New development of discrete differential geometry by discrete integrable systems

• Project/Area Number: 21656027
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Single-year Grants
• Research Field: Engineering fundamentals
• Research Institution: Kyushu University
• Principal Investigator: KAJIWARA Kenji 九州大学, マス・フォア・インダストリ研究所, 教授 (40268115)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,840,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥240,000); Fiscal Year 2011: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2010: ¥900,000 (Direct Cost: ¥900,000); Fiscal Year 2009: ¥900,000 (Direct Cost: ¥900,000)
• Keywords: 離散可積分系 / 離散微分幾何 / 離散ソリトン方程式 / τ関数 / 離散曲線 / 不均一格子 / 双線形形式 / 離散正則関数 / 超幾何関数

Research Abstract

Discrete differential geometry has been considered, which is a mathematical framework describing the dynamics of geometric objects on computers. Theory of discrete integrable systems has been developed, which was successfully applied to discretize the planar curve dynamics on Eucledian plane and to construct explicit formulas in terms of the z functions. As by-products, highaccuracy stable numerical schemes for curve dynamics with adaptive step size control, and seeds of new development to the theory of discrete conformal mappings have been obtained.

Research Products

• [Journal Article] Explicit solutions to the semi妖iscrete modified KdV equation and motion of discrete plane curves (2012)
  • Author(s): J. Inoguchi, K. Kajiwara, N. Matsuura and Y. Ohta
  • Journal Title: J. Phys. A : Math. Theoret Volume: 45 Pages: 45206-45206
• [Journal Article] Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves (2012)
  • Author(s): Jun-ichi Inoguchi, et al
  • Journal Title: Journal of Physics A : Mathematical and Theoretical Volume: 45 Issue: 4 Pages: 45206-45206
  • DOI: 10.1088/1751-8113/45/4/045206
  • Peer Reviewed
• [Journal Article] Motion and Backlund transformations of discrete plane curves (2011)
  • Author(s): J. Inoguchi, K. Kajiwara, N. Matsuura and Y. Ohta
  • Journal Title: to appear in Kyushu J. Math
• [Journal Article] Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves (2011)
  • Author(s): B. F. Feng, J. Inoguchi, K. Kajiwara, K. Maruno and Y. Ohta
  • Journal Title: J. Phys. A : Math. Theoret Volume: 44 Pages: 395201-395201
• [Journal Article] Bilinearization and special solutions to the discrete Schwarzian KdV equation (2011)
  • Author(s): M. Hay, K. Kajiwara and T. Masuda
  • Journal Title: J. Math-for-lnd Volume: 3 Pages: 53-62
  • NAID: 40019161209
• [Journal Article] Semi-discrete modified KdV方程式と平面離散曲線の時間発展 (2011)
  • Author(s): 井ノロ順一, 梶原健司, 松浦望, 太田泰広
  • Journal Title: 九州大学応用力学研究所22-AO-S8「非線形波動研究の新しい展開一現象とモデル化一」 Pages: 75-81
• [Journal Article] Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves (2011)
  • Author(s): B.-F.Feng, J.Inoguchi, K.Kajiwara, K.Maruno and Y.Ohta
  • Journal Title: J. Phys. A : Math. Theor. Volume: 44 Issue: 39 Pages: 19-19
  • DOI: 10.1088/1751-8113/44/39/395201
  • Peer Reviewed
• [Journal Article] Motion and Backlund transformations of discrete plane curves (2011)
  • Author(s): J.Inoguchi, K.Kajiwara, N.Matsuura, Y.Ohta
  • Journal Title: Kyushu Journal of Mathematics Volume: 65(In press)
  • Peer Reviewed
• [Journal Article] Bilinearization and special solutions to the discrete Schwarzian KdV equation (2011)
  • Author(s): M.Hay, K.Kajiwara, T.Masuda
  • Journal Title: Journal of Math-for-Industry Volume: 3(In press)
  • NAID: 40019161209
  • Peer Reviewed
• [Journal Article] Semi-discrete modified KdV方程式と平面離散曲線の時間発展 (2011)
  • Author(s): 井ノ口順一, 梶原健司, 松浦望, 太田泰広
  • Journal Title: 九州大学応用力学研究所研究集会報告 Volume: (In press)
  • Peer Reviewed
• [Journal Article] The discrete potential Boussinesq equation and its multisoliton solutions (2010)
  • Author(s): K. Maruno and K. Kajiwara
  • Journal Title: Applicable Analysis Volume: 89 Pages: 593-609
• [Journal Article] The discrete potential Boussinesq equation and its multisoliton solutions (2010)
  • Author(s): Ken-ichi Maruno, Kenji Kajiwara
  • Journal Title: Applicable Analysis 89(in press)
  • Peer Reviewed
• [Journal Article] Bilinearization and Casorati determinant solutions to non-autonomous 1+ 1 dimensional discrete soliton equations (2009)
  • Author(s): K. Kajiwara and Y. Ohta
  • Journal Title: RIMS Kokyuroku Bessatu Volume: 13 Pages: 53-74
• [Journal Article] Bilinearization and Casorati determinant solutions to non-autonomous 1+1 dimensional discrete soliton equations (2009)
  • Author(s): Kenji Kajiwara, Yasuhiro Ohta
  • Journal Title: RIMS Kokyuroku Bessatsu B13 Pages: 53-74
  • Peer Reviewed
• [Presentation] Discretization of planar curve motions and discrete modified KdV equations (2012)
  • Author(s): K. Kajiwara
  • Organizer: American Mathematical Society 2012 Spring Southeastern Section Meeting
  • Place of Presentation: 南フロリダ大学,アメリカ
  • Year and Date: 2012-03-11
• [Presentation] 相似幾何における平面離散曲線の運動 (2012)
  • Author(s): 黒田利信,梶原健司
  • Organizer: 日本応用数理学会2(11) 12年研究部会連合発表会
  • Place of Presentation: 九州大学
  • Year and Date: 2012-03-09
• [Presentation] Hirota-atsumaの非線形ネットワーク方程式の双線形構造と解 (2012)
  • Author(s): 野見山直之,梶原健司
  • Organizer: 日本応用数理学会2012年研究部会連合発表会
  • Place of Presentation: 九州大学
  • Year and Date: 2012-03-09
• [Presentation] 相似幾何における平面離散曲線の運動 (2012)
  • Author(s): 黒田利信, 梶原健司
  • Organizer: 日本応用数理学会2012年研究部会連合発表会
  • Place of Presentation: 九州大学(福岡)
  • Year and Date: 2012-03-09
• [Presentation] Hirota-Satsumaの非線形ネットワーク方程式の双線形構造と解 (2012)
  • Author(s): 野見山直之, 梶原健司
  • Organizer: 日本応用数理学会2012年研究部会連合発表会
  • Place of Presentation: 九州大学(福岡)
  • Year and Date: 2012-03-09
• [Presentation] 可積分系入門 (2012)
  • Author(s): 梶原健司
  • Organizer: 離散可積分系・離散微分幾何チュートリアル2012
  • Place of Presentation: 九州大学
  • Year and Date: 2012-02-22
• [Presentation] 可積分系入門 (2012)
  • Author(s): 梶原健司
  • Organizer: 離散可積分系・離散微分幾何チュートリアル2012
  • Place of Presentation: 九州大学(福岡)(招待講演)
  • Year and Date: 2012-02-22
• [Presentation] Discretization of planar curve motions and discrete integrable systems (2011)
  • Author(s): 梶原健司
  • Organizer: Winter School for Young Researchers on Mathematical Aspects of Image Processing and Computer Vision
  • Place of Presentation: 東北大学
  • Year and Date: 2011-11-25
• [Presentation] Discretization of planar curve motions and discrete integrable systems (2011)
  • Author(s): 梶原健司
  • Organizer: Winter School for Young Researchers on Mathematical Aspects of Image Processing and Computer Vision
  • Place of Presentation: 東北大学(宮城)(招待講演)
  • Year and Date: 2011-11-25
• [Presentation] 平面離散曲線の運動とRacklund変換 (2011)
  • Author(s): 梶原健司
  • Organizer: 東北大学幾何セミナー
  • Place of Presentation: 東北大学
  • Year and Date: 2011-10-05
• [Presentation] 平面折線の運動その2 : R~ ck1 iznd変換と厳密解 (2011)
  • Author(s): 梶原健司
  • Organizer: 日本数学会2011年度秋季総合分科会
  • Place of Presentation: 名古屋大学
  • Year and Date: 2011-09-24
• [Presentation] 離散平面曲線の時間発展に現れる離散可積分系と離散ホドグラフ変換 (2011)
  • Author(s): 梶原健司
  • Organizer: 日本応用数理学会2011年度年会
  • Place of Presentation: 同志社大学
  • Year and Date: 2011-09-14
• [Presentation] 離散平面曲線の時間発展に現れる離散可積分系と離散ホドグラフ変換 (2011)
  • Author(s): 梶原健司
  • Organizer: 日本応用数理学会2011年度年会
  • Place of Presentation: 同志社大学(京都)
  • Year and Date: 2011-09-14
• [Presentation] An explicit formula for the discrete power function associated with circle patterns of Schramm type (2011)
  • Author(s): K. Kajiwara
  • Organizer: Discrete Integrable Systems
  • Place of Presentation: ローレンツセンター,オランダ
  • Year and Date: 2011-07-20
• [Presentation] An explicit formula for the discrete power function associated with circle patterns of Schramm type (2011)
  • Author(s): Kenji Kajiwara
  • Organizer: Discrete Integrable Systems
  • Place of Presentation: ローレンツセンター,オランダ(招待講演)
  • Year and Date: 2011-07-20
• [Presentation] Motion and Backlund transformations of discrete plane curves (2011)
  • Author(s): K. Kajiwara
  • Organizer: Tropical Geometry and Integrable Systems
  • Place of Presentation: 連合王国
  • Year and Date: 2011-07-06
• [Presentation] Motion and Backlund transformations of discrete plane curves (2011)
  • Author(s): Kenji Kajiwara
  • Organizer: Tropical Geometry and Integrable Systems
  • Place of Presentation: グラスゴー大学,連合王国
  • Year and Date: 2011-07-06
• [Presentation] Discrete integrable systems and dynamics of planar discrete curves, The eve of IMI & ESI (2011)
  • Author(s): K. Kajiwara
  • Organizer: Ito one day workshopBuilding a bridge between mathematics and physics
  • Place of Presentation: 九州大学
  • Year and Date: 2011-03-29
• [Presentation] Discrete integrable systems and dynamics of planar discrete curves (2011)
  • Author(s): Kenji Kajiwara
  • Organizer: The eve of IMI & ESI, Ito one day workshop-Building a bridge between mathematics and physics
  • Place of Presentation: 九州大学大学院数理学研究院(福岡県)
  • Year and Date: 2011-03-29
• [Presentation] Motion and Backlund transformations of discrete plane curves (2011)
  • Author(s): K. Kajiwara
  • Organizer: Integrable Systems Seminar
  • Place of Presentation: テキサス大学パンアメリカン校,アメリカ
  • Year and Date: 2011-02-18
• [Presentation] Motion and Backlund transformations of discrete plane curves (2011)
  • Author(s): Kenji Kajiwara
  • Organizer: Integrabls systems seminar
  • Place of Presentation: テキサス大学パンアメリカン校(アメリカ合衆国)
  • Year and Date: 2011-02-18
• [Presentation] Discretization of integrable systems from nonlinear waves to geometry (2011)
  • Author(s): K. Kajiwara
  • Organizer: Distinguished Lecture, Colloquium
  • Place of Presentation: テキサス大学パンアメリカン校,アメリカ
  • Year and Date: 2011-02-17
• [Presentation] Discretization of integrable systems-from nonlinear waves to geometry (2011)
  • Author(s): Kenji Kajiwara
  • Organizer: Disctinguished lecture, colloquium
  • Place of Presentation: テキサス大学パンアメリカン校(アメリカ合衆国)
  • Year and Date: 2011-02-17
• [Presentation] Motion and Backlund transformations of discrete plane curves (2010)
  • Author(s): K. Kajiwara
  • Organizer: GCOE conference : algebraic and geometric aspects of discrete integrable systems
  • Place of Presentation: 東京大学
  • Year and Date: 2010-12-16
• [Presentation] Semi妖iscrete modified方程式の解と平面離散曲線の連続的運動 (2010)
  • Author(s): 梶原健司
  • Organizer: 九州大学応用力学研究所研究集会「非線形波動研究の新たな展開一現象とモデル化一」
  • Place of Presentation: 九州大学
  • Year and Date: 2010-12-16
• [Presentation] Motion and Backlund transformations of discrete plane curves (2010)
  • Author(s): Kenji Kajiwara
  • Organizer: GCOE conference : algebraic and geometric aspects of discrete integrable systems
  • Place of Presentation: 東京大学大学院数理科学研究科(東京都)
  • Year and Date: 2010-12-16
• [Presentation] Semi-discrete modified KdV方程式の解と平面離散曲線の連続的運動 (2010)
  • Author(s): 梶原健司
  • Organizer: 九州大学応用力学研究所研究集会「非線形波動研究の新たな展開-現象 とモデル化-」
  • Place of Presentation: 九州大学応用力学研究所(福岡県)
  • Year and Date: 2010-10-29
• [Presentation] 平面離散曲線の運動とBacklund変換 (2010)
  • Author(s): 梶原健司
  • Organizer: 東北大学幾何セミナー
  • Place of Presentation: 東北大学大学院理学研究科(宮城県)
  • Year and Date: 2010-10-05
• [Presentation] 平面折線の運動 その2:Backlund変換と厳密解 (2010)
  • Author(s): 梶原健司
  • Organizer: 日本数学会2011年度秋季総合分科会
  • Place of Presentation: 名古屋大学(愛知県)
  • Year and Date: 2010-09-24
• [Presentation] ユークリッド平面上の離散曲線の運動とBacklund変換 (2010)
  • Author(s): 梶原健司
  • Organizer: 日本応用数理学会2010年度年会
  • Place of Presentation: 明治大学
  • Year and Date: 2010-09-08
• [Presentation] ユークリッド平面上の離散曲線の運動とBacklund変換 (2010)
  • Author(s): 梶原健司
  • Organizer: 日本応用数理学会2010年度年会
  • Place of Presentation: 明治大学駿河台キャンパス(東京都)
  • Year and Date: 2010-09-08
• [Presentation] Schwarzian KPMichael Hay, Discrete方程式の双線形化と簡約 (2010)
  • Author(s): 梶原健司
  • Organizer: 日本数学会2010年度年会
  • Place of Presentation: 慶慮義塾大学
  • Year and Date: 2010-03-26
• [Presentation] Discrete Schwarzian KP方程式の双線形化と簡約 (2010)
  • Author(s): 梶原健司, Michael Hay
  • Organizer: 日本数学会2010年度年会
  • Place of Presentation: 慶應義塾大学
  • Year and Date: 2010-03-26
• [Presentation] 可積分系入門 (2010)
  • Author(s): 梶原健司
  • Organizer: 離散可積分系・離散微分幾何チュートリアル
  • Place of Presentation: 九州大学
  • Year and Date: 2010-02-22
• [Book] 離散可積分系・離散微分幾何チユートリアル2012, MIレクチャーノートシリーズ40 (2012)
  • Author(s): 井ノロ順一, 太田泰広, 梶原健司, 松浦望(編)
  • Publisher: 九州大学
• [Book] 離散可積分系・離散微分幾何チュートリアル2012 (2012)
  • Author(s): 井ノ口順一, 他(編)
  • Total Pages: 162
  • Publisher: 九州大学
• [Book] 可視化の技術と現代幾何学 (2010)
  • Author(s): 若山正人(編)
  • Publisher: 岩波書店
• [Remarks]
  • URL: http://gandalf.math.kyushu-u.ac.jp/DISDDG/
• [Remarks]
  • URL: http://gandalf.math.kyushu-u.ac.jp/DISDDG/index.html