Abstract
The nonlinear Schrödinger equation (NLSE) can well identify the development of waves in deep water and optical fibers towards the least-order approximation. This study addresses the Lakshmanan–Porsezian–Daniel (LPD) fractal model which emerges from the application of the Heisenberg spin chain and fiber optics. This paper analyzes three types of nonlinear rules, namely Kerr law, quadratic law, and parabolic law. The variational approach to the combination of the Ritz idea is used to discover the new optical soliton solutions for the LPD-equation. It poses the requisite novel conditions for ensuring the existence of valid solitons. Three- and two-dimensional configurations are demonstrated by choosing the correct values for the parameters. This study focused on the pioneering research boundaries of the LPD-equation and other associated nonlinear evolution models in the field of communications network technology and optical fiber.