Stabilization of small solutions of discrete NLS with potential having two eigenvalues

TL;DR

This paper investigates the long-term dynamics of small solutions to discrete nonlinear Schrödinger equations with a potential having two eigenvalues, demonstrating decomposition into bound states and dispersive waves under certain conditions.

Contribution

It establishes the asymptotic behavior of small solutions in the two-eigenvalue case, including instability of excited states and energy distribution properties.

Findings

Small solutions decompose into bound states and dispersive waves.

Excited states are shown to be unstable.

Generalized equipartition of energy is demonstrated.

Abstract

We study the long time behavior of small (in $l^2$) solutions of discrete nonlinear Schr\"odinger equations with potential. In particular, we are interested in the case that the corresponding discrete Schr\"odinger operator has exactly two eigenvalues. We show that under the nondegeneracy condition of Fermi Golden Rule, all small solutions decompose into a nonlinear bound state and dispersive wave. We further show the instability of excited states and generalized equipartition property.