Figure 2

The stability map for the \({\mathscr{P}}{\mathscr{T}}\)-symmetric solitons maintained by imaginary potential (3), in the case of \(\sigma =1\) and \(\beta =0\) in Eqs (3) and (9). Stable fundamental single-peak solitons are marked by green dots. All unstable solitons are marked by red crosses, irrespective of their structure. Exact soliton solutions, given by Eqs (19) and (20), are indicated by green stars (except for one at \({\gamma }_{0}=2\), which is designated by the red cross, as the exact solutions are unstable at \({\gamma }_{0}\ge 2\)). Green numbers ≥2 in this figure and below denote stable solitons with the same number of peaks. Further, green numbers 1 label stable single-peak solitons with the higher-order radial structure, as in Fig. 1(b). Green numbers 1 or 2, placed close to green dots, imply bistability, i.e., coexistence of stable fundamental single-peak solitons and stable higher-order or double-peak ones. Red crosses placed on top of green dots imply coexistence of fundamental single-peak solitons with some unstable mode. Soliton solutions were not found in white areas.