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The property of matter-wave vector solitons in a spin-1 Bose-Einstein condensate with spin-orbit and Zeeman couplings is investigated by multiscale perturbation method. The excitation spectrum and the corresponding state vectors of the system are obtained analytically, and they can be adjusted by the parameters of the system. The bright and dark vector solitons are formulated by reducing the three-component coupled Gross-Pitaeviskii equations to a standard nonlinear Schrödinger equation, which has the solutions of the bright and dark solitons with positive or negative mass depending on the product of the effective dispersive and nonlinear coefficients. The moving vector solitons are demonstrated by adjusting specific momentum near the energy minimum. Finally, the magnetized features of the vector solitons are discussed by the spin polarization of the system. • The excitation spectrum for the vector solitons, determined by the parameters of the system, is obtained analytically. • The analytical bright and dark vector solitons are formulated and the moving solitons are demonstrated by adjusting specific momentum. • The solitons with positive or negative mass depending on the product of the effective dispersive and nonlinear coefficients are analyzed. • The magnetized features affected by spin-orbit and Zeeman couplings are discussed by the spin polarization of the system. [ABSTRACT FROM AUTHOR]

We investigate the discrete nonlinear Schrödinger model with helicoidal hopping and a power-law nonlinearity, motivated by the tunable nonlinearity in the model of DNA chain and ultra-cold atoms trapped in a helix-shaped optical trap. In the study of modulational instability, we find a successive destabilization along with increasing nonlinear-power. In particular, the critical amplitudes of second-stage instability decrease as nonlinear-power increases. Furthermore, it is shown that information on the stability properties of weakly localized solutions can be inferred from the plane-wave modulational instability results. This link enable us to analytically estimate the critical parameters at which the breather solutions turn unstable, and find these parameters are dramatically influenced by the nonlinear-power. The stability properties of localized breathers perform an obvious change when the nonlinear power crosses a critical value γ c r . It is reflected that at weak nonlinearity the breathers exhibit monostability, while exceeding γ c r the bistability and instability will set in. The interplay between nonlinear-power and long-range hopping on the stability properties of breathers is also discussed in detail. [ABSTRACT FROM AUTHOR]