State transition induced by higher-order effects and background frequency

The state transition between the Peregrine rogue wave and W-shaped traveling wave induced by higher-order effects and background frequency is studied. We find that this intriguing transition, described by an exact explicit rational solution, is consistent with the modulation instability (MI) analysis that involves a MI region and a stability region in a low perturbation frequency region. In particular, the link between the MI growth rate and the transition characteristic analytically demonstrates that the localization characteristic of transition is positively associated with the reciprocal of the zero-frequency growth rate. Furthermore, we investigate the case for nonlinear interplay of multilocalized waves. It is interesting that the interaction of second-order waves in the stability region features a line structure rather than an elastic interaction between two W-shaped traveling waves.