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

Abstract

In this paper, we investigate local and nonlocal reductions of a discrete negative order Ablowitz–Kaup–Newell–Segur equation. By the bilinearization reduction method, we construct exact solutions in double Casoratian form to the reduced nonlocal discrete sine‐Gordon equations. Then, nonlocal semidiscrete sine‐Gordon equations and their solutions are obtained through the continuum limits. The dynamics of soliton solutions are analyzed and illustrated by asymptotic analysis. The research ideas and methods in this paper can be generalized to other nonlocal discrete integrable systems.

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

“…In what follows, for the function F ∶= F(x 1 , x 2 ), we appoint that notation F 𝜎 means F 𝜎 = F(𝜎x 1 , 𝜎x 2 ), where 𝜎 = ±1 (cf. [42]). The notation F 𝜎 = F for 𝜎 = 1 should not be confused with the component F 1 in matrix.…”

“…For the proof of Theorem 2, one can refer to a similar one given in [42]. We call the two equations in (3.5) as matrix equations, in which the first one is the famous Sylvester equation (cf.…”

“…In what follows, for the function , we appoint that notation means , where $$$ \sigma =\pm 1 $$$ (cf. [42]). The notation for $$$ \sigma =1 $$$ should not be confused with the component in matrix.…”

“…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.…”

Integrable coupled bosonic massive Thirring model and its nonlocal reductions

“…The nonlocal NLSE and its various generalizations are considered in the literature [36][37][38][39][40][41][42]. Subsequent developments in the study of nonlocal integrable models lead to new classes of nonlocal derivative NLSE [43,44] and nonlocal sine-Gordon model as well [45]. The nonlocal reductions in case of multi-component generalized version of derivative NLSE and N -wave equations are also considered in the literature [46,47].…”