Studying exploring paraxial wave equation in Kerr media: Unveiling the governing laws, rational solitons and multi-wave solutions and their stability with applications

The rational solitons and other wave solutions demonstrate the existence of wave packets propagating through optical Kerr media, exhibiting non-diffractive and non-dispersive spatiotemporal localization. This paper focuses on constructing rational solitons, multi-wave solutions, and other solutions for a dimensionless time-dependent paraxial wave model (dt-pPWM). These solutions are derived through the application of symbolic computation and the ansatz function method, using both traveling wave and logarithmic transformations. Our approach encompasses three distinct methods: the positive quadratic function approach, the three-wave approach, and the double exponential approach. These wave results find practical applications and contribute to a deeper understanding of the physical phenomena described by this model. We have computed conservative quantities corresponding to the power, momentum, and energy of solitons. To assess the stability of the wave equation, we conducted modulational instability analysis, confirming the stability and accuracy of all soliton solutions. The structure of the solution provides a physical interpretation. By selecting appropriate parametric values, we generated 3D graphics illustrating various multi-wave solutions, including lump waves, rogue waves, and multipeak solitons. Additionally, we observed intriguing phenomena arising from the interaction of multi-waves, which have applications in telecommunications, quantum optics and other related fields.