  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 MI region and stability region in low perturbation frequency region.
In particular, the link between the MI growth rate and transition
characteristic analytically demonstrates that, the size characteristic of
transition is positively associated with the reciprocal of zero-frequency
growth rate. Further, we investigate the case for nonlinear interplay of
multi-localized waves. It is interesting that the interaction of second-order
waves in stability region features a line structure, rather than an elastic
interaction between two w-shaped traveling waves.
