The Rise of Solitons in Sine-Gordon Field Theory: From Jacobi Amplitude to Gudermannian Function

TL;DR

This paper explores how classical sine-Gordon soliton solutions can be derived using elliptic functions, specifically the Jacobi amplitude, linking soliton theory with elliptic integral functions.

Contribution

It introduces a novel expression of sine-Gordon solitons in terms of the Jacobi amplitude, connecting elliptic functions with soliton solutions.

Findings

Solitons can be represented via Jacobi amplitude functions.

The approach generalizes traditional sine-Gordon solutions.

Elliptic functions provide a new perspective on soliton structures.

Abstract

We show how the famous soliton solution of the classical sine-Gordon field theory in $(1+1)$-dimensions may be obtained as a particular case of a solution expressed in terms of the Jacobi amplitude, which is the inverse function of the incomplete elliptic integral of the first kind.