Soliton dynamics in the Lakshmanan–Porsezian–Daniel equation under diverse nonlinear optical laws

Abstract

In this work, a comprehensive analytical investigation of the Lakshmanan–Porsezian–Daniel (LPD) equation is conducted through the application of the Painlevé analysis and the unified method, aimed at constructing exact soliton solutions. Through the application of these analytical tools, explicit forms of soliton solutions are derived with three distinct nonlinear response laws relevant to optical fibers, say, Kerr law, parabolic law, and anti-cubic law, which are associated to various physical regimes and aspects of pulse propagation in nonlinear optical media. For each nonlinearity profile, families of soliton solutions are systematically derived, along with constraint conditions, ensuring their existence, stability, and physical relevance. The novel resulting solutions are then illustrated in three-dimensional surface plots and contour diagrams for suitable parameter values, providing a clearer and more intuitive understanding of the solution dynamics. Finally, a stability analysis of the selected model is performed, confirming that the governing equation exhibits stable behavior under the derived conditions. This study illustrates the versatility of the applied techniques to handle complex nonlinear models, providing rich soliton solutions under various nonlinear laws of optical fibers, and hence contributing to UN Sustainable Development Goals 4, 7 and 9.