无穷维动力系统中Bourgain猜测的证明及其应用

Many kinds of Hamiltonian partial differential equations come from Fluid Dynamics and Quantum Mechanics, studying quasi-periodic solutions of these Hamiltonian partial differential equations has always been hot topics. As we know, parameters play an essential role in investigating the invariant torus of Hamiltonian partial differential equations. From this point of view, it was of great importance for Bourgain and Eliasson to study the existence of invariant torus with tangential frequency along the pre-assigned direction for the finite dimensional Hamiltonian systems, moreover, Bourgain made the conjecture that this conclusion was also valid for infinite dimensional Hamiltonian systems. Berti proved the Bourgain conjecture in 2010 under the bounded perturbation case. This research work plans to .prove the Bourgain conjecture for the infinite dimensional dynamical systems. We specifically investigate (1) a KAM theorem for infinite dimensional Hamiltonian system containing unbounded perturbation with final tangential frequency along the pre-assigned direction,(2) we apply this KAM theorem into KdV equations、Schrodinger equations containing nonlinearities with derivatives,and derive the corresponding invariant tori as well as quasi-periodic solutions with tangential frequency along the fixed direction, (3)we generalize the above results into infinite dimensional reversible systems. This research work will improve our theoretical basis for the studies on dynamical behavior for infinite dimensional systems.

哈密顿偏微分方程广泛存在于流体力学及量子力学等领域中，对其拟周期解的研究一直是动力系统中的热点。众所周知，在研究哈密顿偏微分方程的拟周期解时，参数发挥着至关重要的作用。从需要最少参数的角度看，Bourgain及Eliasson研究有限维哈密顿系统中切频沿给定方向的不变环面的工作颇具价值，随后Bourgain猜测这一结果可以推广到无穷维哈密顿系统中。在有界扰动下，Bourgain猜测已被Berti证明。本课题致力于研究无界扰动下的无界扰动下无穷维系统中的Bourgain猜测。具体包括:(1)带无界扰动的无穷维哈密顿系统中最终切频沿给定方向的KAM定理;(2)将上述KAM定理应用到KdV方程、非线性项带导数的Schrodinger方程，分别得到它们切频沿给定方向的不变环面及相应的拟周期解;(3)将前两步的结果类推到无穷维反转系统中。本课题将为研究无穷维系统的动力学行为提供完善的理论支持。

对Hamilton系统稳定性的研究一直是动力系统领域关心的重点，近年来，随着研究稳定性的重要工具——KAM理论在无穷维系统中的发展，人们转而研究可化为无穷维哈密顿系统的偏微分方程的动力学行为，这其中对于哈密顿偏微分方程拟周期解的存在性研究是一大热点。本课题拟致力于研究带无界扰动的无穷维系统中的Bourgain猜测，即，研究带无界扰动的无穷维哈密顿系统中切频沿给定方向的KAM定理，并借此应用到非线性项带导数的KdV方程等，研究其不变环面及相应的拟周期解的存在性。本课题的开展将为进一步理解无穷维系统，特别是哈密顿偏微分方程的动力学行为提供更多的理论支持。.在2016年，共完成并发表相关论文一篇：.Siqi Xu, Dongfeng Yan(通讯作者), Smooth quasi-periodic solutions for the perturbed mKdV equation, Communications on pure and applied analysis.15(5),2016:1857-1869