Peregrine soliton as a limiting behavior of the Kuznetsov-Ma and Akhmediev breathers
aa r X i v : . [ n li n . PS ] S e p Peregrine soliton as a limiting behavior of theKuznetsov-Ma and Akhmediev breathers
N. Karjanto ∗ Department of Mathematics, University College, Natural Science Campus,Sungkyunkwan University, 2066 Seobu-ro, Suwon, 16419, Republic of Korea
Correspondence*:Corresponding [email protected]
ABSTRACT
This article discusses a limiting behavior of breather solutions of the focusing nonlinearSchr ¨odinger (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 latterbecomes a nonlinear extension of the homoclinic orbit waveform corresponding to an unstablemode in the modulational instability phenomenon. All breathers are prototypes for rogue wavesin nonlinear and dispersive media. We present a rigorous proof using the ǫ - δ argument and showthe corresponding visualization for this limiting behavior. Keywords: nonlinear Schr ¨odinger equation, Kuznetsov-Ma breather, Akhmediev soliton, Peregrine soliton, waves on a non-vanishingand constant background, limiting behavior, modulational instability, rogue waves
The Peregrine soliton, also known as the rational solution, is one of the solutions of the focusingnonlinear Schr¨odinger (NLS) equation. Analyzed and derived for the first time by Peregrine in 1983,its characteristic is localized in both space and time [1]. The Peregrine soliton was successfully observedexperimentally in nonlinear optics [2], water waves [3], and multi-component plasma [4]. Together withthe Kuznetsov-Ma breather and Akhmediev soliton, the Peregrine soliton belongs to the families of solitonsolutions of the NLS equation on a non-vanishing but the constant background, where the plane-wave orcontinuous-wave solution acts for such a pedestal.The NLS equation is a nonlinear evolution equation that models slowly varying envelope dynamics ofa weakly nonlinear quasi-monochromatic wave packet in dispersive media. The model has an infiniteset of conservation laws and belongs to a completely integrable system of nonlinear partial differentialequations through the Inverse Scattering Transform (IST). It has a wide range of applications in variousphysical settings, such as surface water waves, nonlinear optics, plasma physics, superconductivity, andBose-Einstein condensates (BEC).The NLS equation in nonlinear optics was first derived by Kelley in 1965 using a nonlinearelectromagnetic wave Maxwell’s equation introduced by Chiao et al. one year earlier [5, 6]. Furthermore,Karpman and Krushkal in 1969 derived the NLS equation using the Whitham-Lighthill adiabaticapproximation, where the original article in Russian was published one year earlier [7]. Tappert andVarma also derived the NLS equation for heat pulses in solids using the asymptotic theory in 1970 [8].In 1968, Taniuti and Washimi derived the NLS equation describing dispersive hydromagnetic wavespropagating along an applied magnetic field in a cold quasi-neutral plasma using the method of multiple . Karjanto Peregrine soliton scales [9]. To the best of our knowledge, it is this article that mentions, introduces, and hence popularizesthe term “nonlinear Schr¨odinger equation” for the first time. One year later, Taniuti, Yajima, and Asanoderived the NLS equation using the perturbation method in an electron plasma wave system that admitsplane-wave solution with high-frequency oscillation and the nonlinear Klein-Gordon equation [10, 11].The NLS equation was first derived independently in hydrodynamics by Benney and Newell in 1967 forwave packet envelopes propagation and the long time behavior of weakly interacting waves [12] and in1968 by Zakharov for deep-water waves using a spectral method [13]. The NLS equation for gravity waterwaves with uniform and finite depth was derived by Hasimoto and Ono in 1972 using singular perturbationmethods [14].In the field of BEC, the NLS equation with the non-zero potential term is known by another name:the Gross-Pitaevskii equation, where both Gross and Pitaevskii independently derived this equation in1961 [15, 16]. In the field of superconductivity, the time-independent NLS equation resembles somesimilarities with a simplified (1 + 1) -D form of the Ginzburg-Landau equation derived a decade earlier in1950 [17]. For a further overview of the NLS equation, see the encyclopedia articles [18, 19] and bookchapters [20, 21]. For an extensive discussion of the NLS equation, see the monographs [22, 23].The NLS equation admits exact analytical solutions, and some families of these solutions, known as“breather solitons”, are excellent prototypes for rogue wave modeling in various nonlinear media. Anexplicit expression of the breather solitons will be presented in Section 2. In what follows, we provide anoverview of breather solitons and their connection with rogue wave phenomena.There are various excellent reviews on rogue wave phenomena based on the NLS equation as amathematical model and its corresponding breather solitons. Onorato et al. covered rogue waves inseveral physical contexts including surface gravity waves, photonic crystal fibers, laser fiber systems,and 2D spatiotemporal systems [24]. Dudley et al. reviewed breathers and rogue waves in optical fibersystems with an emphasis on the underlying physical processes that drive the appearance of extremeoptical structures [25]. They reasoned that the mechanisms driving rogue wave behavior depend verymuch on the system. Residori et al. presented physical concepts and mathematical tools for rogue wavedescription [26]. They highlighted the most common features of the phenomenon include large deviationsof wave amplitude from the Gaussian statistics and large-scale symmetry breaking. Chen et al. discussedrogue waves in scalar, vector, and multidimensional systems [27] while Malomed and Mihalache surveyedsome theoretical and experimental studies on nonlinear waves in optical and matter-wave media [28].Rogue waves come from and are closely related to modulational instability with resonance perturbationon continuous background [29]. A comparison of breather solutions of the NLS equation withemergent peaks in noise-seeded modulational instability indicated that the latter clustered closely aroundthe analytical predictions [30]. “Superregular breathers” is the term coined indicating creation andannihilation dynamics of modulational instability, and the evidence of the broadest group of thesesuperregular breathers in hydrodynamics and optics has been reported [31]. An interaction betweenbreather and higher-order rogue waves in a nonlinear optical fiber is characterized by a trajectory oflocalized troughs and crests [32].Breather soliton solutions find several applications, among others in beam-plasma interactions [33], inthe transmission line analog of a nonlinear left-handed metamaterials [34], in a nonlinear model describingan electron moving along the axis of a deformable helical molecules [35], and in the mechanismsunderlying the formation of and real-time prediction of extreme events [36]. Additionally, optical rogue . Karjanto Peregrine soliton waves also successfully simulated in the presence of nonlinear self-image phenomenon in the near-fielddiffraction of plane waves from light wave grating, known as the Talbot effect [37].Since the definitions of “rogue waves” and “extreme events” are varied, a roadmap for unifying differentperspectives could stimulate further discussion [38]. Theoretical, numerical, and experimental evidenceof the dissipation effect on phase-shifted Fermi-Pasta-Ulam-Tsingou (FPUT) dynamics in a super wavetank, which is related to modulational instability, can be described by the breather solutions of the NLSequation [39]. Since the behavior of a large class of perturbations characterized by a continuous spectrumis described by the identical asymptotic state, it turns out that the asymptotic stage of modulationalinstability is universal [40]. Surprisingly, the long-time asymptotic behavior of modulationally unstablemedia is composed of an ensemble of classical soliton solutions of the NLS equation instead of thebreather-type solutions [41].There are various techniques to derive the breather solutions of the focusing NLS equation, among othersare the phase-amplitude algebraic ansatz [42–45], the Hirota method [46–49], nonlinear Fourier transformIST [50–54], symmetry reduction methods [55], variational formulation and displaced phase-amplitudeequations [56–58]. Another derivation using IST with asymmetric boundary conditions is given in [59].General N -solitonic solutions of the NLS equation in the presence of a condensate derived using thedressing method describe the nonlinear stage of the modulational instability of the condensate [60].Recently, both theoretical description and experimental observation of the nonlinear mutual interactionsbetween a pair of copropagative breathers are presented and it is observed that the bound state of breathersexhibit a behavior similar to a molecule with quasiperiodic oscillatory dynamics [61].The purpose of this paper is to revisit the connection between the families of the breather solitonsolutions, both analytically and visually. The paper will be presented as follows. After this introduction,Section 2 presents several important exact solutions of the NLS equation. In particular, the focus will beon the breather type of solutions. Section 3 discusses a rigorous proof for the limiting behavior of thebreather wave solutions using the ǫ - δ argument. The limiting behavior will continue in Section 4, wherewe cover it from the visual point of view. We present the corresponding contour plots for various valuesof parameters and the parameterization sketches of the non-rapid oscillating complex-valued breatheramplitudes. Finally, Section 5 concludes our discussion and provide remarks for potential future research. Throughout this article, we adopt the following (1 + 1) -dimension, focusing-type of the NLS equation ina standard form: iq t + q xx + 2 | q | q = 0 , q ( x, t ) ∈ C . (1)Usually, the variables x and t denote the space and time variables, respectively. The simplest-solution iscalled the plane-wave or continuous-wave solution: q ( x, t ) = q ( t ) = e it . Another simple solution witha vanishing background is known as the bright soliton or one-soliton solution, given as follows: q ( x, t ) = q S ( x, t ) = a sech ( ax − abt + θ ) e i ( bx + ( a − b ) t + φ ) , a, b, θ , φ ∈ R . (2)Zakharov and Shabat obtained this solution in the 1970s using the IST [62, 63].Throughout this article, our discussion will be focused on the type of NLS solutions with constant andnon-vanishing background, sometimes also called the families of “breather soliton solutions” [64]. There . Karjanto Peregrine soliton are three families of breathers, and all of them are considered as weakly nonlinear prototypes for freakwaves events. Other solutions of the NLS equation include cnoidal wave envelopes that can be expressedin terms of the Jacobi elliptic functions and can be derived using the Hirota bilinear transformation, thetafunctions, or with some clever algebraic ansatz [45, 65].Otherwise mentioned, our coverage will also follow the historical order of the time when the breathersare found. Thus, when both breathers are discussed, we usually cover the Kuznetsov-Ma breather beforethe Akhmediev soliton. Furthermore, the term “breather” and “soliton” can be used interchangeably inthis article, and they can also appear as a single term “breather soliton”. All of them refer to the sameobject, i.e., the exact analytical solutions of the NLS equation with a non-vanishing, constant pedestal, orbackground of continuous-wave solution.
The first one is called the Kuznetsov-Ma breather, where Kuznetsov, Kawata and Inoue, and Ma derivedit independently in the late 1970s [66–68]. So, perhaps a more accurate name for this breather is theKKIM breather, which stands for the “Kuznetsov-Kawata-Inoue-Ma breather”. However, ever since thebreather dynamics are observed experimentally in optical fibers in 2012 [69], the former name is gettingmore popular even though a similar term “Kuznetsov-Ma soliton” has been introduced earlier [70]. Hence,we adopt and use the terminology “Kuznetsov-Ma breather” throughout this article. We denote it as q M and is explicitly given by q ( x, t ) = q M ( x, t ) = e it (cid:18) µ cos( ρt ) + iµρ sin( ρt )2 µ cos( ρt ) − ρ cosh( µx ) + 1 (cid:19) (3)where ρ = µ p µ . The Kuznetsov-Ma breather does not represent a traveling wave. It is localized inthe spatial variable x and periodic in the temporal variable t , and hence some authors also called it as the“temporal periodic breather” [71].A minor typographical error found in Kawata and Inoue’s paper [67] has been corrected by Gagnon [72].Kawata and Inoue [67], as well as Ma [68], derived the Kuznetsov-Ma breather solution using the IST forfinite boundary conditions at x → ±∞ . The derivation using a direct method of B¨acklund transformationcan be found in [73, 74], where the former analyzed solitary waves in the context of an optical bistablering cavity.Defining the amplitude amplification factor (AF) as the quotient of the maximum breather amplitudeand the value of its background [57], we obtain that the amplitude amplification for the Kuznetsov-Mabreather is always greater than the factor of three and is explicitly given byAF M ( µ ) = 1 + p µ , µ > . (4)The function is bounded below and is increasing as the parameter µ also increases. The plot of this AFcan be found in [57, 58], and different expressions of AF for this breather also appear in [24, 26, 75, 76].The Kuznetsov-Ma breather finds applications as a rogue wave prototype in nonlinear optics [45, 69, 77]and deep-water gravity waves [52, 53, 76, 78]. A numerical comparison of the Kuznetsov-Ma breatherindicated that a qualitative agreement was reached in the central part of the corresponding wave packetand on the real face of the modulation [75]. The stability analysis of the Kuznetsov-Ma breather using . Karjanto Peregrine soliton a perturbation theory based on the IST verified that although the soliton is rather robust with respect todispersive perturbations, damping terms strongly influence its dynamics [79].Dynamics of the Kuznetsov-Ma breather in a microfabricated optomechanical array showed an excellentagreement between theory and numerical calculations [80]. The spectral stability analysis of this breatherhas been considered using the Floquet theory [81]. The mechanism of the Kuznetsov-Ma breather has beendiscussed and two distinctive mechanisms are paramount: modulational instability and the interferenceeffects between the continuous-wave background and bright soliton [82]. New scenarios of rogue waveformation for artificially prepared initial conditions using the Kuznetsov-Ma and superregular breathers insmall localized condensate perturbations are demonstrated numerically by solving the Zakharov-Shabateigenvalue problem [83].A higher-order Kuznetsov-Ma breather can be derived using the Hirota method and utilized in studyingsoliton propagation with the presence of small plane-wave background [84, 85]; or using the bilinearmethod [86].
The second one is called the Akhmediev-Eleonski˘i-Kulagin breather and was found in the 1980s [42–44]. In short, we simply call it the “Akhmediev soliton” and denote it as q A . This breather is localized inthe temporal variable t and is periodic in the spatial variable x , and it can be written explicitly as follows: q ( x, t ) = q A ( x, t ) = e it (cid:18) ν cosh( σt ) + iνσ sinh( σt )2 ν cosh( σt ) − σ cos( νx ) − (cid:19) . (5)Here, the parameter ν , ≤ ν < denotes a modulation frequency (or wavenumber) and σ ( ν ) = ν √ − ν is the modulation growth rate. The colleagues from nonlinear optics prefer calling this solitonas “instanton” instead of “breather” since it breathers only once [87]. Other names for this solution include“modulational instability” [45], “homoclinic orbit” [49, 88], “spatial periodic breather” [71], and “roguewave solution” [53].The amplitude amplification for the Akhmediev soliton is at most of the factor of three and is explicitlygiven by AF A ( ν ) = 1 + p − ν , < ν < . (6)This function is bounded above and below, < AF A < , and is decreasing for an increasing value of themodulation parameter ν . Although the maximum growth rate occurs for ν = √ , the maximum AF occurswhen ν → , 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 oceanwaves where this AF depends on the wave steepness and number of waves under the envelope [89]. Theplot for this AF can be found in [57, 90, 91]. Some variations in the AF expression for this soliton alsoappear in [24, 26, 75, 76, 92, 93].The Akhmediev soliton is rather well-known due to its characteristics being a nonlinear extension oflinear modulational instability. This instability is also known as sideband (or Bespalov-Talanov) instabilityin nonlinear optics [94–96], or Benjamin-Feir instability in water waves [13, 97]. Some authors studied themodulational instability in plasma physics [9, 98–100] and in BEC [101–106]. Modulational instability isdefined as the temporal growth of the continuous-wave NLS solution due to a small, side-band modulation, . Karjanto Peregrine soliton in a monochromatic wave train. A geometric condition for wave instability in deep water waves is givenin [107] and for a historical review of modulational instability, see [108].It has been shown numerically and experimentally that the modulated unstable wave trains grow toa maximum limit and then subside. In the spectral domain, the wave energy is transferred from thecentral frequency to its sidebands during the wave propagation for a certain period, and then it isrecollected back to the primary frequency mode [109–113]. It turns out that the long-time evolution ofthese unstable wave trains leads to a sequence of modulation and demodulation cycles, known as theFPUT recurrence phenomenon [114, 115]. Although the FPUT recurrence using the NLS model has beenobserved experimentally in surface gravity waves in the late 1970s [110], it took more than two decadesfor the phenomenon to be successfully recovered in nonlinear optics [116].Since the modulational instability extends nonlinearly to the Akhmediev soliton, it is no surprise thatthe former is considered as a possible mechanism for the generation of rogue waves while the latteracts as one prototype [117–119]. For wave trains with amplitude and phase modulation, there is acompetition between the nonlinearity and dispersive factors. After the modulational instability occurs, thegrowth predicted by linear theory is exponential, and the nonlinear effect in the form of the Akhmedievsoliton takes over before the wave trains return to the stage similar to the initial profiles with a phase-shift difference [78, 120]. On the other hand, Biondini and Fagerstrom argued that the major cause ofmodulational instability in the NLS equation is not the breather soliton solutions per se, but the existenceof perturbations where discrete spectra are absence [121].Experimental attempts on deterministic rogue wave generation using the Akhmediev solitons suggestedthat the symmetric structure is not preserved and the wave spectrum experiences frequency downshift eventhough wavefront dislocation and phase singularity are visible [57, 122–126]. A numerical calculation ofrogue wave composition can be described in the form of the collision of Akhmediev breathers [127].Another comparison of the Akhmediev breathers with the North Sea Draupner New Year and the Seaof Japan Yura wave signals also show some qualitative agreement [128]. The characteristics of theAkhmediev solitons have also been observed experimentally in nonlinear optics [129].A theoretical, numerical, and experimental report of higher-order modulational instability indicatesthat a relatively low-frequency modulation on a plane-wave induces pulse splitting at different phasesof evolution [130]. Second-order breathers composed of nonlinear combinations of the Kuznetsov-Mabreather and Akhmediev soliton reveal the dependence of the wave envelope on the degenerate eigenvaluesand differential shifts [131]. Similar higher-order Akhmediev solitons visualized in [130, 131] has beenfeatured earlier in [57, 132] and similar illustrations can also be found in [24, 25, 133–138].
The third one is called the Peregrine soliton, also known as the rational solution [1]. This soliton islocalized in both spatial and temporal variables ( x, t ) and is written as follows (denoted as q P ): q ( x, t ) = q P ( x, t ) = e it (cid:18) it )1 + 16 t + 4 x − (cid:19) . (7)This solution is neither a traveling wave nor contains free parameters. Johnson called it a “rational-cum-oscillatory solution” [139], others referred to it as the “isolated Ma soliton” [140], an “explode-decay . Karjanto Peregrine soliton solitary wave” [141], the “rational growing-and-decaying mode” [85], the “algebraic breather” [142], orthe “fundamental rogue wave solution” [136].The amplitude amplification for the Peregrine soliton is exactly of a factor three and this can be obtainedby taking the limit of the parameters toward zero in the previous two breathers:AF P = lim µ → AF M ( µ ) = 3 = lim ν → AF A ( ν ) . (8)Although the other two breather solitons are also proposed as rogue wave prototypes, some authors arguedthat the Peregrine soliton is the most likely freak wave event due to its appearance from nowhere anddisappearance without a trace [91] as well as its closeness to all initial supercritical humps of smalluniform envelope amplitude [143]. Some numerical and experimental studies may support this reasoning.Henderson et al. studied numerically unsteady surface gravity wave modulations by comparing thefully nonlinear and NLS equations [140]. For steep-wave events, their computations produced strikingsimilarities with the Peregrine soliton. On the other hand, Voronovich et al. confirmed numerically thatthe bottom friction effect, even when it is small in comparison to the nonlinear term, could hamperthe formation of breather freak wave at the nonlinear stage of instability [144]. Investigations on linearstability demonstrated that the Peregrine soliton is unstable against all standard perturbations, where theanalytical study is supported by numerical evidence. [145–148].A sequence of experimental studies using the Peregrine soliton demonstrated reasonably goodqualitative agreement with the theoretical prediction. Some discrepancies occur in the modulationalgradients, spatiotemporal symmetries, and for larger steepness values [149], as well as the frequencydownshift [150]. Interestingly, Chabchoub et al. shown further experimentally that the dynamics of thePeregrine soliton and its spectrum characteristics persist even in the presence of wind forcing with highvelocity [151]. By selecting a target location and determining an initial steepness, an experiment using thePeregrine soliton of wave interaction with floating bodies during extreme ocean condition has also beensuccessfully implemented [152].The Peregrine soliton also finds applications in the evolution of the intrathermocline eddies, also knownas the oceanic lenses [153]. It appeared as a special case of stationary limit in the solutions of the spinorBEC model [154], and it was observed experimentally emerging from the stochastic background in surfacegravity deep-water waters [155].Nonlinear spectral analysis using the finite gap theory showed that the spectral portraits of the Peregrinesoliton represent a degenerate genus two of the NLS equation solution [156]. Higher-order Peregrinesolitons in terms of quasi-rational functions are derived in [157]. Higher-order Peregrine solitons up tothe fourth-order using a modified Darboux transformation has been presented with applications in roguewaves in the deep ocean and high-intensity rogue light wave pulses in optical fibers [158]. Super roguewaves modeled with higher-order Peregrine soliton with an amplitude amplification factor of five timesthe background value are observed experimentally in a water-wave tank [134].The following section will show that as the parameter values µ → and ν → , the Kuznetsov-Ma andAkhmediev breathers reduce to the Peregrine soliton. . Karjanto Peregrine soliton
This section provides a rigorous proof of the limiting behavior of breather wave solutions using the ǫ - δ argument. We have the following theorem:T HEOREM The Peregrine soliton is a limiting case for both the Kuznetsov-Ma breather andAkhmediev soliton: lim µ → q M ( x, t ) = q P ( x, t ) = lim ν → q A ( x, t ) . (9)We divide the proof into four parts, and each limit consists of two parts corresponding to the real andimaginary parts of the solitons.P ROOF . The following shows that the limit for the real parts of the Kuznetsov-Ma breather andPeregrine soliton is correct, i.e., lim ν → Re { q M ( x, t ) } = Re { q P ( x, t ) } . For each ǫ > , there exists δ = p ( ǫ + 2) − > such that if < µ < δ , then | Re { q M }− Re { q P } | < ǫ .We know that since cosh µ ( x − x )cos ρ ( t − t ) ≥ for all ( x, t ) ∈ R it then implies ρ cosh µ ( x − x )cos ρ ( t − t ) − µ ≥ ρ − µ. It follows that | Re { q M } − Re { q P } | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µ ρ cosh ν ( x − x )cos ρ ( t − t ) − µ −
41 + 16( t − t ) + 4( x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) µ ρ − µ − (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)p µ + 4 − (cid:12)(cid:12)(cid:12) ≤ p δ + 4 − s(cid:18)q ( ǫ + 2) − (cid:19) + 4 − ǫ. The following verifies that the limit for the imaginary parts of the Kuznetsov-Ma breather and Peregrinesoliton is accurate, i.e., lim ν → Im { q M ( x, t ) } = Im { q P ( x, t ) } . . Karjanto Peregrine soliton
For each ǫ > , there exists δ = q −
10 + 2 p
25 + 4 ǫ/ | t − t | > such that if < µ < δ , then | Im { q M } − Im { q P } | < ǫ . We can write the imaginary parts of q M and q P as followsIm { q M } = µρρ cosh µ ( x − x )sin ρ ( t − t ) − µ cot ρ ( t − t ) ≤ µρ | t − t | ρ − µ Im { q P } = 16( t − t )1 + 16( t − t ) + 4( x − x ) ≤ | t − t | . It follows that | Im { q M } − Im { q P }| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) µρρ cosh µ ( x − x )sin ρ ( t − t ) − µ cot ρ ( t − t ) − t − t )1 + 16( t − t ) + 4( x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) µρ ρ − µ − (cid:12)(cid:12)(cid:12)(cid:12) | t − t | = (cid:12)(cid:12)(cid:12) ( µ + 4) (cid:16)p µ + 4 + 2 (cid:17) − (cid:12)(cid:12)(cid:12) | t − t | < (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) δ + 4 (cid:1) (cid:18) δ (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) | t − t | = (cid:12)(cid:12)(cid:12)(cid:12) δ δ (cid:12)(cid:12)(cid:12)(cid:12) | t − t | = ǫ. In what follows, we present the limit of the real part of the Akhmediev soliton as ν → is indeed thereal part of the Peregrine solitons, i.e., lim ν → Re { q A ( x, t ) } = Re { q P ( x, t ) } . For each ǫ > , there exists δ = p ǫ/ > such that if < ν < δ < , then | Re { q A } − Re { q P } | < ǫ .We know that since − ≤ cos ν ( x − x )cosh σ ( t − t ) ≤ for all ( x, t ) ∈ R it then implies ν − σ ≤ ν − σ cos ν ( x − x )cosh σ ( t − t ) . We also have t − t ) + 4( x − x ) ≥ for all ( x, t ) ∈ R . Furthermore, since ≤ √ − ν ≤ , ≤ − √ − ν ≤ , ≤ − √ − ν ν ≤ , ≤
14 + 2 − √ − ν ν ≤ and − √ − ν ν ≥ δ . Karjanto Peregrine soliton it follows that | Re { q A } − Re { q P } | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ν ν − σ cos ν ( x − x )cosh σ ( t − t ) −
41 + 16( t − t ) + 4( x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) ν ν − σ + 4 (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −√ − ν ν + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) + −√ − ν ν (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) −√ − ν ν (cid:12)(cid:12)(cid:12) ≤ (cid:0) δ (cid:1) = 3 (cid:18)r ǫ (cid:19) = ǫ. In what follows, we demonstrate that the limit of the imaginary part of the Akhmediev soliton becomesthe imaginary part of the Peregrine soliton, i.e., lim ν → Im { q A ( x, t ) } = Im { q P ( x, t ) } . For each ǫ > , there exists δ = p ǫ/ > such that if < ν < δ < , then | Im { q A } − Im { q P } | < ǫ .We can write the imaginary parts of q A and q P as followsIm { q A } = νσ tanh σ ( t − t )2 ν − σ cos ν ( x − x )cosh σ ( t − t ) ≤ νσ | t − t | ν − σ Im { q P } = 16( t − t )1 + 16( t − t ) + 4( x − x ) ≤ | t − t | . Since √ − ν ≤ and (4 − ν ) ≤ δ / | t − t | , it follows that | Im { q A } − Im { q P }| = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) νσ tanh σ ( t − t )2 ν − σ cos ν ( x − x )cosh σ ( t − t ) − t − t )1 + 16( t − t ) + 4( x − x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12) νσ ν − σ − (cid:12)(cid:12)(cid:12)(cid:12) | t − t | = (cid:12)(cid:12)(cid:12) (4 − ν )(2 + p − ν ) − (cid:12)(cid:12)(cid:12) | t − t | < (cid:12)(cid:12)(cid:12)(cid:12) (cid:18) δ | t − t | (cid:19) − (cid:12)(cid:12)(cid:12)(cid:12) | t − t | = 4 δ = 4 (cid:18)r ǫ (cid:19) = ǫ. We have completed the proof. (cid:3)
In the following section, we will visualize the limiting behavior of the breather solutions as theyapproach toward the Peregrine soliton.
In this section, we will visually confirm the limiting behavior of the Kuznetsov-Ma and Akhmedievbreathers toward the Peregrine soliton as both parameter values approach zero. Subsection 4.1 presents . Karjanto Peregrine soliton the contour plots of the amplitude modulus, and Subsection 4.2 discusses the spatial and temporalparameterizations of the breathers. We select several parameter values in sketching the plots. Figure 1displays the chosen parametric values for both breather solutions, where they can be visualized in thecomplex-plane for the parameter pair ( µ, ν ) . ν µ C q A ( x, t ) , < ν < ν = 1 ν =
12 14 ν = q P ( x, t ) µ = µ = µ = 1 √ µ = √ q M ( x, t ) , µ > Figure 1.
Selected parametric values ν and µ = iν displayed in the complex plane for the Kuznetsov-Mabreather and Akhmediev soliton visualized in this section. In this subsection, we observe the contour plots of the amplitude modulus of the breather and howthe changes in the parameter values affect the envelope’s period and wavelength. Similar contour plotshave been presented in the context of electronegative plasmas with Maxwellian negative ions [159]. Inparticular, the contour plot of the Peregrine soliton is also displayed in [149].Figures 2(a)–2(e) display the contour plots of the Kuznetsov-Ma breather for several values ofparameters µ : √ , , , and . Figure 2(e) is a zoom-in version of the same contour plot given inFigure 2(d). Figure 2(f) is the final stop when we let the parameter µ → , for which the Kuznetsov-Ma breather turns into the Peregrine soliton. It is interesting to note that for µ = , the contour plot isnearly identical with the one from the Peregrine soliton, as we can observe by qualitatively comparingpanels (e) and (f) of Figure 2.Parameter values Temporal envelope period µ (exact) µ (decimal) ρ (exact) ρ (approximation) T M (exact) T M (approximation) / √ /
25 0 .
402 50 π/ √
101 15 . / √ / .
031 8 π/ √
17 6 . √ .
236 2 π/ √ . √ √ . π/ √ . Table 1.
Exact values of the temporal envelope period T M and their approximate values for selectedparameter values µ corresponding to the Kuznetsov-Ma breather. . Karjanto Peregrine soliton − . . x − − − − t . . . . . . . . | q M ( x, t ) | (a) µ = √ − . . x − − − − t . . . . . . . . | q M ( x, t ) | (b) µ = 1 − . . x − − − − t . . . . . . . . | q M ( x, t ) | (c) µ = − . . x − − t . . . . . . . . . | q M ( x, t ) | (d) µ = − − − x − − t . . . . . . . . . | q M ( x, t ) | (e) µ = − − − x − − t . . . . . . . . . | q P ( x, t ) | (f) µ = 0 Figure 2.
Contour plots for the moduli of the Kuznetsov-Ma breather for (a) µ = √ , (b) µ = 1 , (c) µ = 0 . , (d) µ = 0 . , (e) also µ = 0 . but a zoom-in version, and (f) µ = 0 , which gives the Peregrinesoliton. Notice that the contour plots (e) and (f) are qualitatively nearly identical.Let T M denote the temporal envelope period for the Kuznetsov-Ma breather, then we know that ingeneral, T M = 2 π/ρ . For µ → ∞ , T M → and vice versa, for µ → , T M → ∞ . For any given value of µ > , T M can be easily calculated. Here are some examples. For µ = √ , T M = π/ √ ≈ . and wedisplay five periods in Figure 2(a) along the temporal axis t . For µ = 1 , T M = 2 π/ √ ≈ . and for thesame time interval as in panel (a), we can only capture three periods along the temporal axis t , as shownin Figure 2(b). Furthermore, for µ = , T M = 8 π/ √ ≈ . and we need to extend almost twice lengthin the time interval in order to capture at least three periods. Figure 2(c) shows this contour plot. Finally,for µ = , T M = 50 π/ √ ≈ . . As we can observe in Figure 2(d), extending the length of timeinterval to around 40 units is sufficient to capture at least three periods, albeit the detail around maximumand minimum is hardly visible. Table 1 displays selected parameter values of the Kuznetsov-Ma breatherand their corresponding temporal envelope periods T M . . Karjanto Peregrine soliton − − − − x − − t . . . . . . . . . | q A ( x, t ) | (a) ν = 1 − − x − − t . . . . . . . . . | q A ( x, t ) | (b) ν = − − −
10 0 10 20 30 x − − t . . . . . . . . . | q A ( x, t ) | (c) ν = − − − x − − t . . . . . . . . . | q P ( x, t ) | (d) ν = 0 Figure 3.
Contour plots for the moduli of the Akhmediev breather for (a) ν = 1 , (b) ν = 0 . , and (c) ν = 0 . , as well as (d) the Peregrine soliton.Figures 3(a)–3(c) display the contour plot of the Akhmediev soliton for selected values of its parameters ν : , , and . Figure 3(d) shows the contour plot of the Peregrine soliton, which occurs as the finaldestination when letting the parameter ν → . Figure 3(d) is identical to Figure 2(f), the only differencelies in the length-scale of both horizontal and vertical axes. Similar to the previous case, zooming-in thecontour plot for ν = in Figure 3(c) will yield a qualitatively nearly identical contour plot with thePeregrine soliton shown in the panel (d). (It is not shown in the figure.)Let L A denote the spatial envelope wavelength for the Akhmediev soliton, then for < ν < , L A =2 π/ν , which gives L A > π . For ν → , L A → π and for ν → , L A → ∞ . Table 2 displays selectedvalues of the parameter ν and their corresponding spatial envelope wavelength L A for the Akhmedievsoliton. For ν = 1 , L A = 2 π and the spatial length of 20 units in Figure 3(a) is sufficient to capture threeenvelope wavelength. For ν = 1 / , L A = 4 π and the spatial length of 40 units in Figure 3(b) is requiredto capture at least three envelope wavelength. For ν = 1 / , L A = 8 π and the spatial length of 60 unitsin Figure 3(c) is needed to capture at least three envelope wavelength. The details around maxima andminima are hardly visible for the latter. . Karjanto Peregrine soliton
Parameter values Spatial envelope wavelength ν (exact) ν (decimal) σ (exact) σ (approximation) L A (exact) L A (approximation) / √ /
16 0 .
496 8 π . / √ / .
968 4 π . √ .
732 2 π . Table 2.
Exact values of the spatial envelope wavelength L A and their approximate values for selectedparameter values ν corresponding to the Akhmediev soliton. In this subsection, we write the breather solutions as q X ( x, t ) = q ( t ) ˜ q X ( x, t ) , where q ( t ) is the plane-wave solution and X = { M , A , P } . Since the plane-wave solution gives a fast-oscillating effect, we onlyconsider the non-rapid oscillating part of the breathers ˜ q X for the parameterization visualization. In thesubsequent 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 [39,45, 76, 77].Figure 4 displays the parameterization of the non-rapid oscillating Kuznetsov-Ma breather ˜ q M in thespatial variable x for different values of the temporal variable t and parameter µ . Different panels indicatedifferent parameter values µ and for each panel, different curves, for which in this particular case, theyare merely straight lines, indicate different time t . For all cases, we consider x ≥ due to the symmetrynature of the breathers. The straight-line trajectories move inwardly focused from the dotted blue circleat x = 0 toward ( − , as x → ∞ . The situation is simply reversed for x < : the path of trajectoriesmove outwardly defocused as x progresses from ( − , at x → −∞ 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 theselected values of t for − T M < − π ≤ t ≤ π < T M . The trajectories in the upper-part and lower-partof the complex-plane correspond to the positive and negative values of t , respectively. We observe that thetrajectories shift faster in space around t = 0 than around t = ± T M = ± πρ .In particular, for t = nπ/ρ , n ∈ Z , ˜ q M reduces to a real-valued function, i.e., Im (˜ q M ) = 0 for all µ > .Hence, the parameterized curve is a straight line at the real-axis. For t = 2 nπ/ρ , n ∈ Z , this is shownby the horizontal solid red line lying on the real axis moving from a point larger than Re (˜ q M ) = 3 toRe (˜ q M ) = − for x > . The represented case t = 0 is displayed in Figure 4 while the case t = π/ρ isnot shown in the figure. Indeed, from (3), we obtain the following limiting values for n ∈ Z : lim x → q M ( x, nπ/ρ ) = 1 + p µ + 4 and lim x → q M ( x, (2 n + 1) π/ρ ) = 1 − p µ + 4 . (10)Additionally, lim x →±∞ q M ( x, nπ/ρ ) = − . Using a similar analysis, vertical straight lines at Re (˜ q M ) = − can be obtained by taking the values of t = ( n + ) π/ρ , for n ∈ Z . The line direction from the positiveand negative regions of Im (˜ q M ) is downward and upward toward ( − , for even and odd values of n ∈ Z ,respectively.Figure 5 displays the sketch of the non-rapid-oscillating Kuznetsov-Ma breather ˜ q 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 M = − πρ < t < πρ = T M . Insteadof a set of straight lines, the trajectories form the shape of elliptical curves. For each x = x ∈ R , the . Karjanto Peregrine soliton
Im(˜ q M ) Re(˜ q M )3 − x = 0 (a) µ = √ Im(˜ q M ) Re(˜ q M )3 − x = 0 (b) µ = 1 Im(˜ q M ) Re(˜ q M )3 − x = 0 (c) µ = 1 / Im(˜ q M ) Re(˜ q M )3 − x = 0 (d) µ = 1 / t − π − π − π − π − π π π π π π Figure 4.
Parameterization of the non-rapid-oscillating complex-valued amplitude of the Kuznetsov-Mabreather ˜ q M in the spatial variable x , x ≥ , for different values of temporal variable t and different valuesof the parameter µ : (a) µ = √ , (b) µ = 1 , (c) µ = 1 / , and (d) µ = 1 / . The selected values of t are t = 0 (solid red), t = ± π/ (dashed green), t = ± π/ (solid purple), t = ± π/ (dash-dottedmagenta), t = ± π/ (solid cyan), and t = ± π/ (dashed orange).ellipse is centered at ( c ( x ) , with semi-minor axis a ( x ) and semi-major axis b ( x ) , where a ( x ) = µρ cosh ( µx ) d ( x ) b ( x ) = ρ cosh ( µx ) p d ( x ) (11) c ( x ) = 2 µ d ( x ) − d ( x ) = 2 cosh (2 µx ) + µ − . (12) . Karjanto Peregrine soliton
Im(˜ q M ) Re(˜ q M )3 − (a) µ = √ Im(˜ q M ) Re(˜ q M )3 − (b) µ = 1 Im(˜ q M ) Re(˜ q M )3 − (c) µ = Im(˜ q M ) Re(˜ q M )3 − (d) µ = x
18 14 12
Figure 5.
Parameterization of the non-rapid oscillating complex-valued amplitude of the Kuznetsov-Mabreather ˜ q M in the temporal variable t ( − π/ρ < t < π/ρ ) for different values of spatial variable x : x = 0 (solid blue), x = 1 / (long-dashed red), x = 1 / (dash-dotted green), x = 1 / (dashed purple), x = 1 (dash-dotted cyan), and x = 2 (solid magenta) and different values of the parameter µ : (a) µ = √ , (b) µ = 1 , (c) µ = 1 / , and (d) µ = 1 / .The special case of a circle is obtained for x = 0 with the radius r = p µ + 4 centered at (1 , . Allcurves move in the counterclockwise direction for increasing t . For x > , the larger the values of x ,the smaller the ellipses become. The situation is the opposite for x < : smaller values of x (but largelynegative in its absolute value sense) correspond to smaller ellipses in the complex plane. Due to its spatial . Karjanto Peregrine soliton
Im(˜ q A ) Re(˜ q A )3 − x = 0 (a) ν = 1 Im(˜ q A ) Re(˜ q A )3 − x = 0 (b) ν = Im(˜ q A ) Re(˜ q A )3 − x = 0 (c) ν = Im(˜ q P ) Re(˜ q P )3 − x = 0 (d) ν = 0 t
116 18 14 12
Figure 6.
Parameterization of the non-rapid oscillating complex-valued amplitude ˜ q in the spatial variable x for different values of temporal variable t : t = 0 (solid red), t = 1 / (long-dashed green), t = 1 / (dash-dotted purple), t = 1 / (dash magenta), t = 1 / (dash-dotted cyan), t = 1 (dashed orange), t = 2 (solid black), and t = 4 (solid red), and modulation frequencies of the Akhmediev solitons (a) ν = 1 ( ≤ x ≤ π ), (b) ν = ( ≤ x ≤ π ), (c) ν = ( ≤ x ≤ π ), and (d) ν = 0 , x ≥ (Peregrine soliton).symmetry, only the plots for positive values of x are displayed. The axis below the figure panels showsthe selected x values for a better overview of the variable scaling: x = 0 , , , , , and x = 2 .Figure 6 displays the sketch in the complex-plane of the non-rapid-oscillating Akhmediev soliton ˜ q A [panels (a)–(c)] and Peregrine soliton ˜ q P [panel (d)] parameterized in the spatial variable x for differentvalues of the temporal variable t and parameter ν . We only display the trajectories corresponding to thepositive values of t , the trajectories for the negative values of t are simply the reflection over the horizontal . Karjanto Peregrine soliton
Im(˜ q A ) Re(˜ q A )3 − (a) ν = 1 Im(˜ q A ) Re(˜ q A )3 − (b) ν = 0 . Im(˜ q A ) Re(˜ q A )3 − (c) ν = 0 . Im(˜ q P ) Re(˜ q P )3 − (d) ν = 0 x π π π π π Figure 7.
Parameterization of the non-rapid oscillating complex-valued amplitude ˜ q in the temporalvariable t ( −∞ < t < ∞ ) for different values of spatial variable x : x = 0 (solid blue), x = π/ (long-dashed red), x = π/ (dash-dotted green), x = π/ (dashed purple), x = π/ (dash-dotted cyan),and x = π (solid magenta), and modulation frequencies of the Akhmediev solitons (a) ν = 1 , (b) ν = 0 . ,(c) ν = 0 . , and (d) ν → (the Peregrine soliton).axis Re (˜ q P ) = 0 . The t -axis below the panels indicate the chosen values of t displayed in the figure. Similarto the trajectories for the Kuznetsov-Ma breather when they are parameterized in the spatial variable x , thetrajectories for the Akhmediev soliton parameterized in x are also collections of straight lines shifting inthe counterclockwise direction for increasing values of t . Different from the previous case, these straightlines are periodic in x . The experimental results of deterministic freak wave generation using the spatialNLS equation showed that instead of straight lines, we obtained non-degenerate Wessel curves, suggestingthat there the periodic lines might be perturbed during the downstream evolution [57, 125]. . Karjanto Peregrine soliton
Re(˜ q P ) x − (a) Re (˜ q P ) vs. x Im(˜ q P ) x − (b) Im (˜ q P ) vs. x t
116 18 14 12
Re(˜ q P ) t − (c) Re (˜ q P ) vs. t Im(˜ q P ) t − (d) Im (˜ q P ) vs. t x π π π π π Figure 8.
Plots of the real and imaginary parts of the non-rapid oscillating complex-valued amplitudefor the Peregrine soliton with respect to the spatial and temporal variables, x (upper panels) and t (lowerpanels), respectively. For upper panels (a) and (b), various curves indicates different time: t = 0 (solidred), t = 1 / (long-dashed green), t = 1 / (dash-dotted purple), t = 1 / (dash magenta), t = 1 / (dash-dotted cyan), t = 1 (dashed orange), and t = 2 (solid black). For lower panels (c) and (d), differentcurves indicates different positions: x = 0 (solid blue), x = π/ (long-dashed red), x = π/ (dash-dottedgreen), x = π/ (dashed purple), x = π/ (dash-dotted cyan), and x = π (solid magenta).For each panel, we only sketch the trajectories for an interval of half the spatial envelope wavelength,i.e., ≤ x ≤ L A = πν . For this limited space interval, the direction of the lines is moving inwardlyfocused, from the dotted-blue outer circle for x = 0 to some values in the left-part of the complex-planenear Re (˜ q A ) = − . As the value of x progresses, L A = πν ≤ x ≤ L A = πν , the trajectories bounce backtoward the initial points by following the identical paths. They then travel in the same manner periodicallyas x → ±∞ . For a decreasing value of the parameter ν , the endpoint of these lines tends to focus aroundthe region near ( − , , as we can observe in Figures 6(a)–6(c). For the Peregrine soliton, the trajectoriesare not periodic as L A → ∞ and they tend to ( − , for x → ±∞ , as can be seen in Figure 6(d).Figure 7 displays the sketch of the non-rapid-oscillating part of the Akhmediev soliton ˜ q A [panels (a)–(c)] and Peregrine soliton ˜ q P [panel (d)] in the complex-plane parameterized in the temporal variable t fordifferent values of the spatial variable x and parameter ν . The values of t run from t → −∞ to t → + ∞ , . Karjanto Peregrine soliton and we only sketch the positive values of x . The plots for the negative values of x are identical and are notshown due to the symmetry property of the soliton. The x -axis below the panels shows the selected valuesof x ranging from x = 0 to x = π . For ˜ q A , the trajectories are composed of circular sectors, ellipticalsectors, and straight lines instead of closed curves like circles or ellipses. Since this soliton is a nonlinearextension of the modulational instability, the trajectories for each value of the parameter ν , < ν < ,are the corresponding homoclinic orbit for an unstable mode, and the presence of a phase shift preventsclosed-path trajectories [39, 45, 49, 88].The circular sectors are attained for x = 0 and the straight lines occur at x = (cid:0) n + (cid:1) πν , n ∈ Z .Trajectories at other locations yield the elliptical sectors. The initial and final points are not identical andthis indicates a phase shift in the soliton. Let φ + ∞ and φ −∞ be the phases for x → ±∞ , respectively. Letalso ∆ φ = φ + ∞ − φ −∞ be the difference between the phase at x = + ∞ and x = −∞ , then we have thefollowing phase relationships: tan φ ±∞ = ± σν − and ∆ φ = 2 arctan (cid:18) σν − (cid:19) . (13)For the Peregrine soliton, the trajectories of time parameterization in the complex-plane are either acircle (for x = 0 ) or ellipses (for other values of x = 0 ). The circle is centered at (1 , with radius r = 2 .Let x = x ∈ R be the position for the Peregrine soliton, then the ellipse has the length of semi-minoraxis a ( x ) , the length of semi-major axis b ( x ) , and is centered at ( c ( x ) , , where a ( x ) = 21 + 4 x , b ( x ) = 2 q x , and c ( x ) = a ( x ) − . (14)Figure 8 should be viewed in connection to Figures 6(d) and 7(d). It displays the plots of the realand imaginary parts of the non-rapid-oscillating complex-valued amplitude for the Peregrine soliton ˜ q 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 ∈ (cid:8) , , , , , , (cid:9) . For the latter, differentcurves correspond to selected values of position x ∈ (cid:8) , π , π , π , π , π (cid:9) . The phase difference in the timeparameterization of ˜ q P is discernible from the behavior of Im (˜ q P ) as t → ±∞ . While lim t →±∞ Re (˜ q P ) = − ,the quantity for lim t →±∞ Im (˜ q P ) takes positive and negative values, respectively. We have considered the exact analytical breather solutions of the focusing NLS equation, where the waveenvelopes at infinity have a nonzero but constant background. These solutions have been adopted asweakly nonlinear prototypes for freak wave events in dispersive media due to their fine agreement withvarious experimental results. We have provided not only a brief historical review of the breathers but alsocovered 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 breatherand Akhmediev soliton. We have verified rigorously using the ǫ - δ argument that as each of the parametervalues from these two breathers is approaching zero, they reduce to the Peregrine soliton. We have alsopresented this limiting behavior visually by depicting the contour plots of the breather amplitude modulus . Karjanto Peregrine soliton for selected parameter values. We displayed the parameterization plots of the non-rapid-oscillatingcomplex-valued breather amplitudes both spatially and temporally.The trajectories for the spatial parameterization in the complex-plane exhibit a set of straight lines for allthe breathers. From x → −∞ to x → + ∞ , the paths are passed twice for the Kuznetsov-Ma breather andare elapsed many times infinitely for the Akhmediev soliton due to its spatial periodic characteristics. Thetrajectories in the complex plane for the parameterization in the temporal variable of the Kuznetsov-Mabreather and Peregrine soliton feature a periodic circle and a set of periodic ellipses due to its temporalsymmetry. For the Akhmediev soliton, on the other hand, the path does not only turn into circle and ellipsesectors but also becomes straight lines as it travels from t → −∞ to t → + ∞ , featuring homoclinic orbitswith a phase shift. CONFLICT OF INTEREST STATEMENT
The authors declare that the research was conducted in the absence of any commercial or financialrelationships that could be construed as a potential conflict of interest.
AUTHOR CONTRIBUTIONS
The author has contributed to completing this article.
FUNDING
This research does not receive any funding.
ACKNOWLEDGMENTS
The author wishes to thank Bertrand Kibler, Amin Chabchoub, and Heremba Bailung for the invitationto contribute to the article collection “Peregrine Soliton and Breathers in Wave Physics: Achievementsand Perspectives”. The author also acknowledges E. (Brenny) van Groesen, Mark Ablowitz, ConstanceSchober, Frederic Dias, Roger Grimshaw, Panayotis Kevrekidis, Boris Malomed, Evgenii Kuznetsov, NailAkhmediev, Alfred Osborne, Miguel Onorato, Gert Klopman, Rene Huijsmans, Andonowati, Stephanvan Gils, Guido Schneider, Anthony Roberts, Shanti Toenger, Omar Kirikchi, Ardhasena Sopaheluwakan,Hadi Susanto, Alexander Iskandar, Agung Trisetyarso, and Defrianto Pratama for fruitful discussion.
SUPPLEMENTAL DATA
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DATA AVAILABILITY STATEMENT
There is no available data for this article.
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