Approximate Analytic Solutions to Coupled Nonlinear Dirac Equations

Abstract

We consider the coupled nonlinear Dirac equations (NLDE's) in 1+1 dimensions with scalar-scalar self interactions $\frac{ g_1^2}{2} ( {\bpsi} ψ)^2 + \frac{ g_2^2}{2} ( {\bphi} ϕ)^2 + g_3^2 ({\bpsi} ψ) ( {\bphi} ϕ)$ as well as vector-vector interactions of the form $\frac{g_1^2 }{2} (\bpsi γ_μ ψ)(\bpsi γ^μ ψ)+ \frac{g_2^2 }{2} (\bphi γ_μ ϕ)(\bphi γ^μ ϕ) + g_3^2 (\bpsi γ_μ ψ)(\bphi γ^μ ϕ). $ Writing the two components of the assumed solitary wave solution of these equations in the form $ψ= e^{-i ω_1 t} \{R_1 \cos θ, R_1 \sin θ\}$, $ϕ= e^{-i ω_2 t} \{R_2 \cos η, R_2\sin η\}$, and assuming that $ θ(x),η(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 Schrödinger equation for which we obtain two exact pulse solutions vanishing at $x \rightarrow \pm \infty$.