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Asymptotic stability of small solitons in one dimension is proved in theframework of a discrete nonlinear Schrodinger equation with septic and higherpower-law nonlinearities and an external potential supporting a simple isolatedeigenvalue. The analysis relies on the dispersive decay estimates fromPelinovsky & Stefanov (2008) and the arguments of Mizumachi (2008) for acontinuous nonlinear Schrodinger equation in one dimension. Numericalsimulations suggest that the actual decay rate of perturbations near theasymptotically stable solitons is higher than the one used in the analysis.

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