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

Abstract

The wave-operator nonlinear Schrodinger equation was introduced in the literature. Further, Nonlocal space–time reverse complex field equations were recently introduced in the literature. Studies in this area were focused on employing the inverse scattering method (ISM) and Darboux transformation (DT). Here, we present an approach to find the solutions of the wave-operator nonlinear Schrodinger equation with space and time reverse (W-O-NLSE-STR). It is based on implementing the unified method (UM) 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 and when they are not-interactive. In the non-interactive case, exact and approximate solutions are obtained, while in the second case exact solutions are only found. In the case of finding approximate analytical solutions, the maximum error (ME) is controlled via an adequate choice of the parameters in the residue terms. It is worth mentioning that the ME, found here, is space and time independent. This is not the case when evaluating the solutions via different numerical techniques. In both two cases, the solutions are evaluated numerically and they are shown graphically. It is observed 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 vise-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 vise-versa) respectively.

Analytic solutions of alpha-beta -time- derivatives complex finance chaotic dynamical system: synchronization and extended center manifold. An explicit approach

“…In the case when α = 0.9 moderate sensitivity occurs. A variety of approximate analytical methods were presented [30][31][32][33][34]. Further relevant works were carried out in [26,[35][36][37][38].…”