Local and nonlocal complex discrete sine-Gordon equation. Solutions and continuum limits

Solutions and continuum limits to nonlocal discrete sine‐Gordon equations: Bilinearization reduction method

“…It is shown that the independent variables of the short pulse models and sG equation that are connected via hodograph transformation are covariant in nonlocal reductions. This gives us the motivation to consider the hodograph transformations between the discrete short pulse models 55 and the discrete sG equations, 36,38 as well as their nonlocal cases.…”

“…What is more, we have shown that the bilinearization reduction method is valid in the study of nonlocal discrete integrable systems. Compared with the Cauchy matrix reduction approach, 36 there is a great advantage of the bilinearization reduction method. The latter method allows one to construct exact solutions of the real reduced equations.…”

“…While in the Cauchy matrix reduction scheme, one cannot achieve that. This is because, in the Cauchy matrix reduction approach, solutions of the original before‐reduction AKNS system should satisfy two Sylvester equations 36 . In the real reduction case, the solvability of these two Sylvester equations usually conflicts with the solvability of the Sylvester equation in the matrix equation set (for more detailed explanations, one can refer to the conclusions in Ref.…”

“…Quite recently, nonlocal complex reduction of a discrete negative order Ablowitz–Kaup–Newell–Segur (AKNS) equation was studied in Ref. 36, where a nonlocal complex discrete sG equation, as well as its exact solutions in Cauchy matrix type, was constructed. We refer the reader to the recent survey 37 and the references therein for more details on the Cauchy matrix solutions of nonlocal complex integrable equations.…”

Solutions and continuum limits to nonlocal discrete modified Korteweg‐de Vries equations

“…In [40], two types of nonlocal H1 equations were revealed, including a reverse‐ $$$ \left(n,m\right) $$$ nonlocal H1 equation and a reverse‐ $$$ n $$$ nonlocal H1 equation. Recently, Xiang et al have systematically investigated exact solutions and continuum limits of the nonlocal discrete sine‐Gordon equations [41, 42], by utilizing the Cauchy matrix reduction approach [43] (see also [22]) and the bilinearization reduction scheme [44, 45], respectively. Compared with the Cauchy matrix reduction approach, the bilinearization reduction scheme has its own preponderance, since it can be used to the survey of real nonlocal integrable systems.…”