Decay of Hamiltonian Breathers under Dissipation

TL;DR

This paper investigates how energy dissipates in a discrete nonlinear Schrödinger system with localized initial energy, showing that energy decay is extremely slow and the system's behavior remains near breather solutions.

Contribution

It provides rigorous asymptotic estimates for energy decay rates and phase space localization near breather solutions under dissipation.

Findings

Energy decreases extremely slowly with dissipation.

System remains localized near breather solutions.

Asymptotic estimates for decay rate and phase space confinement.

Abstract

We study metastable behavior in a discrete nonlinear Schr\"odinger equation from the viewpoint of Hamiltonian systems theory. When there are $n < \infty$ sites in this equation, we consider initial conditions in which almost all the energy is concentrated in one end of the system. We are interested in understanding how energy flows through the system, so we add a dissipation of size $\gamma$ at the opposite end of the chain, and we show that the energy decreases extremely slowly. Furthermore, the motion is localized in the phase space near a family of breather solutions for the undamped system. We give rigorous, asymptotic estimates for the rate of evolution along the family of breathers and the width of the neighborhood within which the trajectory is confined.