Asymptotic stability of small solitons in the discrete nonlinear Schr¨odinger equation in one dimension

Asymptotic stability of small solitons in one dimension is proved in the framework of a discrete nonlinear Schr¨odinger 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 Schr¨odinger 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