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 $\psi =e^{-i{\omega }_{1}t}\{R_1\cos \theta ,R_1\sin \theta \}$, $\varphi =e^{-i{\omega }_{2}t}\{R_2\cos \eta ,R_2\sin \eta \}$, 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 Schrödinger equation for which we obtain two exact pulse solutions vanishing at $x\to \pm \infty$.

11 pages, 10 figures

References in corpus (3)

• Optical analogue of relativistic Dirac solitons in binary waveguide arrays
• Solitary wave in the Nonlinear Dirac Equation with arbitrary nonlinearity
• Formal analogy between the Dirac equation in its Majorana form and the discrete-velocity version of the Boltzmann kinetic equation

Cited by in corpus (2)

• Study of instability of the Fourier split-step method for the massive Gross--Neveu model
• Solitary wave solutions of the 2+1 and 3+1 dimensional nonlinear Dirac equation constrained to planar and space curves