In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Ling et al. 2016 Physica D327, 13–29 (doi:10.1016/j.physd.2016.03.012); Feng et al. 2016 Phys. Rev. E93, 052227 (doi:10.1103/PhysRevE.93.052227)).

在本文中，我们关注聚焦和散焦类型的半离散复式短脉冲（sdCSP）方程，该方程可被视为超短脉冲状态下Ablowitz-Ladik晶格的类似物。通过使用广义的Darboux变换方法，研究了该新可积分半离散方程在零和非零边界条件下的各种孤子解。具体而言，对于聚焦sdCSP方程，推导了多亮度解（零边界条件），多呼吸和高阶流浪波解（非零边界条件），而对于非聚焦sdCSP方程，在零边界条件下，构造了多暗孤子解。我们进一步证明，在连续极限内，等。2016物理学d 327，13-29（DOI：10.1016 / j.physd.2016.03.012）; 冯等人。2016物理 启ê 93，052227（DOI：10.1103 / PhysRevE.93.052227））。