New static solutions of symmetric $\phi^4$ equation

TL;DR

This paper presents new exact static solutions to the symmetric $^4$ equation in 1+1 dimensions, using Jacobi elliptic functions, expanding the set of known solutions and categorizing them by potential parameters.

Contribution

It introduces several novel non-singular solutions to the symmetric $^4$ model, which are expressed in terms of Jacobi elliptic functions and classified by potential parameters.

Findings

New non-singular solutions in terms of Jacobi elliptic functions

Comparison with existing solutions

Categorization of solutions based on potential parameters

Abstract

In this paper, we provide new exact solutions of nonlinear Klein-Gordon ($\phi^4$) equation in $1+1$-dimension. For simplicity, we focus on the static equation and ignore the time-dependence. The symmetric $\phi^4$ equation has played an important role in several areas of physics. We obtain several novel non-singular solutions of the symmetric $\phi^4$ model in terms of the Jacobi elliptic functions and compare them with the well-known solutions. Finally, we categorize these solutions in terms of the potential parameters.