Nonlinear Differential Equations and Related Topics

点击翻译按钮获取中文摘要

Under our research program we studied various types of differential equations, linear and nonlinear, ordinary and partial, and related topics. We state the summary of our results below :1. We studied semi-elliptic operators with coefficients degenerating at a point with infinite order and found sufficient conditions for such operators to be hypoelliptic.2. We found some examples of hypoelliptic operators wich are not micro hypoelliptic3. We studied the local structure of analytic transformations around its fixed point, and obtained several properties of the stable manifold connected with the fixed point4. We studied the geometric structure of a neighborhood of a national curve with a node in a two dimensional complex manifold5. We pursued a long, troublesome numerical experiment to investigate Arnold's tongues and Swallow's tails appearing in 1-dimensional complex dynamical systems, and obtained various interesting figures and pictures.6. In our study of a Gaussian process with nonconstant variance, we obtained a sharp upper bound of the distribution tail.7. We found fine structures of the linear and nonlinear collision integrals appearing in the Boltzmann equation. But the paper is not completed yet. In conclusion, we belive we made important steps in the study of several nonlinear phenomena in mathematics. This research should be continued to obtain further results.