Attractors of nonlinear Hamilton equations

The aim of this project is a rigorous analysis of the long-time asymptotic behaviour of non-linear hyperbolic partial differential equations (PDEs). These investigations are motivated by mathema-tical problems of quantum mechanics. In particular, it is the aim of our investigation to explain certain basic quantum phenomena as inherent mathematical properties of the corresponding hyperbolic PDEs. In this context, for instance, the Bohr transitions between different quantum states should correspond to the convergence to solitary waves of all finite energy solutions of the related PDEs. Analogously, the wave-particle duality (that is, the wave-like behaviour of a system at intermediate times which results in a number of particles in the long-time asymptotics) should correspond to the convergence to a number of solitons and radiation of all finite energy solutions of the related PDEs. Our general goal is to further develop the theory of hyperbolic PDEs and to provide mathematically satisfactory explanations for the above physical problems. Concretely, we want to prove the following three conjectures for the corresponding physically relevant set of hyperbolic PDEs: i) For confining systems with U(1) symmetry all finite energy solutions converge to Solitary Waves. ii) For translation-invariant systems, each finite energy solution asymptotically converges to a finite number of solitons and a scattered diverging free wave. iii) The motion of solitons in a slowly varying external potential is governed by the adiabatic effective dynamics. The mathematical methods used for the investigation include long-time convergence in local energy semi-norms (which are the relevant ones for physical systems with radiation), the Wiener condition on the coupling function (which allows to avoid the usually employed smallness condition), as well as some advanced, modern techniques which have been developed in the last few years by several research groups, among them the main collaborators of this project:symplectic geometry in the Hilbert phase space of nonlinear hyperbolic PDEs, time decay for thelinearized problems, localisation of resonant terms, etc. Therefore, this project will profit significantly from the expertise which its main collaborators have acquired in the relevant methods and techniques over the last years. It will include research teams in Vienna, Munich, Moscow, St. Petersburg, Cambridge, and in the USA. Important contributions to some fundamental problems of mathematical physics can be expected.

The aim of this project is a rigorous analysis of the long-time asymptotic behaviour of non-linear hyperbolic partial differential equations (PDEs). These investigations are motivated by mathema-tical problems of quantum mechanics. In particular, it is the aim of our investigation to explain certain basic quantum phenomena as inherent mathematical properties of the corresponding hyperbolic PDEs. In this context, for instance, the Bohr transitions between different quantum states should correspond to the convergence to solitary waves of all finite energy solutions of the related PDEs. Analogously, the wave-particle duality (that is, the wave-like behaviour of a system at intermediate times which results in a number of particles in the long-time asymptotics) should correspond to the convergence to a number of solitons and radiation of all finite energy solutions of the related PDEs. Our general goal is to further develop the theory of hyperbolic PDEs and to provide mathematically satisfactory explanations for the above physical problems. Concretely, we want to prove the following three conjectures for the corresponding physically relevant set of hyperbolic PDEs: i) For confining systems with U(1) symmetry all finite energy solutions converge to Solitary Waves. ii) For translation-invariant systems, each finite energy solution asymptotically converges to a finite number of solitons and a scattered diverging free wave. iii) The motion of solitons in a slowly varying external potential is governed by the adiabatic effective dynamics. The mathematical methods used for the investigation include long-time convergence in local energy semi-norms (which are the relevant ones for physical systems with radiation), the Wiener condition on the coupling function (which allows to avoid the usually employed smallness condition), as well as some advanced, modern techniques which have been developed in the last few years by several research groups, among them the main collaborators of this project:symplectic geometry in the Hilbert phase space of nonlinear hyperbolic PDEs, time decay for thelinearized problems, localisation of resonant terms, etc. Therefore, this project will profit significantly from the expertise which its main collaborators have acquired in the relevant methods and techniques over the last years. It will include research teams in Vienna, Munich, Moscow, St. Petersburg, Cambridge, and in the USA. Important contributions to some fundamental problems of mathematical physics can be expected.