Discrete solitons in nonlinear Schrödinger lattices with a power-law nonlinearity

We study the discrete nonlinear Schrödinger lattice model with the onsite nonlinearity of the general form, |u|2σu|u|2σu. We systematically verify the conditions for the existence and stability of discrete solitons in the one-dimensional version of the model predicted by means of the variational approximation (VA), and demonstrate the following: monostability of fundamental solitons (FSs) in the case of the weak nonlinearity, 2σ+1<3.682σ+1<3.68; bistability, in a finite range of values of the soliton’s power, for 3.68<2σ+1<53.68<2σ+1<5; and the presence of a threshold (minimum norm of the FS), for 2σ+1≥52σ+1≥5. We also perform systematic numerical simulations to study higher-order solitons in the same general model, i.e., bound states of the FSs. While all in-phase bound states are unstable, stability regions are identified for antisymmetric double solitons and their triple counterparts. These numerical findings are supplemented by an analytical treatment of the stability problem, which allows quantitively accurate predictions for the stability features of such multipulses. When these waveforms are found to be unstable, we show, by means of direct simulations, that they self-trap into a persistent lattice breather, or relax into a stable FS, or sometimes decay completely.