Asymptotics and Attractors of Hyperbolic Equations

The project is aimed at the asymptotic analysis and numerical simulations of linear and nonlinear partial differential equations of hyperbolic type that arise in relativistic quantum mechanics, i.e. quantum mechanics that respect the fact that the Speed of light is not infinite ("special relativity"). Our mathematical research is inspired by fundamental problems of physics, in particular the dualism of waves and particles. The mathematical rigorous link between waves and particles is given by "asymptotic analysis", i.e. the study when a physical parameter of the System tends to zero or inffnity. In many contexts this is called "high frequency limits", i.e. the "geometrical optics limit" when light is considered as "rays" instead of waves. In our context the Limit objects corresponding to particles are the "solitons" which are particular solutions of wave equations. Our research also aims to a better understanding of "quantum transitions" : for bound systems, the stationary states are obtained in a long time asymptotic. Our ueneral goal is to develop the theory of hyperbolic PDEs and to suggest a new dynamical approach to the above mentioned fundamental problems. We study: i) classical limits and ihe long-time behavior of finite energy solutions: the attraction to the Set of all stationary states and the soliton-type asymptotics; ii) the convergence to equilibrium distributions for infinite energy solutions. Previous research in the direction i) concemed linear and nonlinear wave and Maxwell equations. The aim of the project is the extention of the results to the nonlinear Klein-Gordon equations. Previous results in the direction ii) concerned onedimensional crystals, wave and Klein-Gordon equations. The aim of the project is the extention of the results to the Born-Karman model of solid state, to two- and threedimensional crystals with two-temperature initial measures, and translationinvariant coupling of the nonlinear oscillator to the wave field. This project will include several teams in Europe, including Russia, and in the USA and provides also an excellent framework for the international coorperation an state-of-the-art problems of modern mathematical physics, in order to make progress in the theory of Quantum Electro Dynamits (QED).

The overall goal of the project was the asymptotic analysis and numerical simulation of linear and nonlinear partial differential equations of hyperbolic type that arise in relativistic quantum mechanics, i.e. quantum mechanics that respect the fact that the speed of light is not infinite ("special relativity of Einstein"). Nonlinear equations arise from the coupling of matter, described e.g. by the Dirac equation or by the Vlasov equation, to (electromagnetic) fields/potentials, described e.g. by the Maxwell equation. The mathematical rigorous link between waves and particles is provided in particular by "solitons" which are particular solutions e.g. of wave equations. This research is also aimed at a better understanding of "quantum transitions" : for bound systems, the stationary states are obtained in a long time asymptotic of the nonlinear partial differential equations. The project has contributed to develop the theory of hyperbolic PDEs and to a new dynamical approach to better understand fundamental problems in mathematical physics. The main results were obtained in : i) nonrelativistic and classical limits of Klein-Gordon Maxwell and Dirac-Maxwell equations ii) the long-time behavior of finite energy solutions and the attraction to the set of all stationary states and the soliton-type asymptotics; iii) the convergence to equilibrium distributions for infinite energy solutions. The project has succesfully extended previous results on wave equations to the nonlinear Klein-Gordon equations. Also, the project has successfully extended previous results to the Born-Karman model of solid state to higher- dimensional crystals with two-temperature initial measures, and translation-invariant coupling of the nonlinear oscillator to the wave field. The project was carried out by A.Komech and the principial researcher together with several Postdocs and visitors. A total of 33 publications in peer reviewed journals resulted from this project, presentations at international conferences were given, including invitated talks at College de France of the 2 senior project members. Also several international workshops were organized within the frame of this project.