Center manifold theory in the context of infinite one-dimensional lattices

Guillaume James (1) , Yannick Sire (2)

1 CASYS - Calculs Algébriques et Systèmes Dynamiques
2 LATP - Laboratoire d'Analyse, Topologie, Probabilités

Center manifold theory has been used in recent works to analyze small amplitude waves of different types in nonlinear (Hamiltonian) oscillator chains. This led to several existence results concerning traveling waves described by scalar advance-delay differential equations, pulsating traveling waves determined by systems of advance-delay differential equations, and time-periodic oscillations (including breathers) obtained as orbits of iterated maps in spaces of periodic functions. The Hamiltonian structure of the governing equations is not taken into account in the analysis, which heavily relies on the reversibility of the system. The present work aims at giving a pedagogical review on these topics. On the one hand, we give an overview of existing center manifold theorems for reversible infinite-dimensional differential equations and maps. We illustrate the theory on two different problems, namely the existence of breathers in Fermi-Pasta-Ulam lattices and the existence of traveling breathers (superposed on a small oscillatory tail) in semi-discrete Klein-Gordon equations.

Domaines: Systèmes dynamiques [math.DS]

Guillaume James, Yannick Sire. Center manifold theory in the context of infinite one-dimensional lattices. Giovanni Gallavotti. The Fermi-Pasta-Ulam Problem. A Status Report, Springer, pp.207-238, 2008, Lecture Notes in Physics, 978-3-540-72994-5. ⟨10.1007/978-3-540-72995-2_6⟩. ⟨hal-00390032⟩