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Approximate Analytic Solutions to Coupled Nonlinear Dirac Equations (1603.08043v1)

Published 25 Mar 2016 in nlin.PS

Abstract: We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalar-scalar self interactions $\frac{ g_12}{2} ( {\bpsi} \psi)2 + \frac{ g_22}{2} ( {\bphi} \phi)2 + g_32 ({\bpsi} \psi) ( {\bphi} \phi)$ as well as vector-vector interactions of the form $\frac{g_12 }{2} (\bpsi \gamma_{\mu} \psi)(\bpsi \gamma{\mu} \psi)+ \frac{g_22 }{2} (\bphi \gamma_{\mu} \phi)(\bphi \gamma{\mu} \phi) + g_32 (\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_32/ g_22 $ and $g_32/ g_12$. 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 $x \rightarrow \pm \infty$.

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