Optical solitons of improved perturbed nonlinear Schrödinger equation with cubic-quintic-septic and triple-power laws in optical metamaterials

Abstract

Purpose: This paper aims to extract optical solitons of improved perturbed nonlinear Schr¨odinger equations (IP-NLSE) with cubic-quintic-septic (CQS) and a triple-power law (TP-law) using the new Kudryashov and the extended sinh-Gordon equation expansion (eShGEE) methods. 
Methodology: First, we apply a wave transformation to the studied equations to generate the nonlinear ordinary differential equation (NLODE) form. Next, by computing the balancing constant in the NLODE form, we use the new Kudryashov and eShGEE methods to obtain the equation’s solution in the NLODE form. We get an algebraic equation system on the NLODE by replacing the suggested solution function and its derivatives in the NLODE
form. With the help of the solutions of the system, we can determine the appropriate solution sets for unknown parameters. Substituting these sets and wave transforms into the proposed solution functions by the new Kudryashov and eShGEE methods, we get the solutions for the problems under investigation. 
Findings: We have successfully obtained soliton solutions for the considered equations and plotted 3D and 2D graphs of the derived solution functions. In addition to obtaining the soliton solutions, we present some graphical investigation of the impact of the parameters in the considered equations.
Originality: To our best knowledge, the improved perturbed nonlinear Schr¨odinger equations with CQS and a triple-power law have not been studied before. It is also innovative to examine how the equation’s parameters affect the soliton’s behavior. In this regard, the study’s findings are novel, and it is anticipated that they will advance research in the area.

Soliton solutions of the improved perturbed nonlinear Schrödinger equation having parabolic law with non-local nonlinearity in the presence of chromatic and spatio-temporal dispersion terms

“…Investigating analytical solutions of the NLSE provides a deeper understanding of the formation, evolution, and interactions of optical solitons, as well as enabling theoretical predictions about the behavior of specific physical systems [5][6][7][8][9][10][11][12][13]. Therefore, different methods have been improved to produce analytical solutions of the NLSE like the new Kudryashov scheme [14][15][16], the improved generalized Kudryashov method [17], enhanced modified extended tanh application [18][19][20], modified simple equation technique [21,22] and the F-expansion technique [23,24].…”

“…where Θ = 2α + ω − α κ 2 and γ is represented in equation (15). For validity of the solutions in equation (16) and equation (17), the constraint condition is given as λ 3 Θ > 0.…”

Various nonlinear characteristics of breather/rogue waves and controllable interaction phenomena for a new KdV equation with variable coeffcients

“…Nonlinear phenomena exist widely in nature, and nonlinear partial differential equations (NLPDEs) are indispensable tools to describe the nonlinear phenomena. Nowadays, NLPDEs are widely used in water waves [1], elastic mechanics [2], quantum mechanics [3], plasma physics [4,5], nonlinear optics [6,7], communication engineering [8,9], biomedicine [10] and other fields [11][12][13]. It is well known that in some real-world problems, NLPDEs with variable coefficients provide a more realistic perspective on the inhomogeneities of media and non-uniformities of boundaries in comparison than NLPDEs with constant coefficients, and have applications in various fields such as optical fibers [14], propagations of wave and rogue waves [15], and various other physical systems [16][17][18][19].…”