  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.
