Bifurcation analysis, and solution of homoclinic and heteroclinic orbit solution for the paraxial wave model through bifurcation theory

The paraxial wave equation, a simplified version of the full electromagnetic wave formulation, is frequently employed to characterize optical signal transmission in these materials, facilitating an efficient and precise description of light propagation. This study examines the dynamical characteristics of the proposed model using both analytical and dynamical methods, specifically employing a bifurcation approach. Initially, wave transformation is applied to convert the proposed model into its conventional form. The system's dynamic properties are then examined using phase and Hamiltonian portraits, considering different parameter settings. We also find bright periodic, kink, bright-type, and dark-type bell wave, anti-kink, and solitary wave solutions by changing parameters, using their corresponding Hamiltonians, and integrating heteroclinic and homoclinic orbits. We also visually represent various characteristics by carefully choosing the right parameters. Moreover, the physical characteristics and computational results support the robustness and significance of the proposed theory.