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Dynamics and classification of localized solutions in nonlinear lattices

Lattices, or discrete systems, or differential-difference equations, appear naturally in many branches of physics. Indeed, they describe atomic lattices, macromolecules, spin systems, arrays of optical wave­guides, Bose-Einstein condensates in optical lattices, etc. They also appear as a result of discretization of partial differential equations when one studies them numerically.

Our interest is focussed on understanding the phenomenon of energy transfer through lattices, resonant mode interactions, shock waves and solutions with a compact support in lattices. We are also involved in the development of multiple-scale techniques for quantum chains as well as in a theory of interaction of quantum solitons and phonons in weakly nonlinear lattices.

We also address mathematical questions such as the proof of global existence of solutions, their asymptotic behavior at large times, integrable discretization, and possible classification of existing intrinsic localized modes in lattices.

The above studies are carried out in collaboration with groups of the University of Salerno (Italy), Tel Aviv Universty (Israel), University of Massachusetts (USA), Lukin's Institute of Physical Problems (Zelenograd, Russia), CNPq and Universidade Federal do Rio de Janeiro (Brasil).

For the related publications of the group see http://alf1.cii.fc.ul.pt/~konotop/Publications.htm.