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

Abstract

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 Bäcklund 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.

Appendix

Here we present the proof of Proposition 3. From the definition of ϕ_k,u_k and equations (3.18)–(3.19), we have

$$ \begin{array}{@{}rcl@{}} \frac{d\phi_{k}}{d\tau }& =&\mathrm{i}\left( \ln \frac{\bar{f}_{k}g_{k}}{f_{k} \bar{g}_{k}}\right)_{\tau } =\mathrm{i}\left( \ln \frac{\bar{f}_{k}}{\bar{g}_{k}}\right)_{\tau } -\mathrm{i}\left( \ln \frac{f_{k}}{g_{k}}\right)_{\tau }\\ & =&\frac{\mathrm{i}}{2}\left( \frac{\bar{f}_{k}\bar{g}_{k}}{f_{k}g_{k}}-\frac{ f_{k}g_{k}}{\bar{f}_{k}\bar{g}_{k}}\right) =\frac{\mathrm{i}}{2}\left( e^{-\mathrm{i}u_{k}}-e^{\mathrm{i}u_{k}}\right)\\ & =&\sin u_{k}. \end{array} $$

(A.1)

Therefore

$$ \begin{array}{@{}rcl@{}} \sin u_{k+1}-\sin u_{k} & =&\frac{d}{d\tau} (\phi_{k+1}-\phi_{k}) \\ &=&a \Big(\sin\Big(\frac{ \sigma_{k+1}+\sigma_{k}}{2}\Big) - \sin\Big(\frac{\sigma^{\prime}_{k+1}+\sigma^{\prime }_{k}}{2}\Big) \Big) \\ &=&2a\cos \Big(\frac{u_{k+1}+u_{k}}{2}\Big) \sin \Big(\frac{\phi_{k+1}+\phi_{k}}{2}\Big), \end{array} $$

(A.2)

which results in

$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!2 \cos \Big(\frac{u_{k+1} + u_{k}}{2}\Big) \sin \Big(\frac{u_{k+1} - u_{k}}{2}\Big) = 2a\cos \Big(\frac{u_{k+1} + u_{k}}{2}\Big) \sin \Big(\frac{\phi_{k+1} + \phi_{k}}{2}\Big)?? \end{array} $$

(A.3)

i.e.,

$$ a\sin \Big(\frac{\phi_{k+1}+\phi_{k}}{2}\Big)=\sin \Big(\frac{u_{k+1}-u_{k}}{2}\Big). $$

(A.4)

Thus, we complete the proof.