Attractors in nonlinear Hamilton wave-particle systems

The main subject of this project is the long-time behavior of solutions of nonlinear Hamiltonian PDEs (partial differential equations) arising in Atomic Physics. In particular, nonlinear Schrödinger equations and Maxwell-Lorentz equation with rotating charged particle are considered. These equations admit solutions of special form which are called soliton solutions. Solitons are known to be fundamentally important in the study of evolution equations, mainly because they are often easily found numerically, and also because they generally emerge in the long-time asymptotics of solutions of these equations. The main goals of the project are i) to prove the long-time attraction of finite energy solutions to the set of all solitons; ii) to analyze the stability of the solitons. These goals were inspired by the problems of the stability and effective dynamics of elementary particles, because the latter can be identified with solitons of nonlinear field equations. Such an identification is in the spirit of Heisenbergs theory of elementary particles in the context of nonlinear hyperbolic PDEs. The first results in these directions were obtained by numerical simulation in 1965 by N. Zabusky and M. Kruskal for the Korteweg-de Vries equation. In 1967, C. Gardner, J. Greene, M. Kruskal, and R. Miura found that the inverse scattering transform can be used to solve this equation analytically. It was seen that any finite energy solution converges to a finite sum of solitons and a dispersive wave. A little later P. Lax developed a uniform approach to more general integrable equations. However, this approach is not applicable to overwhelming part of fundamental equations of mathematical physics which are non-integrable. Since 1990, this problem is in the center of modern mathematical analysis of nonlinear PDEs by leading experts in the field: V. Bach, M. Esteban, J. Fröhlich, M. Griesemer, P.-L. Lions, I. Rodnianski, E. Séré, W. Schlag, I. Sigal, A. Soffer, H. Spohn, M. Weinstein and others. These investigations strongly influenced the development of Mathematical Physics, theory of PDEs and Functional Analysis. We plan to extend the research to novel equations: the Maxwell-Lorentz equation with rotating charged particle, and the Schrödinger equation coupled to nonlinear oscillators.

The aim of the project is to study long-time behavior of solutions to Hamiltonian nonlinear wave equations. The equations describe the interaction between waves and matter, which is the key object of theoretical physics. The main peculiarity of the problems is that both wave profile and matter parameters are unknown and change during the interaction. Mathematically, this means that the equations are nonlinear. The equations stem from Classical Electrodynamics and Quantum Physics at the beginning of the 20th century, however, many questions concerning the properties of their solutions (in particular, the mathematical description of the Bohr transitions to quantum stationary states) remain open up to now. The special difficulty of the problem is that the friction is absent in all fundamental equations, so the energy is conserved. In particular, Bohr transitions are impossible in any finite-dimensional dynamical system with energy conservation. An intensive mathematical study of the long-time behavior of solutions to these equations started around 1970. The first results on the wave-matter interaction were obtained around 2000 for the Maxwell-Lorentz equations describing a classical particle without rotation coupled to the electromagnetic field. In particular, global attraction to static states has been established, providing a mathematical model of Bohr transitions. The role of friction here is played by the energy radiation to infinity. In our project, we have extended the results to the case of a rotating particle. In this case, the Maxwell-Lorentz equations admit soliton solutions corresponding to particles moving at constant velocity and rotating with constant angular velocity. The solitons provide a mathematical model of a quantum particle with spin. The main results of the project for the Maxwell-Lorentz equations are i) the global attraction of all finite-energy solutions to soliton solutions and ii) the stability of the soliton solutions. In the second part of the project we established similar properties for quantum Klein-Gordon and Dirac equations with nonlinear point interactions. Proving these results required a novel development and introduction of many mathematical methods in the context of the considered equations: Lie-Poincaré calculus, Hamilton-Poisson structure, Casimir invariants, the momentum map, Lyapunov function, symplectic projection in Hilbert spaces, spectral theory of nonselfadjoint operators, and others. The energy radiation to infinity is justified by the dispersive decay of the corresponding linearized equations. Our results are inspired by the problem of stability of elementary particles and provide new mathematical models of the Bohr transitions to quantum stationary states. The developed methods may be useful for further development of stability theory for nonlinear Hamiltonian PDEs.