# Review on the Stability of the Peregrine and Related Breathers

^{1}Departamento de Matemáticas, Universidad de Córdoba, Córdoba, Spain^{2}Dipartimento di Matematica “Guido Castelnuovo”, Universitá di Roma “La Sapienza”, Roma, Italy^{3}CNRS and Departamento de Ingeniería Matemática DIM, Universidad de Chile, Santiago, Chile

In this note, we review stability properties in energy spaces of three important nonlinear Schrödinger breathers: Peregrine, Kuznetsov-Ma, and Akhmediev. More precisely, we show that these breathers are *unstable* according to a standard definition of stability. Suitable Lyapunov functionals are described, as well as their underlying spectral properties. As an immediate consequence of the first variation of these functionals, we also present the corresponding nonlinear ODEs fulfilled by these nonlinear Schrödinger breathers. The notion of global stability for each breather mentioned above is finally discussed. Some open questions are also briefly mentioned.

## Introduction

In this short review, we describe a series of mathematical results related to the stability of the Peregrine breather and other explicit solutions to the cubic nonlinear Schrödinger (NLS) equation, an important candidate to modelize rogue waves. The mentioned model is NLS posed on the real line

We assume a nonzero boundary value condition (BC) at infinity, in the form of a *Stokes wave*

It is well known that Eq. 1 possesses a huge family of complex solutions. Among them, a fundamental role in the dynamics is played by *breathers*. We shall say that a particular smooth solution to Eqs 1 and 2 is a breather if, up to the invariances of the equation, its dynamics show the evolution of some concentrated quantity in an oscillatory fashion. NLS has scaling, shifts, phase, and Galilean invariances: namely, if *u* solves Eq. 1, another solution to Eq. 1 is

In this paper, we review the known results about stability in Sobolev spaces of the Peregrine (*P*) breather^{1} [1]:

and we will also present, with less detail, the analogous properties of the Kuznetsov-Ma (*A*) breathers: (i) if

and (ii) for

Notice the oscillating character of the three above examples. In addition, also notice that

**FIGURE 1**. Left: absolute value

NLS Eq. 1 with nonzero BC Eq. 2 is believed to describe the emergence of rogue or freak waves in the deep sea [1, 5, 6]. Peregrine waves were experimentally observed 10 years ago in Ref. 7. The model itself is also a well-known example of the mechanism known as modulational instability [5, 8]. For an alternative explanation to the rogue wave phenomenon which is stable under perturbations, see Refs. 9, 10.

Along these lines, we will explain that Peregrine and two other breathers are unstable according to a standard definition of stability. It could be the case that a less demanding definition of stability, involving infinite energy solutions, could repair this problem. However, such a question is still an open problem.

This paper is organized as follows. In Section 2, we recall some standard results for NLS with zero and nonzero background and the notion of modulational instability and local well-posedness. Section 3 is devoted to the conservation laws of NLS, and Section 4 to the notion of stability. Sections 5 and 6 review the Peregrine, Kuznetsov-Ma, and Akhmediev breathers’ stability properties. Finally, Section 7 is devoted to a discussion, final comments, and conclusions.

## Modulational Instability

### A Quick Review of the Literature

Let us briefly review the main results involving Eq. 1 in the zero and nonzero BC cases. A much more complete description of the current literature can be found in the papers present in this volume and in Refs. 11–13.

NLS Eq. 1 is a well-known integrable model (see Ref. 14) and describes the propagation of pulses in nonlinear media and gravity waves in the ocean [12]. The local and global well-posedness theory for NLS with zero BC at infinity was initiated by Ginibre and Velo [15]; see also Tsutsumi [16] and Cazenave and Weissler [17]. Finally, see Cazenave [11] for a complete account of the different NLS equations. One should have in mind that Eq. 1 is globally well-posed in

In the zero background cases, one has standard solitons for Eq. 1:

These are time-periodic, spatially localized solutions of Eq. 1 and orbitally stable; see Cazenave-Lions [19], Weinstein [20], and Grillakis-Shatah-Strauss [21]. See also Refs. 22–24 for the case of several solitons.

### Some Heuristics

NLS with nonzero boundary conditions, represented in Eqs 1 and 2, is characteristic of the *modulational instability* phenomenon, which—roughly speaking—says that small perturbations of the exact Stokes solution

There are plenty of works in the literature dealing with this phenomenon, not only in the NLS case. Usually also called Benjamin-Feir mechanism [25], the NLS case has been described in a series of papers [2, 8, 26–28]. See also references therein for more details on the physical literature. Here, we present the standard, simple, but formal explanation of this phenomenon, in terms of a frequency analysis of the linear solution.

To this aim, consider *localized* perturbations of Eqs 1 and 2 of the form

Notice that this ansatz is motivated by Eq. 4. Then Eq. 1 becomes a modified NLS equation with a zeroth-order term, which is real-valued and has the wrong sign:

The associated linearized equation for Eq. 9 is just

This problem has some instability issues, as a standard frequency analysis reveals: looking for a formal standing wave

This phenomenon is similar to the one present in the *bad Boussinesq* equation; see Kalantarov and Ladyzhenskaya [29]. However, in the latter case, the situation is even more complicated, since the linear equation is ill-posed for all large frequencies, unlike NLS Eqs 1 and 2, which is only badly behaved at small frequencies.

### Local Well-Posedness

The above heuristics could lead to thinking that the model Eqs 1 and 2 is not well-posed (in the Hadamard sense [30]) in standard Sobolev spaces (appealing to physical considerations, we will only consider solutions to these models with *finite energy*). Recall that, for

Theorem 2.1. Let

The main feature in the proof of Theorem 2.1 is the fact that if we work in Sobolev spaces, in principle, *there are no**decay estimates for the linear dynamics*. Moreover, one has exponential growth in time of the

Note that *P* in Eq. 4 is always well-defined and has essentially no loss of regularity, confirming in some sense the intuition and the conclusions in Theorem 2.1. Also, note that, using the symmetries of the equation, we have LWP for any solution of Eq. 1 of the form

Two interesting questions are still open: global existence vs. blow-up and ill-posedness of the flow map for lower regularities. Since Eq. 1 is integrable, it can be solved, at least formally, by inverse scattering methods. Biondini and Mantzavinos [33] showed the existence and long-time behavior of a global solution to Eq. 1 in the integrable case, under certain exponential decay assumptions at infinity and a *no-soliton* spectral condition. In this paper, we have decided to present the results stated just in some energy space, with no need for extra decay conditions.

## Conserved Quantities

Being an integrable model, Eqs 1 and 2 possess an infinite number of conserved quantities [14]. Here, we review the most important for the question of stability: mass, energy, and momentum. For both *P*, one has the mass, momentum, and energy,

and the Stokes wave +

In Ref. 34, the mass, energy, and momentum of the *P*Eq. 4 and

We conclude that *P* breathers are zero speed solutions. Note instead that, under a suitable Galilean transformation, they must have nonzero momentum. Note also that *P* has the same energy and mass as the Stokes wave solution (the nonzero background), a property not satisfied by the standard soliton on zero background. Also, compare the mass and energy of the Kuznetsov-Ma breather with the ones obtained in Ref. 36 for the mKdV breather.

Assume now that

A third one, appearing from the integrability of the equation, is given by Ref. 35.

## Orbital Stability

From a physical and mathematical point of view, understanding the stability properties of candidates to rogue waves is of uttermost importance because not all the observed patterns bear the same qualitative and quantitative information.

Since the equation is locally well-posed and does have continuous-in-time solutions, it is possible to define a notion of orbital stability for the Peregrine, Kuznetsov-Ma, and Akhmediev breathers. To study the stability properties of such waves is key to validate them as candidates for explaining rogue waves; see Ref. 36. First, we consider the aperiodic case.

Mathematically speaking, the notion of orbital stability is the one to have in mind. Fix *orbitally stable* in

Here,

Note that no phase correction is needed in Eq. 15: Eq. 9 is no longer *requires an infinite amount of energy*. The same applies for Galilean transformations. If Eq. 15 is not satisfied, we will say that *U* is unstable. Note additionally that condition Eq. 15 requires *w* globally defined; otherwise, *U* is trivially unstable, since *U* is globally defined.

Recall that NLS solitons on a zero background Eq. 7 satisfy Eq. 15 (with an additional phase correction) for

Now, we consider an adapted version of stability for dealing with the Akhmediev breather Eq. 6. We must fix a particular spatial period, which for the latter case will be settled as

By stability in this case, we mean the following. Fix *orbitally stable* in

If Eq. 16 is not satisfied, we will say that *U* is unstable. Note how in this case phase corrections are allowed. This is because they are finite energy perturbations in the periodic case. In other words, any change of the form

## The Peregrine Breather

Recall the Peregrine breather introduced in Eq. 4. Note that it is a polynomially decaying (in space and time) perturbation of the nonzero background given by the Stokes wave

Theorem 5.1. The Peregrine breather Eq. 4 is unstable under small

The proof of this result uses the fact that Peregrine breathers are in some sense converging to the background final state (i.e., they are asymptotically stable) in the whole space norm

Theorem 5.1 is in contrast with other positive results involving breather solutions [36, 38, 47]. In those cases, the involved equations (mKdV, Sine-Gordon) were globally well-posed in the energy space (and even in smaller subspaces), with uniform in time bounds. Several physical and computational studies on the Peregrine breather can be found in Refs. 27, 48 and references therein. A recent stability analysis was performed in Ref. 49 in the case of complex-valued Ginzburg-Landau models. The proof of Theorem 5.1 is in some sense a direct application of the notion of modulational instability together with an asymptotic stability property.

## Sketch of Proof of Theorem 5.1

This proof is not difficult, and it is based on the notion of *asymptotic stability*, namely, the convergence at infinity of perturbations of the breather. Fix *P* in Eq. 4 is orbitally stable, as in Eq. 15. Write

Now consider, as a perturbation of the Peregrine breather, the Stokes wave Eq. 2. One has (Ref. 34)

Therefore, we have two modulationally unstable solutions to Eq. 1 that converge to the same profile as

Although Theorem 5.1 clarifies the stability/instability question for the Peregrine breather, other questions remain unsolved. Is the Peregrine breather stable under less restrictive assumptions on the perturbed data? A suitable energy space for the Peregrine breather could be

endowed with the metric *in the whole space* is key for the instability.

## Variational Characterization

In the following lines, we discuss some improvements of the previous result. In particular, we discuss the variational characterization of the Peregrine breather. For an introduction to this problem in the setting of breathers, see, e.g., Ref. 32. In Ref. 35, the authors quantified in some sense the instability of the Peregrine breather.

Theorem 5.2 (Variational characterization of Peregrine). Let

Moreover, B satisfies the nonlinear ODE

Theorem 5.2 reveals that Peregrine breathers are, in some sense, degenerate. More precisely, contrary to other breathers, the characterization of *P* does not require the mass and the energy, respectively. The absence of these two quantities may be related to the fact that

The proof of Theorem 5.2 is simple and variational and follows previous ideas presented in Ref. 36 for the case of mKdV breathers and Ref. 39 for the case of the Sine-Gordon breather (see also Ref. 47 for a recent improvement of this last result, based in Ref. 38). The main differences are in the complex-valued nature of the involved breathers and the nonlocal character of the *P* breathers.

The following result gives a precise expression for the lack of stability in Peregrine breathers. Recall that

Theorem 5.3 (Direction of instability of the Peregrine breather). Let

From Eq. 20, one can directly check that

where

## The Kuznetsov-Ma and Akhmediev Breathers

Here, we describe the stability properties of the other two important breathers for NLS: the Kuznetsov-Ma (KM) breather Eq. 5 (see Figure 1 and Refs. 2, 4, 52 for details) and the Akhmediev breather Eq. 6 [2].

### Kuznetsov-Ma

Most of the results obtained in the Peregrine case are also available for the Kuznetsov-Ma breather. We start by noticing that

Using a similar argument as in the proof of Theorem 5.1 for the Peregrine case, one can show that Kuznetsov-Ma breathers are unstable [39]. The (formally) unstable character of Peregrine and Kuznetsov-Ma breathers was well known in the physical and fluids literature (they arise from modulational instability); therefore, the conclusions from previous results are not surprising. In water tanks and optic fiber experiments, researchers were able to reproduce these waves [7, 27, 53], if, e.g., the initial setting or configuration is close to the exact theoretical solution.

Floquet analysis has been recently done for the KM breather in Ref. 54. Concerning the variational structure of the KM breather, it is slightly more complicated than the one for Peregrine, but it has the same flavor.

Theorem 6.1 [35]. *Let*

Note that the elliptic equation for the *P* breather Eq. 19 is directly obtained by the formal limit

The variational structure of

Here, *E* and *F* are given by Eqs 11 and 12, respectively. Then, for any

Theorem 6.2 (Absence of spectral gap and instability of the KM breather [35]). Let

Note that classical stable solitons or solitary waves *Q*Eq. 7 easily satisfy the estimate *Q* [20]. Even in the cases of the mKdV breather

The above theorem shows that the *embedded eigenvalue*. This is not true in the case of linear, real-valued operators with fast decaying potentials, but since

### The Akhmediev Breather

Recall the Akhmediev breather Eq. 6. Note that

The instability of

Theorem 6.3 [31]. The Akhmediev breather Eq. 6 is unstable under small perturbations in

The proof of Theorem 6.3 uses Eq. 25 in a crucial way: a modified Stokes wave is an attractor of the dynamics around the Akhmediev breather for a large time.

Remark 6.1. We finally remark that the three above discussed breathers,

## Conclusion

We have reviewed the stability properties of three NLS solutions with nonzero background: Peregrine, Kuznetsov-Ma, and Akhmediev breathers. Working in associated energy spaces, with no additional decay condition, this review also characterizes the spectral properties of each of them. According to the definition of stability, no NLS Eq. 1 breather seems to be stable, not even in larger spaces. The instability is easily obtained from the fact that each breather converges on the whole line, as time tends to infinity, toward the Stokes wave. If the solutions were stable, this would imply that each breather is the Stokes wave itself. Some deeper connections between the stability of breathers and the nonzero background (modulational instability) are highly expected, but it seems that no proof of this fact is in the literature. Maybe Bäcklund transformations, in the spirit of Refs. 38, 41, 47, could help to give preliminary answers, and rigorous IST methods such as the ones in Refs. 42, 43 may help to solve this question. Finally, the dichotomy blow-up/global well-posedness and ill-posedness for large data in NLS Eq. 1 with nonzero background are interesting mathematical open problems to be treated elsewhere.

## Author Contributions

The three authors contributed equally to the conception, research, and writing of the manuscript.

## Funding

We have received partial funding from CMM Conicyt PIA AFB170001 and Fondecyt 1191412.

## Conflict of Interest

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

## Acknowledgments

We thank both referees for several comments, criticisms, and the addition of several unattended references that led to an important improvement of a previous version of this manuscript.

## Footnote

^{1}Or Peregrine soliton, but because of the nature of its variational formulation, it is more a breather than a soliton.

## References

1. Peregrine DH. Water waves, nonlinear Schrödinger equations and their solutions. *J Austral Math Soc Ser B.* (1983). **25**:16–43. doi:10.1016/j.anihpc.2018.10.005

2. Akhmediev N, Korneev VI. Modulation instability and periodic solutions of the nonlinear Schrödinger equation. *Theor Math Phys.* (1986). **69**:1089–93.

3. Klein C, Haragus M. Numerical study of the stability of the Peregrine breather. *Ann Math Sci Appl.* (2017). **2**:217–39.

5. Akhmediev N, Ankiewicz A, Taki M. Waves that appear from nowhere and disappear without a trace. *Phys Lett.* (2009). **373**:675–8. doi:10.1016/j.physleta.2008.12.036

6. Shrira VI, Geogjaev YV. What make the Peregrine soliton so special as a prototype of freak waves? *J Eng Math.* (2010). **67**:11–22. doi:10.1007/s10665-009-9347-2

7. Kibler B, Fatome J, Finot C, Millot G, Dias F, Genty G, et al. The Peregrine soliton in nonlinear fibre optics. *Nature Phys.* (2010). **6**:790–95. doi:10.1038/nphys1740

8. Zakharov VE, Dyachenko AI, Prokofiev AO. Freak waves as nonlinear stage of stokes wave modulation instability. *Eur J Mech B Fluids.* (2006). **5**:677–92.

9. Bona JL, Ponce G, Saut JC, Sparber C. Dispersive blow-up for nonlinear Schrödinger equations revisited. *J Math Pures Appl.* (2014). **102**:782–811. doi:10.1016/j.matpur.2014.02.006

10. Bona JL, Saut JC. Dispersive blowup of solutions of generalized Korteweg-de vries equations. *J Differ Equations.* (1993). **103**:3–57. doi:10.1006/jdeq.1993.1040

11. Cazenave T. *Semilinear Schrödinger equations.* New York, NY; American Mathematical Society (2003). 323 p.

12. Dauxois T, Peyrard M. *Physics of solitons.* Cambridge, UK: Cambridge University Press (2006). 435 p.

13. Yang J. *Nonlinear waves in integrable and nonintegrable systems*. Philadelphia, PA: SIAM Mathematical Modeling and Computation, (2010). 430 p.

14. Zakharov VE, Shabat AB. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. *J Exp Theor Phys Lett.* (1970). **34**:62–9.

15. Ginibre J, Velo G. On a class of nonlinear Schrödinger equations I: the Cauchy problem. *J Funct Anal.* (1979). **32**:1–32. doi:10.1016/0022-1236(79)90076-4

16. Tsutsumi Y. L^{2}-solutions for nonlinear Schrödinger equations and nonlinear groups. *Funkcial Ekvac.* (1987). **30**:115–25.

17. Cazenave T, Weissler F. The Cauchy problem for the critical nonlinear Schrödinger equation in *H*^{s}. *Non Anal.* (1990). **14**:807–36. doi:10.1016/0362-546X(90)90023-A

18. Kenig C, Ponce G, Vega L. On the ill-posedness of some canonical dispersive equations. *Duke Math J.* (2001). **106**:617–33. doi:10.1215/S0012-7094-01-10638-8

19. Cazenave T, Lions PL. Orbital stability of standing waves for some nonlinear Schrödinger equations. *Comm Math Phys.* (1982). **85**:549–61. doi:10.1007/BF01403504

20. Weinstein MI. Lyapunov stability of ground states of nonlinear dispersive evolution equations. *Comm Pure Appl Math.* (1986). **39**:51–68. doi:10.1002/cpa.3160390103

21. Grillakis M, Shatah J, Strauss W. Stability theory of solitary waves in the presence of symmetry. *I J Funct Anal.* (1987). **74**:160–97. doi:10.1016/0022-1236(87)90044-9

22. Kapitula T. On the stability of N-solitons in integrable systems. *Nonlinearity.* (2007). **20**:879–907. doi:10.1088/0951-7715/20/4/005

23. Martel Y, Merle F. Multi-solitary waves for nonlinear Schrödinger equations. *Ann IHP Anal Non.* (2006). **23**:849–64. doi:10.1016/j.anihpc.2006.01.001

24. Martel Y, Merle F, Tsai TP. Stability in H^{1} of the sum of K solitary waves for some nonlinear Schrödinger equations. *Duke Math J.* (2006). **133**:405–466. doi:10.1215/S0012-7094-06-13331-8

25. Bridges TJ, Mielke A. A proof of the Benjamin-Feir instability. *Arch Rat Mech Anal.* (1995). **133**:145–98. doi:10.1007/BF00376815

26. Achilleos V, Diamantidis S, Frantzeskakis DJ, Karachalios NI, Kevrekidis PG. Conservation laws, exact traveling waves and modulation instability for an extended nonlinear Schrödinger equation. *J Phys A Math Theor.* (2015). **48**:355205. doi:10.1088/1751-8113/48/35/355205

27. Brunetti M, Marchiando N, Betti N, Kasparian J. Modulational instability in wind-forced waves. *Phys Lett.* (2014). **378**:3626–30. doi:10.1016/j.physleta.2014.10.017

28. Al Khawaja U, Bahlouli H, Asad-uz-zaman M, Al-Marzoug SM. Modulational instability analysis of the Peregrine soliton. *Comm Nonlinear Sci Num Simul.* (2014). **19**:2706–2714. doi:10.1016/j.cnsns.2014.01.002

29. Kalantarov VK, Ladyzhenskaya OA. The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types, *J Math Sci.* (1978). **10**:53. doi:10.1007/BF01109723.

30. Hadamard J. Sur les problèmes aux dérivées partielles et leur signification physique. *Princeton Univ. Bull.* (1902). **13**:49–52.

31. Alejo MA, Fanelli L, Muñoz C. The Akhmediev breather is unstable. *Sao Paulo J Math Sci.* (2019). **13**:391–401. doi:10.1007/s40863-019-00145-4

32. Muñoz C. Stability of integrable and nonintegrable structures. *Adv Differ Equations.* (2014). **19**:947–996.

33. Biondini G, Mantzavinos D. Long-time asymptotics for the focusing nonlinear Schrödinger equation with nonzero boundary conditions at infinity and asymptotic stage of modulational instability. *Comm Pure Appl Math.* (2017). **70**:2300–65. doi:10.1002/cpa.21701

34. Muñoz C. Instability in nonlinear Schrödinger breathers. *Proyecciones.* (2017). **36**:653–83. doi:10.4067/S0716-09172017000400653

35. Alejo MA, Fanelli L, Muñoz C. Stability and instability of breathers in the *U*(1) Sasa-Satsuma and nonlinear Schrödinger models. Preprint repository name [Preprint] (2019). Available from: https://arxiv.org/abs/1901.10381.

36. Alejo MA, Muñoz C. Nonlinear stability of mKdV breathers. *Comm Math Phys.* (2013). **324**:233–62. doi:10.1007/s00220-013-1792-0

37. Calini A, Schober CM. Dynamical criteria for rogue waves in nonlinear Schrödinger models. *Nonlinearity.* (2012). **25**:R99–R116. doi:10.1088/0951-7715/25/12/R99

38. Alejo MA, Muñoz C. Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers. *Anal Partial Differ Equation.* (2015). **8**:629–74. Doi:10.2140/apde.2015.8.629

39. Alejo MA, Muñoz C, Palacios JM. On the variational structure of breather solutions I: Sine-Gordon equation. *J Math Anal Appl.* (2017). **453**:1111–38. doi:10.1016/j.jmaa.2017.04.056.

40. Alejo MA, Muñoz C, Palacios JM. On the variational structure of breather solutions II: Periodic mKdV equation. *Electron J Differ Equations.* (2017). **2017**:56.

41. Alejo MA, Muñoz C, Palacios JM. On asymptotic stability of the Sine-Gordon kink in the energy space. Preprint repository name [Preprint] (2020). Available from: https://arxiv.org/abs/2003.09358

42. Cheng G, Liu J. Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces. Preprint repository name [Preprint] (2019). Available from: https://arxiv.org/abs/1903.03855

43. Cheng G, Liu J. Soliton resolution for the modified KdV equation. Preprint repository name [Preprint] (2019). Available from: https://arxiv.org/abs/1907.07115

44. Pelinovsky D, Yang J. Stability analysis of embedded solitons in the generalized third-order nonlinear Schrödinger equation. *Chaos.* (2005). **15**:037115. doi:10.1063/1.1929587

45. Gallay T, Pelinovsky DE. Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves. *J Differ Equation.* (2015). **258**:3607–38. doi:10.1016/j.jde.2015.01.018

46. Gallay T, Pelinovsky DE. Orbital stability in the cubic defocusing NLS equation: II. The black soliton. *J Differ Equation.* (2015). **258**:3639–60. doi:10.1016/j.jde.2015.01.019

47. Muñoz C, Palacios JM. Nonlinear stability of 2-solitons of the Sine-Gordon equation in the energy space. *Ann IHP Anal Non.* (2019). **36**:977–1034.

48. Dysthe K, Trulsen K. Note on breather type solutions of the NLS as models for freak waves. *Phys Scr.* (1999). **T82**:48–52. doi:10.1238/Physica.Topical.082a00048

49. Zweck J, Latushkin Y, Marzuola JL, Jones CKRT. The essential spectrum of periodically-stationary solutions of the complex Ginzburg-Landau equation. Preprint repository name [Preprint] (2020). Available from: https://arxiv.org/abs/2005.09176.

50. Gallo C. The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity. *Comm Partial Differ Equation.* (2008). **33**:729–71. doi:10.1080/03605300802031614

51. Gérard P. The Cauchy problem for the Gross-Pitaevskii equation *Ann I H Poincaré AN.* (2006). **23**:765–79. doi:10.1016/j.anihpc.2005.09.004

52. Ma YC. The perturbed plane-wave solutions of the cubic Schrödinger equation. *Stud Appl Math.* (1979). **60**:43–58. doi:10.1002/sapm197960143

53. Kibler B, Fatome J, Finot C, Millot G, Genty G, Wetzel B, et al. Observation of Kuznetsov-Ma soliton dynamics in optical fibre. *Sci Rep.* (2012). **2**:463. doi:10.1038/srep00463.

54. Cuevas-Maraver J, Kevrekidis PG, Frantzeskakis DJ, Karachalios NI, Haragus M, James G. Floquet analysis of Kuznetsov-Ma breathers: A path towards spectral stability of rogue waves. *Phys Rev E.* (2017). **96**:012202. doi:10.1103/PhysRevE.96.012202

55. Ablowitz MJ, Clarkson PA. *Solitons, nonlinear evolution equations and inverse scattering.* Cambridge, UK: Cambridge University Press (1991). 516 p.

Keywords: Peregrine breather, Kuznetsov-Ma breather, Akhmediev breather, stability, nonlinear Schrodinger equation

Citation: Alejo MA, Fanelli L and Muñoz C (2020) Review on the Stability of the Peregrine and Related Breathers. *Front. Phys.* 8:591995. doi: 10.3389/fphy.2020.591995

Received: 05 August 2020; Accepted: 19 August 2020;

Published: 24 November 2020.

Edited by:

Bertrand Kibler, UMR6303 Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), FranceReviewed by:

Dmitry Pelinovsky, McMaster University, CanadaChristian Klein, Université de Bourgogne, France

Copyright © 2020 Alejo, Fanelli and Muñoz. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Claudio Muñoz, cmunoz@dim.uchile.cl

^{†}Present address: Luca Fanelli, Dipartimento di Matematica “Guido Castelnuovo”, Universitá di Roma “La Sapienza”, Roma, Italy