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We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension withscalar-scalar self interaction $\frac{g^2}{k+1} ({\bar \Psi} \Psi)^{k+1}$, aswell as a vector-vector self interaction $\frac{g^2}{k+1} ({\bar \Psi}\gamma_\mu \Psi \bPsi \gamma^\mu \Psi)^{\frac{1}{2}(k+1)}$. We find the exactanalytic form for solitary waves for arbitrary $k$ and find that they are ageneralization of the exact solutions for the nonlinear Schr\"odinger 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 theNLSE, valid when $|\omega-m |\ll 2m$, where $\omega$ is the frequency of thesolitary wave in the rest frame. We discuss the stability and blowup ofsolitary waves assuming the modified NLSE is valid and find that they should bestable for $k < 2$.

Solitary wave in the Nonlinear Dirac Equation with arbitrary nonlinearity