MINI REVIEW article
Review on the Stability of the Peregrine and Related Breathers
- 1Departamento de Matemáticas, Universidad de Córdoba, Córdoba, Spain
- 2Dipartimento di Matematica “Guido Castelnuovo”, Universitá di Roma “La Sapienza”, Roma, Italy
- 3CNRS 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.
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
and we will also present, with less detail, the analogous properties of the Kuznetsov-Ma (
and (ii) for
Notice the oscillating character of the three above examples. In addition, also notice that
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.
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 . The local and global well-posedness theory for NLS with zero BC at infinity was initiated by Ginibre and Velo ; see also Tsutsumi  and Cazenave and Weissler . Finally, see Cazenave  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 , Weinstein , and Grillakis-Shatah-Strauss . See also Refs. 22–24 for the case of several solitons.
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 , 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.
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 . 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.
The above heuristics could lead to thinking that the model Eqs 1 and 2 is not well-posed (in the Hadamard sense ) 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
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  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.
Being an integrable model, Eqs 1 and 2 possess an infinite number of conserved quantities . Here, we review the most important for the question of stability: mass, energy, and momentum. For both
and the Stokes wave +
We conclude that
A third one, appearing from the integrability of the equation, is given by Ref. 35.
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
Note that no phase correction is needed in Eq. 15: Eq. 9 is no longer
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
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
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 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
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
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 .
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 . 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.
Note that the elliptic equation for the P breather Eq. 19 is directly obtained by the formal limit
The variational structure of
Note that classical stable solitons or solitary waves QEq. 7 easily satisfy the estimate
The above theorem shows that the
The Akhmediev Breather
Recall the Akhmediev breather Eq. 6. Note that
The instability of
Theorem 6.3 . 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,
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.
The three authors contributed equally to the conception, research, and writing of the manuscript.
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.
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.
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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), France
Reviewed by:Dmitry Pelinovsky, McMaster University, Canada
Christian 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, email@example.com
†Present address: Luca Fanelli, Dipartimento di Matematica “Guido Castelnuovo”, Universitá di Roma “La Sapienza”, Roma, Italy