Solitary wave in the Nonlinear Dirac Equation with arbitrary nonlinearity

We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{k+1} ({\bar Ψ} Ψ)^{k+1}$, as well as a vector-vector self interaction $\frac{g^2}{k+1} ({\bar Ψ} γ_μΨ\bPsi γ^μΨ)^{\frac{1}{2}(k+1)}$. We find the exact analytic form for solitary waves for arbitrary $k$ and find that they are a generalization of the exact solutions for the nonlinear Schrödinger equation (NLSE) and reduce to these solutions in a well defined nonrelativistic limit. We perform the nonrelativistic reduction and find the $1/2m$ correction to the NLSE, valid when $|ω-m |\ll 2m$, where $ω$ is the frequency of the solitary wave in the rest frame. We discuss the stability and blowup of solitary waves assuming the modified NLSE is valid and find that they should be stable for $k < 2$.

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