Asymptotic stability of small solitons in the discrete nonlinear   Schrodinger equation in one dimension
  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. 
P.G. Kevrekidis, D.E. Pelinovsky and A. Stefanov