P. Kevrekidis, D.E Pelinovsky and A. Stefanov

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

SIAM Journal of Mathematical Analysis 41, 2010-2030 (2009)

Abstract:
Asymptotic stability of small bound states 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 bound states is higher than the one used in the analysis.

Keywords:
Discrete nonlinear Schrodinger equation, bound states, asymptotic stability, dispersive estimates.