Resarchs on Solutions of Nonlinear Elliptic Equations and Numcrical Analysis

首席研究员S.Yotsutani与H.Morishita、N.小林、H.Takaichi和K. Amano一起提出了一种非常精确的数值计算方法，通过使用电荷模拟方法来求解泊松方程。最近，他与森下和安佳诺等人一起，成功地缩短了计算时间，提高了计算精度，这种方法与粒子计算是一致的。另一方面，他与Y.Kabeya和E. Yanagida一起发展了研究径向对称解的形状的数学方法。最近，他发现了系统的变化的变量，以转换的微分方程所产生的椭圆方程的标准形式与E.柳田。这样，各个方程之间的关系就变得非常清楚，而这些理解又促使人们对每一个解的性质有更深的理解。T.lkeda研究了一类非线性反应扩散方程组驻脉冲解的不稳定性。森田表明稳定的涡在金斯堡-朗道方程的可变扩散系数。H.Oka研究了影子系统单调解的连接轨道结构。Yamada证明了一些具有非线性交叉扩散的种群模型的共存态。E.Yanagida提出了抛物型方程组的各种新现象，并给出了证明。

Head investigator S.Yotsutani proposed a very accurate numerical computation method to solve Poisson's equation by using a charge simulation method with H.Morishita, N.Kobayashi, H.Takaichi and K.Amano. Recently, He has succeeded to shorten the time of the calculation considerably and improve the accuracy with H.Morishita and K.Anjano, This method is in the conformity with the paerallel computing. Thus it is possible to know the detailed shape of solutions of nonlinear elliptic systems.On the other hand, he developed the mathematical method of investigate the shape of radially symmetric solutions with Y.Kabeya and E.Yanagida. Recently, he has found the systematic change of variables to transform the differential equations arising from the elliptic equations to canonical form with E.Yanagida. Thus, relations between various equations studied one by one independently become very clear, and the understandings encourage the deep understanding of the propertites of each solution.The reserach results of investigators are asfollows. T.lkeda invesitigate the instabiliz ation of the standing pulse solutions of bistable reaction-diffusion systems. Y.Morita showed stabilization of vorticies in the Ginzburg-Landau equation with a variable diffusion coefficient. H.Oka investigated the connecting orbit structure of monotone solutions in the shadow system. Y.Yamada proved the coexistence states for some population models with nonlinear cross-diffusion. E.Yanagida discoverd various new phenomena of the systems of parabolic equations and gave the proof of them.