(Semi-)Nonrelativisitic Limit of the Nonlinear Dirac Equations

Abstract

We consider  the nonlinear Dirac equation (NLD) with time dependent external electro-magnetic potentials, involving a dimensionless parameter $ε\in(0,1]$ which is inversely proportional to the speed of light. In the nonrelativistic limit regime $ε\ll1$ (speed of light tends to infinity), we decompose the solution into the eigenspaces associated with the 'free Dirac operator' and construct an approximation to the NLD with $O(ε^2)$ error. The NLD converges (with a phase factor) to a coupled nonlinear Schrödinger system (NLS) with external electric potential in the nonrelativistic limit as $ε\to0^+$, and the error of the NLS approximation is first order $O(ε)$. The constructed $O(ε^2)$ approximation is well-suited for numerical purposes.