Differential equations reduced to transcendental equations including complete elliptic integrals and their global solution structure

我们已经发展出了求解微分方程组的原始方法。我们得到了所有的精确解，并导出了与所给出的微分方程的全局分支结构等价的超越方程。我们发展了这种方法，并将其应用于各种问题。特别是，对于一维情况下的交叉扩散方程，我们发现了各种分析超越方程的方法，并得到了显示全局分支结构的方法。此外，我们已经开始研究包括多维情形在内的定常解的稳定性。我们还揭示了平面弹性闭曲线的曲率爆破现象和所有曲线的整体结构。

We have been developed the original method for differential equations. We obtain all exact solutions and derive transcendental equations equivalent to the global bifurcation structure of the given differential equations. We have developed this method and applied it various problems. Especially, as for a cross-diffusion equation in 1 dimensional case, we have discovered various devises to analyze transcendental equations, and obtained the method to show the global bifurcation structure. Furthermore, we have started to investigate the stability of stationary solutions including multidimensional case. We have also reveal the blowup phenomena of curvature for plane elastic closed curves and the global structure of all curves.