A Generalized Sine-Gordon Equation: Reductions and Integrable Discretizations

In this paper, we propose fully discrete analogues of a generalized sine-Gordon (gsG) equation \(u_{t x}=\left( 1+\nu \partial _x^2\right) \sin u\). The key points of the construction are based on the bilinear discrete KP hierarchy and appropriate definition of discrete reciprocal transformations. We derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter limit \(b\rightarrow 0\). In particular, one fully discrete gsG equation is reduced to a semi-discrete gsG equation in the case of \(\nu =-1\) (Feng et al. in Numer Algorithms 94:351–370, 2023). Furthermore, N-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form are presented. Dynamics of one- and two-soliton solutions for the discrete gsG equations are analyzed. By introducing a parameter c, we demonstrate that the gsG equation can reduce to the sine-Gordon equation and the short pulse at the levels of continuous, semi-discrete and fully discrete cases. The limiting forms of the N-soliton solutions to the gsG equation in each level also correspond to those of the sine-Gordon equation and the short pulse equation.