Solitary wave effects of Woods–Saxon potential in Schrödinger equation with 3d cubic nonlinearity

In this research article, we apply the generalized projective Riccati equation method to construct traveling wave solutions of the 3d cubic focusing nonlinear Schrödinger equation with Woods–Saxon potential. The generalized projective Riccati equation method is a powerful and effective mathematical tool for obtaining exact solutions of nonlinear partial differential equations, and it allows us to derive a variety of traveling wave solutions of the 3d cubic focusing nonlinear Schrödinger equation with Woods–Saxon potential. These solutions contain periodic wave solutions, bright and dark soliton solutions. The study of many physical systems, such as Bose–Einstein condensates and nonlinear optics, that give rise to the nonlinear Schrödinger equation. We provide a detailed description of the generalized projective Riccati equation method in the paper, and demonstrate its usefulness in solving the nonlinear Schrödinger equation with Woods–Saxon potential. We present various graphical representations of the obtained solutions using MATLAB software, and analyze their characteristics. Our results provide new insights into the behavior of the 3d cubic focusing nonlinear Schrödinger equation with Woods–Saxon potential, and have potential applications in numerous fields of physics, as well as nonlinear optics and condensed matter physics.