Volume 53, Issue 2
(Semi-)Nonrelativisitic Limit of the Nonlinear Dirac Equations

Yongyong Cai & Yan Wang

J. Math. Study, 53 (2020), pp. 125-142.

Published online: 2020-05

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  • 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.

  • AMS Subject Headings

35Q41, 35Q55, 81Q05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

yongyong.cai@bnu.edu.cn (Yongyong Cai)

wang.yan@mail.ccnu.edu.cn (Yan Wang)

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@Article{JMS-53-125, author = {Cai , Yongyong and Wang , Yan}, title = {(Semi-)Nonrelativisitic Limit of the Nonlinear Dirac Equations}, journal = {Journal of Mathematical Study}, year = {2020}, volume = {53}, number = {2}, pages = {125--142}, 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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v53n2.20.01}, url = {http://global-sci.org/intro/article_detail/jms/16801.html} }
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 -

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.

Yongyong Cai & Yan Wang. (2020). (Semi-)Nonrelativisitic Limit of the Nonlinear Dirac Equations. Journal of Mathematical Study. 53 (2). 125-142. doi:10.4208/jms.v53n2.20.01
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