Peregrine Soliton as a Limiting Behavior of the Kuznetsov-Ma and Akhmediev Breathers

This article discusses a limiting behavior of breather solutions the focusing nonlinear Schrödinger equation. These breathers belong to family solitons on non-vanishing and constant background, where continuous-wave envelope serves as pedestal. The rational Peregrine soliton acts other two solitons, i.e., Kuznetsov-Ma Akhmediev soliton. Albeit with phase shift, latter becomes extension homoclinic orbit waveform corresponding an unstable mode in modulational instability phenomenon. All are prototypes for rogue waves dispersive media. We present rigorous proof using ϵ-δ argument show visualization this behavior.