Numerous analytical wave solutions to the time-fractional unstable nonlinear Schrödinger equation with beta derivative

Fractional nonlinear evolution equations are mathematical representations used to explain a wide range of complex phenomena occurring in nature. By incorporating fractional order viscoelasticity, these equations can accurately depict the intricate behaviour of materials or mediums, requiring fewer parameters compared to classical models. Furthermore, fractional viscoelastic models align with molecular theories and thermodynamics, making them highly compatible. Consequently, the scientific community has shown significant interest in fractional nonlinear evolution equations and their soliton solutions. This study employs the extended Kudryashov method to derive soliton solutions for the time-fractional unstable nonlinear Schrödinger equation, utilizing the Atangana–Baleanu fractional derivative known as the beta derivative. The obtained solitons exhibit various shapes, including V-shaped, periodic, singular periodic, flat kink, and singular bell, under specific conditions. To better understand their physical characteristics, 3D and contour plots are presented by assigning parameter values to certain solutions. Additionally, 2D graphs are generated to observe how the fractional parameter affects the solutions. The Hamiltonian function is determined to further analyse the dynamics of the phase plane. The simulations were conducted using Mathematica and MATLAB software tools. The outcomes of this research contribute to a deeper understanding of the behaviour of fractional nonlinear evolution equations and their soliton solutions, offering insights into the complex dynamics of viscoelastic systems.