Study of singularities of partial differential equations in the complex domain

1.在常微分方程理论中，常采用解析变换的方法来证明两个非线性常微分方程的等价性。本文将该方法推广到偏微分方程的框架中，证明了两个非线性Kowalewskian型偏微分方程的等价性。证明了一类非线性偏微分方程的形式解是多重可和的。这类偏微分方程被认为是由常微分方程的扰动得到的。证明了一类非线性全特征偏微分方程解的唯一性结果。4.研究了一般非线性偏微分方程解的奇异性，构造了具有对数奇异性的解。5.研究了解在奇异点附近的渐近性态，并证明了在某些类偏微分方程中，利用Mellin变换具体给出了解的渐近展开式。6.研究了p-椭圆型方程球对称解的奇异性，通过对Briot-Bouquet型偏微分方程的分析，给出了奇异性的具体形式.

1.In the theory of ODEs, the method of analytic transformation is often used to show the equivalence of two nonlinear ODEs. In this research, this method is extended to the framework of PDEs, and a equivalence of two nonlinear PDEs of Kowalewskian type is proved. The result is applied to the problem of analytic continuation of the solution.2.It is proved that the formal solution of some class of nonlinear PDEs is multi-summable. The PDE in this class is regarded as the one obtained by the perturbation of ODEs. The result is applied to the normal form theory of vector fields.3.A uniqueness result of the solution is proved for a class of nonlinear totally characteristic PDEs. The result is applied to showing the nonexistence of singularities of the solution.4.Singularities of solutions of general nonlinear PDEs are studied and solutions with logarithmic singularities are constructed.5.The asymptotic behaviour of solutions near the singularity is investigated and it is proved that in some classes of PDEs the asymptotic expansion is given concretely by using Mellin transformation. Gevrey type estimate of the remainder term is also obtained.6.Singularities of sphere-symmetric solutions of p-elliptic equation are studied and the form of singularities is given explicitly by the analysis of Briot-Bouquet type PDEs.