2005 Fiscal Year Final Research Report Summary
Formation of Singulatiries in Solutions of Nonlinear Schrodinger Equations and Related Fields
| Project/Area Number |
14340054
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Global analysis
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| Research Institution | Osaka University (2004-2005)
Nagoya University (2002-2003) |
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Principal Investigator |
NAWA Hayato Osaka University, Grad.School of Engineering Science, Professor, 大学院・基礎工学研究科, 教授 (90218066)
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| Co-Investigator(Kenkyū-buntansha) |
SUZUKI Takashi Osaka University, Grad.School of Engineering Science, Professor, 大学院・基礎工学研究科, 教授 (40114516)
OGAWA Toshiyuki Osaka University, Grad.School of Engineering Science, Ass.professor, 大学院・基礎工学研究科, 助教授 (80211811)
ISHIGE Kazuhiro Tohoku University, Graduate school of science, Ass.professor, 大学院・理学研究科, 助教授 (90272020)
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| Project Period (FY) |
2002 – 2005
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| Keywords | Nonlinear Schrodinger Equations / Heat equations / Blow up phenomena / Pattern Formation / Hot spot movement / Bifurcation analysis / Dynamical systems / Gauge theory |
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| Research Abstract |
Nawa and Ishige (collaboration with Ishiwata and Sakajo) organizes a seminar (http://www.gifu-u.ac.jp/~tisiwata/seminar/ma_seminar.html) ; we had many opportunities to have discussions and interactions with researchers from several fields of mathematical sciences through the seminar.
For blowup solutions of the pseudo-conformally invariant nonlinear Schrodinger equation (pc-NLS), Nawa obtain some relation between their asymptotic behavior and blowup rates. This result enable him to launch into a full-dress investigation of blowup rate via the Nelson process behind the solution. He also obtain some analogous results on the derivative nonlinear Schrodinger equation. It turns out that the method developed for pc-NLS is useful for investigating a coupled system of NLS which is modeled on the BCS reduced Hamiltonian in a classical context ; we can see pair-creation in solutions of this classical field equation.
Ishige succeeded to characterize the blowup set for semilinear heat equations with
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Neuman boundary conditions. He also investigated the hot spot movement of solutions of the linear heat equation on the exterior domain of a ball under the Dirichlet or Neuman boundary conditions. In the course of the study, he obtain useful results to make further investigation into the asymptotic behavior of solutions for more general reaction diffusion equations.
Among other his results on several fields of mathematical sciences, Suzuki made an interesting study on self-dual gauge field equations via blowup analysis, and on blowup problem for simplified system of chemotaxis ; continuation of the solution after blowup, blowup in infinite time, etc.
Ogawa developed a theory on the topological verification for numerical analysis via Conley index to obtain rigorous results for formation of localized patterns in solutions to the quintic Swift-Hohenberg equation. He also investigated the oscillatory behavior in an Electrochemical reaction diffusion system ; he conducted a bifurcation analysis with an aid of numerical analysis. Less
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