Exact Solutions and Conservation Laws for the Time-Fractional Nonlinear Dirac System: a Study of Classical and Nonclassical Lie Symmetries

Abstract

Time-fractional Dirac-type systems arise in quantum field theory, plasma physics, and condensed matter systems where fractional calculus captures nonlocal interactions. In this study, we employ classical and nonclassical Lie symmetry methods to analyze the underlying symmetry structure of the system. By deriving infinitesimal generators and performing similarity reductions, we transform the governing fractional partial differential equations (FPDEs) into fractional ordinary differential equations (FODEs). Exact solutions are constructed using the power series method. Furthermore, we establish conservation laws in the fractional setting, ensuring the physical consistency of the system. Our findings offer new insights into the interplay among symmetry, conservation principles, and exact solutions in fractional quantum field models, expanding the analytical toolkit for studying nonlinear relativistic wave equations.