Wavepacket delocalization, self-trapping and fragmentation in discrete chains with relaxing nonlinearity

The discrete nonlinear Schrodinger equation (DNSE) describes wave phenomena in several physical contexts, ranging from electronic transport in crystalline chains to light propagation in nonlinear media and Bose-Einstein condensates. Here, we study the influence of the nonlinear response time on the temporal evolution of a wavepacket initially localized in a single site of a finite closed chain. Distinct long-time wavepacket distributions are identified as a function of the nonlinear strength χ and the characteristic relaxation time τ. Besides the more standard delocalized and self-trapped regimes, we report the occurrence of intermediate phases. In one of them the wavepacket self-focus in the opposite chain site. A phase with asymptotically fragmented wavepackets also develops. A crossover regime on which the ultimate wavepacket distribution is strongly dependent on the precise set of model parameters is also identified. We provide the full phase diagram related to the long-time wavepacket distribution in the (χ, τ) space.