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P. Kevrekidis, D.E Pelinovsky and A. Stefanov

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

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SIAM Journal of Mathematical Analysis 41, 2010-2030 (2009)

**Abstract:**

Asymptotic stability of small bound states 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
bound states is higher than the one used in the analysis.

**Keywords:**

Discrete nonlinear Schrodinger equation, bound states, asymptotic stability,
dispersive estimates.