Asymptotic stability of small solitons in the discrete nonlinear Schrodinger equation in one dimension

TL;DR

This paper proves the asymptotic stability of small solitons in a one-dimensional discrete nonlinear Schrödinger equation with high-power nonlinearities and an external potential, supported by analytical decay estimates and numerical simulations.

Contribution

It extends stability analysis to discrete NLS with septic and higher nonlinearities, combining dispersive decay estimates and numerical validation.

Findings

Proved asymptotic stability of small solitons in the model.

Numerical simulations indicate faster decay rates than theoretical estimates.

Analysis relies on dispersive decay estimates from prior work.

Abstract

Asymptotic stability of small solitons in one dimension is proved in the framework of a discrete nonlinear Schrodinger equation with septic and higher power-law nonlinearities and an external potential supporting a simple isolated eigenvalue. The analysis relies on the dispersive decay estimates from Pelinovsky & Stefanov (2008) and the arguments of Mizumachi (2008) for a continuous nonlinear Schrodinger equation in one dimension. Numerical simulations suggest that the actual decay rate of perturbations near the asymptotically stable solitons is higher than the one used in the analysis.