Exploring conversation laws and nonlinear dynamics of the unstable nonlinear Schrödinger equation: Stability and applications

The nonlinear Schrödinger equation (NLSE) is a fundamental nonlinear model renowned for its accurate description of light pulse propagation in optical fibers. The unstable NLSE, a universal equation in nonlinear integrable systems, governs instabilities in modulated wave trains and characterizes the temporal progression of disturbances in near-stable or unstable media. This article analytically explores the nonlinear dynamics of the uNLSE, employing three symbolic computation methods: the three-wave (TW) approach, the positive-quadratic function (PQF) approach, and the double exponential (DE) approach. Our analysis yields novel exact solutions and multi-wave solutions, including rational solitons, breathers, and various wave types associated with this model. These results are derived using the ansatz function method and symbolic computation, including transformations involving logarithmic and traveling waves. The wave solutions obtained significantly deepen our apprehension of the physical phenomena within this intricate model. Additionally, we computed conserved quantities such as momentum, power, and energy associated with the solitons. The modulational instability (MI) analysis is utilized to evaluate the stability of the uNLSE, which confirmed the exactness and stability of all soliton solutions. Through the selection of appropriate parameter values, we produced 3D and contour visualizations in forms such as lump waves, breather-sort waves, and multi-peak solitons.