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The effect of a small loss or gainin the periodic NLS anomalous wave dynamics. I
F. Coppini 1,4, P. G. Grinevich 2,5, and P. M. Santini 3,6,7
1 PhD Program in Physics, Dipartimento di Fisica, Universita di Roma “LaSapienza”, and Istituto Nazionale di Fisica Nucleare (INFN), Sezione di
Roma, Piazz.le Aldo Moro 2, I-00185 Roma, Italy2 Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina
St., Moscow, 199911, Russia, and L.D. Landau Institute for TheoreticalPhysics, pr. Akademika Semenova 1a, Chernogolovka, 142432, Russia
3 Dipartimento di Fisica, Universita di Roma “La Sapienza”, andIstituto Nazionale di Fisica Nucleare (INFN), Sezione di Roma,
Piazz.le Aldo Moro 2, I-00185 Roma, Italy
4e-mail: [email protected]: [email protected]
6e-mail: [email protected]: [email protected]
October 30, 2019
Abstract
The focusing Nonlinear Schrodinger (NLS) equation is the simplestuniversal model describing the modulation instability (MI) of quasimonochromatic waves in weakly nonlinear media, and MI is consid-ered the main physical mechanism for the appearence of anomalous(rogue) waves (AWs) in nature. Using the finite gap method, two of us(PGG and PMS) have recently solved, to leading order and in termsof elementary functions of the initial data, the NLS Cauchy problemfor generic periodic initial perturbations of the unstable backgroundsolution of NLS (what we call the Cauchy problem of the AWs), inthe case of a finite number of unstable modes. In this paper, concen-trating on the simplest case of a single unstable mode, we study theperiodic Cauchy problem of the AWs for the NLS equation perturbedby a linear loss or gain term. Using the finite gap method and thetheory of perturbations of soliton PDEs, we construct the proper an-alytic model describing quantitatively how the solution evolves, aftera suitable transient, into slowly varying lower dimensional patterns
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(attractors) in the (x, t) plane, characterized by ∆X = L/2 in thecase of loss, and by ∆X = 0 in the case of gain, where ∆X is thex-shift of the position of the AW during the recurrence, and L is theperiod. This process is described, to leading order, in terms of ele-mentary functions of the initial data. Since dissipation can hardly beavoided in all natural phenomena involving AWs, and since a smalldissipation induces O(1) effects on the periodic AW dynamics, gener-ating the slowly varying loss/gain attractors analytically described inthis paper, we expect that these attractors, together with their gen-eralizations corresponding to more unstable modes, will play a basicrole in the theory of periodic AWs in nature.
1 Introduction
The self-focusing Nonlinear Schrodinger (NLS) equation
iut + uxx + 2|u|2u = 0, u = u(x, t) ∈ C, (1)
is the simplest universal model in the description of the propagation of aquasi monochromatic wave in a weakly nonlinear medium; in particular, it isrelevant in water waves [92, 6], in nonlinear optics [81, 19, 75], in Langmuirwaves in a plasma [67], and in the theory of Bose-Einstein condensates [18,77]. Its homogeneous solution
u0(x, t) = a exp(2i|a|2t), a complex constant parameter, (2)
describing Stokes waves [83] in a water wave context, a state of constantlight intensity in nonlinear optics, and a state of constant boson density in aBose-Einstein condensate, is unstable under the perturbation of waves withsufficiently large wave length [15, 65, 13, 92, 98, 84, 78], and this modulationinstability (MI) is considered as the main cause for the formation of anoma-lous (rogue, extreme, freak) waves (AWs) in nature [44, 30, 73, 54, 55, 72].
The integrable nature of the focusing NLS [99] allows one to constructa large zoo of exact solutions, corresponding to perturbations of the back-ground, by degenerating finite-gap solutions [52, 12, 60, 61], when the spec-tral curve becomes rational, or using classical Darboux transformations [68],dressing techniques [100, 96, 97], and the Hirota method [45, 46]. Amongthese basic solutions, we mention the Peregrine soliton [74], rationally local-ized in x and t over the background (2), the so-called Kuznetsov [62] - Kawata
2
- Inoue [49] - Ma [66] soliton, exponentially localized in space over the back-ground and periodic in time, the solution found by Akhmediev, Eleonskiiand Kulagin in [7], periodic in x and exponentially localized in time over thebackground (2), known in the literature as the Akhmediev breather (AB), itselliptic generalizations [9, 8], and its multi-soliton generalizations [52]. Gen-eralizations of these solutions to the case of integrable multicomponent NLSequations, characterized by a richer spectral theory, have also been found[11, 25, 26, 27, 28].
Concerning the NLS Cauchy problems in which the initial condition con-sists of a perturbation of the exact background (2), what we call the Cauchyproblem of the AWs, if such a perturbation is localized, then slowly mod-ulated periodic oscillations described by the elliptic solution of (1) play arelevant role in the longtime regime [16, 17]. The relevance of the Kuznetsov– Kawata - Inoue – Ma solitons and of the superregular solitons (constructedby Zakharov and Gelash [93], see also [94], [95]) in this problem was investi-gated in [37].
If the initial perturbation is x-periodic, numerical and real experimentsindicate that the solutions of NLS exhibit instead time recurrence [89, 63, 90,10, 88, 58, 70, 76], as well as numerically induced chaos [2, 5, 1], in which thealmost homoclinic solutions of Akhmediev type seem to play a relevant role[33, 35, 20, 21, 22]. There are reports of experiments in which the Peregrineand the Akhmediev solitons were observed [23, 57, 90, 87, 58, 70, 76]. Theirrelevance within some classes of localized initial data for NLS, in the smalldispersion regime, was shown in [14, 31]; see also [38, 85] for the investigationof their relevance in ocean waves and fiber optics.
Using the finite-gap method [71, 29, 50, 64, 69, 59] (see [51] for its firstapplication to NLS), two of us (PGG and PMS) have recently solved [39, 40],to leading order and in terms of elementary functions, the Cauchy problemof the AWs for the self-focusing NLS equation
iut + uxx + 2|u|2u = 0, u = u(x, t) ∈ C, (3)
for a generic order ε periodic initial perturbation of the unstable backgroundsolution (2), in the case of a finite number of unstable modes. Namely, thefollowing Cauchy problem was solved:
u(x, 0) = a (1 + εv(x)) , 0 < ε� 1, v(x+ L) = v(x), (4)
3
where
v(x) =∞∑j=1
(cjeikjx + c−je
−ikjx), kj =2π
Lj, (5)
the average of the initial perturbation is assumed, without loss of generality,to be zero, and the period L is assumed to be generic (L/π is not an integer).See also [41] for an alternative approach to the study of the AW recurrence, inthe case of one unstable mode, based on matched asymptotic expansions; see[42] for the study of the numerical instabilities of the AB and of a finite-gapmodel describing them; see [43] for the analytic study of the phase resonancesin the AW recurrence; see [79] and [24] for the analytic study of the AWrecurrence in other NLS type models: respectively the PT-symmetric NLSequation [4] and the Ablowitz-Ladik model [3].
It is well-know that the mode k is linearly unstable if |k| < 2|a|, implyingthat the number N of unstable modes for the problem (3), (4) is finite andgiven by
N =
⌊|a|Lπ
⌋, (6)
where bxc, x ∈ R, denotes the largest integer not greater than x. Moreprecisely, the first N modes {±kj}, 1 6 j 6 N , are linearly unstable,since they give rise to exponentially growing and decaying waves of am-plitudes O(εe±σjt), where the growth rates σj are defined by
σj = kj
√4|a|2 − k2
j , 1 6 j 6 N, (7)
while the remaining modes are linearly stable, since they give rise to smalloscillations of amplitude O(εe±iωjt), where
ωj = kj
√k2j − 4|a|2 , j > N. (8)
For the unstable part of the spectrum it is convenient to introduce thefollowing notation:
φj = arccos
(kj
2|a|
)= arccos
(π
L|a|j
), 0 < φj <
π
2, 1 6 j 6 N, (9)
implying that
kj = 2|a| cosφj, σj = 2|a|2 sin(2φj), 1 6 j 6 N. (10)
4
In particular, if π/|a| < L < 2π/|a|, then N = 1 and the solution iswell approximated by a genus 2 exact solution on a Riemann surface withO(ε) handles, allowing one to describe, to leading order, the following exactrecurrence of AWs in terms of elementary functions of the initial data.Construct the following linear combinations of the Fourier coefficients of theunstable part of the initial perturbation:
α = e−iφ1c1 − eiφ1c−1, β = eiφ1c−1 − e−iφ1c1, (11)
where, hereafter, f is the complex conjugate of f ; then the solution of theCauchy problem to leading order (up to O(ε) corrections), in the finite inter-val 0 ≤ t ≤ T , reads as follows [41]:
u(x, t) =n∑
m=0
A(x, t;φ1, x
(m), t(m))eiρ
(m) − a1−e4inφ11−e4iφ1 e
2i|a|2t, x ∈ [0, L],
(12)where the parameters x(m), t(m), ρ(m), m ≥ 0, are defined in terms of theinitial data by the following elementary functions
x(m) = X(1) + (m− 1)∆X, t(m) = T (1) + (m− 1)∆T,
X(1) = argαk1
+ L4, ∆X = arg(αβ)
k1, ( mod L),
T (1) ≡ 1σ1
log(
2 sin2(2φ1)ε|α|
)= 1
σ1log(
σ21
2|a|4ε|α|
),
∆T = 1σ1
log(
4 sin4(2φ1)ε2|αβ|
)= 1
σ1log(
σ41
4|a|8ε2|αβ|
),
ρ(m) = 2φ1 + (m− 1)4φ1,
n =⌈T−T (1)
∆T+ 1
2
⌉,
(13)
where dxe, x ∈ R denotes the smallest integer not smaller than x, andfunction A is the AB:
A(x, t; θ,X, T ) := a e2i|a|2t cosh[σ(θ)(t− T ) + 2iθ] + sin θ cos[k(θ)(x−X)]
cosh[σ(θ)(t− T )]− sin θ cos[k(θ)(x−X)],
k(θ) = 2|a| cos θ, σ(θ) = k(θ)√
4|a|2 − k2(θ) = 2|a|2 sin(2θ),(14)
exact solution of NLS for all real parameters θ,X, T (see Figure 1).The solution (12)-(14) shows an exact recurrence of AWs described, to
leading order, by the Akhmediev breather, whose parameters change at eachappearance according to (13). It is a very good example of a Fermi-Pasta-Ulam-Tsingou [36] type recurrence without thermalisation (see [40] for the
5
analytic aspects of this recurrence in Fourier space). X(1) and T (1) are respec-tively the position and the time of the first appearance; ∆X is the x-shift ofthe position of the AW between two consecutive appearances, and ∆T is therecurrence time (the time between two consecutive appearances) (see Figure1). Therefore T (1) and ∆T are the characteristic times of the AW recurrence.
0
100
3−3
t
x
Figure 1: The density plot of |u(x, t)| with −L/2 ≤ x ≤ L/2, 0 ≤ t ≤ 100,L = 6, ε = 10−4, a = 1, with a generic initial condition c−1 = 0.3 + 0.3i,c1 = 0.5, obtained using the refined split-step method [53], is in extremelygood quantitative agreement with (12),(13) [39]. Hereafter brighter anddarker colors in the figures correspond to higher and lower values of |u(x, t)|respectively.
We first remark that, although the initial condition is infinite dimensional,the AW recurrence is described, to leading and relevant order, by just thefour real parameters X(1), T (1), ∆X and ∆T . Four free real parametersappear, indeed, in the unstable part of the initial condition (4) (the real andimaginary parts of c1 and c−1, or the real and imaginary parts of α and β).
Then we remark that this recurrence can be predicted from simple qual-itative considerations (see [39, 40]). The unstable mode grows exponentiallyand becomes O(1) at logarithmically large times, when one enters the nonlin-ear stage of MI, and one expects the generation of a transient, O(1), coherentstructure over the unstable background (the AW). Since the AB describes the
6
one-mode nonlinear instability, it is the natural candidate to describe sucha stage, at the leading order. Due again to MI, this AW is expected to bedestroyed in a finite time interval, and one enters the third asymptotic stage,characterized, like the first one, by the background plus an O(ε) perturba-tion. This second linear stage is expected, due again to MI, to give rise to theformation of a second AW (the second nonlinear stage of MI), described againby the Akhmediev breather, but, in general, with different parameters. Andthis procedure iterates forever, in the integrable NLS model, giving rise tothe generation of an infinite sequence of AWs described by different Akhme-diev breathers. Then the AW recurrence is a relevant effect of nonlinear MIin the periodic setting, and the finite gap method is the proper tool to give ananalytic description of it.
We also remark that formulas (11)-(14), in perfect quantitative agreementwith the output of the corresponding numerical experiment [39], were suc-cessfully tested soon after their appearance in a nonlinear optics experiment[76].
We end our remarks observing the the first attempt to apply the finitegap method to solve the NLS Cauchy problem on the segment, for periodicperturbations of the background, was made in [86], but no connection wasestablished between the initial data and the parameters of the θ-functionrepresentation, and no description of the dynamics in terms of elementaryfunctions was given.
Since dissipation can hardly be avoided in all natural phenomena involv-ing AWs, a natural question arises at this point. What is the effect of a smalldissipation on the NLS periodic AW dynamics?
The corresponding Cauchy problem of the AWs becomes
iut + uxx + 2|u|2u = −iνu, 0 < ν � 1, (15)
u(x, 0) = a(1 + εv(x)), a ∈ C, 0 < ε� 1, v(x+ L) = v(x),
and its homogeneous background solution and linear growth rate are, respec-tively [80]
u0(x, t, ν) = a exp(−νt) exp
(i|a|2
ν(1− exp(−2νt))
), (16)
andσ(t, ν) = −ν + k
√4|a|2 exp(−2νt)− k2. (17)
7
It is known [80] that, if the initial perturbation is sufficiently small, a smalldissipation can quench the growth process before the nonlinear effects be-come relevant, stabilizing the MI. In any case, due to (17), instability isalways canceled if the time interval of interest is sufficiently long [80, 56]. Inthis respect we remark that the presence of dissipation introduces anothercharacteristic time (from (17))
Tdiss =1
νlog
(2|a|k
)(18)
(the time at which the unstable mode k becomes stable), and the stabilizingeffect described in [80] takes place when the initial perturbation is sufficientlysmall, and dissipation is strong enough to have Tdiss ≤ T (1).
But what happens in the interesting case in which dissipation is smalland Tdiss � T (1)? And what happens if one is interested in experiments inwhich the time interval is not long enough to allow dissipation to cancel theinstability?
Partial answers to these questions came recently from the following waterwave and numerical experiments. In [58] two results were presented: i) anexperiment in a tank showing that the highly non generic AB initial condition(14) evolves into a recurrence of ABs whose position is shifted by ∆X = L/2(half a period); ii) a numerical experiment showing that the same AB initialcondition evolves, according to (15), into the same pattern, thus interpretingthe result of the real experiment as the effect of dissipation. Soon afterthat, it was shown in [82] that a real sinusoidal initial perturbation of thebackground, evolving numerically according to the focusing NLS equationperturbed by linear loss or gain terms, gives rise to a recurrence of ABs withshifts respectively ∆X = L/2 or ∆X = 0. To the best of our knowledge,no theoretical quantitative explanation involving analytic formulas has beengiven so far to these real and numerical experiments.
In this paper we use some aspects of the exact theory presented in [39, 40],in the simplest case of one unstable mode, together with few aspects of thetheory of perturbations of soliton PDEs, to construct the proper analyticmodel describing quantitatively and in terms of elementary functions theeffect of a small linear loss/gain on the dynamics of the NLS AWs arising froma generic periodic perturbation of the unstable background. In particular,we provide the theoretical explanation of all the above real and numericalexperiments.
8
The paper is organized as follows. In §2 we present and discuss theseanalytic results, and §3 is devoted to their proof.
2 Results
In this Section we present the analytic results describing the O(1) effects of asmall linear loss/gain on the NLS AWs dynamics generated by an O(ε), ε� 1generic periodic perturbation of the unstable background, in the simplestpossible case of one unstable mode.
The Cauchy problem of the AWs for the focusing NLS equation perturbedby a linear loss/gain term studied in this paper reads as follows:
iut + uxx + 2|u|2u = −iνu, ν ∈ R, |ν| � 1, (19)
(if ν > 0 we have a small loss, if ν < 0 we have a small gain), [0, T ] denotesthe time interval in which we construct the solution,
u(x, 0) = a(1 + εv(x)), a ∈ C, 0 < ε� 1, v(x+ L) = v(x), (20)
where
v(x) =∞∑j=1
(cjeikjx + c−je
−ikjx), kj =2π
Lj,
π
|a|< L <
2π
|a|(⇒ N = 1).
(21)We also assume that
|ν|T, |a|2|ν|T 2 � 1, (22)
and the meaning of these conditions can be explained observing that thebackground solution (16) of (19) behaves as follows
u0(t, ν) = a exp(−νt) exp (2i|a|2t (1− νt+O(νt)2)) . (23)
Therefore the amplitude and the oscillation frequency of the backgroundslowly decrease if ν > 0 (loss), and slowly increase if ν < 0 (gain). Thecondition |ν|T � 1 means that we can neglect the slow decay/growth of theamplitudes of the background and of the AWs; the condition |a|2|ν|T 2 � 1means that we can neglect the slow decay/growth of the oscillation frequencyand its effects. In particular, a can be treated as a constant parameter underthe above assumptions.
9
A more complete analytic study of the problem, in which also these ef-fects are taken into account, together with the output of nonlinear opticsexperiments in which the complete theory is tested, will be presented in asubsequent paper.
It is important to remark at this point that the main reason for theO(1) effects on the periodic AW dynamics due to a small loss or gain areconsequence of the fact that the Cauchy problem (19)-(21) involves two smallparameters: ν and ε. As we shall see below, the proper comparison actuallyinvolves ν and ε2.
Under the above hypothesis, the AW recurrence described in (12)-(13)is significantly modified by the small loss/gain in the following way. Thesolution is still described by a recurrence of Akhmediev breathers
u(x, t) =n∑
m=0
A(x, t;φ1, x
(m), t(m))eiρ
(m) − a1−e4inφ11−e4iφ1 e
2i|a|2t, x ∈ [0, L],
(24)where x(1) = x(1), t(1) = t(1) are essentially the same as in (13), but now
∆Xm := x(m+1) − x(m) = arg(Qm)k1
( mod L),
∆Tm := t(m+1) − t(m) = 1σ1
log(
4 sin4(2φ1)ε2|Qm|
)= 1
σ1log(
σ41
4|a|8ε2|Qm|
),
(25)
with
Qm = αβ − ν
ε22 sin(2φ1)
|a|2m = αβ − ν
ε2σ1
|a|4m, m ≥ 1, (26)
α and β are defined in (11), and ρ(m) in (13).From the above elementary formulas (24)-(26), we distinguish the follow-
ing cases (assuming that |αβ|, σ1, |a| = O(1)).
• If |ν| � ε2, then Qm ∼ αβ for every m, and there is no basic dif-ference with the zero-gain/loss case presented in the introduction. Inparticular, if ν = 0, then Qm = αβ for every m, and formulas (24)-(26)coincide with formulas (12)-(13).
• If |ν| is approximately of order ε2, the AW first appearance is essentiallynot affected by loss/gain, but, after it, we have a transient, consistingof a few AW recurrences, in which Qm → −(ν/ε2)(σ1/|a|4)m as mincreases, and the solution tends to one of the two asymptotic states
10
characterized by the following elementary formulas (see the central pic-ture of Figure 2):1)
argQm → π ⇒ ∆Xm → L/2, if ν > 0 (loss),
argQm → 0 ⇒ ∆Xm → 0, if ν < 0 (gain).(27)
2) The recurrence time decreases as m increases according to the ele-mentary formula:
∆Tm →1
σ1
log
(2|a|2 sin3(2φ)
|ν|m
)=
1
σ1
log
(σ3
1
4|a|4|ν|m
). (28)
• If |ν| � ε2, but the conditions |ν|T |, ν|T 2 � 1 are still fulfilled, thenQm ∼ −(ν/ε2)(σ1/|a|4)m, and, after the first AW appearance (essen-tially not affected by loss/gain) and without any transient, the solutionenters immediately one of the above two recurrence patterns (see theright picture of Figure 2):
argQm = π ⇒ ∆Xm = L/2, if ν > 0 (loss),
argQm = 0 ⇒ ∆Xm = 0, if ν < 0 (gain),
∆Tm =1
σ1
log
(σ3
1
4|a|4|ν|m
).
(29)
These two asymptotic states describe lower dimensional recurrence pat-terns depending on just two real parameters defining their position in space-time, unlike the zero-loss/gain case, in which the recurrence depends on fourreal parameters (see the Introduction). In addition, while the difference be-tween two consecutive recurrence times is finite:
|∆Tm+1 −∆Tm| =1
σ1
log
(m+ 1
m
), (30)
the relative difference |∆Tm+1−∆Tm|/∆Tm is small, since ν is small. There-fore we have slowly varying lower dimensional asymptotic states that can beviewed as slowly varying attractors (SVAs), the loss/gain - SVAs, com-pletely ruled by the parameter ν. Similar considerations are valid also inthe case of a finite number of unstable modes, and will be presented in asubsequent paper.
Also special initial conditions are described by formulas (24)-(26):
11
• If the initial condition is the highly non generic Akhmediev breather(14), as in the experiments in [58], then all the NLS spectral gaps areinitially closed [39, 42], β = 0, and Qm = −(ν/ε2)(σ1/|a|4)m is real.It follows that we are basically as in the case ν � ε2, and, after thefirst AW appearance, essentially not affected by loss/gain, the solutionenters immediately the SVAs (29) (see Figure 3). Therefore an initialcondition that theoretically should evolve, according to NLS, into theAB (i.e., with no recurrence), gives rise instead, in the presence of asmall loss or gain, to an AW recurrence described by the above SVAs.The instability of the AB with respect to perturbations of NLS, due tosmall corrective terms or to the numerical scheme approximating NLS,has been already observed (see, for instance, [2, 5, 58, 82, 42]).
• If the initial perturbation is real, then c−1 = c1 and αβ = 4 sin2 φ1|c1|2 >0; consequently the AW dynamics for zero-loss/gain is characterized by∆X = 0. If a small gain is present, Qm = αβ + (|ν|/ε2)(σ1/|a|4)m >0, ∀ m; it follows that ∆Xm = 0, ∀ m, and ∆Tm decreases as m in-creases. If, instead, a small loss is present, the real quantity Qm =αβ − (|ν|/ε2)(σ1/|a|4)m passes from positive values to negative valuesfor a certain m ≥ 1. Correspondingly, ∆Xm = 0 for m < m, and∆Xm = L/2 for m ≥ m; in addition, as m increases, ∆Tm increases ifm < m, and decreases if m ≥ m (see the Figure 4, in which a detailedquantitative comparison between the theoretical formulas (24)-(26) andthe output of a numerical experiment is made).
• If the initial perturbation is purely imaginary, then c−1 = −c1, andαβ = −4 cos2 φ1|c1|2 < 0; consequently the AW dynamics for zeroloss/gain is characterized by ∆X = L/2. If a small loss is present,Qm = −(|αβ| + (|ν|/ε2)(σ1/|a|4)m) < 0, ∀ m; it follows that ∆Xm =L/2, ∀m, and ∆Tm decreases as m increases. If, instead, a small gain ispresent, the real quantity Qm = −|αβ|+(|ν|/ε2)(σ1/|a|4)m passes fromnegative values to positive values for a certain m ≥ 1. Correspondingly,∆Xm = L/2 for m < m, and ∆Xm = 0 for m ≥ m; in addition, as mincreases, ∆Tm increases if m < m, and decreases if m ≥ m.
We end this section with the following important remark.
Since a small dissipation can hardly be avoided in all natural phenomenainvolving AWs, and since a very small dissipation induces O(1) effects on
12
the dynamics of periodic AWs, evolving into slowly varying lower dimensionalasymptotic states (slowly varying loss/gain - attractors) analytically describedin this paper through formulas (29), we expect that these asymptotic states,together with their generalizations corresponding to more unstable modes, willhave to play a basic role in the theory of AWs in nature.
0
100
3−3
t
x
0
100
3−3
t
x
0
100
3−3
t
x
Figure 2: The density plot of |u(x, t)|, with −L/2 ≤ x ≤ L/2, 0 ≤ t ≤ 100,L = 6, ε = 10−4, a = 1, for generic initial data: c1 = 0.5 and c−1 = 0.15−0.2i,obtained using the refined split-step method [53]. From left to right: ν = 0,ν = 10−9 < ε2 = 10−8, and ν = 10−5 � ε2. In the left figure we have theusual AW recurrence described by formulas (12)-(13). In the central figure,the solution tends to the SVA (29) with ∆Xm → L/2, after a relatively longtransient. In the right figure, after the first appearance, the solution enters,without any transient, the SVA (29) with ∆Xm = L/2, m ≥ 1. The firstappearance is essentially the same in all the three cases.
13
0
100
3−3
t
x
0
100
3−3
t
x
Figure 3: The density plot of |u(x, t)| with −L/2 ≤ x ≤ L/2, 0 ≤ t ≤ 100,L = 6, ε = 10−4, a = 1, for the Akhmediev breather initial condition,corresponding to c1 = 0.223417921 + 0.113111515 i, c−1 = −1.7601 · 10−10 +0.250419213 i, obtained using the refined split-step method [53]. If ν = 0, theAW appears theoretically only once. If ν 6= 0, after the first appearance, thesolution enters, without any transient, the SVAs (29). ν = 10−9 in the leftfigure, and ν = −10−9 in the right one. The first appearance is essentially thesame in these two cases, and it essentially coincides with the AB appearancewithout loss/gain.
14
0
100
3−3
t
x
0
100
3−3
t
x
0
100
3−3
t
x
Figure 4: The density plot of |u(x, t)| with −L/2 ≤ x ≤ L/2, 0 ≤ t ≤ 100,L = 6, ε = 10−4, a = 1, for a real initial condition (c−j = cj, ∀j), with c1 =0.3 + 0.4i, obtained using the refined split-step method [53] with quadrupleprecision. Consequently αβ > 0 and Qm is real. Left picture: ν = 0, then∆X = 0. Central picture: ν = 10−9; then, for m = 6, Qm changes itssign, from positive to negative values; correspondingly, ∆Xm switches from0 to L/2. Right picture: ν = 10−5 � ε2; then all Qm are negative and∆Xm = L/2 ∀m. The first appearance is essentially the same in all the threecases. For the central picture we also show the extremely good quantitativeagreement between the theoretical predictions from formulas (24)-(26) andthe numerical output:
t(1) = 5.51209 (theory) t(1) = 5.51208 (numerics)∆T1 = 11.18230 (theory) ∆T1 = 11.18230 (numerics)∆T2 = 11.40337 (theory) ∆T2 = 11.40338 (numerics);∆T3 = 11.77375 (theory) ∆T3 = 11.77376 (numerics);∆T4 = 13.31847 (theory) ∆T4 = 13.31848 (numerics);∆T5 = 11.84989 (theory) ∆T5 = 11.84988 (numerics);∆T6 = 11.44140 (theory) ∆T6 = 11.44142 (numerics);∆T7 = 11.20765 (theory) ∆T7 = 11.20766 (numerics);∆T8 = 11.04319 (theory) ∆T8 = 11.04320 (numerics)
15
3 Proof of the results
In this section we prove the above results. To do it, we make use of thefollowing ingredients. i) Some aspects of the deterministic theory of periodicAWs recently developed in [39, 40] using the finite-gap method; ii) few basicaspects of the theory of perturbations of soliton PDEs (developed in theinfinite line case in [47, 48]), and in the finite-gap case in [32]); iii) theclassical Darboux transformations for NLS [91, 68].
The zero-curvature representation for the self-focusing NLS is:
~Ψx(λ, x, t) = U(λ, x, t)~Ψ(λ, x, t), (31)
~Ψt(λ, x, t) = V (λ, x, t)~Ψ(λ, x, t), (32)
U(λ, x, t) =
[−iλ iu(x, t)
iu(x, t) iλ
]= −iλσ3 + iU(x, t), (33)
σ3 =
[1 00 −1
], U =
[0 u(x, t)
u(x, t) 0
],
V (λ, x, t) =
[−2iλ2 + iu(x, t)u(x, t) 2iλu(x, t)− ux(x, t)
2iλu(x, t) + ux(x, t) 2iλ2 − iu(x, t)u(x, t)
],
where
~Ψ(λ, x, t) =
[Ψ1(λ, x, t)Ψ2(λ, x, t)
].
The linear problem (31) can be rewritten as a spectral problem
L~Ψ(λ, x, t) = λ~Ψ(λ, x, t), (34)
where
L =
[i∂x u(x, t)
−u(x, t) −i∂x
].
It is essential that L is not self-adjoint, and the spectrum of this problemtypically contains complex points.
Equivalently, one can write the following matrix analogue of the spectralproblem (31)
Ψx = −iσ3ΨΛ + iUΨ, Λ =
[λ 00 λ′
], (35)
16
where the first and second columns of Ψ are vector solutions of (31) forrespectively the spectral parameters λ and λ′.
3.1 Finite-gap approximation of the AW Cauchy prob-lem
Here we summarize the aspects of the finite gap theory and of the resultsobtained in [39, 40] used in this work.
If Ψ(λ, x, t) is a fundamental matrix solution of (31),(32) such that Ψ(λ, 0, 0)is the identity, then the monodromy matrix T (λ) is the entire function of λdefined by: T (λ) = Ψ(λ, L, 0). The eigenvalues and eigenvectors of T (λ) aredefined on a two-sheeted covering of the λ-plane. This Riemann surface Γ iscalled the spectral curve and does not depend on time. The eigenvectorsof T (λ) are the Bloch eigenfunctions
~Ψx(γ, x, t) = U(λ(γ), x, t)~Ψ(γ, x, t),
~Ψ(γ, x+ L, t) = eiLp(γ)~Ψ(γ, x, t), γ ∈ Γ, (36)
and λ(γ) denotes the projection of the point γ to the λ-plane.The spectrum is exactly the projection of the set {γ ∈ Γ, Im p(γ) = 0}
to the λ-plane. The end points of the spectrum are the branch points andthe double points (obtained merging pairs of branch points) of Γ, at whicheiLp(γ) = ±1, or, equivalently, trT (λ) = ±2:
~Ψ(γ, x+ L, t) = ±~Ψ(γ, x, t), γ ∈ Γ.
The spectral curve Γ0 corresponding to the background (2) is rational,and a point γ ∈ Γ0 is a pair of complex numbers γ = (λ, µ) satisfying thequadratic equation µ2 = λ2 + |a|2. The corresponding monodromy matrix:
trT0(λ) = 2 cos(µL) (37)
defines the branch points (λ±0 , µ0) = (±i|a|, 0) and the resonant (double)points (λ±n , µn) = (±
√(nπ/L)2 − |a|2, nπ/L), n ∈ Z, n 6= 0. Near the
resonant points:
trT0(λ) = (−1)n[2− λ2
nL4
π2n2(λ− λn)2 +O((λ− λn)4)
]. (38)
17
A generic initial O(ε) perturbation (4) of the background perturbs thisspectral picture. The branch points λ±0 = ±i|a| become E0 = i|a| + O(ε2)and E0 = −i|a| + O(ε2), and all double points λ±n , n ≥ 1, generically splitinto a pair of square root branch points, generating infinitely many gaps. If1 ≤ n ≤ N , where N is the number of unstable modes, λ+
n splits into thepair of branch points (E2n−1, E2n), and λ−n into the pair of branch points(E2n−1, E2n); if n > N , each λn splits into a pair of complex conjugateeigenvalues. In the simplest case in whichN = 1, E1, E2 are the branch pointsobtained through the splitting of the excited mode λ+
1 = i|a| sinφ1 =: λ1, and[39]
El = λ1 + (−1)l ε|a|2
2λ1
√αβ +O(ε2)
= i|a|(
sin(φ1) + (−1)l+1 ε2 sinφ1
√αβ +O(ε2)
), l = 1, 2,
(39)
so that
(E1 − E2)2 = −ε2|a|2αβ
sin2(φ1). (40)
From formulas (13) and (40), one can write the following relations, toleading order, between the unstable gap E1 − E2 and the AW recurrenceperiod ∆T and x-shift ∆X:
∆T = 1σ1
log(
σ41
4|a|6 sin2 φ1|E1−E2|2
),
∆X = arg(−(E1−E2)2)k1
.(41)
In the perturbed case, the analogue of equation (38) is
trT (λ) = trT (λ1) +trT ′′(λ1)
2(λ− λ1)2 +O((λ− λ1)3), (42)
where λ1 is the critical point for trT (λ). It is easy to check that
λ1 − λ1 = O(
max(ε2,(E1 − E2)2
|a|2, νT )
),
trT (λ1)− trT (λ1) = O(
max(ε4,(E1 − E2)4
|a|4, (νT )2)
).
We also have
trT (λ)− trT0(λ) = O(
max(ε2,(E1 − E2)2
|a|2, νT )
).
18
Therefore, to leading order, one can replace in (42) trT (λ1) by trT (λ1),
and trT ′′(λ1) by trT ′′0 (λ1) =2λ21L
4
π2 , obtaining
trT (λ) ∼ trT (λ1) +trT ′′0 (λ1)
2(λ− λ1)2 = trT (λ1) +
λ21L
4
π2(λ− λ1)2. (43)
In [39, 40] it was suggested to approximate the solution of the genericCauchy problem, corresponding to infinitely many gaps (see Figure 5 (leftpicture)), by the finite-gap solution obtained closing all gaps near the realline, since they correspond to the stable modes and contribute to the solutionto O(ε) (see Figure 5 (right picture)). If N = 1, we obtain a genus 2 spectralcurve with two O(ε) handles (see Figure 5 (right picture)).
−λ1
E0
λ1
E1
E2
E3
E5
λ2
λ3
−λ2
−λ3
E0
E1
E2
E5
E3
E6
E4
i|a|
−i|a|
E6
E4
−λ1
λ1
E1
E2
E =0 i|a|
E2
E1
E =0 −i|a|
Figure 5: On the left: the spectral curve generated by a small generic periodicperturbation of the background in case of one unstable mode. On the right:the finite-gap genus 2 spectral curve approximating it.
3.2 Variations
If u evolves according to NLS, the monodromy matrix and the spectral curveare constants of motion. Now we calculate how the monodromy matrix,and, as a corollary, the spectral curve evolve in time in the presence of asmall loss or gain. The calculation of the variation of the monodromy matrixuses a standard formula from ordinary differential equations coming from themethod of variation of constants. A perturbation of the finite-gap potentialU (FG)(x, t) in (31), (33)
U (FG)(x, t)→ U (FG)(x, t) + δU (FG)(x, t)
19
induces the following perturbation of the fundamental matrix solution of (31)
Ψ(FG)(λ, x, t)→ Ψ(FG)(λ, x, t) + δΨ(FG)(λ, x, t),
δΨ(FG)(λ, x, t) = Ψ(FG)(λ, x, t)x∫0
Ψ(FG)−1(λ, x′, t)(iδU (FG)(x′, t))Ψ(FG)(λ, x′, t)dx′
(44)where we have assumed without loss of generality that δΨ(FG)(λ, 0, t) = 0.
Let T (λ, y, x, t) be the transition matrix of (31) with respect to the vari-able y, normalized to be the identity matrix E for y = x [34]:
Ty(λ, y, x, t) = (−iλσ3 + iU (FG)(y, t))T (λ, y, x, t),
T (λ, x, x, t) = E.(45)
ThenT (λ, y, x, t) = Ψ(FG)(λ, y, t)Ψ(FG)−1
(λ, x, t), (46)
where Ψ(FG)(λ, x, t) is any fundamental matrix solution of (31), the followingproperties are satisfied
T (λ, y, x, t) = T−1(λ, x, y, t),
T (λ, z, y, t)T (λ, y, x, t) = T (λ, z, x, t),
T (λ, x+ L,L, t) = T (λ, x, 0, t),
(47)
and the monodromy matrix can be expressed in terms of the transition matrixas follows:
T (λ, t) = T (λ, L, 0, t) = Ψ(FG)(λ, L, t)Ψ(FG)−1(λ, 0, t). (48)
From (44) and (48) it follows that the induced variation of the monodromymatrix reads
δT (λ, t) = iΨ(FG)(λ, L, t)L∫0
Ψ(FG)−1(λ, x, t)(δU (FG)(x, t))Ψ(FG)(λ, x, t)(Ψ(FG))−1(λ, 0, t)dx
= iL∫0
T (FG)(λ, L, x, t)(δU (FG)(x, t))T (FG)(λ, x, 0, t)dx.
(49)Making use of the properties (47), we finally obtain
δ trT (λ, t) = iL∫0
tr[T (λ, x+ L,L, t)T (λ, L, x, t)δU (FG)(x, t)T (λ, x, 0, t)×
×(T )−1(λ, x+ L,L, t)]dx = i
L∫0
tr[T (λ, x+ L, x, t)δU (FG)(x, t)
]dx.
(50)
20
Choosing δU = Utdt = [iσ3(Uxx + 2U3) − νU ]dt, then δ trT = (trT )tdt;substituting these variations into equation (50) and taking into account
thatL∫0
tr[T (λ, x+ L, x, t)σ3
(U
(FG)xx (x, t) + 2(U (FG))3(x, t)
)]dx = 0, it fol-
lows that
(trT )t(λ, t) = −iνL∫
0
tr[T (λ, x+ L, x, t)U (FG)(x, t)
]dx. (51)
At last, integrating this equation over time, from 0 to t, one obtains thevariation of trT in the time interval [0, t]:
∆ trT (λ, t) = −iνt∫
0
dt
L∫0
tr[T (λ, L+ x, x, t)U (FG)(x, t)
]dx
=
= −iνt∫
0
dt
L∫0
[T21(λ, x+ L, x, t)u(FG)(x, t) + T12(λ, x+ L, x, t)u(FG)(x, t)
]dx
.The calculation of the above integral with high genus theta-functions
is very complicated. But, to leading order, this integral can be explicitlycalculated in terms elementary functions using the following properties ofthis solution:
1. Near each AW appearance the solution is well approximated by theAkhmediev breather.
2. Far from the AW appearance, the integral over the x-period tends tozero exponentially in t. Therefore the integral over the finite time in-terval of each AW appearance can be well approximated by the integralover the whole line −∞ < t <∞ of the Akhmediev solution.
We conclude that, to leading order,
∆ trT (λ, t) = nappνJ(λ),
where napp is the number of AW appearances in the time interval [0, t], and
J(λ) = −i+∞∫−∞
dt
L∫0
[T21(λ, x+ L, x, t)u(x, t) + T12(λ, x+ L, x, t)u(x, t)
]dx
,(52)
21
where u(x, t) is the Akhmediev breather and T (λ, x+L, x, t) is its transitionmatrix. Let us recall that, to calculate the variation of the curve we needboth ∆ trT (λ, t) and J(λ) at the point λ = λ1. To compute T (λ, x+L, x, t)it is convenient to use the classical Darboux transformation for NLS.
3.3 Darboux transformation of the constant background
Darboux Transformations (DTs) are used to construct solutions of integrablePDEs from simpler solutions [68]. Here we use the well-known DT of NLS toconstruct the Akhmediev breather and its eigenfunctions from the constantbackground solution.
Proposition [91]. Let ~Ψ0(λ, x, t) be a solution of (31),(32) for U = U0 =[0 u0(x, t)
u0(x, t) 0
], and let Φ0 be a fixed solution of (35) for U = U0, but
with a fixed Λ = Λ1 =
[λ1 00 λ′1
]. Let τ be
τ(x, t) = Φ0(x, t)Λ1Φ−10 (x, t), (53)
implying thatτx = −i[σ3, τ ]τ + i[U0, τ ].
Then
~Ψ(λ, x, t) = [λ− τ(x, t)]~Ψ0(λ, x, t), U(x, t) = U0(x, t)− [σ3, τ(x, t)] (54)
are the Darboux transformed eigenfunction and potential satisfying (31),(32).We apply this transformation on the constant background (2). As we
have already seen, the unperturbed spectral curve Γ0 is rational, and a pointγ ∈ Γ0 is a pair of complex numbers γ = (λ, µ) satisfying the equationµ2 = λ2 + |a|2. On the imaginary interval −|a| ≤ Imλ ≤ |a|, Reλ = 0 wecan write:
λ = i|a| sin(φ), µ = |a| cos(φ), λ+ µ = |a|eiφ.
The Bloch eigenfunctions for the operator L0 can be easily calculatedexplicitly:
~ψ±(γ, x) =
[aei|a|
2t[λ(γ)± µ(γ)
]e−i|a|
2t
]e±iµ(γ)x±2iλ(γ)µ(γ)t, (55)
22
L0ψ±(γ, x) = λ(γ)ψ±(γ, x),
or, in a different normalization,
~ψ±(φ, x) =
[aei|a|
2t∓iφ/2
±|a|e−i|a|2t±iφ/2
]e±i|a| cos(φ)x∓|a|2 sin(2φ)t. (56)
Denote by ~q the special solution of (31), for u = u0, and λ = λ1, obtainedadding up the two vector solutions (56):
~q(x, t) =
[q1
q2
]=
[aei|a|
2t(e−iφ1/2+i|a| cos(φ1)x−|a|2 sin(2φ1)t + eiφ1/2−i|a| cos(φ1)x+|a|2 sin(2φ1)t
)|a|e−i|a|2t
(eiφ1/2+i|a| cos(φ1)x−|a|2 sin(2φ1)t − e−iφ1/2−i|a| cos(φ1)x+|a|2 sin(2φ1)t
) ](57)
= 2
[aei|a|
2t cos(k12x− φ1/2 + iσ1
2t)
i|a|e−i|a|2t sin(k12x+ φ1/2 + iσ1
2t)
], (58)
where
λ1 = i|a| sin(φ1), µ1 = |a| cos(φ1), k1 = 2|a| cos(φ1), σ1 = 2|a|2 sin(2φ1),
and we assume φ1 to be real. The complex symmetry of (31) implies that,
since
[q1
q2
]is solution of (31) for u = u0 and λ = λ1, then
[−q2
q1
]is
solution of (31) for u = u0 and λ = λ1 = −i|a| sin(φ1). Consequently
Φ0 =
[q1 −q2
q2 q1
](59)
solves (35) for U = U0 and Λ1 =
[λ1 0
0 λ1
], and (from (53))
τ(x, t) =λ1
Den(x, t)
[q1(x, t)q1(x, t)− q2(x, t)q2(x, t) 2q1(x, t)q2(x, t)
2q1(x, t)q2(x, t) −q1(x, t)q1(x, t) + q2(x, t)q2(x, t)
],
(60)where
Den(x, t) = q1(x, t)q1(x, t)+q2(x, t)q2(x, t) = 4|a|2 [cosh(σ1t) + sin(φ1) sin(k1x)] .
At last, the dressed potential reads
u(x, t) = ae2i|a|2t − 4λ1Den(x,t)
q1(x, t)q2(x, t) = ae2i|a|2t cosh[σ1t+2iφ1]−sin(φ1) sin(k1x)cosh(σ1t)+sin(φ1) sin(k1x)
.
(61)
23
It coincides with the Akhmediev breather solution (14) after identifying φ1 =θ and introducing suitable free parameters corresponding to the x and ttranslation symmetries of NLS.
If u = u0(x, t), then the corresponding transition matrix reads:
T0(λ, y, x, t) =
[cos(µ(y − x))− iλ
µsin(µ(y − x)) ia
µsin(µ(y − x))e2i|a|2t
iaµ
sin(µ(y − x))e−2i|a|2t cos(µ(y − x)) + iλµ
sin(µ(y − x))
]=
(62)
=1
µ
[|a| cos(µ(y − x)− φ) ia sin(µ(y − x))e2i|a|2t
ia sin(µ(y − x))e−2i|a|2t |a| cos(µ(y − x) + φ)
],
with
∂λT0(λ, y, x, t) =i sin(µ(y − x))
µ3
[−|a|2 −aλe2i|a|2t
−aλe−2i|a|2t |a|2
]+ (63)
+λ(y − x)
µ2
[−|a| sin(µ(y − x)− φ) ia cos(µ(y − x))e2i|a|2t
ia cos(µ(y − x))e−2i|a|2t −|a| sin(µ(y − x) + φ)
].
Consequently:
T0(λn, x+ L, x, t) = (−1)nE,
∂λT0(λ, x+ L, x, t)
∣∣∣∣λ=λn
= (−1)nn iλnπµ3n
[−λn ae2i|a|2t
ae−2i|a|2t λn
].
(64)
If u is the potential (61), dressed from the background u0, then the cor-responding transition matrix reads (from (54))
T (λ, y, x, t) = Ψ(λ, y, t)Ψ−1(λ, x, t) = (λ− τ(y, t))Ψ0(λ, y, t)Ψ−10 (λ, x, t)(λ− τ(x, t))−1
= (λ− τ(y, t))T0(λ, y, x, t)(λ− τ(x, t))−1,(65)
where T0 is given in (62) and τ in (60). To evaluate (65) at y = x + L andλ = λ1, we use the following expansions for λ near λ1:
T0(λ, x+ L, x, t) = T0(λ1, x+ L, x, t) + ∂T0(λ,x+L,x,t)∂λ
|λ=λ1(λ− λ1) +O(λ− λ1)2
= −E − iλ1πµ31
[−λ1 ae2i|a|2t
ae−2i|a|2t λ1
](λ− λ1) +O(λ− λ1)2,
λ− τ(y, t) = λ1−λ1Den(y,t)
[−q2(y, t)
q1(y, t)
][−q2(y, t), q1(y, t)] +O(λ− λ1),
(λ− τ(x, t))−1 = 1(λ−λ1)Den(x,t)
[q1(x, t)q2(x, t)
][q1(x, t), q2(x, t)] +O(1),
(66)
24
and the periodicity property Φ0(λ1, x + L, t) = −Φ0(λ1, x, t). After somealgebra one finally obtains
T (λ1, x+ L, x, t) = −E +2πiλ21µ31
f(x,t)Den2(x,t)
[−q2(x, t)q1(x, t) −q2(x, t)q2(x, t)
q1(x, t)q1(x, t) q1(x, t)q2(x, t)
].
f(x, t) = ae2i|a|2tq22(x, t)− ae−2i|a|2tq2
1(x, t)− 2i|a| sinφ1 q1(x, t)q2(x, t).(67)
Using (67) we finally write explicitely the integral J(λ1), defined in (52),describing the variation of trT (λ1) at each AW appearance:
J(λ1) = − 2π sin2 φ1|a| cos3 φ1
+∞∫−∞
dtL∫0
dxf(x,t)g(x,t)Den2(x,t)
,
g(x, t) = u(x, t)q1(x, t)2 − u(x, t)q2(x, t)2.
(68)
3.4 The final formulas
The following two important simplifications
f(x, t) = ae2i|a|2tq22(x, t)− ae−2i|a|2tq2
1(x, t)− 2i|a| sinφ1 q1(x, t)q2(x, t) = −4a|a|2 cos2(φ1),
g(x, t) = u(x, t)q1(x, t)2 − u(x, t)q2(x, t)2 = 4a|a|2 (cos(2φ1)− sin(φ1) sin(k1x− iσ1t)) ,(69)
lead to the double integral
J(λ1) = 2π|a| sin2 φ1
cosφ1
+∞∫−∞
dt
L∫0
dxcos(2φ1)− sin(φ1) sin(k1x− iσ1t)
(cosh(σ1t) + sin(φ1) sin(k1x))2=
= 2π2 sin2(φ1)
+∞∫−∞
cosh(σ1t)
(cosh2(σ1t)− sin2(φ1))3/2dt (70)
=2π2 sin2(φ1)
σ1
∞∫−∞
d(sinh(σ1t))
(sinh2(σ1t) + cos2(φ1))3/2=
π2 sinφ
|a|2 cos3 φ.
The integration with respect to x has been done using contour integration.The integration wrt t is even more elementary. Therefore the variation∆1(trT (λ1)) of trT (λ1), due to a single appearance of the AW, is givenby
∆1(trT (λ1)) = νJ(λ1) = ν|a|2
π2 sinφ1cos3 φ1
. (71)
25
Evaluating (43) at λ = E1 and recalling that E1 − E2 = 2(E1 − λ1), wehave
trT (λ1) ∼ −2 +sin2(φ1)L4
4π2(E1−E2)2
(= −2− ε2 L
4
4π2αβ, at t = 0
). (72)
Then, due to (71) and (72), after each appearance:
ν
|a|2π2 sinφ1
cos3 φ1
= ∆1(trT (λ1)) =L2 sin2 φ1
4 cos2 φ1
∆1
((E1 − E2)2
). (73)
Therefore we have established that, after each appearance of the AW, thesquare of the gap varies of the same O(ν) quantity:
∆1
((E1 − E2)2
)= 4ν cot(φ1) (74)
with (see (40))
(E1 − E2)2
∣∣∣∣t=0
= −ε2|a|2αβsin2 φ1
. (75)
We conclude that
(E(m)1 − E(m)
2 )2 = −ε2|a|2αβsin2 φ1
+ 4mν cotφ1, m ≥ 0, (76)
where E(m)1 , E
(m)2 are the positions of the branch points of the gap after the
mth AW appearance. This formula implies that, as m increases and the AWdynamics tends to the SVAs (29), the gap tends to become horizontal ifν > 0 (loss), or vertical if ν < 0 (gain) (see Figure 6). Let us remark that,in contrast with [32], the dynamics of the spectral curve is here essentiallydiscrete. The spectral description of the two SVAs is therefore given by theelementary formula
(E(m)1 − E(m)
2 )2 = 4mν cotφ1 (77)
showing that the length of the gap grows through the law
|E(m)1 − E(m)
2 | = 2√m|ν| cotφ1. (78)
26
Figure 6: The figure contains the numerical experiment illustrated in thecentral picture of Figure 2, together with the corresponding time evolutionof the gap E1−E2, due to each AW appearance. It shows how the gap E1−E2
tends to become horizontal as the number of AW appearances increases, inthe case of loss (in the case of gain, the gap would tend to become vertical).The quantitative agreement among the numerical output, the analytic for-mulas (24)-(26) describing the AW dynamics, and the analytic formula (76)describing the position of the gap after each AW appearance is extremelygood.
It is also convenient to introduce the quantity Q such that
ε2∆1Q = − sin2(φ1)∆1((E1 − E2)2)
|a|2, Q
∣∣∣∣t=0
= αβ. (79)
Then also the variation of Q after each appearance of the AW is constant:
∆1Q = Qm+1 −Qm = − ν
|a|2ε22 sin(2φ1) = − νσ1
|a|4ε2, Q0 = αβ (80)
implying (26).
27
4 Acknowledgments
The work of P. G. Grinevich was supported by the Russian Science Foun-dation grant 18-11-00316. P. M. Santini was partially supported by theUniversity La Sapienza, grant 2017.
We acknowledge useful discussions with F. Calogero, A. Chabchoub, C.Conti, E. DelRe, A. Degasperis, A. Gelash, A. Giansanti, D. Pierangeli, and,above all, with F. Briscese, with whom one of us (PMS) shared few years agosome calculations concerning the first linear stage of modulation instabilityand the first appearance of the AW in the presence of a small dissipation.
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