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 many different contexts. If the equation is invariant under the action of some Lie group, then the solitary waves may possess stability properties. The program of research focuses on the stability of solitary waves with minimal energy, which are expected to be of particular importance for applications.
When the amplitude of the waves increases, their interaction with the medium becomes important. This interaction may seriously change the behavior of the waves and lead to the appearance of nonlinear solitary waves, or solitons. Such nonlinear waves describe central phenomena in fiber optics, plasma physics, theory of superconductivity, theory of elementary particles, meteorology, and oceanology. Detailed knowledge of properties of 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 on the scales being inexorably approached by today's electronics and 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 modern Mathematics.
