The phase patterns of higher-order rogue waves

We investigate the phase profiles of rogue wave solutions to the nonlinear Schrödinger equation, all produced via the Darboux transformation scheme. We focus specifically on the second-order rogue wave, in both origin-centred and fissioned form, and extrapolate the results for higher-order structures. In particular, a rogue wave solution of order n can be decomposed into n(n + 1)/2 Peregrine breathers, and each peak applies an additive phase shift of 2π to the underlying plane wave background. Yet it is evident that no evolution path can be phase shifted beyond 2πn. We show that a fused rogue wave arranges its components to avoid any contradiction in this matter. We also show that the phase profile for any structure in the rogue wave hierarchy can be determined by examining phase bifurcations marked by zero-amplitude troughs