On localized solutions of discrete nonlinear Schrodinger equation. An exact result
P. Pacciani, V. V. Konotop, G. Perla Menzala
TL;DR
Focusing on the discrete nonlinear Schrödinger equation $i \dot{q}_n(t) + \Delta q_n(t) - \chi |q_n(t)|^{2(p-1)} q_n(t) = 0$, the paper proves local and global existence of localized solutions for initially localized data with $A_d = \sum_m (1+m^2)^d |a_m|^2 < \infty$ (arbitrary on-site nonlinearity). The analysis reformulates the problem as an integral equation using the linear Green's function and employs a Banach fixed-point argument in the weighted space $X^d(T)$ (with momentum $Q_d(t)$) to establish local existence, and then extends to global existence via a priori bounds and a uniqueness argument based on a weighted energy $Z(t)$. These results extend to various inter-site and saturable nonlinearities, long-range interactions, and DNLS with dissipation, and show that initially localized excitations persist indefinitely with the same degree of localization (or decay faster than $|n|^{-d}$) in time, under a conserved particle number that bounds amplitudes and with finite group velocity preventing collapse. The work elucidates the localization mechanism via the lattice spectrum with a cut-off frequency, yielding well-posedness and ruling out collapse, while noting open questions for more general inter-site nonlinearities and for multi-dimensional or multi-atomic lattices.
Abstract
Local and global existence of localized solutions of a discrete nonlinear Schrodinger (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.