New perspectives on dispersive equations

关于色散方程的新观点

• 批准号: 1068815
• 负责人: Gigliola Staffilani
• 金额: $ 32.69万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068815&HistoricalAwards=false
• 关键词: New; perspectives; dispersive; equations; New; perspectives; dispersive; equations

Dispersive partial differential equations (PDE) model certain wave propagation phenomena in nature. Their solutions are waves that spread out in space as time evolves but that conserve energy. Probably the best known equation within the class of dispersive PDE is the nonlinear Schrodinger equation, which models a variety of physical waves, from signals in fiber optics to the macroscopic dynamics of the Bose-Einsten condensate. In the last fifteen years enormous progress has been made in settling fundamental questions on existence of solutions to the equation, their long-time behavior, and singularity formation. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena that have been studied with sophisticated tools from nonlinear Fourier analysis, analytic number theory, and geometry. The principal investigator's main line of research has revolved around this aspect of mathematics for along time. More recently, a growing interest has been shown in incorporating nondeterministic points of view into the field of dispersive PDE. This has become one of the principal investigator's favorite lines of work in the last couple of years, and one that will be pursued in this project. Another new element in the project is the study of the "process" used by mathematical physicists to pass from a complex system of particles interacting with one another to a macroscopic wave function that is able to describe the most important features of the system as a whole. More precisely, the principal investigator will study the effective evolution equations arising as an appropriate limit of many body quantum dynamics. From a mathematical point of view the types of problems that the principal investigator intends to explore in this project lie at the intersection of Fourier analysis, analytic number theory, numerical analysis, geometry, probability, and mathematical physics. Often it is difficult to "communicate" in so many different mathematical languages, but the time seems ripe to start such a conversation. The description of the problems that the principal investigator proposes in her research program should testify to this fact. Although she conducts her research mostly in the realm of pure mathematics, the solutions to the problems she considers have the potential to see very concrete and very diverse consequences in real life. For example, understanding the most efficient way to send a signal through a fiber optic cable or being able to anticipate the properties of a gas when the temperature approaches absolute zero are two very different phenomena in nature. On the other hand, they are both aspects of solutions to the same dispersive equation, the nonlinear Schrodinger equation that is one of the foci of this project, and they can be studied using the same mathematical language. In recent years, the world has witnessed incredible advances in technology. As scientists and engineers seek to take even greater strides in the technological arena, they are guided by experiments, but also by mathematical models, such as those provided by research on the Schrodinger equation. There is no doubt that solid predictions on the mathematical side will help target the experimental component in the most efficient and cost-effective way possible, thus saving resources as well as valuable time.

分散性偏微分方程（PDE）模型某些自然界的某些波传播现象。他们的解决方案是随着时间的流逝，它们在空间中散布的波浪，但可以节省能量。分散PDE等级中最著名的方程可能是非线性Schrodinger方程，该方程模拟了各种物理波，从光纤信号到玻色剂冷凝物的宏观动力学。在过去的十五年中，在解决有关方程，长期行为和奇异性形成的解决方案的基本问题方面取得了巨大进展。这项工作的推力主要集中在波浪现象的确定性方面，这些方面已通过非线性傅立叶分析，分析数理论和几何形状从精致的工具进行了研究。主要研究者的主要研究线围绕着数学的这一方面围绕着。最近，人们对将非确定性的观点纳入分散PDE领域的兴趣越来越大。在过去的几年中，这已成为主要研究者最喜欢的工作线之一，并且将在该项目中追求。 该项目中的另一个新元素是数学物理学家使用的“过程”的研究，从相互相互作用的复杂粒子系统传递到宏观波函数，该系统能够描述整个系统的最重要特征。更确切地说，主要研究者将研究有效的进化方程，这是许多人体量子动力学的适当限制。从数学角度来看，主要研究者打算在该项目中探索的问题的类型在于傅立叶分析，分析数理论，数值分析，几何，几何，概率和数学物理学的交集。通常，很难用许多不同的数学语言“交流”，但是开始这样的对话的时候似乎已经成熟了。主要研究者在其研究计划中提出的问题的描述应证明这一事实。尽管她主要在纯数学领域进行研究，但她认为的问题的解决方案有可能在现实生活中看到非常具体且非常多样化的后果。例如，了解通过光纤电缆发送信号的最有效方法，或者当温度接近绝对零是两个截然不同的现象时，可以预测气体的性质。另一方面，它们都是相同分散方程的解决方案的方面，即是该项目的焦点之一的非线性Schrodinger方程，可以使用相同的数学语言进行研究。近年来，世界已经见证了技术的令人难以置信的进步。当科学家和工程师试图在技术领域取得更大的进步，因此他们受到实验的指导，也以数学模型为指导，例如关于Schrodinger方程的研究提供的模型。毫无疑问，数学方面的扎实预测将以最有效，最具成本效益的方式来瞄准实验组成部分，从而节省资源以及宝贵的时间。