Chaotic ring-shaped soliton structures in the nonlinear refractive index associated with the Milnor polynomial in an atomic medium

This manuscript investigates the coherent control of chaotic ring-shaped soliton structures and examines the nonlinear refractive index associated with the Milnor polynomial in an atomic medium. The slow and fast light phenomena of the ring-shaped soliton involve self-reinforcing wave packets that preserve a circular or necklace-like structure during propagation at the group velocity. Slow light ring-shaped soliton ( v g < c ) are observed in the region of normal dispersion, where the group velocity of a wave packet is less than the speed of light in a vacuum. In addition to fast light ring-shaped soliton ( v g > c , v g < 0 ) , the anomalous dispersion region is characterized by group velocities that exceed the speed of light in vacuum or are negative. Fast light ring-shaped solitons commonly occur in microring resonators due to the cross-kerr non-linearity, which arises from optical susceptibility. The Milnor polynomial is examined in the context of nonlinear dispersion and absorption spectra by applying coupling fields within the atomic medium. The density matrix formalism is employed to calculate the probe field and its associated optical susceptibility, as well as to analyze nonlinear phenomena using the Milnor polynomial. Linear and non-linear probe fields are designed, cross-Kerr non-linearity is implemented using four-wave mixing, and dispersion and absorption are managed through the Milnor polynomial. Slow and fast light ring-shaped solitons are used in optical communication buffers and delays, optical switching, advanced telecommunications, filtering, soliton-based cryptography, security, and quantum information processing. We emphasize that the proposed model enables controllable formation and manipulation of complex structured solitons, demonstrates the influence of the Milnor polynomial on the nonlinear refractive index and beam dynamics, and provides a theoretical framework for future applications in structured-light engineering, nonlinear photonics, and quantum optical systems.