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Mathematical Analysis of helical waves arising in some-reaction diffusion systems

Research Project

Project/Area Number 12440032
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionRyukoku University

Principal Investigator

IKEDA Tsutomu  Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (50151296)

Co-Investigator(Kenkyū-buntansha) NINOMIYA Hirokazu  Ryukoku University, Faculty of Science and Technology, Associated Professor, 理工学部, 助教授 (90251610)
MORITA Yoshihisa  Ryukoku University, Faculty of Science and Technology, Professor, 理工学部, 教授 (10192783)
IKEDA Hideo  Toyama University, Faculty of Science, Professor, 理学部, 教授 (60115128)
NAGAYAMA Masaharu  Kyoto University, Research Institute for Mathematical Sciences, Research Assistant, 数理解析研究所, 助手 (20314289)
SAKAI Kazushige  Ryukoku University, Faculty of Science and Technology, Research Assistant, 理工学部, 助手 (00288664)
Project Period (FY) 2000 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥11,300,000 (Direct Cost: ¥11,300,000)
Fiscal Year 2002: ¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2001: ¥5,000,000 (Direct Cost: ¥5,000,000)
Fiscal Year 2000: ¥3,400,000 (Direct Cost: ¥3,400,000)
Keywordsself-propagating high-temperature syntheses / planar traveling wave / planar pulsating wave / regular helical wave / irregular helical wave / wave patterns / apparent activation energy / bifurcation theory
Research Abstract

A helical wave is observed in self-propagating high-temperature syntheses (SHS), for instance. One can create a high-quality uniform product by the SHS when a combustion wave keeps its profile and propagates at a constant velocity. When experimental conditions are changed, however, the planar traveling wave may lose its stability and give place some non-uniform ones. Actually, a planar pulsating wave appears through the Hopf bifurcation of planar traveling wave. Moreover, we observe a wave that propagates in the form of spiral encircling the cylindrical sample with several reaction spots. This wave is called a helical wave since it has been shown by our 3D numerical simulation that the isothermal surface of the wave has some wings and it helically rotates down as time passes on. Similar helical waves are observed also in propagation fronts of polymerizations in laboratory and they are obtained also by numerical simulation of some autocatalytic reactions as well as the SHS.
We have been studied the existing condition of helical wave and the transition process of wave patterns from traveling mode to pulsating mode and/or helical mode, and we have obtained the following results:
1. A stable helical wave can bifurcate directly from a planar traveling wave.
2. Even if a traveling wave is stable in R, the corresponding planar traveling wave can be unstable in the band domain as well as in the cylindrical domain, and a helical wave takes the place of planar traveling wave.
3. There are no stable helical wave when the band width L is small or the radius R of cylindrical domain is small.
4. Helical waves with different numbers of reaction spots can coexist stably.

Report

(4 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (45 results)

All Other

All Publications (45 results)

  • [Publications] 池田 勉: "燃焼合成反応におけるヘリカル波"応用数理. 11. 40-48 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Jimbo: "Ginzburg-Landau Equation with Magnetic Effect in a Thin Domain"Calc.Var.Partial Differential Equations. 15. 325-352 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y.Kabeya: "Imperfect bifurcations arising from elliptic boundary value problems"Nonlinear Anal.Ser.A : Theory Methods. 48. 663-684 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Sakai: "Phase field model for phase transformations of multi-phase and multi-component alloys"Journal of Crystal Growth. 237-239. 144-148 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Nagayama: "Three-dimensional numerical simulation of helically propagating combustion waves"J.of Material Synthesis and Processing. 9. 153-163 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Nishida: "Pattern formation of heat convection problems"Lecture Notes in Computational Sciences and Engineering. 19. 209-218 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Ikeda: "Helical Combustion waves in self-propagating high-temperature syntheses (in Japanese)"Bulletin of the Japan Society for Industrial and Applied Mathematics. 11-2. 40-48 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Jimbo: "Ginzburg-Landau Equation with Magnetic Effect in a Thin Domain"Calc. Var. Partial Differential Equations. 15. 325-352 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Kabeya: "Imperfect bifurcations arising from elliptic boundary value problems"Nonlinear Anal. Ser. A: Theory Methods. 48. 663-684 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Sakai: "Phase field model for phase transformations of multi-phase and multi-component alloys"Journal of Crystal Growth. 237-239. 144-148 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Nagayama: "Three-dimensional numerical simulation of helically propagating combustion waves"J. of Material Synthesis and Processing. 9. 153-163 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Nishida: "Pattern formation of heat convection Problems"Lecture Notes in Computational Sciences and Engineering. 19. 209-218 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Ikeda: "Multiple travelling wave solutions of three-component systems with competition and diffusion"Methods and Applications of Analysis. 8. 479-496 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Morita: "Symmetry Breaking Homoclinic Bifurcations in Diffusively Coupled Equations"J. Dynamics and Differential Equations. 13. 613-649 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] H. Ninomiya: "Stability of traveling curved fronts in a curvature flow with driving force"Methods Appl. Anal.. 8-3. (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] D. Hilhorst: "Imperfect bifurcations arising from elliptic boundary value problems"Nonlinear Anal., Ser. A: Theory Methods. 48-5. 663-684 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] Y. Hayashima: "A camphor oscillates while breaking Symmetry"Journal of Physical Chemistry B. 105-22. 5353-5357 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. I. Kohira: "Synchronized self-motion of two camphor boats"Langmuir. 17. 7124-7129 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Nagayama: "On the interior layer appearing in the similarity solutions of the Navier-Stokes equations"Jpn. J. Indast. Appl. Math.. 19-2. 277-300 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.-I. Ei: "Pulse-pulse interaction in Reaction-Diffusion system"Physics D. 165. 176-198 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M. Mimura: "Non-annihilation of traveling pulses in reaction-diffusion systems"Methods and Applications of Analysis. (to appear in).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Nishida: "Pattern formation of heat convection problems"Lecture Notes in Computational Sciences and Engineering. 19. 209-218 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Jimbo: "Ginzburg-Landau Equation with Magnetic Effect in a Thin Domain"Calc.Var.Partial Differential Equations. 15. 325-352 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Kabeya: "Imperfect bifurcations arising from elliptic boundary value problems"Nonlinear Anal.Ser.A : Theory Methods. 48. 663-684 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Sakai: "Phase field model for phase transformations of multi-phase and multi-component alloys"Journal of Crystal Growth. 237-239. 144-148 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Hayashima: "Self-motion of a camphoric acid boat sensitive to chemical environment"Phys.Chem.Chem.Phys.. 4. 1386-1392 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.-I.Ei: "Pulse-pulse interaction in Reaction-Diffusion system"Physica D. 165. 176-198 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 池田 勉: "燃焼合成反応におけるヘリカル波"応用数理. 11・2. 40-48 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masaharu Nagayama: "Three-dimensional numerical simulation of helically propagating combustion waves"J. of Material Synthesis and Processing. 9・3. 153-163 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hideo Ikeda: "A singular perturbation problem arising in Oseen's spiral flows"Japan J. Indust. Appi. Math.. 18. 393-403 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hideo Ikeda: "Multiple travelling wave solutions of three-component systems with competition and diffusion"to appear in Methods and Applications of Analysis. (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Hideo Ikeda: "Global bifurcation phenomena of standing pulse solutions for three-component systems with competition and diffusion"to appear in Hiroshima Math. J.. 32. (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Waichiro Matsumoto: "On the Cauchy-Kowalevskaya theorem of Nagumo type for systems"to appear in "Hyperboic differential operators and related problems"(Ed. V. Ancina et al.), Marcel Dekker.

    • Related Report
      2001 Annual Research Report
  • [Publications] Daniel Hilhorst: "A competition-diffusion system approximation to the classical two-phase Stefan problem"Japan Journal of Industrial and Applied Mathematics. 18・2. 161-180 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Kazushige Sakai: "Phase field model for phase transformations of multi-phase and multi-component alloys"to appear in "Proceeding of The Thirteenth International Conference on Crystal Growth".

    • Related Report
      2001 Annual Research Report
  • [Publications] Yoshikazu Yamagishi: "Cantor bouquet of holomorphic stable manifolds for a periodic indeterminate point"Nonlinearity. 14. 113-120 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Yuko Hayashima: "A camphor oscillates while breaking symmetry"Journal of Physical Chemistry B. 105・22. 5353-5357 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masahiro Kohira: "Synchronized self-motion of two camphor boats"Langmuir. 17. 7124-7129 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Masaharu Nagayama: "On the interior layer appearing in the similarity solutions of the Navier-Stokes equations"to appear in Japan Journal of Industrial and Applied Mathematics.

    • Related Report
      2001 Annual Research Report
  • [Publications] Masayasa Mimura: "Dynamics of travelling breathers arising in reaction-diffusion systems-ODE modeling approach-"Hiroshima Math.J.. 30. 221-256 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Hideo Ikeda: "A singular perturbation problem arising in Oseen's spiral flows"Japan J.Indust .Appl.Math.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Yoshihisa Morita: "Symmetry Breaking Homoclinic Bifurcations in Diffusively Coupled Equations"J.Dyn.Diff.Eqns.. (to appear).

    • Related Report
      2000 Annual Research Report
  • [Publications] Hirokazu Ninomiya: "Stability of traveling curred fronts in a curvature flow with driving force"IMS Workshop on Reaction-Diffusion Systems. (掲載予定).

    • Related Report
      2000 Annual Research Report
  • [Publications] kazushige Sakai: "A Mathematical Model of solidification Dynamics of Binary Alloys."Japan Journal of Industrial and Applied Mathematics. 17・1. 43-58 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] kazushige Sakai: "Phase Fierd Models for solidification of Alloys."INFORMATION. 3・4. 447-457 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2000-04-01   Modified: 2016-04-21  

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