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Peregrine soliton as a limiting behavior of the Kuznetsov-Ma and Akhmediev breathers

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