Integrable semi-discretizations and self-adaptive moving mesh method for a generalized sine-Gordon equation

In the present paper, two integrable and one non-integrable semi-discrete analogues of a generalized sine-Gordon (sG) equation are constructed. The keys of the construction are the Backlund transformation of bilinear equations and appropriate definition of the discrete hodograph transformation. We construct N-soliton solutions for the semi-discrete analogues of the generalized sG equation in the determinant form. In the continuous limit, we show that the semi-discrete generalized sG equations converge to the continuous generalized sG equation. Furthermore, we propose four self-adaptive moving mesh methods for the generalized sG equation, two are integrable and two are non-integrable. Integrable and non-integrable self-adaptive moving mesh methods are proposed and used for simulations of regular, irregular and loop soliton while comparing with the Crank-Nicolson (C-N) scheme. The numerical solutions show that the proposed self-adaptive moving methods perform better than the C-N scheme.