Some optical soliton solutions with bifurcation analysis of the paraxial nonlinear Schrödinger equation

The paraxial nonlinear Schrödinger equation finds diverse applications in the fields of nonlinear optics, optical communication systems, plasma physics, and mathematical physics. The outcomes of this research endeavor are directed toward achieving three principal objectives. Firstly, one of our key aims is to uncover novel soliton solutions for the model, thereby making a valuable addition to the existing body of research. Secondly, we delve into the dynamics of these solutions using the modified extended auxiliary equation mapping approach, a methodology that provides valuable insights into the system's behavior. Our final objective involves an examination of the model's dynamic characteristics, achieved through bifurcation and stability analyses, as well as the identification of the associated Hamiltonian function. The presentation of three-dimensional and contour plots, with the parameter values having been thoughtfully chosen, is done for the purpose of ensuring the physical validity of our findings. These findings shed light on the practicability, efficacy, and computing speed of the methodologies that were utilized, delivering answers that are comprehensive and trustworthy. This research significantly enhances the field by advancing our comprehension of soliton solutions within the model, introducing innovative investigative techniques, and conducting a comprehensive exploration of the system's bifurcation and stability aspects. As a direct outcome of this study, new avenues have emerged for further exploration and potential application in the realms of nonlinear optics, optical communication systems, and related fields.