Bifurcation analysis, modulation instability and optical soliton solutions and their wave propagation insights to the variable coefficient nonlinear Schrödinger equation with Kerr law nonlinearity

Abstract

The present paper aims to explore bifurcation analysis, modulation instability, and optical soliton solutions in nonlinear media with third-order dispersion terms. A variable coefficient third-order nonlinear Schrödinger’s equation (PNLSE) with truncated M-fractional derivative is considered. We also discuss a few assets that the derivative satisfies. Initially, a novel evaluation method known as bifurcation analysis is employed to look at the complex model's dynamic behavior. Figure 1 provides an analytical and graphical study of the observed mechanism of static soliton through a saddle-node bifurcation in the nonlinear Schrödinger problem using a matching technique. Secondly, By implementing novel two analytic techniques such as unified solver and generalized unified techniques to offer insights into wave propagation and optical soliton conduct in nonlinear optics, optical communications, quantum mechanics, plasma physics, and engineering. The underlying idea of these two approaches is to transform the equation with partial derivatives into a version of the equation with ordinary derivatives, which are first required to incorporate a new wave definition. These techniques permit us to generate innovative soliton solutions that can be formulated in terms of rational, hyperbolic, and trigonometrical functions. The collected outcomes have the potential to facilitate an understanding and elucidation of the physical characteristics of waves moving within a dispersive substance. We additionally estimated conservative values related to solitons, such as energy, momentum, and power. By selecting appropriate parameter values, the graphical shapes in 3D, density, and 2D are generated using relevant parameter values to visually present the obtained results, including periodic wave soliton, interaction of kink and bell shape soliton, periodic breather wave soliton, double periodic soliton through unified solver method and interaction of kink and periodic lump wave soliton and also with the periodic soliton, double periodic wave soliton, periodic wave soliton with lump soliton, multi periodic wave with kink shape soliton, periodic wave soliton, periodic breather wave soliton, periodic wave soliton, multi periodic wave soliton, periodic lump wave soliton, breather wave soliton multi periodic wave through generalized unified technique, providing an insightful visualization of the discovered solutions. In a two-dimensional graph, we show the effect of truncated M-fractional parameters for [g=0.1,0.5,0.9]. Additionally, the modulation instability spectrum can be expressed utilizing a linear analysis technique, and the modulation instability bands are shown to be influenced by the third-order and group velocity dispersion. The findings indicate that the modulation instability disappears for negative values of the fourth order in a typical dispersion regime. Consequently, it was shown that the techniques mentioned previously could be an effective tool to generate unique, precise soliton solutions for numerous uses, which are crucial to nonlinear optics, optical communications, and engineering