Approximate Analytic Solutions to Coupled Nonlinear Dirac Equations

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 $x \rightarrow \pm \infty$.