Field and reverse field solitons in wave-operator nonlinear Schrödinger equation with space-time reverse: Modulation instability

Abstract

The wave-operator nonlinear Schrödinger equation was introduced in the literature. Further, nonlocal space–time reverse complex field equations were also recently introduced. Studies in this area were focused on employing the inverse scattering method and Darboux transformation. Here, we present an approach to find the solutions of the wave-operator nonlinear Schrödinger equation with space and time reverse (W-O-NLSE-STR). It is based on implementing the unified method together with introducing a conventional formulation of the solutions. Indeed, a field and a reverse field may be generated. So, for deriving the solutions of W-O-NLSE-STR, it is evident to distinguish two cases (when the field and its reverse are interactive or not-interactive). In the non-interactive and interactive cases, exact and approximate solutions are obtained. In both cases, the solutions are evaluated numerically and they are displayed graphically. It is observed that the field exhibits solitons propagating essentially (or mainly) on the negative space variable, while those of the reverse field propagate on the other side (or vice versa). These results are completely novel, and we think that it is an essential behavior that characterizes a complex field system with STR. On the other hand, they may exhibit right and left cable patterns (or vice versa). It is found that the solutions of the field and its reverse exhibit self-phase modulation by solitary waves. In the interactive case, the pulses of the field and its reverse propagate in the whole space. The analysis of modulation stability shows that, when the field is stable, its reverse is unstable or both are stable. This holds whenever the polarization of the medium is self-defocusing.