Pure-cubic form of the Nonlinear Schrödinger Model with the Polynomial Laws Excluding the Chromatic Dispersion via the New Kudryashov's Integration Algorithm

Abstract

In this study, we aim to derive exact analytical solutions, specifically bright soliton and singular solutions, for the pure-cubic nonlinear Schr & ouml;dinger equation with third-order dispersion, incorporating cubic-quintic-septic nonlinearities, and to investigate its dynamic behavior through modulation instability analysis. This model, characterized by the dominance of third-order dispersion and the absence of the standard group-velocity dispersion term, accurately reflects realistic wave evolution in ultrafast optics and supercontinuum generation. The equation was analyzed using the new Kudryashov's scheme, which is highly effective for solving high-order nonlinear Schr & ouml;dinger-type equations. To complement this, a detailed modulation instability analysis was performed. As a result, bright soliton and singular solutions were successfully obtained. Furthermore, the conditions under which the model exhibits modulation instability were precisely identified. The findings were illustrated through 2D and 3D graphical representations. The primary novelty of this work lies in the successful application of the new Kudryashov method and the subsequent detailed modulation instability analysis to this particular, highly-nonlinear form of the Schr & ouml;dinger equation. The derived analytical solutions serve as crucial benchmark data for future studies involving external perturbations.