A generalized Sylvester equation and discrete Ablowitz–Kaup–Newell–Segur type equations

A generalized Sylvester equation is introduced to revisit the Cauchy matrix schemes of the discrete negative-order Ablowitz–Kaup–Newell–Segur (AKNS) equation and the discrete third-order AKNS equation. Starting from the generalized Sylvester equation, we introduce a master function $\boldsymbol{S}^{(i,j)}$ that admits a recurrence relation under a constraint relation. By imposing the shifts on matrices $\boldsymbol{r}$ and $\,^\mathrm{t}\! {\boldsymbol{s}}$, the shifts of the master function $\boldsymbol{S}^{(i,j)}$ are derived. By introducing the dependent variables, the above two discrete AKNS equations are constructed as closed forms. For two different choices of the coefficient matrices in the Sylvester equation that preserve the constraint condition, exact solutions in asymmetric and symmetric cases are presented, with one-soliton, two-soliton, and the simplest Jordan-block solutions given explicitly. Continuum limits to the semidiscrete and continuous AKNS-type equations as well as the corresponding exact solutions are also discussed.