The new closed form for soliton solutions to the unstable nonlinear Schrödinger model in mathematical physics

The unstable nonlinear Schrödinger equation (UNLSE) is an important paradigm for studying nonlinear phenomena such as periodic waves, explosive waves, solitons, and modulation instability in domains including optics, plasma physics, and fluid dynamics. In this work, we introduce the closed form approach for solving the UNLSE based on the well-known extended tanh method. These solutions are very critical to explain several complex phenomena in applied science. The suggested approach has several advantages, including reducing time-consuming and complicated calculations. This method could possibly be used as a box solver for mathematicians, engineers, and physicists. The two-dimensional (2D) and three-dimensional (3D) graphs show the behavior of the proposed solutions for various free parameter values. In addition, we demonstrate how the physical properties impact the behavior of the solutions. The UNLSE serves as both a key partial differential equation in applied mathematics and a predictive model in physics, offering profound insights into universal mechanisms of wave amplification, collapse, and nonlinear coherence. This study enhances our understanding of mathematical strategies for resolving complex models, which can be applied to nonlinear wave equations in a range of scientific disciplines.