Abstract
This article discusses a limiting behavior of breather solutions of the focusing nonlinear Schrödinger equation. These breathers belong to the family 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.
Highlights
The study of wave phenomena traces its history back to the time of Pythagoras, research on nonlinear and rogue waves has attracted great scientific interest recently, both theoretically and experimentally
Theoretical, numerical, and experimental evidence of the dissipation effect on phase-shifted FPUT dynamics in a super wave tank, which is related to modulational instability, can be described by the breather solutions of the nonlinear Schrödinger (NLS) equation [164]
We present the limit of the real part of the Akhmediev soliton as ] → 0 is the real part of the Peregrine soliton, i.e., limRe qA(x, t)
Summary
The study of wave phenomena traces its history back to the time of Pythagoras, research on nonlinear and rogue waves has attracted great scientific interest recently, both theoretically and experimentally. The purpose of this article is to provide an overview of the relationship between these soliton solutions in this context It fills the gap in the details of limiting behavior. While the connection is well-known, the rigorous proof seems to be absent, and the visualizations found in the literature are incomplete. We will present this connection of the limiting behavior both analytically and visually. This introduction section covers a brief history of the NLS equation, exact solutions of the NLS equation, and a literature review on rogue waves
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