Dynamical analysis of the nonlinear Schrödinger equation: insights into sensitivity and chaotic phenomena

This study aims to explore the intricate dynamics of light propagation as described by a coupled nonlinear Schrödinger equation. By analysing the derived optical solutions, we will uncover the underlying phenomena influencing wave interactions in nonlinear media. We demonstrate numerous soliton solutions using two analytical techniques: the modified Khater method and the Sardar subequation method. These methods employ an innovative approach to parameter transformation in nonlinear equations, enabling the extraction of diverse single solutions by leveraging the structure of the governing equations. Through these methods, we will elucidate the complex behaviours of light in nonlinear media and contribute to the broader understanding of wave interactions. The investigation of nonlinear wave phenomena reveals intricate interactions that are essential for advancing our knowledge in fields such as mathematics, ocean engineering and physics. The graphical representation of various analytical solutions, including trigonometric, exponential and hyperbolic functions, effectively illustrates the dynamics of wave structures. This exploration not only enhances theoretical understanding but also has practical implications for real-world applications. We examine the dynamics of the phase portrait, sensitivity and chaotic behaviour of the planer dynamical system, which is transformed using the Galilean transformation. We execute the simulations using the software Mathematica and MATLAB. Ultimately, this article utilizes appropriate parameter values to illustrate 3D and 2D graphs of some derived solutions. Our study underscores the significance of visual tools in enhancing theoretical insights and fostering advancements in applied sciences.