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 in the frame of the focusing nonlinear Schrödinger equation. We demonstrate that such drifting MI can be achieved by using either the local periodization process of general breathers or numerical simulations starting from initial modulation with asymmetric sidebands. In particular, we identify a defining characteristic of drifting MI—namely, its correspondence to two distinct asymmetric bands (Band I and Band II) in the inverse scattering transform (IST) spectrum. We show that the relative phase between the sidebands has no effect on the IST spectral portrait but changes the characteristic quantities (amplitude peak, group velocity, and oscillating period) of the drifting MI. A nontrivial result is that we provide a way to identify the valid general breather from the IST spectrum bands whose characteristic quantities match those observed in drifting MI patterns.