Exploration conversations laws, different rational solitons and vibrant type breather wave solutions of the modify unstable nonlinear Schrödinger equation with stability and its multidisciplinary applications

The modified unstable nonlinear Schrodinger equation governs specific instabilities observed in modulated wave-trains and also describes the time evolution of disturbances in marginally stable or unstable media. In this article, we explore the modified unstable dynamical model analytically using three symbolic computational techniques: the positive quadratic function approach, the three-wave approach, and the double exponential approach. As a result of our analysis, we have derived novel exact solutions, including rational solitons, multi-wave solutions, and other types of wave solutions for this dynamical model. These solutions are obtained through symbolic computation and the ansatz function method, incorporating both traveling wave and logarithmic transformations. The derived wave solutions hold practical significance, contributing significantly to understanding the physical phenomena of this complex model. We also computed conservative quantities such as power, momentum, and energy associated with solitons. To evaluate the stability of this modified unstable equation, we conducted a comprehensive modulational instability analysis, confirming the stability and exactness of all soliton solutions. By selecting appropriate parameter values, we generated 3D visual representations in various forms, including breather-type waves, lump waves, multi-peak solitons, etc. Furthermore, our observations revealed intriguing phenomena arising from the interactions among these multi-waves, with applications spanning a wide range of scientific and engineering disciplines.