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On dispersive stability of Hamiltonian systems on lattices
Author(s) -
Patz Carsten
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700847
Subject(s) - linearization , hamiltonian (control theory) , hamiltonian system , linearity , dispersion relation , stability (learning theory) , mathematics , nonlinear system , linear system , physics , mathematical physics , statistical physics , mathematical analysis , quantum mechanics , computer science , mathematical optimization , machine learning
We study the long‐time dynamics of oscillations in lattices of infinitely many particles interacting via certain non‐linear potentials. The aim is to proof dispersive stability of such Hamiltonian systems analogously to results known for PDEs. To do so we first recapitulate the dynamics of linear Hamiltonian systems on an infinite chain and give optimal decay rates based on the dispersion relation. Based on this we proof that if the non‐linearity is weak enough, the non‐linear system shows a similar behaviour like its linearization. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)