Drifting Modulation Instability: Periodization of General Breathers, Asymmetric Sideband Modulation, and IST Spectral Portrait

We study drifting modulation instability (MI) that exhibits asymmetric discrete spectrum recurrences in the nonlinear stage within the framework of the focusing nonlinear Schrödinger equation. We demonstrate that such drifting MI can arise either from the local periodization of general breathers or from numerical simulations starting with an initial modulation containing asymmetric sidebands. In particular, we identify a defining characteristic of drifting MI—namely, its correspondence to two distinct asymmetric bands (Bands I and II) in the inverse scattering transform (IST) spectrum. We show that the relative phase between sidebands does not effect the IST spectral portrait but does alter the characteristic quantities (i.e., amplitude peak, group velocity, and oscillating period) of the drifting MI. A nontrivial result is that we provide a method to identify the valid general breather from the IST spectrum bands whose characteristic quantities match those observed in drifting MI patterns.