Geometric-stair-shaped super-regular breathers induced by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric nonlinearity

We numerically discover an alternative type of super-regular breather (SRB) that exhibits an asymmetric geometric stair structure in its spatiotemporal amplitude distribution. This kind of geometric-stair-shaped SRB is induced by the $\mathcal{PT}$-symmetric nonlocal nonlinearity and is effectively excited by localized periodic perturbations on a plane wave. By adjusting the amplitude and phase differences of the initial perturbations, we can control the direction and rate of its ascent or descent. Interestingly, due to the presence of $\mathcal{PT}$-symmetric nonlinearity, applying a spatial offset to the initial wave can excite an SRB with a significantly larger peak value, which can reach over 100 times the amplitude of the plane wave. Our results provide a method for exciting SRBs with spatial asymmetry and can offer different insights for the experimental observation of extremely asymmetric modulation instability phenomena.