Investigation of new techniques for energy and its applications

In this article, we present a unified mathematical solver that is widely used in applied nonlinear differential equations based on the well-known tanh–coth method. We employ this solver to provide new solutions for the unstable and modified unstable nonlinear Schrödinger models. We also introduce He’s frequency technique to examine the dynamical behavior properties. The presented solutions include various mathematical structures, such as solitons, dissipative super-waves, bright and dark envelope waves, and explosive envelope waves. The behavior of the presented solutions for suitable free parameter values is depicted by two-dimensional and three-dimensional graphs. The proposed approach offers various benefits, such as eliminating time-consuming and difficult calculations. The results presented here can be utilized to characterize a wide range of physical phenomena in numerous disciplines of practical study, such as pulse propagation in optical fibers and quantum field. Finally, this study increases the current understanding of mathematical methods for solving nonlinear differential equations, which may be applied to other nonlinear wave equations in other scientific areas.