PT-symmetric ring-like superposed KM solitons in 2D PT coupled waveguide

Employing a ring approximation and the improved Darboux transformation, vector ring-like PT-symmetric and anti-symmetric solutions are derived in a two-dimensional coupled waveguide system with balanced gain and loss. This family of solutions encompasses the ring-like KM soliton, its superposed second-order form, and a “3-1” type superposed structure. A comparison between the real and imaginary parts of PT-symmetric and anti-symmetric solutions is presented. These hierarchical solutions are characterized by radial (r-direction) localization and periodic propagation axial (z-direction) recurrence of their rogue wave constituents. In three-dimensional (xyz) space, they manifest as localized rings in the transverse (xy) plane, which recur periodically along the propagation (z) axis, forming single or double-layered structures. The ring’s spatial extent amplifies with increasing radius parameter R (constant thickness parameter T), while its thickness grows with larger T (constant R).