Approximate analytic solutions to coupled nonlinear Dirac equations

We consider the coupled nonlinear Dirac equations (NLDEs) in 1+1 dimensions with scalar-scalar self-interactions g122(ψ¯ψ)2+g222(ϕ¯ϕ)2+g32(ψ¯ψ)(ϕ¯ϕ) as well as vector-vector interactions of the form g122(ψ¯γμψ)(ψ¯γμψ)+g222(ϕ¯γμϕ)(ϕ¯γμϕ)+g32(ψ¯γμψ)(ϕ¯γμϕ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e−iω1t{R1cos⁡θ,R1sin⁡θ}, ϕ=e−iω2t{R2cos⁡η,R2sin⁡η}, and assuming that θ(x),η(x) have the same functional form they had when g3=0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g32/g22 and g32/g12. 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→±∞.