Harmonic Analysis and Nonlinear Hamiltonian Equations

该项目建议在谐波分析和非线性哈密顿方程领域进行研究。奇异积分算子分析的一个研究方向。目前的研究方向包括奇异和非奇异正则关系的傅里叶积分算子的勒贝格空间估计。这种估计为分析非线性哈密顿系统解的性质：全局存在性、光滑性和稳定性提供了重要的工具。另一个研究方向涉及非线性哈密顿系统孤立波解的特殊性质。这种类孤子解被证明存在于许多不同的环境中。如果方程在李群作用下是不变的，则孤波可能具有稳定性。本项目的研究重点是最小能量孤立波的稳定性，这对应用具有特别重要的意义。当波的振幅增加时，它们与介质的相互作用就变得重要了。这种相互作用可能会严重改变波的行为，并导致非线性孤立波或孤子的出现。这种非线性波描述了光纤、等离子体物理、超导理论、基本粒子理论、气象学和海洋学中的中心现象。这些波的特性的详细知识，特别是他们的稳定性或不稳定性，将允许预测天气突然变化的机会，发展光波导，并描述量子效应的尺度是不可避免地接近今天的电子和芯片制造商，更不用说实验物理学。自然现象对复杂的相关领域，谐波分析和非线性方程理论提出了新的挑战，并刺激了现代数学工具的进一步完善。

The project suggests investigations in the area of Harmonic Analysis and Nonlinear Hamiltonian Equations. One direction of the research in the analysis of singular integral operators.Current interests include Lebesgue space estimates on Fourier integral operators associated to singular and nonsingular canonical relations. Such estimates provide important tools for the analysis of properties of solutions to nonlinear Hamiltonian systems: global existence, smoothness, and stability. The other direction of the research concerns specific properties of solitary wave solutions to nonlinear Hamiltonian systems.Such soliton-like solutions were proved to exist in manydifferent contexts. If the equation is invariant under theaction of some Lie group, then the solitary waves may possessstability properties. The program of research focuses on thestability of solitary waves with minimal energy, which areexpected to be of particular importance for applications.When the amplitude of the waves increases, their interaction with the medium becomes important. This interaction may seriouslychange the behavior of the waves and lead to the appearance ofnonlinear solitary waves, or solitons. Such nonlinear wavesdescribe central phenomena in fiber optics, plasma physics,theory of superconductivity, theory of elementary particles,meteorology, and oceanology. Detailed knowledge of propertiesof such waves, and in particular their stability or instability,will allow to predict the chances of sudden weather changes,develop optical waveguides, and describe quantum effects onthe scales being inexorably approached by today's electronicsand chip manufacturers, let alone the Experimental Physics.Natural phenomena pose new challenges to the intricately related fields, Harmonic Analysis and the Theory of Nonlinear Equations, and stimulate further refinement of the tools of modernMathematics.