耦合非线性Dirac方程的近似解析解
Approximate Analytic Solutions to Coupled Nonlinear Dirac Equations
摘要：
考虑1+1维耦合非线性Dirac方程（NLDE's），方程中既有标量-标量自相互作用$\frac{ g_1^2}{2} ( {\bpsi} ψ)^2 + \frac{ g_2^2}{2} ( {\bphi} ϕ)^2 + g_3^2 ({\bpsi} ψ) ( {\bphi} ϕ)$，也有矢量-矢量相互作用$\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). $，将方程的孤立波解的两个分量写成$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\}$的形式，并假设$ θ(x),η(x)$具有当$g_3$ =0时的{ \it same}函数形式，这是一个近似符合守恒定律，我们然后找到近似的解析解$R_i(x)$是有效的小值的$g_3^2/ g_2^2 $和$g_3^2/ g_1^2$。在非相对论极限下，我们证明了这两个耦合模型都可以转化为同一个耦合非线性薛定谔方程，并得到了两个在$x \rightarrow \pm \infty$处消失的精确脉冲解。
英文摘要：
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 $x \rightarrow \pm \infty$.
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