Harmonic Analysis and Nonlinear Hamiltonian Equations

该项目建议在调和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。目前的兴趣包括与奇异和非奇异正则关系有关的傅立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整体存在性、光滑性和稳定性提供了重要的工具。研究的另一个方向是关于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。如果方程在某个李群的作用下是不变的，则孤立波可能具有稳定性。研究的重点是能量最小的孤立波的稳定性，当孤立波的振幅增大时，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，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。对这种波的性质，特别是它们的稳定性或不稳定性的详细了解，将使我们能够预测天气突然变化的可能性，开发光波导，并描述当今电子和芯片制造商无情地接近的尺度上的量子效应，更不用说实验物理学了。自然现象给复杂相关的领域--谐和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.