Program in Nonlinear Waves, Kinetic Theory and Hamiltonian Partial Differential Equations-Fields Institute, Spg 04

项目编号：dms -0352061项目负责人：Nicholas M. ercolani2004年春季学期在菲尔兹研究所举办的“非线性波、动力学理论和哈密顿偏微分方程（PDE）”专题项目学期科研活动。本学期的重点将集中在由非线性波动理论、动力学理论和哈密顿系统驱动的PDE研究领域。哈密顿偏微分方程是一类线性和非线性偏微分方程，它们具有这样的性质，即它们可以用具有无限多个自由度的哈密顿系统的形式，使用各种各样的、有时是非经典的辛结构来表示。主要的例子包括非线性波动方程、非线性薛定谔方程和水波的euler方程。与动力系统的类比提出了一些基本问题，这些问题构成了本学期关注哈密顿偏微分方程的部分动机。特别值得注意的是，在来自统计力学的动力学方程理论和产生于其宏观极限的非线性偏微分方程系统之间建立的联系。范例是玻尔兹曼方程的流体动力学缩放极限，但有许多新兴领域与此分析相关，包括非线性波湍流理论的数学严谨基础的开始。组织者希望这个专题项目能成为一个非常动态的研究焦点，研究偏微分方程对各种物理现象的建模，如行星运动、激光和光纤系统、海浪和流体湍流。该项目的特点是广泛的国际性，它为年轻数学家提供了一个接触的机会。短期课程系列，四个研讨会和专题讨论会特别适合发展研究数学家或物理学家在他们的职业生涯早期参加。组委会相信，该项目将促进各学科参与者之间的进一步互动和持久合作。此外，将积极鼓励妇女和代表性不足的少数民族的参与。

AbstractAward: DMS-0352061Principal Investigator: Nicholas M. ErcolaniThis proposal is for support of junior US based mathematicians toparticipate in scientific activities at the Fields Instituteduring the thematic program semester in Nonlinear Waves, KineticTheory and Hamiltonian Partial Differential Equations (PDE) thatwill take place during the Spring semester of 2004. The focus ofthe semester will concern areas of research on PDE that aremotivated by nonlinear wave theory, kinetic theory, andHamiltonian systems. Hamiltonian PDE form a class of linear andnonlinear partial differential equations which share the propertythat they can be written in the form of a Hamiltonian system withinfinitely many degrees of freedom, using various and sometimesnonclassical symplectic structures. Principal examples includenonlinear wave equations, nonlinear Schroedinger equations andEuler's equations for water waves. The analogy with dynamicalsystems raises a number of basic questions, which form a part ofthe motivation for this semester of focus on Hamiltonian PDE. Ofparticular note is the connection that has been establishedbetween the theory of kinetic equations coming from statisticalmechanics, and the nonlinear systems of PDE which arise in theirmacroscopic limits. The paradigm is the fluid dynamical scalinglimit of the Boltzmann equation, but there are numerous emergingareas of relevance for this analysis, including the beginnings ofa mathematically rigorous foundation for the theory of nonlinearwave turbulence.The organizers are expecting this thematic program to be a verydynamic focus of research on Partial Differential Equations thatmodel a variety of physical phenomena, such as planetary motions,lasers and optical fiber systems, ocean waves and fluidturbulence. The character of the program is broadlyinternational, and it represents an opportunity for exposure foryoung mathematicians. The short course series, the four workshopsand the symposia are especially appropriate for participation bydeveloping research mathematicians or physicists early in theircareer. The organizing committee believes that the program willengender further interaction and lasting collaboration amongparticipants from various disciplines. Additionally, theparticipation of women and under-represented minorities will beactively encouraged.