Two-Parameter Family of Nonlinear Dirac Equations With Scalar-Scalar plus Vector-Vector Interactions

RESUMO

We obtain exact solutions of the nonlinear Dirac equation in $1+1$ dimensions of the form $\psi(x,t) = e^{-i\omega t} \psi(x)$ for scalar-scalar (SS) plus vector-vector (VV) interaction with interaction Lagrangian given by $$L_{I} = \frac{g^2}{\kappa+1}[(\bar{\psi} \psi)^{\kappa+1} +\frac{1}{p} (\bar{\psi} \gamma_\mu \psi \bar{\psi} \gamma^{\mu} \psi)^{\kappa+1}]$$ where $p > 0$ but arbitrary otherwise. We look for solutions with $0 < \omega < m$ where $\omega, m$ are frequency and mass, respectively. We find solutions for all values of $\omega$ in this range. We compute the charge $Q$ and the energy $E$ for each of the solitary wave solutions and explore the region in the ($p, \kappa$) parameter space in which solitary wave bound states exist (i.e., for which $E/Q < m$). We show that for all the cases while both $E$ and $Q$ depend on the coupling constant $g$, their ratio $E/Q$ is independent of $g$. We further find that in case $p\kappa \le 1$, the charge density for all the solitary waves have only single hump while for $p \kappa > 1$ there is a transition from double to single hump and we determine it as a function of $\omega/m$. We notice that for all $p$ there is a transition at $\kappa=2$ in the behavior of $E/Q$ as a function of $\omega$ which we speculate is related to the onset of instability of the solutions at $\kappa=2$. We obtain the nonrelativistic reduction of the two-parameter family to a non-relativistic modified nonlinear Schrödinger equation (NLSE) and discuss the stability of the single hump olitary waves in the domain of validity of the modified NLSE.