Nonlinear stage of modulation instability in the Kundu–Eckhaus equation: excitation of super-regular breathers

Abstract

We investigate the nonlinear stage of modulation instability (MI) in the Kundu–Eckhaus (KE) equation, specifically focusing on the complex dynamics triggered by localized perturbations upon a continuous-wave background. By utilizing the inverse scattering transform, we quantitatively relate these perturbations to a non-empty discrete spectrum associated with the generation of super-regular breathers (SRBs). In stark contrast to the symmetric evolution observed in the standard nonlinear Schrödinger (NLS) equation, the presence of higher-order nonlinear effects inherently breaks the spatial parity symmetry of the wave evolution. This symmetry breaking induces a pronounced and collective spatial tilt in both the diverging SRBs and the internal oscillation structures. Based on the MI growth of the localized seed, we derive an analytical estimate for the SRB boundary velocities. We show that the SRB boundary velocities consist of a common parameter-dependent shift, which causes the collective tilt, and opposite MI-driven components that govern the relative separation of the two SRB branches. The analytical predictions are in good agreement with direct numerical simulations. These findings provide a clear physical understanding of how higher-order nonlinearities govern the asymmetric wave evolution during the nonlinear stage of MI.