Self-steepening, Raman scattering and self-phase modulation-interactions via the perturbed Chen–Lee–Liu equation with an extra dispersion. Modulation insability and spectral analysis

Abstract

AbstractIt is worth mentioning that the perturbed Chen–Lee–Liu equation (PCLLE) exhibits the effects of self-steepening (SS), Raman scattering (RS) and self-phase modulation (SPM). Our attention is focused, here, to inspect the challenge between these phenomena may lead to a dominant one among them. On the other hand, we investigate the dominant phenomena produced due to these interactions. Furthermore, the structure of the configuration of pulses propagation in optical fibers are depicted. These phenomena are illustrated, here, via studying the PCLLE with an extra dispersion. This equation is formulated and the exact solutions of this new equation are found by using the unified method (UM). The importance of the UM stems from the fact that, in the applications, we have found that the UM is of low time cost in symbolic computation. So, we think that it prevails the known methods in the literature. On the other hand, the solutions for the complex envelope field equations, found in the literature, are always considered with real wave amplitude. Here, a transformation based on using complex wave amplitude is introduced. Indeed, in this case, solutions describe the waves that result from soliton- periodic wave collision, which may reveal novel phenomena. The solutions obtained are evaluated numerically and represented in graphs. It is shown that pulses compression occurs which may be due to self-phase modulation. Also, dispersive shock wave can be produced which may be argued to the presence of the extra dispersion and self-sdteepening. It is worthy to mention that, self-steepening arises for a small value of the related coefficient. The results obtained, here, are novel. The modulation instability is analyzed and it is found that it triggers at a critical values of SS and RS coefficients. At this stage, shock wave may occur. It is observed that the spectrum shows soliton with periodic waves background.