Formation of Singulatiries in Solutions of Nonlinear Schrodinger Equations and Related Fields

Nawa and Ishige（与Ishiwata和Sakajo合作）组织研讨会（http://www.gifu-u.ac.jp/~tisiwata/seminar/ma_seminar.html）；通过研讨会，我们有很多机会与来自数学科学多个领域的研究人员进行讨论和互动。对于伪共形不变非线性薛定谔方程（pc-NLS）的爆破解，Nawa得到了其渐近性与爆破率之间的关系。这一结果使他能够通过解决方案背后的纳尔逊过程展开对爆炸率的全面调查。他还在导数非线性薛定谔方程上得到了一些类似的结果。结果表明，为pc-NLS开发的方法可用于研究经典情况下以BCS约简哈密顿量为模型的NLS耦合系统；我们可以在这个经典场方程的解中看到对的产生。Ishige成功地描述了具有更多诺伊曼边界条件的半线性热方程的爆破集。他还研究了在狄利克雷或诺伊曼边界条件下，球外域线性热方程解的热点运动。在研究过程中，他得到了一些有用的结果，为进一步研究更一般的反应扩散方程解的渐近行为提供了依据。在数学科学的许多领域中，铃木对自对偶规范场方程的爆破分析和简化化趋化系统的爆破问题进行了有趣的研究；ogawa发展了一种通过Conley指数进行数值分析的拓扑验证理论，得到了五次型Swift-Hohenberg方程解中局部模式形成的严格结果。他还研究了电化学反应扩散体系中的振荡行为；他借助数值分析进行了分岔分析。少

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 … More 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