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In this paper, we investigate the properties of modulation instability (MI) in the fractional Schrödinger equation with cubic–quintic nonlinearity (SECN), particularly focusing on some novel dynamical behaviors different from those in the conventional SECN. First, we apply the linear-stability theory to derive an analytical expression for instability gain in such novel fractional physical system. Just like the case of the conventional SECN, our analysis shows that the dynamical behavior of MI in the fractional SECN still depends on the various combinations of the signs of cubic and quintic nonlinearities. In particular, MI is found to still occur even in the fractional Schrödinger equation (SE) with self-defocusing cubic nonlinearity if the quintic nonlinearity is positive. Moreover, we further find that the Lévy indices related to the fractional effects mainly act to increase not only the fastest growth frequency but also the bandwidth, when compared to the case in the conventional SECN. However, the corresponding maximum gain is completely independent on such Lévy indices. The theoretical predictions are confirmed by the numerical results from the split-step Fourier method. Our findings suggest that the novel fractional systems with cubic–quintic nonlinearity provide some possibilities to control the dynamical behavior of MI and new solitary waves.

The additional dispersive magnetic permeability in metamaterials (MMs) described by the Drude model is able to generate the anomalous self-steepening (SS) effect which can be positive, negative even zero. This provides more great flexibility for us to control ultrashort pulse propagation, when compared with the corresponding case in naturally positive-index materials (PIMs). In this paper, we theoretically investigate the self-accelerating control of Airy pulse through such anomalous SS effect in MMs with a cubic nonlinearity. The emphasis is given to the counterintuitive dynamical behavior different from those in PIMs. It is shown that positive SS in MMs acts to promote the self-acceleration of such pulse and the corresponding spectral features exhibits a broad redshifted behaviors, as it does in PIMs; while the negative SS gives rise to the appearance of the opposite situation. That is to say, the self-acceleration of Airy pulse will be delayed through broadening the spectrum toward blue side. Meanwhile, the effect of the specific truncation coefficient imposed on Airy pulses in MMs also be discussed under the roles of positive and negative SS, respectively. Our findings show that the SS effect in MMs can provide a more powerful tool in controlling the self-accelerating of Airy pulse when compared with the corresponding case in PIMs.