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Abstract: The head-on collisions of two ring dark solitons in inhomogeneous Bose–Einstein condensates (BECs) with thin disk-shaped potential are studied by the extended Poincaré–Lighthill–Kuo (PLK) perturbation method. The result shows that the system admits a solution with two concentric ring solitons, one moving inwards and the other moving outwards, which in small-amplitude limit, are described by two modified cylindrical KdV equations in the respective reference frames. In particular, the analytical phase shifts induced by the head-on collisions between two ring dark solitary waves are derived, and the result shows that the phase shifts change with the radial coordinate r according to the law (where ), which are quite different with the homogeneous case. [Copyright &y& Elsevier]

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]