Existence and uniqueness solution analysis of time-fractional unstable nonlinear Schrödinger equation

•The study employs the powerful sine-Gordon expansion (SGE) method to achieve exact solutions for time-fractional unstable nonlinear Schrödinger (NLS) equations with the time-fractional β-time derivative.•The research establishes the existence and uniqueness of solutions to the time-fractional problem using a fixed-point argument.•The study employs 3D plots generated through software tools like MAPLE and MATLAB to graphically analyze the specific values of obtained solution parameters.