Asymptotic Stability of Small Bound States in the Discrete Nonlinear Schrdinger Equation

Asymptotic stability of small bound states in one dimension is proved in the framework of a discrete nonlinear Schrdinger 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 and Stefanov [J. Math. Phys., 49 (2008), 113501] and the arguments of Mizumachi [J. Math. Kyoto Univ., 48 (2008), pp. 471497] for a continuous nonlinear Schrdinger 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.