Solitary wave in the Nonlinear Dirac Equation with arbitrary nonlinearity

TL;DR

This paper derives exact solitary wave solutions for nonlinear Dirac equations with arbitrary nonlinearity in 1+1 dimensions, connecting them to nonlinear Schrödinger solutions and analyzing their stability and corrections.

Contribution

It provides the first exact analytic solitary wave solutions for NLDE with arbitrary nonlinearity and explores their nonrelativistic limit and stability properties.

Findings

Exact solitary wave solutions for arbitrary nonlinearity k.

Reduction to nonlinear Schrödinger equation in nonrelativistic limit.

Stability analysis indicating stability for k<2.

Abstract

We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction $\frac{g^2}{k+1} ({\bar \Psi} \Psi)^{k+1}$, as well 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 exact analytic form for solitary waves for arbitrary $k$ and find that they are a generalization 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 the NLSE, valid when $|\omega-m |\ll 2m$, where $\omega$ 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 $k < 2$.