Modulation instability in fractional Schrödinger equation with cubic–quintic nonlinearity

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.