Integrable discretizations for a generalized sine-Gordon equation and the reductions to the sine-Gordon equation and the short pulse equation

TL;DR

This paper develops fully discrete and semi-discrete integrable analogues of a generalized sine-Gordon equation, constructs explicit soliton solutions, and explores reductions to the classical sine-Gordon and short pulse equations.

Contribution

It introduces new fully discrete and semi-discrete integrable models for the generalized sine-Gordon equation and derives their soliton solutions and reduction properties.

Findings

Constructed N-soliton solutions in determinant form.

Derived reductions to sine-Gordon and short pulse equations.

Analyzed soliton dynamics with numerical plots.

Abstract

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 bilinear equations of the discrete KP hierarchy and the proper definition of discrete hodograph transformations are the keys to the construction. Then we derive semi-discrete analogues of the gsG equation from the fully discrete gsG equation by taking the temporal parameter $b\rightarrow0$. Especially, one full-discrete gsG equation is reduced to a semi-discrete gsG equation in the case of $\nu=-1$ (Feng {\it et al. Numer. Algorithms} 2023). Furthermore, $N$-soliton solutions to the semi- and fully discrete analogues of the gsG equation in the determinant form are constructed. Dynamics of one- and two-soliton solutions for the discrete gsG equations are discussed with plots. We also investigate the reductions to the sine-Gordon (sG) equation and the short pulse (SP) equation. By introducing an important parameter $c$, we demonstrate that the gsG equation reduces to the sG equation and the SP equation, and the discrete gsG equation reduces to the discrete sG equation and the discrete SP equation, respectively, in the appropriate scaling limit. The limiting forms of the $N$-soliton solutions to the gsG equation also correspond to those of the sG equation and the SP equation.