In this article, we use the unified transform method to analyze the initial-boundary value problem for the coupled higher-order nonlinear Schrödinger equations on the half-line. Suppose that the solution $\{q_1(x,t),q_2(x,t)\}$ exists, we show that it can be expressed in terms of the unique solution of a matrix Riemann–Hilbert problem formulated in the plane of the complex spectral parameter $\lambda$ .

In this paper, based on gauge transformation of Lax pairs, we construct an N-fold Darboux transformation for the discrete Ragnisco–Tu equation which is a typical member in the Ragnisco–Tu hierarchy. By using the N-fold Darboux transformation, new multi-soliton solutions of the discrete Ragnisco–Tu equation can be directly constructed starting from a seed solution.

By considering a new discrete isospectral eigenvalue problem, a hierarchy of integrable positive and negative lattice models is derived. It is shown that they correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. And the equation in the resulting hierarchy is integrable in Liouville sense. Further, a Darboux transformation is established for the typical equations by using gauge transformations of Lax pairs, from which the exact solutions are given.