Methods of Hamiltonian Mechanics for Nonlinear Wave Equations

该提案中描述的研究是关于一类线性和非线性偏微分方程（PDE），它们都具有可以写成具有无限多个自由度的汉密尔顿系统形式的特性。主要的例子包括非线性波动方程，非线性薛定谔方程和欧拉方程的水波。 与动力系统的类比激发了许多基本问题。(1)非线性偏微分方程不变环面的存在性问题涉及到KAM理论到无穷维情形的推广。这些结果的含义包括非线性发展方程的解的存在性，存在的所有时间和经常性。已开发的PDE的技术也有关系的几个开放的问题，在经典的动力系统有关的持久性共振环面。(2)一个正规形式变换的哈密顿偏微分方程给出了详细的基本非线性，它有应用Nekhoroshev稳定性的结果和证明阿诺德条件的真正的非线性。(3)从正规形变换出发，非线性偏微分方程的长时间存在性结果在数学物理经典渐近机制的严格分析中是非常重要的。此外，提议者将解决一些相关的问题，在偏微分方程的理论。(4)线性和非线性Schroeinger方程的色散平滑估计依赖于双特征的散射性质，这种形式的估计在低平滑数据的初值问题的研究中发挥了作用。(5)二维水波中的驻波和三维水波中的行波的经典问题是PDE中的问题的实例，其表现出小因子，并且分析结果涉及收敛的微妙问题。 天体力学是描述行星体运动的问题，一直吸引着数学家，并激发了他们的发现。本世纪初，法国数学家H.庞加莱和他的美国同时代人G.D.伯克霍夫和G.W.希尔，他们的工作从根本上彻底改变了这一主题，使其概念清晰，并引入了强大的新分析技术。这项工作在数学领域中产生了一种新的范式，并促成了许多现代数学分支的诞生。一个重要的问题是，仍然是n体问题，问题的稳定性的运动n行星在其相互吸引力。事实上，在20世纪60年代，庞加莱去世50年后，三位数学家，A.N. Kolmogorov，V.I. Arnold和J. Moser（现在称为KAM理论）对该领域做出了重大贡献。在其发展的时候，它在回旋加速器中的高能粒子的稳定性，以及我们太阳系的稳定性或其他重要的力学系统的经典问题中有物理应用。描述n个物体运动的方程是常微分方程的例子。这是一个问题时的发展KAM理论是否这种稳定性的结果可以扩展到偏微分方程。这些描述了连续介质（如流体、气体、电磁场或弹性固体）随时间的演化。力学和连续统运动之间的类比在数学上是相当优美的。然而，n体问题是一个有限维的，而问题的偏微分方程与此类似的考虑是固有的无限维，这总结了基本的数学困难的扩展方法。本研究计划的主要内容是将哈密顿力学的分析技术的某些方面扩展到物理上重要的偏微分方程。这些结果将在各种不同的物理应用中产生一些重要的影响，包括海浪的研究，光纤中信号脉冲的传播，以及在

The research described in this proposal is on a class of linear and nonlinear partial differential equations (PDE)which share the property that they can be written in the form of a Hamiltonian system with infinitely many degrees of freedom. Principal examples include nonlinear wave equations, nonlinear Schroedinger equations and Euler's equations for water waves. The analogy with dynamical systems motivates a number of basic questions. (1) The existence of invariant tori for the nonlinear partial differential equations involves extensions of KAM theory to infinite dimensional settings. Implications of these results include the existence of solutions of nonlinear evolution equations which exist for all time and are recurrent. Techniques that have been developed for PDE also have bearing on several open problems in classical dynamical systems concerning the persistence of resonant tori. (2) A normal forms transformation of a Hamiltonian PDE gives details of its essential nonlinearities, and it has applications to Nekhoroshev stability results and to a proof of the Arnold condition of genuine nonlinearity. (3) Starting from a normal forms transformation, long time existence results for nonlinear PDE are important in a mathematically rigorous analysis of the classical asymptotic regimes of mathematical physics. In addition the proposer will address a number of related questions in the theory of PDE. (4) Estimates of dispersive smoothing for linear and nonlinear Schroeinger equations depend upon the scattering properties of bicharacteristics, and estimates of this form play a role in the study of the initial value problem for data of low smoothness. (5) The classical problems of standing waves in two dimensional water waves and traveling waves in three dimensional water waves are instances of problems in PDE which exhibit small divisors, and analytic results involve delicate questions of convergence. Celestial mechanics, the problem of describing the motion of planetary bodies, has always intrigued mathematicians and motivated their discoveries. At the beginning of this century, the French mathematician H. Poincar' and his American contemporaries G.D. Birkhoff and G.W. Hill, essentially revolutionizedthe subject with their work, bringing it conceptual clarity and introducing powerful new analytic techniques. This work gave rise to a new paradigm in its field, and contributed to the birth of many modern branches of mathematics. An important question was and remains the n-body problem; the problem of stabilityof the motion of n planets in their mutual gravitational attraction. Indeed in the 1960's, 50 years after the death of Poincar'e, the development of a theory by three mathematicians, A.N. Kolmogorov, V.I. Arnold and J. Moser (it is now called KAM theory) made a significant contribution to the field. At the time ofits development there were physical applications to the stability of high energy particles in a cyclotron, as well as to the classical questions of the stability of our solar system or to other important systems in mechanics. The equations which describe the motion of n bodies are examples of ordinary differential equations. It was a question at the time of the development of KAM theory whether such stability results could be extended to partial differential equations. These describe the evolution in time of a continuous medium such as a fluid, a gas, and electromagnetic field or an elastic solid. The analogy between mechanics and the motion of a continuum is mathematically quite elegant. However the n-body problem is a finite dimensional one, while problems of partial differential equations viewed with this analogy in mind are inherently infinite dimensional, and this summarizes the essential mathematical difficulty of extending the methods. The principal content of this research program is the extension of some aspects of the analytic techniques of Hamiltonian mechanics to physically important partial differential equations. These results will have some significant consequences in a variety of diverse physical applications, including the study of ocean waves, the propagation of signal pulses in optical fibers, and in the