Self-similarity for V-shaped field potentials - further examples

TL;DR

This paper explores new models with V-shaped potentials in scalar fields across different dimensions, discovering explicit self-similar solutions that illustrate phenomena like string freezing in static states.

Contribution

It introduces three novel scalar field models with V-shaped potentials and provides explicit self-similar solutions demonstrating interesting dynamical behaviors.

Findings

Explicit self-similar solutions found for each model

Models exhibit phenomena like string freezing

Solutions describe dynamic processes in scalar fields

Abstract

Three new models with V-shaped field potentials $U$ are considered: a complex scalar field $X$ in 1+1 dimensions with $U(X)= |X|$, a real scalar field $\Phi$ in 2+1 dimensions with $U(\Phi) = |\Phi|$, and a real scalar field $\phi$ in 1+1 dimensions with $U{\phi) = \phi \Theta(\phi)$ where $\Theta$ is the step function. Several explicit, self-similar solutions are found. They describe interesting dynamical processes, for example, `freezing' a string in a static configuration.