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• The vertical grain displacement model is presented. • The acquired results are given by some figures. • The modulation analysis is given. • Bright, dark and singular optical solitons are presented. The paper concerns envelope soliton propagating through the nonlinear transverse dust grain displacement in dusty plasma crystals. The work found its motivation from the recent paper of the authors [7]. To reach the goal of this study we establish the linear dispersion and the nonlinear Schrödinger equation (NLSE) by employing the quasi discrete approximation [7,13–15]. Contrary to ref. [7] , it is observed one additional regime of the formation of the dark soliton while choosing the gap frequency value (ω g ≈ 150 sec −) which is close to the experimental value given by Motcheyo et al. [7] , Tsopgue et al. To corroborate the prediction made analytically to group velocity, was confirmed by numerical and experimental studies. As, it is established that in general manner the modulated waves describing by the NLSE can be stable or unstable to the perturbation, the modulation instability of steady state have been done. Despite, that the bandwidth of the model has been reduced compared to [7,13–15] , we can say without hesitation that this work will be a capital contribution in dusty plasma crystal. [ABSTRACT FROM AUTHOR]

The detection of the solitons whose the propagation is spatially localized on the array is an interesting feature for experimental purposes. Here, we investigate the effect of position-modulated self-Kerr nonlinearity on discrete soliton propagating in an optomechanical array. The optomechanical cells are coupled to their neighbors by optical channels, and each cell supports the self-Kerr nonlinear term. Modulational Instability (MI) together with numerical simulations are carried out to characterize the behavior of the solitonic waves. It results that the nonlinear term shortens the transient regime of the temporal solitonic waves, which allows the wave propagation to be confined along specific cells. As the nonlinear term increases, the pulsed shape of soliton waves get sharped and highly peaked. For a strong enough value of the nonlinear term, the waves feature chaos-like motion. Owing to these results, the position-modulated self-Kerr nonlinear term manifests itself as being an energy source for the solitonic waves, allowing to the generated solitons to propagate during a long time while acquiring energy. These results shed light on the fact that nonlinear optomechanical platforms could be used to generate long-lived temporal localized solitons and even chaotic solitonic waves, which are good prerequisites for information processing purposes. [ABSTRACT FROM AUTHOR]