Approximate-analytic optical soliton solutions of a modified-Gerdjikov–Ivanov equation: modulation instability

Abstract

The Gerdjikov–Ivanov equation (GIE) occupied a remarkable area of research in the literature. In the present work, a modified GIE (MGIE) is considered which is new and was not studied in the literature. Also, the modified-unified method (MUM) is used to obtain approximate analytic solutions (AASs) of MGIE. Up to our knowledge, no AASs for non-integrable complex field equation were found up to now. Thus the AASs found, here, are novel. The UM addresses finding the exact solutions to integrable equations. In this sense as no exact solution for MGIE exists, consequently, it is not integrable. So, here, approximate analytic optical soliton solutions are invoked. The UM stands for expressing the solution of nonlinear evolution equations in polynomial and rational forms in an auxiliary function (AF) with an appropriate auxiliary equation. For finding exact solutions by the UM, the coefficients of the AF, with all powers, are set equal to zero, For a non-integrable equation, only approximate solutions are affordable. In this case, we are led to utilizing the MUM. Herein, non-zero coefficients (residue terms (RTs)) are considered as errors, which are space and time-independent. It is worth mentioning that, this is in contrast to the errors found by the different numerical methods, where they are space and time-dependent. Further, in the present case, the maximum error is controlled via an adequate choice of the parameters in the RTs. These solutions are displayed in graphs. Breather soliton, chirped soliton and M-shape soliton, among others, are observed. Furthermore, modulation instability (MI) is studied and it is found MI triggers when the coefficient of the nonlinear dispersion exceeds a critical value.

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

“…It unifies all known methods such as, the tanh, modified [46], and extended versions, the F-expansion [47], the exponential, the G'/G expansion method [48], the Lie symmetries [49] and the Kerdyashov method [50]. On the other hand, in the applications, it is established that the UM [51,52] is of low time cost in symbolic computations. Thus, we think that it prevails the use of the Lie group to construct the symmetries of NLPDEs, as it is required to carry a hierarchy of long steps.…”

Exploring the dynamic interplay of intermodal and higher order dispersion in nonlinear negative index metamaterials

“…In [26], modulation instability influenced by nonlinear dispersion term in an optical metamaterial with the accompanying of coherent and partially coherent ultrashort pulses discussed in detail. In the realm of scientific exploration, where curiosity and innovation intertwine, a multitude of pioneering works have emerged, each unveiling the tantalizing novelty of optical soliton solutions across an array of mathematical models [27,[42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. Several studies contribute significantly to the field of nonlinear wave equations and soliton theory, but they also reveal specific gaps in understanding soliton dynamics, stability under varying conditions, and the influence of higher-order nonlinearities and dispersions.…”