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

Nawa和Ishige(与石田和Sakajo合作)组织了一个研讨会(http://www.gifu-u.ac.jp/~tisiwata/seminar/ma_seminar.html)；我们有很多机会通过研讨会与来自多个数学科学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得到了它们的渐近行为和爆破速度之间的一些关系。这一结果使他能够通过解决方案背后的纳尔逊过程启动对井喷率的全面调查。关于导数的非线性薛定谔方程，他也得到了一些类似的结果。结果表明，用PC-NLS建立的方法对于研究耦合的NLS系统是有用的，它是建立在经典背景下的BCS约化哈密顿量上的；我们可以在这个经典场方程解中看到对的产生。Ishige成功地用…刻画了半线性热方程的爆破集更多的诺伊曼边界条件。他还研究了球外部区域上的线性热方程解在Dirichlet或Neuman边界条件下的热点运动。在研究过程中，他得到了一些有用的结果来进一步研究更一般的反应扩散方程的解的渐近行为。在他的其他几个数学科学领域的结果中，铃木通过爆破分析对自对偶规范场方程进行了有趣的研究，并对简化的趋化系统的爆破问题进行了有趣的研究；爆破后解的连续性，无限时间爆破等。小川发展了一种通过Conley指数进行数值分析的拓扑验证的理论，得到了关于五次Swift-Hohenberg方程解的局部化模式形成的严格结果。他还研究了电化学反应扩散系统的振荡行为；他借助数值分析进行了分叉分析。较少

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