| The dynamics of a spatially confined (finite) classical Hamiltonian system disentangles into exact normal mode excitations in the harmonic approximation. Each separate excitation at a fixed energy is a uniquely defined periodic orbit (PO). These POs are continued into the anharmonic domain when normal modes interact. The resulting q-breathers are time-periodic solutions which exponentially localize in the normal mode space (Phys. Rev. Lett. 95, 064102 (2005)). The properties of q-breathers and small perturbations of them account for all major ingredients of the famous 51 year old Fermi-Pasta-Ulam (FPU) problem of (non)equipartition in a nonlinear atomic chain (Phys. Rev. E 73, 036618 (2006)). By construction q-breathers are obtained in two- and three-dimensional lattices as well (Phys. Rev. Lett. 97, 025505 (2006)) and may become relevant objects to study the energy locking and flow between normal modes of finite systems. Further scaling of q-breathers to infinitely large system sizes (nlin.PS/0607019)) provides rigorous results which need to be interpreted in the light of the theory of anharmonic lattice vibrations of crystals. We find that at any finite nonzero energy density (temperature) and nonlinearity coefficient an effective nonlinearity parameter defines a wave vector range around the band edges of the harmonic phonon dispersion where the phonon concept breaks down completely due to strong interaction, resonances and complete delocalization in normal mode space. In particular these findings may explain scaling properties of anomalous heat conductivity studies in large FPU chains where one needs to overcome the ballistic regime of long wavelength phonons. If time allows, I will discuss perspectives which include i) additional discrete symmetries (which induce degeneracies between harmonic normal mode frequencies and lead to q-breather structures with a vortex like flow of energy among a few normal modes of the system); ii) extensions of the concept to (discrete) nonlinear Schroedinger equations; iii) quantization of q-breathers; iv) spatial disorder. |
|