Exact solutions for a complex scalar field under discrete symmetry

Abstract

We report on the presence of families of exact solutions for a complex scalar field that behaves according to the rules of discrete $Z_N$ symmetry. Since the family of models is exactly solved, the results appear to be of interest to integrability, to build junctions and networks of localized structures and to describe scalar dark matter in high energy physics.

Figures (3)

• Figure 1: The phase $\Theta(x)$\ref{['theta']} for $N=3$ (top), $4$ (middle top), $5$ (middle bottom) and $6$ (bottom), with $n=0$, $x_0=0$ and $\alpha=0.1$ (red), $0.3$ (orange) $0.9$ (blue).
• Figure 2: The solutions \ref{['gsol']} for $N=3,4,5,6$, with $n=0,1,\dots, N-1$, $x_0 = 0$ and $\alpha=1$. We used $n=0$ (red), $1$ (orange), $2$ (yellow), $3$ (green), $4$ (blue), and $5$ (purple).
• Figure 3: The energy density \ref{['rhox']} for $N=3,4,5,6$, with $n=0$, $x_0=0$ and $\alpha=0.2$ (red), $0.4$ (orange), $0.6$ (yellow) and $0.8$ (green).