On localized solutions of discrete nonlinear Schrِdinger equation: An exact result

Local and global existence of localized solutions of a discrete nonlinear Schrödinger (DNLS) equation, with arbitrary on-site nonlinearity, is proved. In particular, it is shown that an initially localized excitation persists localized during infinite time. Moreover, if initial localization is stronger than | n | − d with any power d, it maintains itself as such during infinite time. The results are generalized to various types of inter-side and saturable nonlinearities, to lattices with long range interactions, as well as DNLS with dissipation.