New traveling wave solutions for paraxial wave equation via two integrating techniques

The purpose of the presented work is to obtain the exact solutions of the Paraxial wave equation using two different methods, the modified auxiliary equation method and ( G'/G,1/G) -expansion method. Results obtained by these methods are unique. The paraxial wave equation is a partial differential equation that describes the propagation of optical waves in the paraxial approximation. This equation is derived from Maxwell’s equations, which describe the behavior of electromagnetic waves. The paraxial wave equation is a fundamental tool in optics for analyzing the propagation of optical waves in the paraxial approximation. The application of integrating techniques allows researchers and engineers to obtain solutions and predict the behavior of optical systems in a wide range of applications. As a result of modified auxiliary equation method and ( G'/G,1/G) -expansion method, new soliton solutions of the given model are extracted. These solutions include hyperbolic solutions and periodic solutions. The necessary constraint conditions for the existence of solutions are also provided. Moreover, for appropriate parametric values, acquired findings are displayed via contour, 2D and 3D graphics that show the physical relevance and dynamical behaviors of the proposed equation. It is noticed that proposed methods are well organized and efficient in obtaining accurate solutions for many nonlinear evolution equations.