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

Abstract

This article discusses a limiting behavior of breather solutions of the focusing nonlinear Schr ödinger (NLS) equation. These breathers belong to the families of solitons on a nonvanishing 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 ǫ-δ argument and show the corresponding visualization for this limiting behavior.

On spatial Fourier spectrum of rogue wave breathers

“…Proof. By taking b = 1 and n = 0 in Lemma 4, that is, Equation (19), we obtain the result immediately.…”

“…A visual exploration of the limiting behavior of these NLS breathers has been discussed extensively, where the trajectories of breather dynamics are depicted in the complex plane for various parameter values. 19 Although the discussion of these breathers in the spatial-temporal domain has reached maturity, this article fills a gap in covering their features in the wavenumber-temporal domain by observing their spectra. Note that this "spectrum" should not be confused with the spectrum from the IST but rather the physical spectrum that can be calculated by switching from the space domain into the wavenumber domain by means of the spatial Fourier transform.…”

Fourier spectrum and related characteristics of the fundamental bright soliton solution

“…The former exists in the anomalous dispersion regime, modeled by the focusing NLS equation ( 1), whereas the latter occurs under the normal dispersion regime, which is governed by the defocusing NLS equation [15,16]. This fundamental bright soliton solution occurs as limiting cases of a family of stationary periodic wave solutions, another family of periodic solutions, in which both involve Jacobi elliptic functions, and the Kuznetsov-Ma breather family [17][18][19][20]. Although the possibility for the formation of the bright soliton was suggested as early as 1973, it was not until 1980 that its appearance was observed experimentally in optical fibers [21][22][23][24].…”

Theoretical and numerical evidence for the potential realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates

“…Rogue waves are extreme wave events that emerge out of nowhere and disappear without a trace [3,4]. Under appropriate approximations, they can be mathematically described by solutions of the nonlinear Schrödinger equation (NLS) [5][6][7][8][9]. This mathematical description of rogue waves allowed to extrapolate these phenomena to a large variety of nonlinear physical systems, other than oceanic waves, ranging from nonlinear optics [10][11][12][13] to plasmas [14][15][16], and from liquid helium [17] to Bose-Einstein condensates (BECs) [18,19] (see also the reviews of Refs.…”