TY - JOUR
T1 - (Semi-)Nonrelativisitic Limit of the Nonlinear Dirac Equations
AU - Cai , Yongyong
AU - Wang , Yan
JO - Journal of Mathematical Study
VL - 2
SP - 125
EP - 142
PY - 2020
DA - 2020/05
SN - 53
DO - http://doi.org/10.4208/jms.v53n2.20.01
UR - https://global-sci.org/intro/article_detail/jms/16801.html
KW - Nonlinear Dirac equation, nonrelativistic limit, error estimates.
AB - <p style="text-align: justify;">We consider&nbsp; 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 &#39;free Dirac operator&#39; 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.</p>