Chaotic behavior and traveling wave solutions of the paraxial nonlinear Schrödinger equation

The paraxial nonlinear Schrödinger equation is used in many domains such as mathematical physics, optical communication systems, nonlinear optics, and plasma physics. In this work, we aim to investigate the dynamic properties of the model employing bifurcation theory and determining the corresponding Hamiltonian function. By applying the bifurcation and planar dynamical system theories, we obtain the traveling wave solutions with the corresponding phase portraits. The purpose of the carefully selected parameter values and the resulting plot presentations is to guarantee the physical validity of our results. These results provide answers that are thorough and reliable by illuminating the applicability, effectiveness, and computing speed of the employed scheme. In addition, chaotic behavior and sensitive analysis of the model under consideration are studied. This work has directly opened up new directions for investigation and possible applications in other nonlinear science fields.