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Dynamics of patterns and their interactions for nonlinear evolutional equations

Research Project

Project/Area Number 16540200
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionKyushu University

Principal Investigator

EI Shin-ichiro  Kyushu University, Faculty of Mathematics, Professor (30201362)

Co-Investigator(Kenkyū-buntansha) YANAGIDA Eiji  Tohoku University, Faculty of Science, Professor (80174548)
FUJII Kazuyuki  Yokohama City University, Department of Mathematical Sciences, Professor (00128084)
SHIRAISHI Takaaki  Yokohama City University, Department of Mathematical Sciences, Professor (50143160)
MIZUMACHI Tetsu  Kyushu University, Faculty of Mathematics, Associate Professor (60315827)
Project Period (FY) 2004 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,970,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsReaction diffusion systems / Pulse interaction / Interfacial dynamics / Traveline front / 反応拡散方程式 / 平面解 / 4階の方程式 / チューリング不安定性 / 局在解
Research Abstract

In the period of this project, we completely classified the bifurcation structures of 1 dimensional traveling front solutions and analyzed the dynamics of traveling front solutions in inhomogeneous media from the dynamical system point of view. By the analysis, we can know how the solutions go through or reflect by obstacles with words of invariant manifold theory. The interaction of two front solutions is also treated. In general, the treatment of pulse solution is difficult. Applying the results of the interaction of two front solutions, we construct a pulse solution as the combination of two front solutions, which makes easier to treat pulse solutions. Second main result is to establish a general framework to analyze the boundary spike solutions in two dimensional domain. We proved a spike solution moves along the boundary towards the point with maximal curvature of the boundary. We also showed the existence of stable stationary solutions with two peaks in the neighborhood of a point on the boundary with maximal curvature. The results are applied to the Gierer-Meinhardt model and its shadow system. The extension of these results to higher dimensional spaces is still open. For the interfacial dynamics of reaction-diffusion systems in two dimensional spaces including spiral motion, the rigorous treatment has not been completed yet. But almost any necessary preparation in order to study such an interfacial problem such as visualization soft ware and the search program of the motion of a spiral core has been done. At the beginning of the project, we did not know explicit direction to solve this kind of interfacial problems except ambiguous expectations. At the end of the period of this project, now we have explicit ways to reach the goal. The results obtained during the period of the project are inherited to the next project as indispensable and important results.

Report

(5 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (41 results)

All 2008 2007 2006 2005 2004

All Journal Article (29 results) (of which Peer Reviewed: 5 results) Presentation (8 results) Book (4 results)

  • [Journal Article] Dynamics of front solutions in a specific reaction-diffusion system in one dimension2008

    • Author(s)
      Ei, et. al.
    • Journal Title

      Japan J.Ind.Appl.Math 25

      Pages: 117-147

    • NAID

      10021073097

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Dynamics of front solutions in a specific reaction-diffusion system in one dimension2008

    • Author(s)
      Ei, et. al.
    • Journal Title

      Japan J.Ind.Appl.Math. 25

      Pages: 117-147

    • NAID

      10021073097

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] General Solution of the Quantum Damped Harmonic Oscillator2008

    • Author(s)
      R., Endo, K., Fujii, T., Suzuki
    • Journal Title

      to appear in International Journal of Geometric Methods in Modern Physics Vol. 5, No. 5

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Flow Representation of the Bose-Hubbard Hamiltonian : General Case2008

    • Author(s)
      K., Fujii, T., Suzuki
    • Journal Title

      to appear in International Journal of Geometric Methods in Modern Physics Vol. 5, No. 3

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Dynamics of front solutions in a specific reaction-diffusion system in one dimension2008

    • Author(s)
      S. -I. Ei, et. al.
    • Journal Title

      Japan J. Ind. Appl. Math. 25

      Pages: 117-147

    • NAID

      10021073097

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Neuron phase shift adaptive to time delay in locomotor control2008

    • Author(s)
      K. Ohgane, S.-I. Ei and H. Mahara
    • Journal Title

      Appl.Math.Modelling (In press)

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic stability of small solitons for 2D Nonlinear Schr\"{o}dinger equations with potential2007

    • Author(s)
      Tetsu, Mizumachi
    • Journal Title

      Journal of Mathematics of Kyoto University 47(3)

      Pages: 599-620

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Instability of vortex solitons for 2D focusing NLS2007

    • Author(s)
      Tetsu, Mizumachi
    • Journal Title

      Advances in Differential Equations 12(3)

      Pages: 241-264

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Irregular behavior of solutions for Fisher's equation2007

    • Author(s)
      E., Yanagida
    • Journal Title

      J. Dyn. Diff. Eqs. 19

      Pages: 895-914

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On certain one-dimensional elliptic systems under different growth conditions at respective infinities2007

    • Author(s)
      M., Iida, K., Nakashima, E., Yanagida
    • Journal Title

      Advanced Studies in Pure Mathematics 19 No. 4

      Pages: 895-914

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] More on the isomorphism $SU(2)otimes SU(2) cong SO(4)2007

    • Author(s)
      Fujii et al.
    • Journal Title

      International Journal of Geometric Methods in Modern Physics (In press)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] On the magic matrix by Makhlin and the B-C-H formula in SO(4)2007

    • Author(s)
      Fujii et al.
    • Journal Title

      International Journal of Geometric Methods in Modern Physics (In press)

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Interacting Spots in reaction diffusion systems2006

    • Author(s)
      Ei, et. al.
    • Journal Title

      DCDS 14

      Pages: 31-62

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Interacting Spots in reaction diffusion systems2006

    • Author(s)
      Ei et al.
    • Journal Title

      DCDS 14

      Pages: 31-62

    • Related Report
      2006 Annual Research Report 2005 Annual Research Report
  • [Journal Article] Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems2006

    • Author(s)
      Yanagida et al.
    • Journal Title

      J. Diff. Eqs. 230

      Pages: 446-464

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Instability of vortex solitons for 2D focusing NLS2006

    • Author(s)
      Mizumachi
    • Journal Title

      Advances in Differential Equations 12(3)

      Pages: 241-264

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A remark on linearly unstable standing wave solutions to NLS2006

    • Author(s)
      Mizumachi
    • Journal Title

      Nonlinear Analysis 64

      Pages: 657-676

    • Related Report
      2006 Annual Research Report 2005 Annual Research Report
  • [Journal Article] 'Initial state'coordinations reproduce the instant flexibility for human walking2005

    • Author(s)
      Ei, et. al.
    • Journal Title

      Biol Cybern 93

      Pages: 426-435

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Initial state' coordinations reproduce the instant flexibility for human walking2005

    • Author(s)
      Ei, et. al.
    • Journal Title

      Biol Cybern. 93

      Pages: 426-435

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A Variational Approach to Singular Perturbation Problems in Reaction-Diffusion Systems2005

    • Author(s)
      Ei et al.
    • Journal Title

      Physica D 207

      Pages: 171-219

    • Related Report
      2005 Annual Research Report
  • [Journal Article] ‘Initial state' coordinations reproduce the instant flexibility for human walking2005

    • Author(s)
      Ohgane, Ohgane, Ei, Mahara, Ohtsuki
    • Journal Title

      Biol Cybern. (In press)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A Liouville property and quasiconvergence for a semilinear heat equation2005

    • Author(s)
      Yanagida et al.
    • Journal Title

      J.Diff.Eqs. 208

      Pages: 194-214

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Standard and Non-Standard Quantum Models : A Non-Commutative Version of the Classical System of SU(2) and SU(1,1) Arising from Quantum Optics2005

    • Author(s)
      Fujii
    • Journal Title

      Int.J.Geom.Meth.Mod.Phys. 2

      Pages: 783-822

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Interacting Spots in reaction diffusion systems2005

    • Author(s)
      S.-I.Ei et al.
    • Journal Title

      DCDSに掲載予定

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Emergence of adaptability to time delay in bipedal locomotion2004

    • Author(s)
      K.Ohgane, S.-I.Ei et al.
    • Journal Title

      Biol.Cybern. 90(2)

      Pages: 125-132

    • Related Report
      2004 Annual Research Report
  • [Journal Article] 周期解の取り扱いに関する一考察,応用数理(岩波)2004

    • Author(s)
      栄 伸一郎
    • Journal Title

      応用数理(岩波書店) 14(1)

      Pages: 35-47

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation2004

    • Author(s)
      P.Polacik, E.Yanagida
    • Journal Title

      Diff.Int.Eqs. 17

      Pages: 535-548

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Cavity QED and Quantum Computation in the Weak Coupling Regime2004

    • Author(s)
      K.Fujii et al.
    • Journal Title

      Journal of Optics B : Quantum and Semiclassical Optics 6

      Pages: 502-509

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Asymptotic stability of solitary wave solutions to the regularized long wave equation2004

    • Author(s)
      Tetsu Mizumachi
    • Journal Title

      Journal of Differential Equations 200

      Pages: 312-341

    • Related Report
      2004 Annual Research Report
  • [Presentation] 周期解の位相ダイナミクスの導出法とその応用2008

    • Author(s)
      栄 伸一郎
    • Organizer
      日本数学会応用数学特別セッション
    • Place of Presentation
      近畿大学
    • Year and Date
      2008-03-25
    • Related Report
      2007 Annual Research Report
  • [Presentation] Asymptotic stability of lattice solitons in the energy space2008

    • Author(s)
      T., Mizumachi
    • Organizer
      Nonlinear Evolution Equations of Mathematical Physics
    • Place of Presentation
      Baton, USA
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] The dynamics of boundary spikes for reaction-diffusion systems in space dimension 22007

    • Author(s)
      S.-I., Ei
    • Organizer
      Mathematical modeling and analysis in biological and chemical systems Orsay
    • Place of Presentation
      France
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Moving singularities of solutions to a semilinear parabolic equation2006

    • Author(s)
      E., Yanagida
    • Organizer
      Third Euro-Japanese Workshop on Blowup El Escorial
    • Place of Presentation
      Spain
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] 反応拡散系におけるパルスの相互作用たついて2005

    • Author(s)
      栄 伸一郎
    • Organizer
      日本数学会企画特別講演
    • Place of Presentation
      岡山大学
    • Year and Date
      2005-09-21
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dynamics of front solutions of reaction-diffusion systems with bistable nonlinearity2005

    • Author(s)
      S.-I., Ei
    • Organizer
      Mathematical analysis of complex phenomena in life sciences
    • Place of Presentation
      Tokyo, Japan
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dynamics of front solutions in heterogeneous media2005

    • Author(s)
      S.-I., Ei
    • Organizer
      Dynamics of nonlinear waves
    • Place of Presentation
      Groningen, Holland
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Presentation] Dynamics of Turing patterns for reaction-diffusion systems a cylindrical domain on 2D2004

    • Author(s)
      S.-I., Ei
    • Organizer
      Hyperbolic prootems, Theory, Numerics, in Applications
    • Place of Presentation
      Oosaka, Japan
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] フーリエ解析+偏微分方程式2007

    • Author(s)
      藤原, 栄
    • Total Pages
      198
    • Publisher
      裳華房
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] Fourier analysis + partial differential equations2007

    • Author(s)
      K., Fujiwara, S.-I., Ei
    • Total Pages
      198
    • Publisher
      Shokabou
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] フーリエ解析+偏微分方程式2007

    • Author(s)
      藤原 栄
    • Total Pages
      198
    • Publisher
      裳華房
    • Related Report
      2007 Annual Research Report
  • [Book] パターン形成とダイナミクス2006

    • Author(s)
      三村, 上山, 西浦, 長山, 栄
    • Total Pages
      149
    • Publisher
      東大出版会
    • Related Report
      2006 Annual Research Report 2005 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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