Long Time Behavior for Hamilton PDEs

The project is the continuation of my Lise Meitner FWF project M1329-N13. The main goal of the project is the long-time behavior of solutions to nonlinear Hamilton PDEs. We plan to prove the dispersion long time decay in weighted norms for solutions to discrete Schrödinger, wave and Dirac equations with potentials having a non-compact support. We will deduce the decay proving the limiting absorption principle and the generalized Puiseux expansions of the corresponding re- solvents at the thresholds. Recently we have obtained similar dispersion decay for continuous Schrödinger and Klein-Gordon equations with magnetic potentials and for 1D and 2D Dirac equations. Further, we plan to apply the dispersion decay for the statistical stabilization for discrete Schrödinger, wave and Dirac equations. Namely, we are going to consider the equations with random translation invariant initial data satisfying the mixing condition of the Rosenblatt or Ibragimov-Linnik type. We will prove that the distribution of the solution converges, in the long time limit, to a Gaussian measure. This result is a generalization of the Central Limit Theorem for the discrete PDEs. Our next goal is a long time behavior of solutions to 1D Klein-Gordon and Dirac equations coupled to a nonlinear oscillator, and to 3D Klein-Gordon equations with concentrated nonlinearities. First, we plan to prove the asymptotic stability of soliton solutions. Namely, we plan to prove that the solitary manifold S attracts all finite energy solutions with the initial data close to the stable part of S. The convergence holds in weighted norms. Second, we plan to obtain a soliton scattering asymptotics which means that the solutions with the initial data close to the stable part of S asymptotically is a sum of some solitary wave and dispersive part which is a solution to the corresponding free equations. The remainder converges to zero in a global norm. The proofs will rely on the dispersion decay for the corresponding linearized equations. Finally, we plan to prove a convergence to S for all finite energy solutions without the assumption that the initial data are close to S for 3D Klein-Gordon equation with concentrated nonlinearities. The convergence means that S is a global attractor for the equations. This global attraction is caused by nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersion radiation. The convergence will be established in local energy seminorms. The results and methods of the project could be useful for application in Spectral Theory, Scattering Theory, Quantum and Statistical Physics. Asymptotic stability of solitary waves should clarify the stability of elementary particles in the spirit of the Heisenberg program. Statistical stabilization may explain the role of Gibbs canonical equilibrium measures in Statistical Physics.

We obtained new results on long time behaviour of solutions of linear and nonlinear Hamilton PDEs: dispersion decay, asymptotic stability of solitary waves, soliton scattering asymptotics, global attraction to solitary manifold, etc. We have proved dispersion decay in weighted norms for Klein-Gordon equations with scalar and magnetic potentials, and for discrete wave, Schr odinger and Dirac equations. These results will be useful in further development of stability theory for nonlinear Hamilton PDEs. We established global attraction to solitary manifold for wave, Klein-Gordon and Dirac equa- tion with concentrated nonlinearities, and for wave equation coupled with non-relativistic particle. These results give novel mathematical models of Bohr's transitions to quantum stationary states. We have proved asymptotic stability of stationary states and scattering asymptotics for wave equation coupled to a non-relativistic particle. One of the key ingredient in the proofs is our result on the dispersion decay for the corresponding linearized dynamics. These results were inspired by the problem of stability of elementary particles. We established asymptotic stability of the linearized dynamics for the Schr odinger-Poisson- Newton equations describing infinite crystals with a cubic lattice and one ion per cell. This is the first result on linear stability for infinite crystals. We constructed global dynamics and proved orbital stability of ground states with periodic arrangement of ions for nonlinear Schr odinger-Poisson-Newton equations for finite crystals with moving ions under periodic boundary conditions and under novel Jellium and Wiener-type conditions on the ion charge density. We have established spectral representation for solutions to linear Hamilton equations with non- negative energy in Hilbert spaces. The representation is indispensable in the proof of asymptotic stability of solitary waves for nonlinear Hamilton equations. As a principal application of these re- sults, we justify the eigenfunction expansion for linearized relativistic Ginzburg-Landau equations. We have proved well-posedness, global attractor and the Bogolyubov averaging principle for 2D damped driven nonlinear Schr odinger equation with almost periodic pumping in a bounded region. The results and methods of the project could be useful for application in Quantum Physics and Solid State Physics. The dispersion decay could be applied to various stability problems. Asymptotic stability of solitary waves might clarify the stability of elementary particles.