In this article we obtain generalized semi-discrete short pulse equation through discretization of the associated linear eigenvalue problem of the short pulse equation. Through implication of different symmetry reductions on the generalized semi-discrete short pulse equation, new integrable equations are assembled such as complex and the $\mathcal{PT}$-symmetric nonlocal semi-discrete short pulse equation. Darboux transformation is applied to construct multi-soliton solutions and we obtain a generalized formula to compute higher-order nontrivial solutions of the generalized semi-discrete short pulse equation. To demonstrate the dynamics of solutions, explicit expressions of nontrivial solutions up-to second-order are investigated for the $\mathcal{PT}$-symmetric nonlocal semi-discrete short pulse equation. It is interesting to add that at every stage the solutions obtained in this work are reduced to already obtained results of reverse space–time $\mathcal{PT}$-symmetric short pulse equation under continuum limit. Interactions between loops of attractive and repulsive types and first-order breather solution under local and nonlocal reductions are illustrated first time in this article for local and $\mathcal{PT}$-symmetric nonlocal semi-discrete short pulse equation.

在本文中，我们通过离散化短脉冲方程的线性特征值问题，获得了广义的半离散短脉冲方程。通过将不同的对称性折减包含在广义半离散短脉冲方程上，组合了新的可积方程，例如复数方程和$\mathcal{PT}$对称非局部半离散短脉冲方程 运用Darboux变换构造多孤子解，我们得到了一个广义公式来计算广义半离散短脉冲方程的高阶非平凡解。为了证明解的动力学，研究了非平凡解的表达式，直到二阶。$\mathcal{PT}$对称非局部半离散短脉冲方程 有趣的是，在每个阶段，在这项工作中获得的解决方案都简化为已经获得的反向时空结果。$\mathcal{PT}$连续极限下的非对称短脉冲方程。本文首次说明了局部和非局部减少下有吸引力和排斥类型的回路与一阶呼吸解之间的相互作用。$\mathcal{PT}$对称非局部半离散短脉冲方程