无界系统的KAM理论和Birkhoff正规形理论及其应用

Many scientific problems can be described by differntial equations. The existence and the long time stabiltiy of solutions to these equations are hot research topics. The KAM theory and Birkhoff normal form theory are imporant tools to study the existence and the long time stability of solutions to these equations. One can show the existence and the long time stability of solutions to some differential equations by the KAM theory and Birkhoff normal form theory. But a wide class of these differential equations can be transformed as unbounded nearly integrable Hamiltonian systems and reversible systems which have good mathematical structures and symmetries, but research on the KAM theory and Birkhoff normal form theory of these systems is still lacking..In this project, we will mainly study two topics,(1)studying the KAM theory of unbounded infinite dimensional nearly integrable reversible system. (2)studying the Birkhoff normal form theory of Hamiltonian systems. In topic (1), we will try to get the KAM theory of reversible systems with frequencies satisfying only weak asymptotic conditions, and we will apply this result to a class of wave equations whose nonlinear term contains derivative with respect to the time variable and prove the existence of quasi-periodic solutions with small amplitude. In topic (2), we will construct a bounded symplectic transformation and get the normal form of the unbounded infinite dimensional Hamiltonian systems. We will also apply this Birkhoff normal form to the derivative nonlinear Schrodinger equation and show its long time stability.

很多科学问题需要用微分方程描述。这些方程解的存在性和有效稳定性是人们关心的热点问题。而KAM理论和Birkhoff正规形理论是刻画微分方程解的存在性和有效稳定性的重要工具，并得到了许多方程解的存在性和有效稳定性。但实际应用中很多方程只能转化为无界近可积反转系统或无界近可积哈密顿系统。到目前为止无界反转系统的KAM理论尚不完善，无界哈密顿系统的Birkhoff正规形理论尚未建立。.本项目将主要研究两个课题，(1)无界反转系统的KAM理论；(2)无界哈密顿系统的Birkhoff正规形理论。在(1)中，我们力争得到法向频率满足弱渐近逼近条件时无界反转系统的KAM理论并应用该理论得到非线性项带有关于时间变量导数的波方程的小振幅拟周期解的存在性。在课题(2)中，我们将重点研究构造有界辛变换来得到无界哈密顿系统的Birkhoff正规形理论，以及通过该理论研究非线性项带导数的薛定谔方程解的有效稳定性。

1，建立了适合无界的无穷维哈密顿系统在平衡点附近的Birkhoff正规形。即当该系统的频率满足强非共振条件时，对于任意给定的正整数r*找到了有界的Lie变换（辛变换）把原来哈密顿系统变换为新的哈密顿系统。该新哈密顿系统具有r*阶Birkhoff正规形，其余项为具有比r*更高阶的零点或是关于指标高阶的变量的次数大于3. .2，研究了一组(其中位势是属于某个具有正测定的集合)非线性项含有关于空间变量x的导数但不直接依赖变量x的薛定谔方程在周期边界条件下，其解的动力学性态。特别的，给出了对大部分的位势对应的非线性薛定谔方程，当初值的范数在指标为s的索伯利夫空间下足够小的情况下，其相应的解在足够长的时间也不会远离初值(即，等到该方程的有限时间稳定)并给出该稳定时间的估计。.3，给出了扰动的哈密顿KdV方程在周期边界条件下，解得长时间稳定问题。 扰动KdV方程解的性态一直非常受关注。本项目给出了在小初值下，解的长时间稳定结论。该问题中的KdV方程线性部分没有位势所以没有外部参数，不能得到任意高阶的Birkhoff正规形但仍可以等到4阶的Birkhoff正规形，这使得扰动KdV方程的解的长时间稳定只能到达初值范数的-5/2次幂。特别地，该结论适用于扰动项直接周期依赖于空间变量x，且本身系统的周期是该周期的整数倍(该整数不能被3整除)。.4，建立了哈密顿扰动下的KdV方程在周期边界条件下的解的均值原理。即，当初值的s指标范数很小时，相应的解与解得均值的差在时间长度小于初值的-5/2次幂时仍然很小。