の離散化 (2018)
  • Author(s): 松浦望
  • Organizer: 九州可積分系セミナー (九州大学)
• [Presentation] 曲線短縮方程式の離散化 (2018)
  • Author(s): 松浦望
  • Organizer: 幾何学のスペクトル (北海道大学)
• [Presentation] Lotka-Volterra flow on discrete centroaffine plane curves (2018)
  • Author(s): 朴炯基, 梶原健司, 松浦望
  • Organizer: 日本応用数理学会2018年度年会 (名古屋大学)
• [Presentation] A construction method for discrete indefinite affine spheres (2017)
  • Author(s): Shimpei Kobayashi and Nozomu Matsuura
  • Organizer: The Tenth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
  • Place of Presentation: University of Georgia
  • Year and Date: 2017-03-31
  • Int'l Joint Research
• [Presentation] dNLS flow on discrete space curves (2017)
  • Author(s): Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura and Yasuhiro Ohta
  • Organizer: The Tenth IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory
  • Place of Presentation: University of Georgia
  • Year and Date: 2017-03-30
  • Int'l Joint Research
• [Presentation] Explicit formulas for area-preserving deformations of plane curves in the equicentroaffine geometry (2017)
  • Author(s): 梶原健司・丸野健一・松浦望・中西和音・朴炯基
  • Organizer: 日本応用数理学会第13回研究部会連合発表会
  • Place of Presentation: 電気通信大学
  • Year and Date: 2017-03-06
• [Presentation] 離散非固有アフィン球面の表現公式 (2017)
  • Author(s): 松浦望
  • Organizer: 九州可積分系セミナー
• [Presentation] Explicit formula for mKdV flow on centroaffine plane curves (2017)
  • Author(s): 梶原健司・黒瀬俊・松浦望・朴炯基
  • Organizer: 日本応用数理学会2017年度年会
• [Presentation] ミンコフスキ平面上および2次元ド・ジッター空間上の曲線から等積中心アフィン平面曲線へのある変換 (2017)
  • Author(s): 黒瀬俊・松浦望
  • Organizer: 日本数学会2017年度秋季総合分科会
• [Presentation] 平面曲線の等積変形に対する明示公式 (2016)
  • Author(s): 松浦望
  • Organizer: 福岡大学微分幾何セミナー
  • Place of Presentation: 福岡大学
  • Year and Date: 2016-11-24
• [Presentation] 等積幾何における平面離散曲線の変形 (2016)
  • Author(s): 松浦望
  • Organizer: 意匠設計のための微分幾何学・離散微分幾何
  • Place of Presentation: 九州大学
  • Year and Date: 2016-09-29
• [Presentation] 離散空間曲線の運動に対する行列式解とPfaffian解 (2016)
  • Author(s): 廣瀬三平・井ノ口順一・梶原健司・松浦望・太田泰広
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-19
• [Presentation] 渦糸方程式の離散化 (2016)
  • Author(s): 廣瀬三平・井ノ口順一・梶原健司・松浦望・太田泰広
  • Organizer: 日本数学会2016年度年会
  • Place of Presentation: 筑波大学
  • Year and Date: 2016-03-16
• [Presentation] 空間離散曲線の等距離等周変形II: 渦糸方程式の離散化との関係 (2016)
  • Author(s): 廣瀬三平・井ノ口順一・梶原健司・松浦望・太田泰広
  • Organizer: 第12回日本応用数理学会研究部会連合発表会
  • Place of Presentation: 神戸学院大学
  • Year and Date: 2016-03-05
• [Presentation] 空間離散曲線の等距離等周変形I: lattice Landau-Lifschitz方程式による変形 (2016)
  • Author(s): 廣瀬三平・井ノ口順一・梶原健司・松浦望・太田泰広
  • Organizer: 第12回日本応用数理学会研究部会連合発表会
  • Place of Presentation: 神戸学院大学
  • Year and Date: 2016-03-05
• [Presentation] Integrable discrete models of vortex filaments (2016)
  • Author(s): Nozomu Matsuura
  • Organizer: Transformations and Singularities
  • Place of Presentation: Tokyo Institute of Technology
  • Year and Date: 2016-02-23
  • Int'l Joint Research / Invited
• [Presentation] 渦糸方程式の離散化 (2016)
  • Author(s): 松浦望
  • Organizer: 測地線及び関連する諸問題2016
  • Place of Presentation: 熊本大学
  • Year and Date: 2016-01-11
  • Invited
• [Presentation] 渦糸方程式の離散化 (2015)
  • Author(s): 廣瀬三平・井ノ口順一・梶原健司・松浦望・太田泰広
  • Organizer: 可積分系が拓く現象数理モデル
  • Place of Presentation: 明治大学
  • Year and Date: 2015-11-06
• [Presentation] 渦糸方程式の離散化 (2015)
  • Author(s): 廣瀬三平・井ノ口順一・梶原健司・松浦望・太田泰広
  • Organizer: 日本応用数理学会2015年度年会
  • Place of Presentation: 金沢大学
  • Year and Date: 2015-09-09
• [Funded Workshop] The 13th International Conference on Symmetries and Integrability of Difference Equations (SIDE13) (2018)