Figure 4

The evolution of solitons with different propagation constants under weak perturbations is shown. Panel (a) plots the maximum amplitude of solitons during propagation, where the red line corresponds to solitons with \(\beta = 0.2\) and the blue line to solitons with \(\beta = -0.2\). Panels (b) through (e) provide snapshots of solitons at different propagation distances (\(z = 100\) for panels (b) and (c), \(z = 250\) for panels (d) and (e) for the four cases with \(\beta = 0.2\) and \(\beta = -0.2\) under commensurate (\(\theta = \arctan (3/4)\)) and incommensurate (\(\theta = \pi /6\)) conditions. The figures show that stable solitons retain their shape and peak amplitude during propagation, while unstable solitons break up, forming multiple peaks and failing to preserve their initial profiles.