In this paper, we investigate a semi-discrete two-component integrable system on zigzag-runged ladder lattice and find a variety of two-component localized waves. First of all, we investigate the modulational instability to reveal the formation mechanism of rogue waves from three distinct plane-wave solutions for this system. Secondly, the discrete generalized \((m,N-m)\)-fold Darboux transformation is first constructed to solve this system. As applications, three kinds of plane-wave seed solutions are chosen to exhibit discrete two-component localized wave solutions including rogue wave solutions, breather solutions, periodic wave solutions and interaction solutions. Moreover, the dynamical behaviors of such localized wave solutions are discussed via numerical simulation. Finally, we summarize a few mathematical features to obtain various two-component localized waves through the discrete generalized\((m,N-m)\)-fold Darboux transformation. It is found that there are simultaneously bright–dark, dark–bright and bright–bright two-component rogue wave solutions in this discrete two-component system, and moreover, the shapes of bright–dark rogue waves in two components appear alternately with the increase of order number of Darboux transformation when some special seed solutions are selected, which have not been reported before. It is hoped that the results obtained in this paper may be used to understand some related physical phenomena.

在本文中，我们研究了锯齿形梯形格子上的半离散两分量可积系统，并发现了各种两分量局部波。首先，我们研究调制的不稳定性，以从该系统的三个不同的平面波解中揭示流氓波的形成机理。其次，离散广义\（（m，Nm）\）折叠式Darboux变换首先被构造来解决该系统。作为应用，选择了三种平面波种子解以显示离散的两分量局部波解，包括流浪解，呼吸解，周期波解和交互作用解。此外，通过数值模拟讨论了这种局部波解的动力学行为。最后，我们总结了一些数学特征，可通过离散广义\（（m，Nm）\）获得各种二分量局部波倍的Darboux变换。结果发现，在这个离散的两组分系统中，同时存在明暗，暗亮和明亮两组分流氓波解，而且，两个组分中的明暗流氓波的形状与交变交替出现。选择某些特殊的种子解决方案时，Darboux变换的阶数增加，以前没有报道过。希望本文获得的结果可用于了解一些相关的物理现象。