Study of Diophantine phenomena and exact asymptotic analysis of singular partial differential equations

给出了处理向量场范式理论中形式变换散度的不需要丢番图条件的解析方法。我们的方法基于渐近分析，扩展了所谓的Borel-Laplace方法，并给出了某一子域上的发散级数的真实意义。此外，利用Borel恢复方法，给出了解析不可积性和光滑可积性同时存在的哈密顿系统的发散第一积分的意义。

We showed analytical methods without the Diophantine condition in treating divergence of formal transformations in the normal form theory of a vector field. Our methods, based on asymptotic analysis, extend the so-called Borel-Laplace method, and give a real meaning to a divergent series in certain subdomain. Moreover, we give a meaning, by a Borel resummation method, to divergent first integrals of Hamiltonian systems , for which analytic non-integrability and smooth integrability occur similtaneously.