Qualitative study with optical solitons and vortex structures of two-dimensional vector nonlinear Schrödinger equation in weakly nonlinear waveguides

This study investigates the wave structures and qualitative assessment of a two-dimensional vector nonlinear Schrödinger equation (2D VNLSE) that occurs in Bose–Einstein condensates (BECs) and nonlinear optics. We convert the vector PDE form into two linked scalar PDEs to yield optical vortex structures of the governing model. We create different optical soliton solutions using the Kudryashov and the sine-Gordon expansion methods. We present their 3D, 2D, and density maps to allow for a comprehensive and understandable representation of their underlying physical patterns. Furthermore, we assess the system’s qualitative behavior by analyzing its obvious characteristics, such as modulation instability, sensitivity, bifurcation, and chaos detection, and we give accompanying illustrations to support our conclusions. This study’s accomplishment allows researchers to examine the dynamics and growth of optical solitons and vortices in nonlinear systems in greater detail, which opens up important possibilities with broad significance in many scientific and technological fields.