Approximate Analytic Solutions to Coupled Nonlinear Dirac Equations

TL;DR

This paper develops approximate analytic solutions for coupled nonlinear Dirac equations in 1+1 dimensions, exploring scalar and vector interactions, and shows their reduction to coupled nonlinear Schrödinger equations in the nonrelativistic limit.

Contribution

It introduces an approximation method for solving coupled nonlinear Dirac equations with scalar and vector interactions, valid for small coupling ratios, and connects these solutions to nonlinear Schrödinger equations.

Findings

Derived approximate solutions for coupled NLDEs with scalar and vector interactions.

Showed reduction to coupled nonlinear Schrödinger equations in the nonrelativistic limit.

Identified exact pulse solutions for the reduced equations.

Abstract

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalar-scalar self interactions $\frac{ g_1^2}{2} ( {\bpsi} \psi)^2 + \frac{ g_2^2}{2} ( {\bphi} \phi)^2 + g_3^2 ({\bpsi} \psi) ( {\bphi} \phi)$ as well as vector-vector interactions of the form $\frac{g_1^2 }{2} (\bpsi \gamma_{\mu} \psi)(\bpsi \gamma^{\mu} \psi)+ \frac{g_2^2 }{2} (\bphi \gamma_{\mu} \phi)(\bphi \gamma^{\mu} \phi) + g_3^2 (\bpsi \gamma_{\mu} \psi)(\bphi \gamma^{\mu} \phi ). $ Writing the two components of the assumed solitary wave solution of these equations in the form $\psi = e^{-i \omega_1 t} \{R_1 \cos \theta, R_1 \sin \theta \}$, $\phi = e^{-i \omega_2 t} \{R_2 \cos \eta, R_2\sin \eta \}$, and assuming that $ \theta(x),\eta(x)$ have the {\it same} functional form they had when $g_3$=0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for $R_i(x)$ which are valid for small values of $g_3^2/ g_2^2 $ and $g_3^2/ g_1^2$. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schr\"odinger equation for which we obtain two exact pulse solutions vanishing at $x \rightarrow \pm \infty$.