Discrete second-order Ablowitz–Kaup–Newell–Segur equation and its modified form

By introducing shift relations satisfied by a matrix $\boldsymbol{r}$, we propose a generalized Cauchy matrix scheme and construct a discrete second-order Ablowitz–Kaup–Newell–Segur equation. A modified form of this equation is given. By applying an appropriate skew continuum limit, we obtain the semi-discrete counterparts of these two discrete equations; in the full continuum limit, we derive continuous nonlinear equations. Solutions, including soliton solutions, Jordan-block solutions, and mixed solutions, of the resulting discrete, semi-discrete, and continuous Ablowitz–Kaup–Newell–Segur-type equations are presented. The reductions to discrete, semi-discrete, and continuous nonlinear Schrödinger equations and modified nonlinear Schrödinger equation are also discussed.