Jacobi Elliptic Solutions of $\lambda\phi^4$ Theory in a Finite Domain

TL;DR

This paper derives and analyzes Jacobi elliptic function solutions for a scalar field in a finite domain, establishing boundary conditions, uniqueness, and physical quantities like energy and charge, with implications for kink solutions.

Contribution

It provides a comprehensive analysis of Jacobi elliptic solutions to the $ ext{}\lambda ext{-} ext{phi}^4$ theory in finite domains, including boundary conditions, uniqueness, and physical properties.

Findings

Elliptic cn-type solutions satisfy vacuum boundary conditions in finite domains.

Uniqueness of elliptic sn-type solutions with Dirichlet boundary conditions.

Existence of a minimal mass for solutions in a finite box.

Abstract

The general static solutions of the scalar field equation for the potential $V(\phi)= -1/2 M^2\phi^2 + \lambda/4 \phi^4$ are determined for a finite domain in $(1+1)$ dimensional space-time. A family of real solutions is described in terms of Jacobi Elliptic Functions. We show that the vacuum-vacuum boundary conditions can be reached by elliptic cn-type solutions in a finite domain, such as of the Kink, for which they are imposed at infinity. We proved uniqueness for elliptic sn-type solutions satisfying Dirichlet boundary conditions in a finite interval (box) as well the existence of a minimal mass corresponding to these solutions in a box. We define expressions for the ``topological charge'', ``total energy'' (or classical mass) and ``energy density'' for elliptic sn-type solutions in a finite domain. For large length of the box the conserved charge, classical mass and energy density of the Kink are recovered. Also, we have shown that using periodic boundary conditions the results are the same as in the case of Dirichlet boundary conditions. In the case of anti-periodic boundary conditions all elliptic sn-type solutions are allowed.