Nonlocal complex short pulse equation in $${\mathcal{P}\mathcal{T}}$$-symmetry like symmetry breaking, breather–grammian interactions and soliton solutions

Research on [Formula: see text]-symmetry and spontaneous symmetry breaking captivates contemporary scholars due to its extensive applicability in several fields, including microwave propagation and nonlinear optics. This article studies the nonlocal complex short pulse (NL-CSP) equation in which we discuss how under certain symmetry reduction general complex short pulse equation turns into NL-CSP equation. We construct the binary Darboux transformation for the reverse space-time NL-CSP equation and derive its quasi-grammian solutions. Further, we obtain explicit expressions for spontaneous symmetry-breaking and symmetry-preserving breather, interaction of breather with grammian and also the soliton solutions. It is concluded that the existence of both symmetry-breaking and symmetry-preserving solutions for NL-CSP equation. Finally, to verify the theoretical results, we illustrate the dynamics of these solutions using surface and contour plots.