Orbitally but not asymptotically stable ground states for the discrete NLS

TL;DR

This paper demonstrates the existence of discrete nonlinear Schrödinger ground states that are orbitally stable but not asymptotically stable, due to internal modes that prevent energy leakage and lead to perpetual oscillations.

Contribution

It provides examples of discrete NLS ground states with internal modes that decouple from continuous modes, showing a novel stability behavior unlike Euclidean space cases.

Findings

Ground states are orbitally but not asymptotically stable.

Internal modes do not exchange energy with continuous modes.

Discrete modes exhibit perpetual oscillations.

Abstract

We consider examples of discrete nonlinear Schroedinger equations in Z admitting ground states which are orbitally but not asymptotically stable in l ^2(Z). The ground states contain internal modes which decouple from the continuous modes. The absence of leaking of energy from discrete to continues modes leads to an almost conservation and perpetual oscillation of the discrete modes. This is quite different from what is known for nonlinear Schroedinger equations in eucliedean spaces. We do not investigate connections with work on quasi periodic solutions as in M.Johansson & S.Aubry and of Bambusi & Vella