Spontaneous symmetry breaking and \(\mathcal{P}\mathcal{T}\) -symmetry attracts the modern researcher due to its implementation in many fields such as microwave propagation, nonlinear optics. This article studies the \(\mathcal{P}\mathcal{T}\) -symmetric semi-discrete short pulse equation ( \(\mathcal{P}\mathcal{T}\) -sdSPE) that can be viewed as a cognate to the Ablowitz-Ladik lattice in the ultra-short-pulse regime. The Lax pair of the system is constructed and demonstrated that one can obtain a variety of new integrable models by symmetry reductions. Furthermore, quasi-grammian solutions of \(\mathcal{P}\mathcal{T}\) -sdSPE are presented using the binary Darboux transformation. Finally, as an explicit example, symmetry preserving and non-preserving grammians, rogue, breather and soliton solutions are celebrated.
