Abstract
The Ablowitz-Ladik lattice has a two-parameter family of travelling breathers. We derive a necessary condition for their persistence underperturbations of the system. From this we deduce non-persistence for a varietyof examples of perturbations. In particular, we show that travelling breathersdo not persist under many reversible perturbations unless an additional symmetry is preserved, and we address the case of Hamiltonian perturbations.
Highlights
The phenomenon of discrete breathers, i.e. spatially localised periodic oscillations in nonlinear lattices, has attracted much recent interest
One largely unresolved topic in this context is the rigorous mathematical status of travelling discrete breathers: such objects have repeatedly been observed in numerical simulations (e.g. [3, 5, 10]), their existence has been proved for only a few rather special models ([1, 5])
In this note we discuss a necessary condition for smooth persistence of travelling discrete breathers under perturbations of the Ablowitz–Ladik (AL) lattice
Summary
The phenomenon of discrete breathers, i.e. spatially localised periodic oscillations in nonlinear lattices, has attracted much recent interest (see e.g. the review articles [2, 4, 6, 8]). In search of a better mathematical understanding, a natural first question is whether the known travelling breathers persist under certain perturbations of the respective underlying system. In this note we discuss a necessary condition for smooth persistence of travelling discrete breathers under perturbations of the Ablowitz–Ladik (AL) lattice.
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