One-Parameter Family of 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).

Figures (7)

• Figure 1: Potential in Eq. (1) for three values of the elliptic function modulus $m$. Note that for $m >1/2$ the minima and maxima are split into two each (green curve)
• Figure 2: Kink solution given by Eq. (25) for $m=1/4$.
• Figure 3: Potential given by Eq. (28) for $m=1/4$.
• Figure 4: Kink solution given by Eq. (32) for $m=1/2$.
• Figure 5: Potential given by Eq. (33) for $m=1/2$.