Generalized PT-symmetric nonlinear Dirac equation: Exact solitary waves solutions, stability, and conservation laws.

We investigate the interplay between gain-loss dynamics and nonlinearity in the PT-symmetric nonlinear Dirac equation with a scalar-scalar power-law interaction (|Ψ¯Ψ|kΨ,k>0). Exact solitary wave solutions reveal that the PT-transition point is determined solely by the existence condition and is independent of the nonlinearity exponent k. Despite the presence of gain and loss, the energy remains conserved. Although charge and canonical momentum are not conserved, a PT-symmetric modification of the canonical momentum yields a new conserved quantity. This modified conservation law gives rise to unusual kinematic properties, including nonzero momentum for stationary solutions and zero-momentum states for moving solutions at specific values of the gain-loss parameter. Finally, linear stability analysis and numerical simulations show that gain-loss effects and higher-order nonlinearities reduce the stability region of solitary waves: Solutions remain stable for k ≤ 2 and become unstable in part of the parameter space for k > 2.