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 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.

1 Introduction

Although 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. In particular, the focusing nonlinear Schrödinger (NLS) equation and its exact analytical solutions that belong to the family of soliton on constant background have been adopted as models and prototypes for rogue wave phenomena. The purpose of this article is to provide an overview of the relationship between these soliton solutions in this context. It also 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.

1.1 A Brief Historical Background of the Nonlinear Schrödinger Equation

The NLS equation is a nonlinear evolution equation that models slowly varying envelope dynamics of a weakly nonlinear quasi-monochromatic wave packet in dispersive media. The model has an infinite set of conservation laws and belongs to a completely integrable system of nonlinear partial differential equations. It has a wide range of applications in various physical settings, such as surface water waves, nonlinear optics, plasma physics, superconductivity, and Bose-Einstein condensates (BEC) [1–8].

Among early derivations of the NLS equation were found in nonlinear optics [9, 10], plasma physics [11–15], and hydrodynamics [16–18]. In BEC, the NLS equation with the non-zero potential term is known as the Gross-Pitaevskii equation [19, 20]. In superconductivity, the time-independent NLS equation resembles some similarities with a simplified $\left(1+1\right)$-D form of the Ginzburg-Landau equation [21]. A further overview and extensive discussion of the NLS equation can be found in [7, 22–26].

1.2 Exact Solutions of the Nonlinear Schrödinger Equation

There are various techniques to derive analytical solutions of the NLS equation, among others, are the phase-amplitude algebraic ansatz [27–30], the Hirota method [31–34], nonlinear Fourier transform of inverse scattering transform (IST) [6, 35–40], symmetry reduction methods [41], variational formulation and displaced phase-amplitude equations [42–44]. Another derivation using IST with asymmetric boundary conditions is given in [45].

Throughout this article, we adopt the following $\left(1+1\right)$D, focusing-type of the NLS equation in a standard form:

$i{q}_t+q_{xx}+2{\left|q\right|}^{2}q=0,q\left(x,t\right)\in C.(1)$

Usually, the variables x and t denote the space and time variables, respectively. The simplest-solution is called the “plane-wave” or “continuous-wave” solution: $q\left(x,t\right)=q_0\left(t\right)=e^{2it}$. Another simple solution with a vanishing background is known as the “bright soliton” or “one-soliton solution”, given as follows:

$q\left(x,t\right)=q_S\left(x,t\right)=a\text{sech}\left(ax-2abt+{\theta }_0\right)e^{i\left(bx+\left(a^2-b^2\right)t+{\varphi }_0\right)},a,b,{\theta }_0,{\varphi }_0\in ℝ.(2)$

We focus our discussion on the family of exact solutions with constant and non-vanishing background, also called “breather soliton solutions” [46]. There are three types of breather, and all of them are considered as weakly nonlinear prototypes for freak waves. Other solutions of the NLS equation include cnoidal wave envelopes that can be expressed in terms of the Jacobi elliptic functions and can be derived using the Hirota bilinear transformation, theta functions, or with some clever algebraic ansatz [30, 47].

In this subsection, the coverage follows the historical order of the time when the breathers were found. Furthermore, the term “breather” and “soliton” can be used interchangeably in this article, and they can also appear as a single term “breather soliton”. All of them refer to the same object, i.e., the exact analytical solutions of the NLS equation with a non-vanishing, constant pedestal, or background of continuous-wave solution.

1.2.1 The Kuznetsov-Ma Breather

The first found solution is called the “Kuznetsov-Ma breather”, where Kuznetsov derived it for the first time in the 1970s [48]. The original Russian version of his paper was published as a preprint in 1976 by the Institute of Automation and Electrometry of the Siberian Branch of the USSR (now Russian) Academy of Sciences in Novosibirsk. This preprint was then reproduced in English and appeared in the Proceedings of the 13th International Conference on Phenomena in ionized Gases and Plasma, held in East Berlin, German Democratic Republic, on 12–17 September 1977 [49].

Although some authors stated that Kawata and Inoue, as well as Ma, also derived this solution independently, understanding the history behind its development might change our perspective [50, 51]. Between 1976 and 1977, Evgenii A. Kuznetsov met Tutomu Kawata (also spelled Tsutomu, 川田 勉) many times because the latter was a postdoctoral researcher in the Landau Institute for Theoretical Physics in Chernogolovka, near Moscow, under the mentorship of Professor Vladimir E. Zakharov. Kuznetsov personally gave Kawata his preprint on the soliton solution of the NLS equation. Furthermore, although Yan-Chow Ma has never really met with Kuznetsov, Ma was surely aware of Kuznetsov’s paper. In his work [51], Ma has cited another Kuznetsov’s paper that was written together with Alexander V. Mikhailov on the stability of stationary waves using the Korteweg-de Vries (KdV) equation [49, 52].

Although the term “Kuznetsov-Ma soliton” has been introduced earlier [53], we will adopt and use the terminology “Kuznetsov-Ma breather” throughout this article. When the breather dynamics were observed experimentally for the first time in optical fibers by Kibler and collaborators, this term is getting popular since then [54]. We denote it as $q_{\text{M}}$, and it is explicitly given by

$q\left(x,t\right)=q_{\text{M}}\left(x,t\right)=e^{2it}\left(\frac{{\mu }^3\cos \left(\rho t\right)+i\mu \rho \sin \left(\rho t\right)}{2\mu \cos \left(\rho t\right)-\rho \cosh \left(\mu x\right)}+1\right),(3)$

where $\rho =\mu \sqrt{4+{\mu }^2}$. The Kuznetsov-Ma breather does not represent a traveling wave. It is localized in the spatial variable x and periodic in the temporal variable t, and hence some authors also called it as the “temporal periodic breather” [55].

A minor typographical error found in Kawata and Inoue’s paper [50] has been corrected by Gagnon [56]. Kawata and Inoue [50], as well as Ma [51], derived the Kuznetsov-Ma breather solution using the IST for finite boundary conditions at $x\to \pm \infty$. The derivation using a direct method of Bäcklund transformation can be found in [57, 58], where the former analyzed solitary waves in the context of an optical bistable ring cavity.

Defining the amplitude amplification factor (AF) as the quotient of the maximum breather amplitude and the value of its background [43], we obtain that the amplitude amplification for the Kuznetsov-Ma breather is always larger than the factor of three, and it is explicitly given by

${\text{AF}}_{\text{M}}\left(\mu \right)=1+\sqrt{4+{\mu }^2},\mu >0.(4)$

The function is bounded below and is increasing as the parameter μ also increases. The plot of this AF can be found in [43, 44], and different expressions of the AF for this breather also appear in [59–62].

The Kuznetsov-Ma breather finds applications as a rogue wave prototype in nonlinear optics [30, 54, 63] and deep-water gravity waves [38, 39, 61, 64]. A numerical comparison of the Kuznetsov-Ma breather indicated that a qualitative agreement was reached in the central part of the corresponding wave packet and on the real face of the modulation [59]. The stability analysis of the Kuznetsov-Ma breather using a perturbation theory based on the IST verified that although the soliton is rather robust with respect to dispersive perturbations, damping terms strongly influence its dynamics [65].

The dynamics of the Kuznetsov-Ma breather in a microfabricated optomechanical array showed an excellent agreement between theory and numerical calculations [66]. The spectral stability analysis of this breather has been considered using the Floquet theory [67]. The mechanism of the Kuznetsov-Ma breather has been discussed and two distinctive mechanisms are paramount: modulational instability and the interference effects between the continuous-wave background and bright soliton [68]. New scenarios of rogue wave formation for artificially prepared initial conditions using the Kuznetsov-Ma and superregular breathers in small localized condensate perturbations are demonstrated numerically by solving the Zakharov-Shabat eigenvalue problem [69].

A higher-order Kuznetsov-Ma breather can be derived using the Hirota method and utilized in studying soliton propagation with an influence of small plane-wave background [70, 71]; or using the bilinear method [72].

1.2.2 The Akhmediev Soliton

The second one is called the Akhmediev-Eleonskiĭ-Kulagin breather and was found in the 1980s [27–29]. In short, we simply call it the “Akhmediev soliton” and denote it as $q_{\text{A}}$. This breather is localized in the temporal variable t and is periodic in the spatial variable x, and it can be written explicitly as follows:

$q\left(x,t\right)=q_{\text{A}}\left(x,t\right)=e^{2it}\left(\frac{{\nu }^3\cosh \left(\sigma t\right)+i\nu \sigma \sinh \left(\sigma t\right)}{2\nu \cosh \left(\sigma t\right)-\sigma \cos \left(\nu x\right)}-1\right).(5)$

Here, the parameter ν, $0\le \nu <2$ denotes a modulation frequency (or wavenumber) and $\sigma \left(\nu \right)=\nu \sqrt{4-{\nu }^2}$ is the modulation growth rate. The colleagues from nonlinear optics prefer calling this soliton “instanton” [73, 74] instead of “breather” since it breathes only once [75]. Other names for this solution include “modulational instability” [30], “homoclinic orbit” [34, 76], “spatial periodic breather” [55], and “rogue wave solution” [39].

The amplitude amplification for the Akhmediev soliton is at most of the factor of three, and it is explicitly given by

${\text{AF}}_{\text{A}}\left(\nu \right)=1+\sqrt{4-{\nu }^2},0<\nu <2.(6)$

This function is bounded above and below, $1<{\text{AF}}_{\text{A}}<3$, and is decreasing for an increasing value of the modulation parameter ν. Although the maximum growth rate occurs for $\nu =\sqrt{2}$, the maximum AF occurs when $\nu \to 0$, when the Akhmediev breather becomes the Peregrine soliton. To the best of our knowledge, this expression was introduced by Onorato et al. in their study on freak wave generation in random ocean waves where this AF depends on the wave steepness and number of waves under the envelope [77]. The plot for this AF can be found in [43, 78, 79]. Some variations in the AF expression for this soliton also appear in [59–62, 80, 81].

The Akhmediev soliton is rather well-known due to its characteristics being a nonlinear extension of linear modulational instability. This instability is also known as sideband (or Bespalov-Talanov) instability in nonlinear optics [82–84], or Benjamin-Feir instability in water waves [17, 85]. Some authors studied the modulational instability in plasma physics [11, 86–88] and in BEC [89–94]. Modulational instability is defined as the temporal growth of the continuous-wave NLS solution due to a small, side-band modulation, in a monochromatic wave train. A geometric condition for wave instability in deep water waves is given in [95] and for a historical review of modulational instability, see [96].

It has been shown numerically and experimentally that the modulated unstable wave trains grow to a maximum limit and then subside. In the spectral domain, the wave energy is transferred from the central frequency to its sidebands during the wave propagation for a certain period, and then it is recollected back to the primary frequency mode [97–101]. It turns out that the long-time evolution of these unstable wave trains leads to a sequence of modulation and demodulation cycles, known as the Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence phenomenon [102, 103]. Although the FPUT recurrence using the NLS model has been observed experimentally in surface gravity waves in the late 1970s [98], it took more than 2 decades for the phenomenon to be successfully recovered in nonlinear optics [104].

Since the modulational instability extends nonlinearly to the Akhmediev soliton, it is no surprise that the former is considered as a possible mechanism for the generation of rogue waves while the latter acts as one prototype [105–107]. For wave trains with amplitude and phase modulation, there is a competition between the nonlinearity and dispersive factors. After the modulational instability occurs, the growth predicted by linear theory is exponential, and the nonlinear effect in the form of the Akhmediev soliton takes over before the wave trains return to the stage similar to the initial profiles with a phase-shift difference [64, 108]. On the other hand, Biondini and Fagerstrom argued that the major cause of modulational instability in the NLS equation is not the breather soliton solutions per se, but the existence of perturbations where discrete spectra are absence [109].

Experimental attempts on deterministic rogue wave generation using the Akhmediev solitons suggested that the symmetric structure is not preserved and the wave spectrum experiences frequency downshift even though wavefront dislocation and phase singularity are visible [43, 110–114]. A numerical calculation of rogue wave composition can be described in the form of the collision of Akhmediev breathers [115]. Another comparison of the Akhmediev breathers with the North Sea Draupner New Year and the Sea of Japan Yura wave signals also show some qualitative agreement [116]. The characteristics of the Akhmediev solitons have also been observed experimentally in nonlinear optics [117].

A theoretical, numerical, and experimental report of higher-order modulational instability indicates that a relatively low-frequency modulation on a plane-wave induces pulse splitting at different phases of evolution [118]. Second-order breathers composed of nonlinear combinations of the Kuznetsov-Ma breather and Akhmediev soliton reveal the dependence of the wave envelope on the degenerate eigenvalues and differential shifts [119]. Similar higher-order Akhmediev solitons visualized in [118, 119] have been featured earlier in [43, 120] and similar illustrations can also be found in [60, 121–127].

1.2.3 The Peregrine Soliton

The third one is called the “Peregrine soliton”, also known as the “rational solution” [128]. This soliton is localized in both spatial and temporal variables $\left(x,t\right)$ and is written as follows (denoted as $q_{\text{P}}$):

$q\left(x,t\right)=q_{\text{P}}\left(x,t\right)=e^{2it}\left(\frac{4\left(1+4it\right)}{1+16t^2+4x^2}-1\right).(7)$

This solution is neither a traveling wave nor contains free parameters. Johnson called it a “rational-cum-oscillatory solution” [129]. Others referred to it as the “isolated Ma soliton” [130], an “explode-decay solitary wave” [131], the “rational growing-and-decaying mode” [71], the “algebraic breather” [132], or the “fundamental rogue wave solution” [124].

The amplitude amplification for the Peregrine soliton is exactly of factor three, and this can be obtained by taking the limit of the parameters toward zero in the previous two breathers:

${\text{AF}}_{\text{P}}=\underset{\mu \to 0}{\lim }{\text{AF}}_{\text{M}}\left(\mu \right)=3=\underset{\nu \to 0}{\lim }{\text{AF}}_{\text{A}}\left(\nu \right).(8)$

Although the other two breather solitons are also proposed as rogue wave prototypes, some authors argued that the Peregrine soliton is the most likely freak wave event due to its appearance from nowhere and disappearance without a trace [79] as well as its closeness to all initial supercritical humps of small uniform envelope amplitude [133]. Some numerical and experimental studies may support this reasoning.

Henderson et al. studied numerically unsteady surface gravity wave modulations by comparing the fully nonlinear and NLS equations [130]. For steep-waves, their computations produced striking similarities with the Peregrine soliton. On the other hand, Voronovich et al. confirmed numerically that the bottom friction effect, even when it is small in comparison to the nonlinear term, could hamper the formation of a breather freak wave at the nonlinear stage of instability [134]. Investigations on linear stability demonstrated that the Peregrine soliton is unstable against all standard perturbations, where the analytical study is supported by numerical evidence [135–138].

An important breakthrough in the study of rogue waves is the observation of the Peregrine soliton in nonlinear media. In nonlinear optics, the existence of strongly localized temporal and spatial peaks on a non-vanishing background, which indicates near-ideal Peregrine soliton characteristics, was successfully implemented for the first time in optical fiber generating femtosecond pulses in 2010 [139]. Not long after that, the Peregrine soliton was also observed experimentally for the first time in a water wave tank [140]. A comparison between the predictions from the theoretical model and the measurement results exhibits an excellent qualitative agreement in terms of wave signal pattern, its amplification factor, and its symmetric structure. Another successful experimental observation of the Peregrine solitons is reported in ion-acoustic waves of a multicomponent plasma with negative ions when the density of negative ions is equal to the critical value [141].

A sequence of related experimental studies using the Peregrine soliton demonstrated reasonably good qualitative agreement with the theoretical prediction. Some discrepancies occur in the modulational gradients, spatiotemporal symmetries, and for larger steepness values [142], as well as the frequency downshift [143]. Interestingly, Chabchoub et al. shown further experimentally that the dynamics of the Peregrine soliton and its spectrum characteristics persist even in the presence of wind forcing with high velocity [144]. By selecting a target location and determining an initial steepness, an experiment using the Peregrine soliton of wave interaction with floating bodies during extreme ocean condition has also been successfully implemented [145].

The Peregrine soliton also finds applications in the evolution of the intrathermocline eddies, also known as the oceanic lenses [146]. It appeared as a special case of stationary limit in the solutions of the spinor BEC model [147], and it was observed experimentally emerging from the stochastic background in deep-water surface gravity waves [148].

Nonlinear spectral analysis using the finite gap theory showed that the spectral portraits of the Peregrine soliton represent a degenerate genus two of the NLS equation solution [149]. Higher-order Peregrine solitons in terms of quasi-rational functions are derived in [150]. Higher-order Peregrine solitons up to the fourth-order using a modified Darboux transformation has been presented with applications in rogue waves in the deep ocean and high-intensity rogue light wave pulses in optical fibers [151]. Super rogue waves modeled with higher-order Peregrine soliton with an amplitude amplification factor of five times the background value are observed experimentally in a water-wave tank [122].

1.3 A Literature Review on Rogue Waves

There are various excellent reviews on rogue wave phenomena based on the NLS equation as a mathematical model and its corresponding breather solitons. Onorato et al. covered rogue waves in several physical contexts including surface gravity waves, photonic crystal fibers, laser fiber systems, and 2D spatiotemporal systems [60]. Dudley et al. reviewed breathers and rogue waves in optical fiber systems with an emphasis on the underlying physical processes that drive the appearance of extreme optical structures [125]. They reasoned that the mechanisms driving rogue wave behavior depend very much on the system. Residori et al. presented physical concepts and mathematical tools for rogue wave description [62]. They highlighted the most common features of the phenomenon include large deviations of wave amplitude from the Gaussian statistics and large-scale symmetry breaking. Chen et al. discussed rogue waves in scalar, vector, and multidimensional systems [152] while Malomed and Mihalache surveyed some theoretical and experimental studies on nonlinear waves in optical and matter-wave media [153].

Rogue waves come from and are closely related to modulational instability with resonance perturbation on continuous background [154]. A comparison of breather solutions of the NLS equation with emergent peaks in noise-seeded modulational instability indicated that the latter clustered closely around the analytical predictions [155]. “Superregular breathers” is the term coined indicating creation and annihilation dynamics of modulational instability, and the evidence of the broadest group of these superregular breathers in hydrodynamics and optics has been reported [156]. An interaction between breather and higher-order rogue waves in a nonlinear optical fiber is characterized by a trajectory of localized troughs and crests [157].

Breather soliton solutions find several applications, among others in beam-plasma interactions [158], in the transmission line analog of nonlinear left-handed metamaterials [159], in a nonlinear model describing an electron moving along the axis of deformable helical molecules [160], and in the mechanisms underlying the formation and real-time prediction of extreme events [161]. Additionally, optical rogue waves also successfully simulated in the presence of nonlinear self-image phenomenon in the near-field diffraction of plane waves from lightwave grating, known as the Talbot effect [162].

Since the definitions of “rogue waves” and “extreme events” are varied, a roadmap for unifying different perspectives could stimulate further discussion [163]. 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 NLS equation [164]. Since the behavior of a large class of perturbations characterized by a continuous spectrum is described by the identical asymptotic state, it turns out that the asymptotic stage of modulational instability is universal [165]. Surprisingly, the long-time asymptotic behavior of modulationally unstable media is composed of an ensemble of classical soliton solutions of the NLS equation instead of the breather-type solutions [166].

General N-solitonic solutions of the NLS equation in the presence of a condensate derived using the dressing method describe the nonlinear stage of the modulational instability of the condensate [167]. Rogue waves on a periodic background in the form of cnoidal functions that exhibit modulational instability not only generalize the Peregrine’s soliton but also potentially stimulate further discussion [168]. Recently, both theoretical description and experimental observation of the nonlinear mutual interactions between a pair of copropagative breathers are presented and it is observed that the bound state of breathers exhibits a behavior similar to a molecule with quasiperiodic oscillatory dynamics [169].

The paper will be presented as follows. After this introduction, Section 2 discusses rigorous proof for the limiting behavior of the breather wave solutions using the ϵ-δ argument. The limiting behavior will continue in Section 3, where we cover it from the visual viewpoint. We present the corresponding contour plots for various values of parameters and the parameterization sketches of the non-rapid oscillating complex-valued breather amplitudes. Finally, Section 4 concludes our discussion and provide remarks for potential future research.

2 Limiting Behavior

This section provides rigorous proof of the limiting behavior of breather wave solutions using the ϵ-δ argument. We have the following theorem:

Theorem 1. The Peregrine soliton is a limiting case for both the Kuznetsov-Ma breather and Akhmediev soliton:

$\underset{\mu \to 0}{\lim }{q}_{\text{M}}\left(x,t\right)=q_{\text{P}}\left(x,t\right)=\underset{\nu \to 0}{\lim }{q}_{\text{A}}\left(x,t\right).(9)$

We split the proof into four parts, and each limit consists of two parts corresponding to the real and imaginary parts of the solitons.

Proof. The following shows that the limit for the real parts of the Kuznetsov-Ma breather and Peregrine soliton is correct, i.e.,

$\underset{\nu \to 0}{\lim }\mathrm{Re}\left\{q_{\text{M}}\left(x,t\right)\right\}=\mathrm{Re}\left\{q_{\text{P}}\left(x,t\right)\right\}.$

For each $\epsilon >0$, there exists $\delta =\sqrt{{\left(\epsilon +2\right)}^2-4}>0$ such that if $0<\mu <\delta$, then $|\mathrm{Re}\left\{q_{\text{M}}\right\}-\mathrm{Re}\left\{q_{\text{P}}\right\}|<\epsilon$. We know that since

$\frac{\cosh \mu \left(x-x_0\right)}{\cos \rho \left(t-t_0\right)}\ge 1\text{for}\text{all}\left(x,t\right)\in {ℝ}^2,$

it then implies

$\rho \frac{\cosh \mu \left(x-x_0\right)}{\cos \rho \left(t-t_0\right)}-2\mu \ge \rho -2\mu ,$

It follows that

$\begin{array}{l}\left|\mathrm{Re}\left\{q_{\mathrm{M}}\right\}-\mathrm{Re}\left\{q_{\mathrm{P}}\right\}\right|=\left|\frac{{\mu }^3}{\rho \frac{\cosh \nu \left(x-x_0\right)}{\cos \rho \left(t-t_0\right)}-2\mu }-\frac{4}{1+16{\left(t-t_0\right)}^2+4{\left(x-x_0\right)}^2}\right|\\ \le \left|\frac{{\mu }^3}{\rho -2\mu }-4\right|=\left|\sqrt{{\mu }^2+4}-2\right|\\ \le \sqrt{{\delta }^2+4}-2=\sqrt{{\left(\sqrt{{\left(\epsilon +2\right)}^2-4}\right)}^2+4}-2=\epsilon .\end{array}$

The following verifies that the limit for the imaginary parts of the Kuznetsov-Ma breather and Peregrine soliton is accurate, i.e.,

$\underset{\nu \to 0}{\lim }\mathrm{Im}\left\{q_{\text{M}}\left(x,t\right)\right\}=\mathrm{Im}\left\{q_{\text{P}}\left(x,t\right)\right\}.$

For each $\epsilon >0$, there exists $\delta =\sqrt{-10+2\sqrt{25+4\epsilon /|t-t_0|}}>0$ such that if $0<\mu <\delta$, then $\left|\mathrm{Im}\left\{q_{\text{M}}\right\}-\mathrm{Im}\left\{q_{\text{P}}\right\}\right|<\epsilon$. We can write the imaginary parts of $q_{\text{M}}$ and $q_{\text{P}}$ as follows

$\begin{array}{l}\mathrm{Im}\left\{q_{\text{M}}\right\}=\frac{\mu \rho }{\rho \frac{\cosh \mu \left(x-x_0\right)}{\sin \rho \left(t-t_0\right)}-2\mu \cot \rho \left(t-t_0\right)}\le \frac{\mu {\rho }^2\left|t-t_0\right|}{\rho -2\mu },\\ \text{Im}\left\{q_{\text{P}}\right\}=\frac{16\left(t-t_0\right)}{1+16{\left(t-t_0\right)}^2+4{\left(x-x_0\right)}^2}\le 16\left|t-t_0\right|,\end{array}$

It follows that

$\begin{array}{l}\left|\mathrm{Im}\left\{q_{\mathrm{M}}\right\}-\mathrm{Im}\left\{q_{\mathrm{P}}\right\}\right|=\left|\frac{\mu \rho }{\rho \frac{\cosh \mu \left(x-x_0\right)}{\sin \rho \left(t-t_0\right)}-2\mu \cot \rho \left(t-t_0\right)}-\frac{16\left(t-t_0\right)}{1+16{\left(t-t_0\right)}^2+4{\left(x-x_0\right)}^2}\right|\\ \le \left|\frac{\mu {\rho }^2}{\rho -2\mu }-16\left|\right|t-t_0\right|=|\left({\mu }^2+4\right)\left(\sqrt{{\mu }^2+4}+2\right)-16\left|\right|t-t_0|\\ <|\left({\delta }^2+4\right)\left(\frac{{\delta }^2}{4}+4\right)-16\left|\right|t-t_0|=|\frac{{\delta }^4}{4}+5{\delta }^2\left|\right|t-t_0|=\epsilon .\end{array}$

In what follows, we present the limit of the real part of the Akhmediev soliton as $\nu \to 0$ is indeed the real part of the Peregrine soliton, i.e.,

$\underset{\nu \to 0}{\lim }\mathrm{Re}\left\{q_{\text{A}}\left(x,t\right)\right\}=\mathrm{Re}\left\{q_{\text{P}}\left(x,t\right)\right\}.$

For each $\epsilon >0$, there exists $\delta =\sqrt{\epsilon /3}>0$ such that if $0<\nu <\delta <2$, then $\left|\mathrm{Re}\left\{q_{\text{A}}\right\}-\mathrm{Re}\left\{q_{\text{P}}\right\}\right|<\epsilon$. We know that since

$-1\le \frac{\cos \nu \left(x-x_0\right)}{\cosh \sigma \left(t-t_0\right)}\le 1\text{for all}\left(x,t\right)\in {ℝ}^2,$

it then implies

$2\nu -\sigma \le 2\nu -\sigma \frac{\text{cos}\nu \left(x-x_0\right)}{\text{cosh}\sigma \left(t-t_0\right)}.$

We also have $1+16{\left(t-t_0\right)}^2+4{\left(x-x_0\right)}^2\ge 1$ for all $\left(x,t\right)\in {\mathrm{ℝ}}^2$. Furthermore, since $0\le \sqrt{4-{\nu }^2}\le 2$, $0\le 2-\sqrt{4-{\nu }^2}\le 2$,

$\begin{array}{l}\frac{1}{4}\le \frac{2-\sqrt{4-{\nu }^2}}{{\nu }^2}\le \frac{1}{2},\\ \frac{1}{2}\le \frac{1}{4}+\frac{2-\sqrt{4-{\nu }^2}}{{\nu }^2}\le \frac{3}{4},\\ \mathrm{and}\frac{2-\sqrt{4-{\nu }^2}}{4{\nu }^2}\ge \frac{1}{4{\delta }^2},\end{array}$

it follows that

$\begin{array}{l}\left|\mathrm{Re}\left\{q_{\mathrm{A}}\right\}-\mathrm{Re}\left\{q_{\mathrm{P}}\right\}\right|=\left|\frac{{\nu }^3}{2\nu -\sigma \frac{\cos \nu \left(x-x_0\right)}{\cosh \sigma \left(t-t_0\right)}}-\frac{4}{1+16{\left(t-t_0\right)}^2+4{\left(x-x_0\right)}^2}\right|\\ \le \left|\frac{{\nu }^3}{2\nu -\sigma }+4\right|=\left|\frac{1}{\frac{\left(2-\sqrt{4-{\nu }^2}\right)}{{\nu }^2}}+\frac{1}{\frac{1}{4}}\right|=\frac{\left|\frac{1}{4}+\frac{\left(2-\sqrt{4-{\nu }^2}\right)}{{\nu }^2}\right|}{\left|\frac{\left(2-\sqrt{4-{\nu }^2}\right)}{4{\nu }^2}\right|}\\ \le \frac{3}{4}\left(4{\delta }^2\right)=3{\left(\sqrt{\frac{\epsilon }{3}}\right)}^2=\epsilon .\end{array}$

In what follows, we demonstrate that the limit of the imaginary part of the Akhmediev soliton becomes the imaginary part of the Peregrine soliton, i.e.,

$\underset{\nu \to 0}{\lim }\mathrm{Im}\left\{q_{\text{A}}\left(x,t\right)\right\}=\mathrm{Im}\left\{q_{\text{P}}\left(x,t\right)\right\}.$

For each $\epsilon >0$, there exists $\delta =\sqrt{\epsilon /4}>0$ such that if $0<\nu <\delta <2$, then $\left|\mathrm{Im}\left\{q_{\text{A}}\right\}-\mathrm{Im}\left\{q_{\text{P}}\right\}\right|<\epsilon$. We can write the imaginary parts of $q_{\text{A}}$ and $q_{\text{P}}$ as follows

$\begin{array}{l}\mathrm{Im}\left\{q_{\text{A}}\right\}=\frac{\nu \sigma \tanh \sigma \left(t-t_0\right)}{2\nu -\sigma \frac{\cos \nu \left(x-x_0\right)}{\cosh \sigma \left(t-t_0\right)}}\le \frac{\nu {\sigma }^2\left|t-t_0\right|}{2\nu -\sigma },\\ \mathrm{Im}\left\{q_{\text{P}}\right\}=\frac{16\left(t-t_0\right)}{1+16{\left(t-t_0\right)}^2+4{\left(x-x_0\right)}^2}\le 16\left|t-t_0\right|.\end{array}$

Since $2+\sqrt{4-{\nu }^2}\le 4$ and $\left(4-{\nu }^2\right)\le 4+{\delta }^2/\left|t-t_0\right|$, it follows that

$\begin{array}{l}\left|\mathrm{Im}\left\{q_{\text{A}}\right\}-\mathrm{Im}\left\{q_{\text{P}}\right\}\right|=\left|\frac{\nu \sigma \tanh \sigma \left(t-t_0\right)}{2\nu -\sigma \frac{\cos \nu \left(x-x_0\right)}{\cosh \sigma \left(t-t_0\right)}}-\frac{16\left(t-t_0\right)}{1+16{\left(t-t_0\right)}^2+4{\left(x-x_0\right)}^2}\right|\\ \le \left|\frac{\nu {\sigma }^2}{2\nu -\sigma }-16\left|\right|t-t_0\right|=\left|\left(4-{\nu }^2\right)\left(2+\sqrt{4-{\nu }^2}\right)-16\left|\right|t-t_0\right|\\ <\left|4\left(4+\frac{{\delta }^2}{|t-t_0|}\right)-16\left|\right|t-t_0\right|=4{\delta }^2=4{\left(\sqrt{\frac{\epsilon }{4}}\right)}^2=\epsilon .\end{array}$

We have completed the proof.

In the following section, we will visualize the limiting behavior of the breather solutions as they approach toward the Peregrine soliton.

3 Limiting Behavior Visualized

In this section, we will visually confirm the limiting behavior of the Kuznetsov-Ma and Akhmediev breathers toward the Peregrine soliton as both parameter values approach zero. Subsection 3.1 presents the contour plots of the amplitude modulus, and Subsection 3.2 discusses the spatial and temporal parameterizations of the breathers. We select several parameter values in sketching the plots. Figure 1 displays the chosen parametric values for both breather solutions, where they can be visualized in the complex-plane for the parameter pair $\left(\mu ,\nu \right)$.

FIGURE 1

FIGURE 1. Selected parametric values ν and $\mu =i\nu$ displayed in the complex plane for the Kuznetsov-Ma breather and Akhmediev soliton visualized in this section.

3.1 Contour Plot

In this subsection, we observe the contour plots of the amplitude modulus of the breather and how the changes in the parameter values affect the envelope’s period and wavelength. Similar contour plots have been presented in the context of electronegative plasmas with Maxwellian negative ions [170]. In particular, the contour plot of the Peregrine soliton is also displayed in [142].

Figures 2A-E display the contour plots of the Kuznetsov-Ma breather for several values of parameters μ: $\sqrt{2},1,1/2$, and $1/5$. Figure 2E is a zoom-in version of the same contour plot given in Figure 2D. Figure 2F is the final stop when we let the parameter $\mu \to 0$, for which the Kuznetsov-Ma breather turns into the Peregrine soliton. It is interesting to note that for $\mu =1/5$, the contour plot is nearly identical to the one from the Peregrine soliton, as we can observe by qualitatively comparing panels (E) and (F) of Figure 2.

FIGURE 2

FIGURE 2. Contour plots for the moduli of the Kuznetsov-Ma breather for (A)$\mu =\sqrt{2}$, (B)$\mu =1$, (C)$\mu =0.5$, (D)$\mu =0.2$, (E) also $\mu =0.2$ but a zoom-in version, and (F)$\mu =0$, which gives the Peregrine soliton. Notice that the contour plots (e) and (f) are qualitatively nearly identical.

Let $T_{\text{M}}$ denote the temporal envelope period for the Kuznetsov-Ma breather, then we know that in general, $T_{\text{M}}=2\pi /\rho$. For $\mu \to \infty$, $T_{\text{M}}\to 0$ and vice versa, for $\mu \to 0$, $T_{\text{M}}\to \infty$. For any given value of $\mu >0$, $T_{\text{M}}$ can be easily calculated. Here are some examples. For $\mu =\sqrt{2}$, $T_{\text{M}}=\pi /\sqrt{3}\approx 1.814$ and we display five periods in Figure 2A along the temporal axis t. For $\mu =1$, $T_{\text{M}}=2\pi /\sqrt{5}\approx 2.81$ and for the same time interval as in panel (A), we can only capture three periods along the temporal axis t, as shown in Figure 2B. Furthermore, for $\mu =1/2$, $T_{\text{M}}=8\pi /\sqrt{17}\approx 6.1$ and we need to extend almost twice the length in the time interval in order to capture at least three periods. Figure 2C shows this contour plot. Finally, for $\mu =1/5$, $T_{\text{M}}=50\pi /\sqrt{101}\approx 15.63$. As we can observe in Figure 2D, extending the length of time interval to around 40 units is sufficient to capture at least three periods, albeit the detail around maximum and minimum is hardly visible. Table 1 displays selected parameter values of the Kuznetsov-Ma breather and their corresponding temporal envelope periods $T_{\text{M}}$.

TABLE 1

TABLE 1. Exact values of the temporal envelope period $T_{\text{M}}$ and their approximate values for selected parameter values μ corresponding to the Kuznetsov-Ma breather.

Figures 3A–C display the contour plot of the Akhmediev soliton for selected values of its parameters ν: 1, $1/2$, and $1/4$. Figure 3D shows the contour plot of the Peregrine soliton, which occurs as the final destination when letting the parameter $\nu \to 0$. Figure 3D is identical to Figure 2F, the only difference lies in the length-scale of both horizontal and vertical axes. Similar to the previous case, zooming-in the contour plot for $\nu =1/4$ in Figure 3C will yield a qualitatively nearly identical contour plot with the Peregrine soliton shown in the panel (D). (It is not shown in the figure.)

FIGURE 3

FIGURE 3. Contour plots for the moduli of the Akhmediev breather for (A)$\nu =1$, (B)$\nu =0.5$, and (C)$\nu =0.25$, as well as (D) the Peregrine soliton.

Let $L_{\text{A}}$ denote the spatial envelope wavelength for the Akhmediev soliton, then for $0<\nu <2$, $L_{\text{A}}=2\pi /\nu$, which gives $L_{\text{A}}>\pi$. For $\nu \to 2$, $L_{\text{A}}\to \pi$, and as $\nu \to 0$, $L_{\text{A}}\to \infty$. Table 2 displays selected values of the parameter ν and their corresponding spatial envelope wavelength $L_{\text{A}}$ for the Akhmediev soliton. For $\nu =1$, $L_{\text{A}}=2\pi$ and the spatial length of 20 units in Figure 3A is sufficient to capture three envelope wavelength. For $\nu =1/2$, $L_{\text{A}}=4\pi$ and the spatial length of 40 units in Figure 3B is required to capture at least three envelope wavelength. For $\nu =1/4$, $L_{\text{A}}=8\pi$ and the spatial length of 60 units in Figure 3C is needed to capture at least three envelope wavelength. The details around maxima and minima are hardly visible for the latter.

TABLE 2

TABLE 2. Exact values of the spatial envelope wavelength $L_{\text{A}}$ and their approximate values for selected parameter values ν corresponding to the Akhmediev soliton.

3.2 Parameterization in Spatial and Temporal Variables

In this subsection, we write the breather solutions as $q_{\text{X}}\left(x,t\right)=q_0\left(t\right){\tilde{q}}_{\text{X}}\left(x,t\right)$, where $q_0\left(t\right)$ is the plane-wave solution and $\text{X}=\left\{\text{M,A,P}\right\}$. Since the plane-wave solution gives a fast-oscillating effect, we only consider the non-rapid oscillating part of the breathers ${\tilde{q}}_{\text{X}}$ for the parameterization visualization. In the subsequent figures, we present both spatial and temporal parameterizations of the Kuznetsov-Ma breather, Akhmediev, and Peregrine solitons. A similar description has been briefly covered and discussed in [30, 61, 63, 164]. This article does not only complements and supplements but also provides detailed explanations to those references. Additionally, note that the unit circle centered at the origin appeared in each panel of Figures 4–7 (dotted black circle) corresponds to the phase of the continuous-wave pedestal, i.e., the manifold of the breathers for $x\to \pm \infty$ or $t\to \pm \infty$ [30].

FIGURE 4

FIGURE 4. Parameterization of the non-rapid-oscillating complex-valued amplitude of the Kuznetsov-Ma breather ${\tilde{q}}_{\text{M}}$ in the spatial variable x, $x\ge 0$, for different values of temporal variable t and different values of the parameter μ: (A)$\mu =\sqrt{2}$, (B)$\mu =1$, (C)$\mu =1/2$, and (D)$\mu =1/5$. The selected values of t are $t=0$ (solid red), $t=\pm \pi /64$ (dashed green), $t=\pm \pi /32$ (solid purple), $t=\pm \pi /16$ (dash-dotted magenta), $t=\pm \pi /8$ (solid cyan), and $t=\pm \pi /4$ (dashed orange).

Figure 4 displays the parameterization of the non-rapid oscillating Kuznetsov-Ma breather ${\tilde{q}}_{\text{M}}$ in the spatial variable x for different values of the temporal variable t and parameter μ. Different panels indicate different parameter values μ and for each panel, different curves, for which in this particular case, they are merely straight lines, indicate different time t. For all cases, we consider $x\ge 0$ due to the symmetry nature of the breathers. The straight-line trajectories move inwardly focused from the dotted blue circle at $x=0$ toward $\left(-\mathrm{1,0}\right)$ as $x\to \infty$. The situation is simply reversed for $x<0$: the path of trajectories move outwardly defocused as x progresses from $\left(-\mathrm{1,0}\right)$ at $x\to -\infty$ toward the dotted blue circle at $x=0$. At the bottom of these four panels, we also present the t-axis and corresponding values of the selected values of t for $-T_{\text{M}}/2<-\pi /4\le t\le \pi /4<T_{\text{M}}/2$. The trajectories in the upper-part and lower-part of the complex-plane correspond to the positive and negative values of t, respectively. We observe that the trajectories shift faster in space around $t=0$ than around $t=\pm T_{\text{M}}/2=\pm \pi /\rho$.

In particular, for $t=n\pi /\rho$, $n\in \mathbb{Z}$, ${\tilde{q}}_{\text{M}}$ reduces to a real-valued function, i.e., Im$\left({\tilde{q}}_{\text{M}}\right)=0$ for all $\mu >0$. Hence, the parameterized curve is a straight line at the real-axis. For $t=2n\pi /\rho$, $n\in \mathbb{Z}$, this is shown by the horizontal solid red line lying on the real axis moving from a point larger than Re$\left({\tilde{q}}_{\text{M}}\right)=3$ to $\text{Re}\left({\tilde{q}}_{\text{M}}\right)=-1$ for $x>0$. The represented case $t=0$ is displayed in Figure 4 while the case $t=\pi /\rho$ is not shown in the figure. Indeed, from (3), we obtain the following limiting values for $n\in \mathbb{Z}$:

$\begin{array}{l}\underset{x\to 0}{\lim }{q}_{\mathrm{M}}\left(x,2n\pi /\rho \right)=1+\sqrt{{\mu }^2+4},\\ \mathrm{and}\underset{x\to 0}{\lim }{q}_{\mathrm{M}}\left(x,\left(2n+1\right)\pi /\rho \right)=1-\sqrt{{\mu }^2+4}.\end{array}(10)$

Additionally, $\underset{x\to \mathrm{\pm }\infty }{\lim }{q}_M\left(x,n\pi /\rho \right)=-1$. Using a similar analysis, vertical straight lines at $\mathrm{Re}\left({\tilde{q}}_{\text{M}}\right)=-1$ can be obtained by taking the values of $t=\left(n+1/2\right)\pi /\rho$, for $n\in \mathbb{Z}$. The line direction from the positive and negative regions of Im$\left({\tilde{q}}_{\text{M}}\right)$ is downward and upward toward $\left(-\mathrm{1,0}\right)$ for even and odd values of $n\in \mathbb{Z}$, respectively.

Figure 5 displays the sketch of the non-rapid-oscillating Kuznetsov-Ma breather ${\tilde{q}}_{\text{M}}$ in the complex-plane parameterized in the temporal variable t for different values of the spatial variable x and parameter μ. For each case, t is taken for one temporal envelope period, i.e., $-T_{\text{M}}/2=-\pi /\rho <t<\pi /\rho =T_{\text{M}}/2$. Instead of a set of straight lines, the trajectories form the shape of elliptical curves. For each $x=x_0\in ℝ$, the ellipse is centered at $\left(c\left(x_0\right),0\right)$ with semi-minor axis $a\left(x_0\right)$ and semi-major axis $b\left(x_0\right)$, where

$a\left(x_0\right)=\frac{\mu \rho \cosh \left(\mu x_0\right)}{d\left(x_0\right)}(11)$

$b\left(x_0\right)=\frac{\rho \cosh \left(\mu x_0\right)}{\sqrt{d\left(x_0\right)}}(12)$

$c\left(x_0\right)=\frac{2{\mu }^2}{d\left(x_0\right)}-1(13)$

$d\left(x_0\right)=2\cosh \left(2\mu x_0\right)+{\mu }^2-2.(14)$

FIGURE 5

FIGURE 5. Parameterization of the non-rapid oscillating complex-valued amplitude of the Kuznetsov-Ma breather ${\tilde{q}}_{\text{M}}$ in the temporal variable t ($-\pi /\rho <t<\pi /\rho$) for different values of spatial variable x: $x=0$ (solid blue), $x=1/8$ (long-dashed red), $x=1/4$ (dash-dotted green), $x=1/2$ (dashed purple), $x=1$ (dash-dotted cyan), and $x=2$ (solid magenta) and different values of the parameter μ: (A)$\mu =\sqrt{2}$, (B)$\mu =1$, (C)$\mu =1/2$, and (D)$\mu =1/5$.

The special case of a circle is obtained for $x_0=0$ with the radius $r=\sqrt{{\mu }^2+4}$ centered at $\left(\mathrm{1,0}\right)$. All curves move in the counterclockwise direction for increasing t. For $x>0$, the larger the values of x, the smaller the ellipses become. The situation is the opposite for $x<0$: smaller values of x (but largely negatives in its absolute value sense) correspond to smaller ellipses in the complex plane. Due to its spatial symmetry, only the plots for positive values of x are displayed. The axis below the figure panels shows the selected x values for a better overview of the variable scaling: $x=0$, $1/8$, $1/4$, $1/2$, 1, and $x=2$.

In Figure 4, the starting points of the trajectories for $x=0$ are shrinking as μ decreases, as indicated by the dotted blue exterior circles. For the same interval of time t, these initial points also tend to be absorbed toward the right-hand side of the exterior circles while they focus toward $\left(-\mathrm{1,0}\right)$. As we can observe in panels (A) and (B), the trajectories at $t=\pm \pi /4$ originate from the left-hand side of the exterior circles for $\mu \ge 1$. A similar pattern was no longer observed as the values of μ get smaller, as we can see in panels (C) and (D). Meanwhile, the circles and ellipses are getting smaller in Figure 5 for decreasing values of μ. Except for the circles that are always centered at $\left(\mathrm{1,0}\right)$, the centers of the ellipses shift toward the left-hand side of the blue exterior circle near $\left(-\mathrm{1,0}\right)$ as μ decreases.

Figure 6 displays the sketch in the complex-plane of the non-rapid-oscillating Akhmediev soliton ${\tilde{q}}_{\text{A}}$ [panels (A)‐(C)] and Peregrine soliton ${\tilde{q}}_{\text{P}}$ [panel (D)] parameterized in the spatial variable x for different values of the temporal variable t and parameter ν. We only display the trajectories corresponding to the positive values of t, the trajectories for the negative values of t are simply the reflection over the horizontal axis $\mathrm{Re}\left({\tilde{q}}_{\text{P}}\right)=0$. The t-axis below the panels indicate the chosen values of t displayed in the figure. Similar to the trajectories for the Kuznetsov-Ma breather when they are parameterized in the spatial variable x, the trajectories for the Akhmediev soliton parameterized in x are also collections of straight lines shifting in the counterclockwise direction for increasing values of t. Different from the previous case, these straight lines are periodic in x. The experimental results of deterministic freak wave generation using the spatial NLS equation showed that instead of straight lines, we obtained non-degenerate Wessel curves, suggesting that the periodic lines might be perturbed during the downstream evolution [43, 113].

FIGURE 6

FIGURE 6. Parameterization of the non-rapid oscillating complex-valued amplitude $\tilde{q}$ in the spatial variable x for different values of temporal variable t: $t=0$ (solid red), $t=1/16$ (long-dashed green), $t=1/8$ (dash-dotted purple), $t=1/4$ (dash magenta), $t=1/2$ (dash-dotted cyan), $t=1$ (dashed orange), $t=2$ (solid black), and $t=4$ (solid red), and modulation frequencies of the Akhmediev solitons (A)$\nu =1$$\left(0\le x\le \pi \right)$, (B)$\nu =1/2$$\left(0\le x\le 2\pi \right)$, (C)$\nu =1/4$$\left(0\le x\le 4\pi \right)$, and (D)$\nu =0$, $x\ge 0$ (Peregrine soliton).

For each panel, we only sketch the trajectories for an interval of half the spatial envelope wavelength, i.e., $0\le x\le L_{\text{A}}/2=\pi /\nu$. For this limited space interval, the direction of the lines is moving inwardly focused, from the dotted-blue exterior circle for $x=0$ to some values in the left-part of the complex-plane near $\mathrm{Re}\left({\tilde{q}}_{\text{A}}\right)=-1$. As the value of x progresses, $L_{\text{A}}/2=\pi /\nu \le x\le L_{\text{A}}=2\pi /\nu$, the trajectories bounce back toward the initial points by following the identical paths. They then travel in the same manner periodically as $x\to \pm \infty$. For a decreasing value of the parameter ν, the endpoint of these lines tends to focus around the region near $\left(-\mathrm{1,0}\right)$, as we can observe in Figures 6A–C. For the Peregrine soliton, the trajectories are not periodic as $L_{\text{A}}\to \infty$, and they tend to $\left(-\mathrm{1,0}\right)$ for $x\to \pm \infty$, as can be seen in Figure 6D.

In Figure 6, a prominent difference in the trajectories for different values of the parameter ν is its lengths. The length of the trajectories is increasing for decreasing values of ν. While the starting points for the Kuznetsov-Ma breathers are shrinking, for this family of Akhmediev solitons, they are expanding as $\nu \to 0$ until the dotted blue exterior circle reaches a radius of 2 unit length. Moreover, the endpoints for larger values of ν stop at some points where their real values become negative but still larger than $-1$. These endpoints eventually approach $\left(-\mathrm{1,0}\right)$ as $\nu \to 0$.

Figure 7 displays the sketch of the non-rapid-oscillating part of the Akhmediev soliton ${\tilde{q}}_{\text{A}}$ [panels (A)‐(C)] and Peregrine soliton ${\tilde{q}}_{\text{P}}$ [panel (D)] in the complex-plane parameterized in the temporal variable t for different values of the spatial variable x and parameter ν. The values of t run from $t\to -\infty$ to $t\to +\infty$, and we only sketch the positive values of x. The plots for the negative values of x are identical and are not shown due to the symmetry property of the soliton. The x-axis below the panels shows the selected values of x ranging from $x=0$ to $x=\text{π}$. For ${\tilde{q}}_{\text{A}}$, the trajectories are composed of circular sectors, elliptical sectors, and straight lines instead of closed curves like circles or ellipses. Since this soliton is a nonlinear extension of the modulational instability, the trajectories for each value of the parameter ν, $0<\nu <2$, are the corresponding homoclinic orbit for an unstable mode, and the presence of a phase shift prevents closed-path trajectories [30, 34, 76, 164].

FIGURE 7

FIGURE 7. Parameterization of the non-rapid oscillating complex-valued amplitude $\tilde{q}$ in the temporal variable t$\left(-\infty <t<\infty \right)$ for different values of spatial variable x: $x=0$ (solid blue), $x=\pi /8$ (long-dashed red), $x=\pi /6$ (dash-dotted green), $x=\pi /4$ (dashed purple), $x=\pi /2$ (dash-dotted cyan), and $x=\pi$ (solid magenta), and modulation frequencies of the Akhmediev solitons (A)$\nu =1$, (B)$\nu =0.5$, (C)$\nu =0.25$, and (D)$\nu \to 0$ (the Peregrine soliton).

The circular sectors are attained for $x=0$ and the straight lines occur at $x=\left(n+1/2\right)\pi /\nu ,n\in \mathbb{Z}$, $n\in \mathbb{Z}$. Trajectories at other locations yield the elliptical sectors. The initial and final points are not identical, and this indicates a phase shift in the soliton. Let ${\varphi }_{+\infty }$ and ${\varphi }_{-\infty }$ be the phases for $x\to \pm \infty$, respectively. Let also $\Delta \varphi ={\varphi }_{+\infty }-{\varphi }_{-\infty }$ be the difference between the phases at $x=+\infty$ and $x=-\infty$, then we have the following phase relationships:

$\begin{array}{l}\tan {\varphi }_{\pm \infty }=\pm \frac{\sigma }{{\nu }^2-1},\end{array}(15)$

$\begin{array}{l}\mathrm{and}\Delta \varphi =2\arctan \left(\frac{\sigma }{{\nu }^2-1}\right).\end{array}(16)$

For the Peregrine soliton, the trajectories of time parameterization in the complex-plane are either a circle (for $x=0$) or ellipses (for other values of $x\ne 0$). The circle is centered at $\left(\mathrm{1,0}\right)$ with radius $r=2$. Let $x=x_0\in ℝ$ be the position for the Peregrine soliton, then the ellipse has the length of semi-minor axis $a\left(x_0\right)$, the length of semi-major axis $b\left(x_0\right)$, and is centered at $\left(c\left(x_0\right),0\right)$, where

$\begin{array}{l}a\left(x_0\right)=\frac{2}{1+4x_0^2},\end{array}(17)$

$\begin{array}{l}b\left(x_0\right)=\frac{2}{\sqrt{1+4x_0^2}},\end{array}(18)$

$\begin{array}{l}\mathrm{and}c\left(x_0\right)=a\left(x_0\right)-1.\end{array}(19)$

Nearly all trajectories in Figure 7 follow the right-hand side paths instead of the left-hand side. For decreasing values of ν, the trajectories are generally expanding in size, except for the curve at $x=\pi$ that becomes a straight line when the parameter value changes from $\nu =1$ to $\nu =1/2$. When the values of ν is further decreased, the trajectories at $x=\pi$ become an elliptical sector and an ellipse for $\nu =1/4$ and $\nu =0$, respectively.

Figure 8 should be viewed in connection to Figures 6D, 7D. It displays the plots of the real and imaginary parts of the non-rapid-oscillating complex-valued amplitude for the Peregrine soliton ${\tilde{q}}_{\text{P}}$ with respect to x and t, which are presented in the upper and lower panels, respectively. For the former, different curves correspond to selected values of time $t\in \left\{0,1/16,1/8,1/4,1/2,\mathrm{1,2}\right\}$. For the latter, different curves correspond to selected values of position $x\in \left\{0,\pi /8,\pi /6,\pi /4,\pi /2,\pi \right\}$. The phase difference in the time parameterization of ${\tilde{q}}_{\text{P}}$ is discernible from the behavior of Im$\left({\tilde{q}}_{\text{P}}\right)$ as $t\to \pm \infty$. While $\underset{x\to \pm \infty }{\lim }\mathrm{Re}\left({\tilde{q}}_P\right)=-1$, the quantity for $\underset{x\to \pm \infty }{\lim }\mathrm{Im}\left({\tilde{q}}_P\right)$ takes positive and negative values, respectively.

FIGURE 8

FIGURE 8. Plots of the real and imaginary parts of the non-rapid oscillating complex-valued amplitude for the Peregrine soliton with respect to the spatial and temporal variables, x (upper panels) and t (lower panels), respectively. For upper panels (A) and (B), various curves indicates different time: $t=0$ (solid red), $t=1/16$ (long-dashed green), $t=1/8$ (dash-dotted purple), $t=1/4$ (dash magenta), $t=1/2$ (dash-dotted cyan), $t=1$ (dashed orange), and $t=2$ (solid black). For lower panels (C) and (D), different curves indicates different positions: $x=0$ (solid blue), $x=\pi /8$ (long-dashed red), $x=\pi /6$ (dash-dotted green), $x=\pi /4$ (dashed purple), $x=\pi /2$ (dash-dotted cyan), and $x=\pi$ (solid magenta).

4 Conclusion

We have considered the exact analytical breather solutions of the focusing NLS equation, where the wave envelopes at infinity have a nonzero but constant background. These solutions have been adopted as weakly nonlinear prototypes for freak waves in dispersive media due to their fine agreement with various experimental results. We have provided not only a brief historical review of the breathers but also covered some recent progress in the field of rogue wave modeling in the context of the NLS equation.

In particular, we have discussed the Peregrine soliton as a limiting case of the Kuznetsov-Ma breather and Akhmediev soliton. We have verified rigorously using the ε-δ argument that as each of the parameter values from these two breathers is approaching zero, they reduce to the Peregrine soliton. We have also presented this limiting behavior visually by depicting the contour plots of the breather amplitude modulus for selected parameter values. We displayed the parameterization plots of the non-rapid-oscillating complex-valued breather amplitudes both spatially and temporally.

The trajectories for the spatial parameterization in the complex-plane exhibit a set of straight lines for all the breathers. From $x\to -\infty$ to $x\to +\infty$, the paths are passed twice for the Kuznetsov-Ma breather and are elapsed many times infinitely for the Akhmediev soliton due to its spatial periodic characteristics. The trajectories in the complex plane for the parameterization in the temporal variable of the Kuznetsov-Ma breather and Peregrine soliton feature a periodic circle and a set of periodic ellipses due to its temporal symmetry. For the Akhmediev soliton, on the other hand, the path does not only turn into circle and ellipse sectors but also becomes straight lines as it travels from $t\to -\infty$ to $t\to +\infty$, featuring homoclinic orbits with a phase shift.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author Contributions

The author confirms being the sole contributor of this work and has approved it for publication.

Dedication

The author would like to dedicate this article to his late father Zakaria Karjanto (Khouw Kim Soey, 許金瑞) who not only taught him the alphabet, numbers, and the calendar in his early childhood but also cultivated the value of hard work, diligence, discipline, perseverance, persistence, and grit. Karjanto Senior was born in Tasikmalaya, West Java, Japanese-occupied Dutch East Indies on 1 January 1944 (Saturday Pahing) and died in Bandung, West Java, Indonesia on 18 April 2021 (Sunday Wage).

Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The author wishes to thank Bertrand Kibler, Amin Chabchoub, and Heremba Bailung for the invitation to contribute to the article collection “Peregrine Soliton and Breathers in Wave Physics: Achievements and Perspectives” and Marlena Radomska from the Production Team of Frontiers in Physics for her excellent and outstanding administrative assistance. The author also acknowledges E. (Brenny) van Groesen, Mark Ablowitz, Constance Schober, Frederic Dias, Roger Grimshaw, Panayotis Kevrekidis, Boris Malomed, Evgenii Kuznetsov, Nail Akhmediev, Alfred Osborne, Miguel Onorato, Gert Klopman, René Huijsmans, Andonowati, Stephan van Gils, Guido Schneider, Anthony Roberts, Shanti Toenger, Omar Kirikchi, Ardhasena Sopaheluwakan, Hadi Susanto, Alexander Iskandar, Agung Trisetyarso, Ade Irma Suriajaya, and Defrianto Pratama for fruitful discussion.

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