New Challenges in Nonlinear PDEs.

这个项目涉及调和分析、非线性偏微分方程组、几何学和概率论之间的研究。一方面，它从不确定的角度研究频散非线性波动现象。在过去的二十年里，在解决色散方程解的存在性、其长期行为和奇性形成等问题方面取得了巨大的进展。这项工作的主旨主要集中在波动现象的确定性方面，在这些方面，来自非线性傅立叶分析、几何和解析数论的复杂工具发挥了关键作用。然而，仍然存在一些根本性的障碍。克服它们的一个自然方法是从不确定的角度考虑演化方程，并从概率的角度将其纳入分析工具。这个项目要研究的一些问题包括随机化在适定理论中的作用，解的几乎确定(在概率意义上)在时间上的全局存在性，相关的Gibbs度量的存在性和动力学性质，以及规范变换下的统计系综的行为。另一方面，主要研究者将研究某些双曲型(或非椭圆型)非线性薛定谔方程和系统的特殊类型解的存在性和长期动力学。其目的是对与涡旋细丝理论、铁磁性和非线性光学中的当前工作(例如，研究光脉冲在正常色散光学介质中的演化)有关的模型进行严格的数学分析，这些模型引起了物理界的注意。物理中的波动现象，如光、声和重力，都是使用偏微分方程式进行数学建模的。非线性波动模型出现在量子力学、铁磁性、振动系统、半导体和光纤中。色散方程模拟自然界中重要的波传播现象。他们的解决方案是在保持能量或质量不变的情况下，随着时间的推移在空间中传播的波。最著名的色散方程是控制量子粒子(例如电子)运动的非线性薛定谔方程、玻色-爱因斯坦凝聚体的宏观动力学以及光纤中的信号。这个项目专注于对物理和工程中自然产生的色散方程进行严格的数学分析。傅立叶分析、概率、几何和解析数论的协同作用为研究波相互作用并产生新的、修改的传播模式的非线性效应提供了一个适应良好且功能强大的工具箱。首席研究员将研究的问题对深层流体中的长内重力波的研究、涡丝和空气动力学的理论，以及在当今电信系统和互联网通信中至关重要的非线性光纤的当前工作特别感兴趣。数学无处不在的作用是通过严谨的研究为最佳预测奠定基础，在此基础上，我们每天享受的技术进步和工程应用可以有效地实现。对学生和初级研究人员的培训是该项目的一个组成部分。

This project deals with research at the interface of harmonic analysis, nonlinear partial differential equations, geometry, and probability. On the one hand, it is concerned with the study of dispersive nonlinear wave phenomena from a nondeterministic viewpoint. In the last two decades enormous progress has been made in settling questions on existence of solutions to dispersive equations, their long-time behavior, and singularity formation. The thrust of this body of work has focused primarily on deterministic aspects of wave phenomena, where sophisticated tools from nonlinear Fourier analysis, geometry, and analytic number theory have played crucial roles. Yet there remain some fundamental obstacles. A natural approach to overcome them is to consider evolution equations from a nondeterministic point of view and to incorporate into the analysis tools from probability. Some of the issues to be investigated in this project are the role of randomization in the well-posedness theory, the almost sure (in the sense of probability) global-in-time existence of solutions, the existence and dynamical properties of associated Gibbs measures, and the behavior of statistical ensembles under gauge transformations. On the other hand, the principal investigator will study the existence and long-time dynamics of special types of solutions to certain hyperbolic (or nonelliptic) nonlinear Schrodinger equations and systems. The aim is to develop a rigorous mathematical analysis of models arising in connection with the theory of vortex filaments, ferromagnetism, and current work in nonlinear optics (e.g., examining the evolution of optical pulses in normally dispersive optical media), models that have attracted the attention of the physics community. Wave phenomena in physics such as light, sound, and gravity are mathematically modeled using partial differential equations. Nonlinear wave models arise in quantum mechanics, ferromagnetism, vibrating systems, semiconductors, and optical fibers. Dispersive equations model important wave propagation phenomena in nature. Their solutions are waves that spread out in space as time evolves while conserving energy or mass. The best known dispersive equations are the nonlinear Schrodinger equations that govern the motion of quantum particles (e.g., electrons), the macroscopic dynamics of the Bose-Einstein condensate, and signals in fiber optics. This project focuses on the rigorous mathematical analysis of dispersive equations that arise naturally in physics and engineering. The synergy of Fourier analysis, probability, geometry, and analytic number theory provides a well-adapted and powerful toolbox to study the nonlinear effects that allow waves to interact and produce new, modified propagation patterns. The problems that the principal investigator will study are of particular interest in the study of long internal gravity waves in deep stratified fluids, the theory of vortex filaments and aerodynamics, and in current work on nonlinear fiber optics that is of fundamental importance in today's telecommunication systems and internet traffic. The ubiquitous role of mathematics is to lay the foundations through rigorous research for the best predictions, based on which the technological advances and engineering applications we enjoy every day, can be efficiently enabled. The training of students and junior researchers is an integral part of the project.