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.