Harmonic Analysis and Nonlinear Hamiltonian Equations

该项目建议在谐波分析和非线性汉密尔顿方程领域进行研究。 奇异积分算子分析的研究方向之一，目前主要研究奇异和非奇异典型关系下Fourier积分算子的Lebesgue空间估计。这样的估计提供了重要的工具，分析非线性哈密顿系统的解的性质：整体存在性，光滑性和稳定性。另一个研究方向是非线性Hamilton系统的孤立波解的特殊性质，这类类孤立波解在许多不同的背景下都被证明是存在的.如果方程在李群作用下是不变的，则孤立波可能具有稳定性。该研究计划的重点是具有最小能量的孤立波的稳定性，预计这对应用特别重要。当波的振幅增加时，它们与介质的相互作用变得重要。这种相互作用可能会严重改变波的行为，导致非线性孤立波或孤子的出现。这种非线性波描述了纤维光学、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。详细了解这种波的性质，特别是它们的稳定性或不稳定性，将有助于预测天气突变的可能性，发展光波导，并在当今电子和芯片制造商不可阻挡地接近的尺度上描述量子效应，更不用说实验物理学了。自然现象对错综复杂的相关领域提出了新的挑战，谐波分析和非线性方程理论，并促进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.