Dynamics of Computational Solitons: Modulation Instability, Bifurcation, Chaotic Nature With Different Chaos-Detecting Tools, and Influence of Multiplicative Noise Intensity

This study presents the dynamic analysis of stochastic fractional unstable nonlinear Schrödinger equation (SFuNLSE), focusing on the analysis of bifurcation, chaotic nature, return map, recurrence plots, strange attractor, basin attractor, power spectrum, chaotic nature and also finds the exact optical soliton solution with multiplicative noise intensity. The unstable NLSE is more important to describe diverse complex phenomena in plasma physics, optical fiber communication, quantum mechanics, superfluidity, and numerous others. We first insert a wave variable to transform the SFuNLSE problem into an ordinary differential equation (ODE). This adaption facilitates a thorough bifurcation analysis, wherein the modified equations are articulated as dynamical systems via a Galilean transformation. The dynamical features of the suggested equation are assessed through this bifurcation by depicting phase diagrams. This research also uncovers essential equilibrium states and associated transitional behaviors. Subsequently, we do a comprehensive examination of the proposed equation utilizing an array of nonlinear dynamics methods, encompassing bifurcation, chaotic characteristics, return maps, recurrence plots, weird attractors, basin attractors, and power spectrum analysis. These strategies offer profound insights into the system’s intricate behavior. Ultimately, we employ the function transformation method to provide innovative optical soliton solutions. The obtained solutions are articulated in the forms of Jacobian elliptic, exponential, hyperbolic, and trigonometric functions. Two-dimensional and three-dimensional profiles were plotted using suitable values for the permitted parameters to analyze the physical features of the resultant solution. This method illustrates the efficacy of graphical simulations in depicting the behavior and interaction of these solutions in practical contexts. The comparison results indicate that the level of multiplicative noise and the fractional parameter significantly affect the derived solutions. We examine the modulation instability (MI) of the unstable NLSE, which is beneficial for managing severe wave events, enhancing optical communication, and exploring nonlinear wave dynamics in diverse physical systems. All analyses and the diagram are combined with the computer software MATLAB 2023. This work introduces novel phenomena to enhance the field of nonlinear optical research and communication technologies.