Investigation of the optical solitons for the Lakshmanan–Porsezian–Daniel equation having parabolic law

This paper investigates the optical soliton solutions of the Lakshmanan–Porsezian–Daniel equation, a nonlinear evolution equation modeling the propagation of optical soliton waves in materials influenced by parabolic law nonlinearity and spatio-temporal dispersion. Employing both the Kudryashov and new Kudryashov methods, the study captures various soliton waves, deriving both dark and bright soliton solutions. The research delves into the effects of the model parameters on these solutions, presenting the findings through detailed 3D and 2D simulation images. Practical and rapid results achieved using both Kudryashov methods are highlighted, positioning this work as a valuable reference for peers. The study’s distinct combination of the Lakshmanan–Porsezian–Daniel equation and the chosen analytical techniques not only introduces novelty but also emphasizes practicality and efficiency. This innovative approach, coupled with its significant contribution to the knowledge base, underscores the research’s relevance in the field.