One-Parameter Family of Elliptic Sine-Gordon Equations

TL;DR

This paper introduces a one-parameter family of elliptic sine-Gordon equations, analyzing their properties and solutions, bridging the gap between sine-Gordon and sine hyperbolic-Gordon equations as the parameter varies.

Contribution

It presents a novel continuous family of elliptic sine-Gordon equations parameterized by modulus m, connecting two well-known integrable equations.

Findings

Derived kink solutions for various modulus values

Showed the transition from sine-Gordon to sine hyperbolic-Gordon equations as m varies

Analyzed properties of the elliptic sine-Gordon equations

Abstract

We introduce a continuous one-parameter family of elliptic sine-Gordon equations (SGE) characterized by the modulus $0 \le m \le 1$ of Jacobi elliptic functions and analyze some of its properties and obtain its kink solution for various values of modulus $m$. These elliptic SGE have the novel property that while in the limit $m = 0$ they go over to the integrable sine-Gordon equation, in the $m = 1$ limit they go over to the integrable sine hyperbolic-Gordon equations (SHGE).