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.