Bifurcation and solitary wave solutions of a time-dependent paraxial equation using improved modified extended tanh function method

This paper discusses an elaborate investigation of the dimensionless time-dependent paraxial equation based on its varied soliton solutions obtained through an improved modified extended tanh function method. The approach is capable of producing kink, bell-shaped, singular wave, periodic, singular periodic wave, bright and dark solitary wave, breather soliton, singular bell, M-shaped, W-shaped, and V-pattern solitons expressed as hyperbolic and trigonometric functions. Visualization via three dimensional (3D) surface plots, two dimensional (2D) cross-sections and contour plots allows a thorough examination of the wave morphology. Nonlinear dynamics are investigated using bifurcation, chaotic, sensitivity, and stability analyses that uncover complex solution behaviors and stability regimes. Modulation instability analysis verifies the stability of soliton structures under perturbations. The work demonstrates the effectiveness of the improved modified extended tanh function method for solving nonlinear partial differential equations and enhances theoretical knowledge of paraxial wave phenomena.