Study of connection problem for non-integrable Hamiltonian systems

• Project/Area Number: 26400118
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Hiroshima University
• Principal Investigator: Yoshino Masafumi 広島大学, 理学研究科, 教授 (00145658)
• Project Period (FY): 2014-04-01 – 2017-03-31
• Project Status: Completed (Fiscal Year 2016)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
• Keywords: ボレル総和法 / ハミルトン系 / 非可積分性 / 接続問題 / モノドロミー / Fuchsian equation / 特異摂動 / モーメント総話法 / semi formal solution / parametric Borel sum / semi-formal solution / Stokes phenomenon / Lotka Volterra / nonintegrability

Outline of Final Research Achievements

We study the structure of non-integrability of a Hamiltonian system and the global　properties of solutions by using a singular first integral.The difficulty is to give the meaning to the divergent formal first integral or to the divergence of the normalizing transformation. We prove the Borel summability with respect to a certain parameter in the equation for first order system of partial differential equations. As for the non-integrability we show the semi-global non-integrability of a Hamiltonian system which is not analyzed by differential Galois theory. Because the global property of the first integral in the argument is not well understood, the result has the possibility to improve.

Research Products

• [Journal Article] Monodromy of confluent hypergeometric system of Okubo type (2016)
  • Author(s): Masafumi Yoshino
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B57 Pages: 281-296
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Nonintegrability of Hamiltonian system perturbed from integrable system with two singular points (2016)
  • Author(s): Masafumi Yoshino and Yoshikatsu Sasaki
  • Journal Title: Math. Zeitschrift Volume: 284 Issue: 3-4 Pages: 1005-1020
  • DOI: 10.1007/s00209-016-1684-z
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Monodromy of confluent hypergeometric system of Okubo type (2016)
  • Author(s): Masafumi Yoshino
  • Journal Title: 数理研講究録別冊 Volume: 62 Pages: 255-262
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Analytic continuation of Borel sum of formal solution of semilinear partial differential equation (2015)
  • Author(s): Masafumi Yoshino
  • Journal Title: Asymptotic Analysis Volume: 92 Issue: 1-2 Pages: 65-84
  • DOI: 10.3233/asy-141270
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Semi-formal solution and monodromy of some confluent hypergeometric equations (2015)
  • Author(s): Masafumi Yoshino
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: B4x Pages: 255-262
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Borel summability of some semilinear system of partial differential equations (2015)
  • Author(s): Hiroshi Yamazawa and Masafumi Yoshino
  • Journal Title: Opuscula Mathematica Volume: 35, no. 6 Pages: 825-845
  • Peer Reviewed / Acknowledgement Compliant
• [Presentation] Blowup of wave equation and Birkoff theory of some Hamiltonian system (2017)
  • Author(s): Masafumi Yoshino
  • Organizer: A Life in Mathematics Generalized functions, Microlocal analysis, PDEs and Dynamical systems
  • Place of Presentation: トリノ大学（イタリア）
  • Int'l Joint Research / Invited
• [Presentation] Application of Borel summability to small denominator problem (2016)
  • Author(s): Masafumi Yoshino
  • Organizer: 日本数学会年会
  • Place of Presentation: 筑波大学(茨城県・つくば市)
  • Year and Date: 2016-03-16
• [Presentation] Parametric Borel summability of first order partial differential equation not satisfying the Poincare condition (2016)
  • Author(s): Masafumi Yoshino
  • Organizer: Formal and Analytic Solutions of Partial Differential Equations" (FASPDE)
  • Place of Presentation: リスボン大学（ポルトガル）
  • Int'l Joint Research / Invited
• [Presentation] Moving singularity and monodromy of some Hamiltonian system (2016)
  • Author(s): Masafumi Yoshino
  • Organizer: 日本数学会
  • Place of Presentation: 関西大学
• [Presentation] Semi-global non integrability of Hamiltonian system and Borel summability (2015)
  • Author(s): Masafumi Yoshino
  • Organizer: Algebraic Analytic Methods, (RIMS)
  • Place of Presentation: RIMS (Kyoto)
  • Year and Date: 2015-12-11
  • Int'l Joint Research / Invited
• [Presentation] Monodromy of some resonant Hamiltonian system (2015)
  • Author(s): Masafumi Yoshino
  • Organizer: Microlocal Analysis and Singular Perturbation Theory, (RIMS)
  • Place of Presentation: RIMS (Kyoto)
  • Year and Date: 2015-10-05
  • Int'l Joint Research / Invited
• [Presentation] Borel summability of formal solutions of first order system of PDE (2015)
  • Author(s): Masafumi Yoshino
  • Organizer: Analytic, Algebraic and Geometric Aspects of Differential Equations 2015, Banach center (Poland)
  • Place of Presentation: Banach center (Poland)
  • Year and Date: 2015-09-16
  • Int'l Joint Research / Invited
• [Presentation] 形質進化を伴う３種系の挙動 (2015)
  • Author(s): 吉野正史，田中 嘉成
  • Organizer: 日本数学会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] ある半線形偏微分方程式系の形式解のボレル総和法 (2015)
  • Author(s): 吉野正史，山澤浩司
  • Organizer: 日本数学会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] Nonintegrability of Hamiltonian system perturbed from integrable system with two singular points (2015)
  • Author(s): 吉野正史，佐々木良勝
  • Organizer: 日本数学会
  • Place of Presentation: 明治大学
  • Year and Date: 2015-03-21 – 2015-03-24
• [Presentation] Semi-formal theory and formula of monodromy (2014)
  • Author(s): Masafumi Yoshino
  • Organizer: FASFE
  • Place of Presentation: Valladolid, Spain
  • Year and Date: 2014-09-01 – 2014-09-05
  • Invited
• [Remarks] http://home.hiroshima-u.ac.jp/yoshinom/
• [Remarks] Research and Papers (研究成果報告)
  • URL: http://home.hiroshima-u.ac.jp/yoshinom/
• [Remarks] 吉野正史のホームページ
  • URL: http://home.hiroshima-u.ac.jp/yoshinom/