PT-symmetry, interaction-induced PT-symmetry breaking, and phase scattering in multi-ring solitons with balanced gain–loss waveguides

The propagation of light in parity-time (PT) symmetric waveguides with balanced gain and loss reveals unexpected nonlinear structures. Vector ring solitons are investigated in a two-dimensional PT-symmetric coupled waveguide system with balanced gain and loss. Through reduction to invariant manifolds and azimuthal averaging, the dynamics collapse to an integrable radial nonlinear Schrödinger equation for which we construct a Lax pair and apply an improved Darboux transformation. Three seed backgrounds: zero, Bessel, and plane wave, yield distinct ring soliton families: bright rings, hybrid linear–nonlinear rings, and dark rings with Peregrine-type rogue waves. A Galilean transformation preserving the Lax pair structure gives each ring an independent velocity, enabling multi-ring collisions with exact scattering phase shifts. The central finding is interaction-induced PT-symmetry breaking: the critical gain–loss threshold for two-ring configurations drops below the single-ring value of unity, governed by an analytical formula in terms of ring separation and width. Linear stability analysis via Pöschl–Teller potentials proves that one-ring solutions are spectrally stable with purely imaginary eigenvalues, confirmed by direct numerical evaluation. These results provide a unified framework for ring solitons in non-Hermitian waveguides with rigorous stability criteria and testable experimental predictions.