We list the main works on long-term behavior of solutions and attractors for nonlinear dissipative partial differential equations, beginning with the seminal work of L. Landau in 1944. We recall the main stages in the emergence of scattering theory fornonlinear Hamiltonian partial differential equations and formulate a general conjectureon the global attractors for such equations,invariant with respect to some Lie group. Furthermore, we listthe main results presented in this monograph: (1) the results onglobal attraction to stationary states in the case of a trivial symmetry group, to solitons in the case of the translation group, and to stationary orbits in the case of unitary and rotation groups; (2)the results on asymptotic stability of solitons and their effective adiabatic dynamics in weak external fields; (3) the results on numerical simulation of global attraction to solitons; and (4) the results on dispersive decay. In conclusion, we comment on the connection between the theory of attractors and quantum mechanics and the theory of elementary particles.

In this chapter we present in detail the first resultsonglobal attractionto stationary orbits obtained for a 1D Klein--Gordon equationcoupled to a nonlinear oscillator. The proofs rely on the concept of the omega-limit trajectory and a nonlinear analog of the Kato theorem on the absence of embedded eigenvalues, and on new theory of multipliers in the space of quasimeasures and a novel application of the Titchmarsh convolution theorem. Besides the formal proof, we give an informal explanation of the nonlinear radiative mechanism, which causes theglobal attraction: nonlinear energy transfer from lower to higher harmonics and subsequent dispersive radiation of energy to infinity. In conclusion, we specifythe general conjecture on global attractors, which summarizes all results obtained thus far.

Here we discuss the possible relation ofour generalconjecture on global attractors ofnonlinear Hamiltonian PDEs todynamicaltreatment of Bohr's postulates and of wave--particle duality, which are fundamental postulates of quantum mechanics, in the context of couplednonlinear Maxwell--SchrödingerandMaxwell--Dirac equations. The problem of adynamicaltreatment was the main inspiration for our theoryof global attractors ofnonlinear Hamiltonian PDEs.

We present in detailthe resultsonglobal attraction to stationary states for nonlinear Hamiltonian PDEs in infinite space: for 1Dwave equations coupled to one nonlinear oscillator (the "Lamb system"), for a 1D wave equation coupled to several nonlinearoscillators andfor a 1D wave equation coupled to a continuum of nonlinear oscillators, and for a 3Dwave equation and Maxwell’s equationscoupled to a charged relativistic particle with density of charge satisfying the Wiener condition. In particular, the radiation damping in classical electrodynamics is rigorously proved for the first time. The proofs rely on calculation of energy radiation to infinity and use the concept of omega-limit trajectories and the Wiener Tauberian theorem. The last sectionconcerns3D wave equations with concentrated nonlinearities. The key step in the proof is an investigation of a nonlinear integro-differential equation.

In this chapterwe present thefirst results on global attraction tosolitons forthe scalar wave field and the Maxwell field coupled to the charged relativistic particlewith density of charge satisfying the Wiener condition. In particular, the radiation damping in classical electrodynamics is rigorously proved for the first time (in this chapter, for the translation-invariant case). The proofs rely on calculation of energy radiation to infinity and canonical transformation to the comoving frame and use energy bound from below and the Wiener Tauberian theorem.