Analytical solutions of differential-difference sine-Gordon equation

In modern textile engineering, non-linear differential-difference equations are often used to describe some phenomena arising in heat/electron conduction and flow in carbon nanotubes. In this paper, we extend the variable coefficient Jacobian elliptic function method to solve non-linear differential-difference sine-Gordon equation by introducing a negative power and some variable coefficients in the ansatz, and derive two series of Jacobian elliptic function solutions. When the modulus of Jacobian elliptic function approaches to 1, some solutions can degenerate into some known solutions in the literature.