Newly modified unified auxiliary equation method and its applications

Abstract

This study aims to propose a new modified unified auxiliary equation method for the first time. To illustrate the validity of the new modified method, we consider the dimensionless time -dependent paraxial equation which is investigated previously in Tarla and Yilmazer (2022) using the unified auxiliary equation method. As a result, some different and more general solutions are obtained such as singular, periodic, hyperbolic, dark-bright, trigonometric, exponential, and Jacobi elliptic functions. The obtained results are compared with the solutions obtained by the unified auxiliary equation method in Tarla and Yilmazer (2022) in the result and discussion section. We conclude that our findings are novel and distinctive, and we give the comparison in the section on results and discussion. It is demonstrated that the modified unified auxiliary equation method provides a more efficient mathematical technique for solving nonlinear partial differential equations. In addition, we draw 3D profiles and counter plots for some obtained solutions with the assistance of a computer program by giving a specific value for the involved parameters.