On the discrete Kuznetsov–Ma solutions for the defocusing

Ablowitz-Ladik equation with large background amplitude

E. C. Boadi1 , E. G. Charalampidis2,3 , P. G. Kevrekidis4 , N. J. Ossi1 , B. Prinari1
1
Department of Mathematics, State University of New York, Buffalo, NY 14260
2
Department of Mathematics and Statistics, San Diego State University, San Diego,
CA 92182
3
Computational Science Research Center, San Diego State University, San Diego, CA
92182
4
Department of Mathematics and Statistics, University of Massachusetts, Amherst,
arXiv:2501.00121v1 [nlin.SI] 30 Dec 2024

MA 01003

Abstract
The focus of this work is on a class of solutions of the defocusing Ablowitz-Ladik lattice on an
arbitrarily large background which are discrete analogs of the Kuznetsov-Ma (KM) breathers of the
focusing nonlinear Schrödinger equation. One such solution was obtained in 2019 as a byproduct of
the Inverse Scattering Transform, and it was observed that the solution could be regular for certain
choices of the soliton parameters, but its regularity was not analyzed in detail. This work provides
a systematic investigation of the conditions on the background and on the spectral parameters that
guarantee the KM solution to be non-singular on the lattice for all times. Furthermore, a novel KM-
type breather solution is presented which is also regular on the lattice under the same conditions.
We also employ Darboux transformations to obtain a multi-KM breather solution, and show that
parameters choices exist for which a double KM breather solution is regular on the lattice. We
analyze the features of these solutions, including their frequency which, when tending to 0, renders
them proximal to rogue waveforms. Finally, numerical results on the stability and spatio-temporal
dynamics of the single KM breathers are presented, showcasing the potential destabilization of the
obtained states due to the modulational instability of their background.

1 Introduction
The Ablowitz-Ladik (AL) model:
dqn 1
i = 2 (qn+1 − 2qn + qn−1 ) − σ|qn |2 (qn+1 + qn−1 ), σ = ±1, n ∈ Z, t ∈ R (1)
dt h
was introduced in [2, 3] as an integrable spatial discretization of the focusing (σ = −1) and defocusing
(σ = 1) nonlinear Schrödinger (NLS) equation:

iqt = qxx − 2σ|q|2 q,

to which they reduce as the lattice spacing h → 0 with nh → x.

Besides being used as numerical schemes for their continuous counterparts, the AL equations describe
the quantum correlation function of the XY -model of spins [21, 24]. When σ = −1, the equation plays
a significant role in the study of anharmonic lattices [35] and lossless nonlinear electric lattices [27], and
it is gauge equivalent to an integrable discretization of the Heisenberg spin chain [20]. Additionally, the
focusing AL hierarchy describes the integrable motions of a discrete curve on the sphere [17]. Importantly,
in addition to its intrinsic value, the AL model provides a springboard towards exploring solutions in the
non-integrable, yet more widespread discrete variant of the nonlinear Schrödinger model, namely the so
called discrete NLS equation [22]. Indeed, this has been leveraged for solitary wave dynamics in early
works, such as, e.g., [11], but also for more complex structures including periodic and rogue waves, e.g.,
in the more recent work of [33]. This justifies a multi-faceted interest in the periodic (and rogue) wave
solutions of the Ablowitz-Ladik model [5], from a theoretical point of view in their own right, but also
from a computational and applications point of view, as well.

1
 In this work we are interested in characterizing special solutions of the defocusing AL equation on a
nontrivial background. Specifically, we will consider the equation:
dQn
i = Qn+1 + Qn−1 + 2r2 Qn − |Qn |2 (Qn+1 + Qn−1 ), r2 = Q2o − 1 > 0 (2)
dτ
with boundary conditions:

lim Qn (τ ) = Q± ≡ Qo eiθ± Qo > 1, θ± ∈ R. (3)

n→±∞

Modulo a rescaling of dependent and independent variables (Qn = hqn and τ = t/h−2 ), the above
2
equation is gauge-equivalent to Eq. (1) (the gauge transformation Qn → Qn = e−2iQo τ has simply the
effect of allowing for constant boundary conditions as n → ±∞).
As mentioned above, the AL equations are integrable, and their initial-value problem can be solved
by means of the Inverse Scattering Transform (IST). The IST for Eq. (2) was initially developed under
the assumption that the background amplitude Qo satisfies a “small norm” condition, 0 < Qo < 1
[1, 14, 36]. This condition is actually not just a technical assumption: in [28] it was shown that the
defocusing AL equation, which is modulationally stable for 0 ≤ Qo < 1, becomes unstable if Qo > 1,
and exhibits discrete rogue wave solutions, some of which are regular for all times. This finding sparked
renewed interest in the defocusing AL equation on a background. In [29] the IST was generalized to the
case of arbitrarily large background Qo > 1. As a by-product of the IST, explicit solutions were also
obtained in [29], which are the discrete analog of the celebrated Kuznetsov–Ma [23, 26], Akhmediev [7, 8],
Tajiri-Watanabe [34] and Peregrine solutions [5, 6, 30], and which mimic the corresponding solutions for
the focusing AL equation [9, 10, 31, 32].
The recent work [15] further investigated the stability of background solutions of the AL equations in
the periodic setting, confirming in particular that, unlike its continuous counterpart, in the defocusing
case (AL− , in the terminology of [15]) the background solution is unstable under a monochromatic
perturbation of any wave number if the amplitude of the background is greater than 1. A discrete
Kuznetsov-Ma (KM) breather solution was presented in [29], and it was observed that the solution could
be regular for certain choices of the soliton parameters, but its regularity was not analyzed in detail.
The goal of the present work is to provide a systematic investigation of the conditions on the back-
ground and on the spectral parameters (discrete eigenvalue and associated norming constant) that guar-
antee the KM solution to be non-singular for all n ∈ Z and all τ ∈ R. Furthermore, a novel KM-type
breather solution will be presented which is also regular on the lattice under the same conditions. We also
employ the Darboux transformations derived in [15] to analyze Akhmediev breathers, and, by generaliz-
ing the construction to the case of KM breathers, we obtain multi-KM breather solutions. In particular,
we show that parameters choices exist for which a double KM breather is regular on the lattice. Finally,
we corroborate the analytical findings by numerical computations where for the obtained exact solutions,
a spectral stability analysis is performed and dynamical evolution of the lattice model with such wave-
forms as initial data is explored. The latter will often lead to unbounded growth of the solution due to
the manifestation of the modulational instability of its background.
Our presentation is structured as follows. First, we provide some relevant background of the model,
including its Lax pair and associated discrete spectrum. We then move to the construction of the KM
breathers, followed by an analysis of the conditions under which the relevant waveforms are regular. We
then proceed to provide a construction of the solutions via Darboux transformations, before exploring
the stability and direct numerical dynamics of the associated waveforms. We close with a discussion of
the conclusions of our findings and some directions for future research.

2 Lax pair and discrete spectrum

The Lax pair for the AL system (2) is given by [2]:

dvn
vn+1 = Ln vn , = Mn vn , (4a)
 dτ
  
z 0 0 Qn
Ln = Z + Qn , Z= , Qn = , (4b)
0 1/z Q∗n 0
 
1
Mn = iσ3 Qn Qn−1 − Q2o I − (z − 1/z)2 I − Qn Z−1 + Qn−1 Z , (4c)
2

2
 Sheet I: λ=ξ+ 2

.ξ
ξ−1

Im ξ Im ξ

bI
0 |λ|=1

r2 -- 1 1 Re ξ
r1 |λ|=1 aI

~
r

θ1 Sheet II: λ=ξ− 2

ξ−1

.
−1
θ2 θ .1 Im ξ
Re ξ

aII
0 |λ|=1

-- 1 1 Re ξ
|λ|=1 b II

Figure 1: Left: The choice of branch cut in the complex ξ-plane, ξ = (z + 1/z)/2ir. Right: The 2-sheeted
covering defined by λ = ξ + (ξ 2 − 1)1/2 .

where vn is a 2-component vector, z ∈ C is the scattering parameter, and Qn (τ ) is the AL potential.

If Qn (τ ) satisfies (3) as n → ±∞, the asymptotic behavior of the simultaneous solutions of the Lax
pair is given by λn (ir)n eiΩτ or λ−n (ir)−n e−iΩτ , where λ as a function of z is defined by
p
ir(λ + 1/λ) = z + 1/z , r = Q2o − 1 ∈ R+ , (5)

and the dispersion relation is given by:

i ir 2
Ω(z) = − r(z − 1/z)(λ − 1/λ) ≡ − (z − 1)(λ2 − 1) . (6)
2 2λz
Clearly, λ = λ(z) has branching, with 4 branch points on the imaginary axis located at:

z = i(r ± Qo ) , z = i(−r ± Qo ) . (7)

Note that −r − Qo < −1 < r − Qo < 0 < Qo − r < 1 < Qo + r, which specifies the order of the branch
points. It is convenient to introduce the variable ξ = (z + 1/z)/2ir, and define a 2-sheeted Riemann
surface cut across the segment (−1, 1) on the real ξ-axis, with local coordinates on each sheet given as
in the left panel of Fig. 1:

ξ − 1 = r1 eiθ1 , ξ + 1 = r2 eiθ2 0 ≤ θ1 , θ2 < 2π (8)

iθ
ξ = r̃e 0 ≤ θ < 2π (9)
p
such that on Sheet I/II λ(z) = ξ ± ξ 2 − 1. The IST developed in [29] is formulated in terms of a
uniformization variable
ζ(z) = λ(z)/z , (10)
for which one has
ζ − ir ζ − ir ζ − ir
z2 = , λ2 = ζ , zλ = , (11)
ζ(irζ − 1) irζ − 1 irζ − 1
showing that all these quantities, as well as all quantities which are even functions of λ and z, are mero-
morphic functions of ζ. It is worth noticing that |λ2 (ζ)| ≶ 1 for ζ ∈ D± , where D± = {ζ ∈ C : |ζ − ir| ≶ Qo },
and |λ(ζ)|2 = 1 for ζ ∈ Σ, where Σ = {ζ ∈ C : |ζ − ir| = Qo }. We refer the reader to [29] for the details
about the IST, while noting that in Appendix B we provide a short errata to correct some typos and
small inaccuracies in [29]. For the purposes of this work, it suffices to recall that the solitons/breathers of
the AL equation are characterized by special values of the spectral parameter (z, or ζ) for which bounded
eigenfunctions of the Lax pair exist. In terms of the uniformization variable ζ such discrete eigenvalues
ζk∗ +ir
n o
∗ irζk −1
appear in symmetric quartets: ζk , irζ ∗ +1 , −1/ζk , ζk −ir , with the first two in D− and the last two
k
in D+ . It is convenient to then group the eigenvalues as follows:
∗
 2J ζk−J + ir
ζk , ζ̄k k=1
, ζk = ∗ for k = J + 1, . . . , 2J (12)
irζk−J + 1

3
and ζ̄k = −1/ζk∗ for k = 1, . . . , 2J. As is typical, the discrete eigenvalue specifies the velocity and
amplitude of the corresponding soliton/breather. To each discrete eigenvalue ζ̄k ∈ D+ is associated a
norming constant C̄k :

∗ (Q∗+ )2
Ck = − C̄k /ζ̄k2 , Ck+J = − C̄k k = 1, . . . , J (13)
(ζ̄k − ir)2
which in turn characterizes the soliton/breather position and phase.
The general discrete breather solution of Eq. (2) on the nontrivial background (3) is given in Eq. (89)
in [29], and it is an analog of the discrete Tajiri-Watanabe (TW) breather of the focusing AL equation
obtained in [31]. The solution is given in terms of a discrete eigenvalue ζ̄1 ∈ D+ and its associated
norming constant C̄1 :
(δ1∗ + δ2∗ ) cos η − i(δ1∗ − δ2∗ ) sin η + δ3∗ e−ξ
 
ir
Qn (τ ) = Q+ 1 + √ , (14)
2Q∗+ δ4 cosh (ξ − ξo ) + α cos(η − ηo )

where
ζ̄1 − ir Q2+ ζ̄1∗ |γ|2 (ζ̄1 − ir)|ζ̄2 − ζ̄1 |2 (ζ̄1 + ζ̄1∗ )
δ1 ± δ2 = C̄1 ± C̄1∗ , δ3 = − , (15a)
1 − irζ̄1 (1 + irζ̄1∗ )2 Q∗+ (ζ1 − ir)(1 + |ζ̄1 |2 )|λ̄21 − 1|2
|γ|2 |ζ̄1 |2 |ζ̄1 − ir|2 (|1 − irζ̄1 |2 − Q2o )2 Q∗+ C̄1 γ λ̄21
δ4 = , γ= , Ã1 + γ = , (15b)
r2 Q2o |λ̄21 − 1|2 (1 + |ζ̄1 |2 )2 |1 − irζ̄1 |2 1 − irζ̄1 λ̄21 − 1
ξ = n log ρ + 2(Im Ω̄1 )τ , η = nϕ + 2(Re Ω̄1 )τ, (15c)
p
α = |Ã1 + γ|/ δ4 > 1 , (15d)
p Re(Ã1 + γ) Im(Ã1 + γ)
ξo = log δ4 , cos ηo = , sin ηo = − , (15e)
|Ã1 + γ| |Ã1 + γ|
r2 (ζ̄12 + 2iζ̄1 /r + 1)(ζ̄12 − 2irζ̄1 + 1)
Ω̄1 ≡ Ω(ζ̄1 ) = . (15f)
2ζ̄1 (ζ̄1 − ir)(1 − irζ̄1 )

According to (11), λ2 (ζ) = ζ(ζ − ir)/(irζ − 1) and

λ̄21 = λ2 (ζ̄1 ) =: eiϕ /ρ. (16)

Also, ζ̄2 = (ζ̄1∗ + ir)/(irζ̄1∗ + 1) and ζ1 = −1/ζ̄1∗ , and we recall that |λ2 (ζ)| < 1 for any ζ ∈ D+ (hence,
ρ > 1). It is also important to point out that α > 1 for any choice of the soliton parameters ζ̄1 and C̄1 ,
which means that in general the TW solution is singular. This singularity, as well as ways to avoid it, is
crucial for our considerations that follow next. We refer the reader to [29] for the details of the derivation
of Eq. (14), while noting that we corrected a typo in the expressions for ξ, η (see also Appendix B).

3 Kuznetsov-Ma (KM) breathers

The discrete analog of the Kuznetsov-Ma (KM) solutions is obtained when the discrete eigenvalue ζ̄1 ∈
D+ \ {0, r} is purely imaginary, i.e., ζ̄1 = iκ with
p p
κ ∈ (− 1 + r2 + r, 1 + r2 + r) \ {0, r} ,
p
where we recall r = Q2o − 1 > 0. In this case, the parameter λ̄21 = λ2 (ζ̄1 ) ∈ R, but note that it is not
necessarily positive. Specifically, one has
κ(κ − r)
λ̄21 = (17)
1 + rκ
√
and one can check that −1/r < − 1 + r2 + r for any r, therefore
p p
λ̄21 > 0 for − r2 + 1 + r < κ < 0 and r < κ < r2 + 1 + r
λ̄21 < 0 for 0 < κ < r.

In [29] the implicit assumption was made that λ̄21 > 0, but in the parametrization λ̄21 = eiϕ /ρ one needs
to take ϕ = π if λ̄21 < 0, while indeed ϕ = 0 when λ̄21 > 0. This will not affect the regularity of the

4
solution (see below), but it will give rise to a different solution if one chooses values of r, κ for which
λ̄21 < 0. Based on the above discussion, we have:

1 + κr
ρ= > 1, (18)
κ(κ − r)

and
p p
ϕ=0 for − 1 + r2 + r < κ < 0 or r < κ < r2 + 1 + r (19)
ϕ=π for 0 < κ < r. (20)

Now, considering (15) when ζ̄1 = iκ, one can immediately see that δ3 = 0 and

r(rκ2 + 2κ − r)(−κ2 + 2rκ + 1)

Ω̄1 = ∈R [corrected a typo in Eq. (94) in [29]] . (21)
2κ(κ − r)(1 + κr)

The remaining coefficients (15) in the TW solution reduce to

|C̄1 |2 κ2 (κ − r)2 (rκ2 + 2κ − r)2

δ4 = > 0, (22a)
(1 + κ2 )2 (1 + κr)2 (κ2 − 2κr − 1)2
κ−r κ
δ1 ± δ2 = iC̄1 ∓ iC̄1∗ Q2+ , (22b)
1 + κr (1 + κr)2
Q∗+ κ(κ − r)
Ã1 + γ = C̄1 , (22c)
(1 + rκ)(κ2 − 2rκ − 1)

and
p
ξ = n log ρ , η = nϕ + 2Ω̄1 τ , ξo = log δ4 , (23a)
√
|Ã1 + γ| (1 + κ2 ) 1 + r2
α= √ ≡ > 1, (23b)
δ4 |rκ2 + 2κ − r|
Re(Ã1 + γ) Im(Ã1 + γ)
cos ηo = ≡ −sgn(λ̄21 ) cos(χ − θ) , sin ηo = − ≡ sgn(λ̄21 ) sin(χ − θ) , (23c)
|Ã1 + γ| |Ã1 + γ|
iχ iθ
where
iθ
√ we have written the norming constant C̄1 = c e , c > 0, and the background as Q2+ = Qo e ≡
e 2
1 + r , and to simplify the expressions of cos ηo , sin ηo we have taken into account that κ −2rκ−1 < 0
for κ in the allowed range, and that the remaining factor in Ã1 + γ is simply λ̄21 . Note the additional
term nϕ in the expression (23a) of η compared to [29], which, we reiterate, is only present if 0 < κ < r.
For simplicity, we can assume below that θ+ = 0, i.e., Q+ = Qo , and also take the norming constant
to be real and positive, i.e., χ = 0 and C̄1 = c > 0. Then, if r, κ are chosen so that λ̄21 > 0, ηo = π and
from (14) we obtain

ir (δ1∗ + δ2∗ ) cos (2Ω̄1 τ ) − i (δ1∗ − δ2∗ ) sin (2Ω̄1 τ )

Qn (τ ) = Qo + √ . (24)
2 δ4 cosh(n log ρ − ξo ) − α cos (2Ω̄1 τ )

Conversely, still assuming that the phases of Q+ and C̄1 are zero, if r, κ are chosen so that λ̄21 < 0, then
ηo = 0, and (14) yields

ir(−1)n (δ1∗ + δ2∗ ) cos (2Ω̄1 τ ) − i (δ1∗ − δ2∗ ) sin (2Ω̄1 τ )

Qn (τ ) = Qo + √ . (25)
2 δ4 cosh(n log ρ − ξo ) + (−1)n α cos (2Ω̄1 τ )

Eq. (24) was already obtained in [29], and we only corrected a typo. In the remainder of this work,
we will refer to this solution as KM1. On the other hand, to the best of our knowledge Eq. (25) is a
novel KM breather solution of the defocusing AL equation with background Qo > 1, which we will refer
to as KM2. Some prototypical case examples of both families of solutions are shown in Fig. 2. There,
it can be seen that the examples of KM2 breathers, carrying the staggered spatial structure evident in
Eq. (25), feature peaks that bear an alternating structure in time, contrary to what is observed for the
KM1 breathers.

5
 r = 4, k = 4.5, c= 101.562

(a) (b) (c) (d) (e)

0 0 0 0
2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 2 4 6 8 10 0
2 4 6 8 10

-5 -5 -5 -5

log10 max En(τ) log10 max En(τ) log10 max En(τ) log10 max En(τ) log10 max En(τ)
-5

n∈ℤ n∈ℤ n∈ℤ n∈ℤ n∈ℤ

-10 -10 -10 -10
-10

-15 -15 -15 -15

-15

-20 -20 -20 -20 -20

Figure 2: Regular Kuznetsov-Ma solutions corresponding to (a) r = 0.8, κ = −0.3, c=1.23 (Period:
3.44946); (b) r = 0.8, κ = 0.7, c = 257.5 (Period: 0.530412); (c) r = 2, κ = 3.5, c = 2.32339 (Period:
1.62646); (d) r = 2, κ = 0.09, c = 13.5884 (Period: 0.261321); (e) r = 4, κ = 4.5, c = 101.562 (Period:
0.0466168). (a), (c), and (e) are KM1 solutions, i.e., given by (24), while (b) and (d) are KM2 solutions,
i.e., corresponding to (25). To confirm that the solutions presented are correct, below each panel is a
log-plot of the error maximized over n ∈ Z as a function of τ .

4 Regularity conditions for the KM breathers

Note that α > 1 for any κ ̸= r [see Eq. (23b)], which means that the denominators in Eq. (24) and

Eq. (25) will generically have zeros, and hence the solution will be singular unless, for a given r (specified
by Qo ), one can choose the soliton parameters κ and C̄1 so that no integer falls in the interval where
cosh(n log ρ − ξo ) ≤ α.
Indeed, the minimum value √ for the cosh term in the denominator is 1, and it is achieved when
x = ξo / log ρ, where ξo = log δ4 . The interval where cosh is less than or equal to α is given by

(ξo − arccosh α)/ log ρ < x < (ξo + arccosh α)/ log ρ, (26)

and the width of this interval is 2 arccosh α/ log ρ, which is independent of the norming constant C̄1 ,
while the center of the interval, which is the center of the soliton, ξo / log ρ, is determined by |C̄1 | via δ4 .
A necessary condition in order for the above interval (26) not to contain any integer is that the width
of the interval be strictly less than 1, and hence a necessary condition for regularity of the solution is
that " √ #
(1 + κ2 ) 1 + r2 1 1 + κr
f (r, κ) := arccosh 2
− log < 0. (27)
|rκ + 2κ − r| 2 κ(κ − r)
The latter
p is an inequality forp the two parameters r, κ, and one can find pairs of values ro , κo with ro > 0
and − ro2 + 1 + r < κo < ro2 + 1 + ro , κo ̸= 0, ro for which this necessary condition for regularity is
satisfied; see, in particular, the representation of the left panel of Fig. 3. Note that the set of loci of
(r, κ) for which (27) is satisfied is certainly not empty, since log ρ in the second term is positive (recall
ρ > 1), and it becomes arbitrarily large as κ → r from either side. The contour plot of f (r, κ) is shown
on the left panel in Fig. 3. A stronger regularity condition, i.e., requiring the width of the interval where
cosh(n log ρ − ξo ) − α ≤ 0 to be smaller than 1/2, can be imposed to ensure a larger lower bound on the
denominator of the solution. The corresponding contour plot is shown on the right panel in Fig. 3.
Once ro , κo are chosen to satisfy (27) so that the interval has width less than 1, we need to choose
the center of the interval so that it does not contain any integer. We can fix n̄ ∈ Z arbitrarily, and place
the center the soliton in the interval (n̄, n̄ + 1) by requiring:

ξo
n̄ + ∆ < < n̄ + 1 − ∆, (28)
log ρ
where the half width of the interval, ∆, is
√
arccosh (1 + κ2 ) 1 + r2 |rκ2 + 2κ − r|−1
 
arccosh α 1
∆= = < . (29)
log ρ log 1+κr 2
κ(κ−r)

6
 6 (a) 6 (b)

5 5

4 4

κ =r κ =r
3 3
κ

κ
2 2

1 κ = (Qo − 1)/r 1 κ = (Qo − 1)/r

0 0
κ =0 κ =0

-1 -1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0
r r

√
Figure 3: (a) Contour plot of f (r, κ) = arccosh α − log ρ as a function of r (horizontal axis) and κ
√ √
(vertical axis) for r − r2 + 1 < κ < r + r2 + 1. The blue shaded regions correspond to f (r, κ) < 0
(where the regularity condition is satisfied) and the red shaded regions correspond to f (r, κ) > 0 (where
the regularity condition is not satisfied).
√ The dashed red lines correspond √ to κ = 0, (Qo − 1)/r, r, where
f (r, κ) is singular. Note that if r − r2 + 1 < κ < 0 or r < κ < r + r2 + 1 the solution is a KM1,
while if 0 < κ < r the solution is a KM2. (b) Same for g(r, κ) = arccosh α − log(ρ)1/4 . In order for the
stronger regularity condition to be satisfied, (r, κ) should be chosen in the blue regions.






[Note the width of the above interval is exactly 1 − 2∆ > 0.] The smaller the value of ∆, the further
away from the singularities one can choose the closest integers, and therefore the smaller the value of
Qn (τ ). √
Recalling that ξo = log δ4 , the inequality (28) can be written as
p
log ρn̄+∆ < log δ4 < log ρn̄+1−∆ ⇔ βρn̄+∆ < |C̄1 | < βρn̄+1−∆ , (30)
√
where we have used the definition of δ4 in terms of C̄1 and

|C̄1 | (1 + κ2 )(1 + κr)(κ2 − 2rκ − 1)

β=√ ≡ . (31)
δ4 κ(κ − r)(rκ2 + 2κ − r)

Any choice of c = |C̄1 | in the above intervals guarantees that for the corresponding values of r, κ the
denominator of Qn (τ ) will not vanish at the integers. For instance, if one sets n̄ = 0, then n = 0 and
n = 1 are the integers where the denominator will be the smallest, and one would want to choose c in
each range so that n = 0 and n = 1 are equidistant from the soliton center. Recall that the soliton center
is ξo / log ρ, and the choice of c in the above intervals guarantees that 0 < ∆ < ξo / log ρ < 1 − ∆ < 1.
The plots in Fig. 4 show cosh (n log ρ − ξo ) − α at the integer values n = 0, 1. It is clear that the optimal
choice for c in each case is the value at the intersection of the two curves.
Note that for any fixed r the frequency, Ω̄1 , and log ρ, which controls the width of the breather in each
period, become singular at κ = 0, r (cf Fig 5). As it is clear from the contour plots in Fig 3, solutions
are regular in left and right neighborhoods of κ = 0, r, but in order to obtain solutions whose frequency
is not too large and width not too small, one wants to keep sufficiently far from those singular values.
On the other hand, the further away one moves from κ = 0, r in the intervals where the solution remains
bounded for all n ∈ Z, the closer ∆ is to its limiting value of 1/2, and correspondingly the larger the
solution is going to be. So a careful balance is required in the choice of parameters in order to obtain
more meaningful KM solutions which are regular for all n ∈ Z, τ ∈ R. In Table 1 we provide some
such examples, for various values of r > 0. For the parameters in the Table, we have chosen n̄ = 0 for
simplicity.

7
 0.7

0.6 1.5

0.5

0.4 1.0

0.3

0.2 0.5

0.1

6 7 8 9 10 11 12 80 100 120 140

Figure 4: Plots of cosh (n log ρ − ξo ) − α at the integer vales n = 0 (blue) and n = 1 (orange) for r = 1
and κ = −0.13 (left), κ = 0.06 (right) as functions of c = |C̄1 | in the range of allowed values, i.e.,
6.1 ≤ c ≤ 11.9 (left) and 70 ≤ c ≤ 156. The optimal choice for c in each case is at the intersection point,
i.e., c̃ = 8.52544 and c̃ = 104.164, respectively.
Ω log ρ
30

4
20

10 3

κ
1 2 3 4 2

-10
1

-20

κ
1 2 3 4

√ √
Figure 5: Plots of Ω̄1 (left) and log ρ (right) as functions of κ ∈ (− r2 + 1 + r, r2 + 1 + r) for r = 2.

Table 1: Soliton parameters for regular KM breathers

r range for κ κ ∆ Ω̄1 log ρ range for c optimal c inf n∈Z |denom|
1/2 (-0.618, 0) -0.08 0.37 -3.37 3.0 [90, 193] 131.7418 0.67
(0, 0.0665) 0.05 0.44 4.53 3.8 [652,1005] 809.7477 0.64
(0.4196, 1/2) 0.44 0.44 -4.61 3.8 [780,1231] 980.1981 0.67
(1/2, 1.61803) 0.6 0.37 3.5 3.1 [130,287] 193.2707 0.71
0.8 (-0.480625, 0) -0.1 0.32 -4.0 2.3 [19,41] 27.6189 0.45
(0, 0.09407) 0.08 0.46 4.7 2.9 [125,159] 141.1548 0.24
(0.6565247, 0.8) 0.69 0.43 -5.3 3.0 [188,285] 231.2327 0.4
(0.8, 2.08062) 1.2 0.37 1.7 1.4 [9,13] 10.8274 0.12
1 (-0.414, 0) -0.13 0.31 -3.5 1.8 [7,11] 8.5254 0.26
(0, 0.10717) 0.1 0.48 4.75 2.5 [62,68] 65.008 0.08
(0.806396, 1) 0.82 0.48 -4.8 2.5 [102,115] 108.565 0.1
(1, 2.4) 1.5 0.32 2.0 0.8 [6.7, 9.9] 8.14424 0.1
√
5/2 (-0.38, 0) -0.14 0.30 -3.5 1.6 [3.8,7.1] 5.18209 0.2
(0, 0.113275) 0.1 0.46 5.4 2.4 [45,53] 48.7821 0.13
(0.892, 1.118) 0.9 0.48 -5 2.3 [80.84] 82.087 0.04
(1.118, 2.61803) 1.5 0.3 3.4 1.5 [12,21] 15.7055 0.2
2 (-0.236068, 0) -0.13 0.2 -5 1.0 [0.7,1.2] 0.9242 0.1
(0, 0.135249) 0.1 0.4 10.9 1.8 [10.3, 15.3] 12.5187 0.19
(1.46773, 2) 1.6 0.38 -11.4 1.9 [37,57] 45.8334 0.2
(2, 4.2) 3 0.2 4.2 0.8 [5.1,8.2] 6.4804 0.08

Table
√ 2: Parameters√ for regular KM breathers. Recall that, for fixed r, one needs to choose κ ∈
(− r2 + 1 − r, r2 + 1 + r) \ {0, r} in such a way that the regularity condition (29) holds. The interval
for c = |C̄1 | is then chosen so that (30) holds; for the values in this table, n̄ = 0. The estimate for the
lower bound of the denominator is obtained by evaluating cosh(n log ρ − ξo ) − α for the chosen r, κ, c = c̃
at n = 0, 1. Finally, choices of 0 < κ < r yield KM2 solutions, while for all other admissible values of κ
a KM1 solution is obtained.

8
 Im z Im z
Sheet I Sheet II Im ζ
2 2
λ=ξ+ ξ−1 λ=ξ− ξ−1 D-
ξ=(z+1/z)/(2ir) i(r+Qo) ξ=(z+1/z)/(2ir) i(r+Qo)
ζ+o

Σ
|λ|=1
Qo
|z|=1
|z|=1
D+
ir
i(-r+Qo) i(-r+Qo) λ+o
-1 1
-1 0 1 Re z -1 0 1
Re z
0
Re ζ
i(r-Qo) i(r-Qo) −
ζo
|z|=1
-i/r
Qo /r

i(-r-Qo) i(-r-Qo)

−
λo

p
Figure
p 6: Left: Sheet I of z-plane, where λ = ξ + ξ 2 − 1. Center: Sheet II of z-plane, where λ =
2
ξ − ξ − 1. Right: ζ-plane; the dashed/solid red and blue circle |ζ − ir| = Qo corresponds to the
continuous spectrum (where |λ| = 1), and the corresponding oriented contour Σ identifies the regions
D+ and D− . The points ζo± = i(r ± Qo ) are the images of the four branch points z = i(±r ± Qo ), and
λ±
o = −i(1 ∓ Qo )/r; the dashed/solid purple circle C = {ζ ∈ C : |irζ − 1| = Qo } corresponds to |z| = 1.

5 Darboux transformations for KM breathers

A Darboux transformation is meant to take a known solution of an integrable PDE or ODE into a
new solution. In [15], the Darboux transformation for the focusing and defocusing AL equations (AL± ,
respectively) on a nontrivial background were obtained to derive and study the stability of periodic
Akhmediev breather solutions. The construction of the Darboux matrix (DM) for the Lax pair (4a)
follows the one proposed in [18, 25] for the case of rapidly decaying solutions (see also [13], where the
Darboux transformation is used to obtain rogue wave solutions of the focusing AL equation superimposed
[0]
to standing periodic waves). The DM Dn (z, τ ) takes a fundamental solution Ψn (z, τ ) of the Lax pair
[0] [0]
(4a) with potential Qn (τ ) into a new fundamental solution Ψn (z, τ ) = Dn (z, τ )Ψn corresponding to a
[0]
potential Qn (τ ). The requirement that both Ψn and Ψn satisfy the equations in the Lax pair implies
that the DM is
dDn
Dn+1 L[0]n = Ln Dn , = Mn Dn − Dn Mn[0] . (32)
dτ
Furthermore, the symmetries of the Lax pair and of the boundary conditions require that
 
0 1
Dn (z, τ ) = σ1 Dn∗ (1/z ∗ , τ )σ1 , σ1 = .
1 0

The details of the construction of the DM are given in [15]; in Appendix B we report, in our notations,
what is needed below in order to reproduce the KM solutions presented in Sec 3, as well as some examples
of double KM breathers.
Below, we will apply the DM to the case of KM breathers. In order to compare the solutions in Sec 3
with the ones one can derive using the DM, we need to take into account that the latter is given in terms
of the original spectral parameter z (and λ(z), as defined in (5)), while the former is expressed in terms
of the uniform variable ζ (cf. Fig 6). For this reason, we start here by converting the purely imaginary
discrete eigenvalues ζ̄j = iκj ∈ D+ into the corresponding values for zj and λj .
Consider purely imaginary values of ζ = iκ, for all κ in the interval: (−Qo + r, Qo + r) \ {0, r}. It
follows from (11) that
r−κ r−κ r−κ
z2 = ∈ R, λ2 = −κ ∈ R, zλ = i ∈ iR ,
κ(rκ + 1) rκ + 1 rκ + 1
showing that√z 2 > 0 in the interval 0 < κ < r where λ2 < 0, whereas z 2 < 0 for −Qo + r < κ < 0 and
r < κ < r + 1 + r2 where λ2 > 0. This implies that when z ∈ R then λ ∈ iR and vice versa. Note also
that the third equation above allows one to determine the relative signs of z and λ.
Specifically, we obtain the following.
For 0 < κ < r: r
r−κ
z = ±µ, λ = ±iκµ with µ = ∈ R+ . (33)
κ(1 + rκ)

9
 Since µ → 0+ as κ → r− and µ → +∞ as κ → 0+ , considering both signs in z = ±µ we see that
the entire real z-axis corresponds to the segment i[0, r] in the ζ-plane.
For −Qo + r < κ < 0:
r
r−κ
z = ±iµ , λ = ±|κ|µ with µ = ∈ R+ . (34)
|κ|(1 + rκ)

As κ → (−Qo + r)+ , one has µ → Qo + r, and µ → +∞ as κ → 0− , so in this case, considering

both signs for z = ±iµ, the segment i[−Qo + r, 0] in the ζ-plane corresponds to the semilines on
the imaginary z-axis: z ∈ i(−∞, −(Qo + r)] ∪ i[Qo + r, +∞).
For r < κ < r + Qo : s
|r − κ|
z = ±iµ , λ = ∓κµ with µ = ∈ R+ . (35)
κ(1 + rκ)
And since µ → Qo − r as κ → (r + Qo )− and µ → 0 as κ → r+ , it follows, again considering both
signs in z = ±iµ it follows that the segment i[r, r + Qo ] in the ζ-plane corresponds to segment
i[−Qo + r, Qo − r] along the imaginary z-axis.

For the purpose of the DM, one wants to parametrize both z and λ above using one single parameter,
and we can use the same parametrization for λ which was used for our KM breathers, namely:
 
−1/2 iϕ/2
√
− log ρ+iϕ/2 ±n ±iΩτ √ nϕ
λ=ρ e ≡e , λ e = exp ∓n log ρ ∓ (Im Ω)τ ± i ± i(Re Ω)τ ,
2

and further simplify the latter to:

 
√ nϕ
λ±n e±iΩτ = exp ∓n log ρ ± i ± iΩτ
2

based on the fact that for KM breathers ρ > 1, ϕ/2 = 0, π, ±π/2 depending on which of the above cases
for κ we are considering, and also that Im Ω = 0. As far as z is concerned, that can be obtained from
(5), and the appropriate sign chosen according to the cases indicated before. We need to establish the
range of values of ρ > 1. This can be done by looking at each interval of values for κ below, determine
what the corresponding interval of values for ρ should be by solving (18) for ρ in terms of κ. For any
fixed r, computing ∂κ ρ when 0 < κ < r yields two stationary points, located at κ = (±Qo − 1)/r.
These correspond to ζ = iκ coinciding with the points λ±o (see Fig. 6); and κ = (Qo − 1)/r ∈ (0, r). In
conclusion, for each given r, one has:

r2
min0<κ<r ρ(r, κ) = ρ(r, (Qo − 1)/r) = > 1. (36)
(Qo − 1)2

On the other hand, when −Qo + r < κ < 0 or r < κ < Qo + r, one finds that ∂κ ρ = (−r2 κ2 − 2κ +
r)/κ2 (κ − r)2 , with the same two stationary points κ = −iλ± o . In these cases, however, the stationary
points are both outside the ranges of values for κ, so ρ is a monotonic function of κ. Furthermore, clearly
ρ → +∞ as either κ → 0− or as κ → r− , while ρ → 1 as κ → r ± Qo , confirming that in both cases ρ
can take any value greater than 1.

0 < κ < r:
The KM breathers in this case have λ purely imaginary and z real, and z, −iλ with the same sign.
Based on the fact that we expect |λ| < 1 and taking, without loss of generality, λ in the upper
half-plane, we have
√ √ √ √
q
λ = i/ ρ ≡ ie− log ρ ⇔ z = r sinh log ρ ± r2 sinh2 log ρ − 1. (37)

Now, it is not obvious that z above is indeed real for any choice of ρ > 1; the condition that the
argument of the square root be non-negative can be written as r2 (ρ + 1/ρ + 2) − 4Q2o ≥ 0, which,
in turn is equivalent to
r2 ρ2 − 2(1 + Q2o )ρ + r2 ≥ 0.

10
 The roots of the above quadratic equation are ρ = (1 ± Qo )2 /r2 , and the inequality is satisfied
if either ρ ≤ (Qo − 1)2 /r2 or ρ ≥ (Qo + 1)2 /r2 and the latter can be shown to coincide with the
minimum value of ρ in the interval 0 < κ < r, as computed in (36), so indeed z as given in (37)
is real. Note that both choices for the sign of the square root in (37) would correspond to positive
z (equivalently, to positive −iλz), and therefore for a given ρ greater or equal than the minimum
value in (36), one could have two different KM breathers.
−Qo + r < κ < 0:
The KM breathers in this case have λ real and z purely imaginary, and, as before, λ, −iz have the
same sign. Using similar parametrization as before:
√ √ √ √
q
λ = 1/ ρ ≡ e− log ρ ⇔ z = ir cosh log ρ + i r2 cosh2 log ρ + 1 (38)

and here the positive sign of the square root is the only one that ensures the correct sign of z.
r < κ < Qo + r:
The only difference with respect to the previous case is that −iz and λ should have opposite signs.
If we want to keep z in the upper half plane, then we should have
√ √ √
q
λ = −e− log ρ ⇔ z = −ir cosh log ρ + i r2 cosh2 log ρ + 1 (39)

In all cases Ω, as a function of z, λ is given in Eq. (6), which is actually independent of the signs since
it depends on z 2 and λ2 .
[0]
In order to build the single KM and double KM breathers using the DM, we take Qn (τ ) = Qo , i.e,
the background solution. In this case, one can take the fundamental matrix solution of the Lax pair to
be !
Qo −n −iΩ(ρ)τ
[0] n
λ(ρ)n eiΩ(ρ)τ ir/λ(ρ)−z(ρ) λ(ρ) e
Ψn (ρ, τ ) = (ir) Qo , (40)
n iΩ(ρ)τ
irλ(ρ)−1/z(ρ) λ(ρ) e λ(ρ)−n e−iΩ(ρ)τ

where λ(ρ) and z(ρ) are given by (37), (38), or (39) depending on the desired range of values for κ; and
Ω(ρ) is defined by (6).
Following the construction of the Darboux transformation outlined in Appendix A, after simplification
one can obtain explicit single KM solutions like those given in Section 3. These will depend on a single
real parameter ρ1 that determines z1 ≡ z(ρ1 ), λ1 ≡ λ(ρ1 ), and Ω1 ≡ Ω(ρ1 ); as well as a complex
proportionality constant γ1 ≡ |γ1 |eiφ (defined in (56)) which plays the role of the norming constant.
First, by parametrizing z(ρ) and λ(ρ) through either (38) or (39) (i.e. enforcing that z1 = −z1∗ and
λ1 = λ∗1 ), one can obtain the solution

r (A − A−1 ) cos(η − φ) − i(A + A−1 ) sin(η − φ)

Qn (τ ) = Qo + (λ1 − λ−1
1 )B , (41)
2 cosh(ξ − ξo ) + B sin(η − φ)

where ξ = n log ρ1 , η = 2Ω1 τ , and

Qo (z1 + z1−1 )A √
A= −1 , B = 2 −1 √ , ξo = − log |γ1 | ρ1 . (42)
irλ1 − z1 |z1 − A z1 | ρ1

If instead z(ρ) and λ(ρ) are parametrized by (37) with either choice of the sign of the square root (i.e.
z1 = z1∗ and λ1 = −λ∗1 ), the solution is the KM2 breather, in the form

r (A − A−1 ) cos(η − φ) − i(A + A−1 ) sin(η − φ)

Qn (τ ) = Qo + (−1)n (λ1 − λ−1
1 )B . (43)
2 cosh(ξ − ξo ) + (−1)n B sin(η − φ)

For the purpose of comparing the solutions (41) and (43) with the previous forms (24) and (25), one
can set the phase of γ1 to be φ = π/2 (which is apparently equivalent to setting the phase of C̄1 to zero
as was done in Section 3). With this, (41) can be rewritten as

ir (A + A−1 ) cos η − i(A − A−1 ) sin η

Qn (τ ) = Qo + (λ1 − λ−1
1 )B , (44)
2 cosh(ξ − ξo ) − B cos η

11
from which by comparison with (24) one can directly identify the coefficients in terms of the uni-
formization variable (parametrized by κ) with the parameters in terms of the original spectral parameter
(parametrized by ρ1 ) in the following way:

δ1∗ ± δ2∗
(λ1 − λ−1
1 )B(A ± A
−1
) ←→ √ , B ←→ α . (45)
δ4
Finally, by comparing the two expressions for the center of the KM breather ξo , one finds a way to relate
the norming constant C̄1 to the proportionality constant γ1 ; namely, |γ1 |2 ρ1 = 1/δ4 (recall that δ4 is
proportional to |C̄1 |2 ).
The parameter B as defined above is real and could be positive or negative depending on the choice
of parametrization. Regardless, it is true that |B| > 1 for any ρ1 , in which case the solutions above are
generically singular. Following the same logic as in Section 4, we seek parameters ρ1 and γ1 such that
no integer lies in the interval where cosh(ξ − ξo ) ≤ |B|. A necessary condition for regularity that ensures
that the width of this interval is less than 1 is
1
arccosh |B| − log ρ1 < 0 . (46)
2
As before, it may be deemed to be useful to impose a stronger regularity condition, for example by
replacing the 1/2 in (46) by 1/4 1 . Furthermore, we can center the solution between two lattice sites n̄
and n̄ + 1 by choosing |γ1 | in the interval

∆−n̄− 32 −∆−n̄− 12 arccosh |B|

ρ1 < |γ1 | < ρ1 , ∆= . (47)
log ρ1
Additionally, the Darboux transformation can be used to construct a double KM solution according
to the formulae provided in Appendix A. We remark that if two parameter sets {ρ1 , γ1 } and {ρ2 , γ2 } are
chosen according to the regularity conditions (46) and (47) such that each set separately characterizes a
well-behaved single KM solution, the resulting double KM solution is not guaranteed to remain regular
(even with centers n̄1 and n̄2 chosen to be sufficiently separated). However, one can find combinations of
parameters for which the double KM solution is well-behaved, examples of which are displayed in Fig. 7.

6 Instability & numerical blow-up

In this section, we study numerically and show respective results for a few representative cases of regular
single KM breathers associated with Fig. 2. Specifically, we attempt herein to shed light on the spectral
stability of KM breathers that are introduced in this work. In doing so, we obtain first the respective
time-periodic yet numerically exact solution of Eq. (2) by using Newton’s method which is subsequently
used in the variational equations towards deriving the monodromy matrix. The eigenvalues of the mon-
odromy matrix are the Floquet multipliers which provide information about the stability of the pertinent
structures, see [33] for the formulation of the stability analysis problem. Briefly, and as far as Newton’s
method is concerned, time-periodic solutions of period Tb = π/Ω̃1
km
X
Qn = Qn,m e2imΩ̃1 τ , (48)
m=−km

where Qn,m are the Fourier coefficients, and 2km + 1 are the total Fourier modes in time we consider. In
this work, we use km = 61 and thus 123 Fourier collocation points are used for the temporal discretiza-
tion. Upon plugging Eq. (48) into Eq. (2) where periodic (in space) boundary conditions are imposed
therein, we obtain a nonlinear algebraic system of equations for Qn,m that we solve with Newton’s
method; see [33] for the formulation of the relevant root-finding problem and [12] for a recent yet rele-
vant application of the method. We mention in passing that the initial guesses to the solver are provided
directly by the exact solutions we aim to study, and Newton’s method converges in 1 iteration (within
machine precision). However, we have noticed that the solver may need to perform more iterations, e.g.,
3-4, to converge for the same number of Fourier collocation points employed. One such case is strongly
1 However, as regards the practical manifestation of the instability, as illustrated in the next section, this was not found

to modify the observed dynamics significantly.

12
 (a) (b)
(b) (c)

0 0 0
0.5 1.0 1.5 2.0 0.05 0.10 0.15 0.20 0.5 1.0 1.5 2.0 2.5 3.0

log10 max E(τ) log10 max E(τ) log10 max E(τ)

-5 log10 max
n∈ℤ E (τ)
n -5 log10 max
n∈ℤE (τ)
n -5 log10 max
n∈ℤ E (τ)
n
n∈ℤ n∈ℤ n∈ℤ

-10 -10 -10

-15 -15 -15

-20 -20 -20

Figure 7: (a) Double KM1 solution with r = 2 and z1 and z2 parametrized by (38), corresponding to
ρ1 = 6, γ1 = 10−4 , ρ2 = 3, γ2 = 104 . (b) Double KM2 solution with r = 4 and z1 and z2 parametrized
by (37) with the positive sign, corresponding to ρ1 = 8, γ1 = 10−4 , ρ2 = 12, γ2 = 104 . (c) Double KM1
solution with r = 2 and z1 and z2 parametrized by (38), corresponding to ρ1 = 2.2588611606273092,
γ1 = 10−4 , ρ2 = 3.9647690662942265, γ2 = 104 ; chosen such that the period of the left wave is π/8 and
the period of the right wave is π/4.



reminiscent of Fig. 2(c) but with r = 2, κ = 3, and c = 6.48041 where the solver converged in 3 iterations
(if km = 81, i.e., 163 collocation points in time, the solver convergences in 1 iteration). There are also
other cases, such as the one with r = 4, κ = 8, and c = 0.253 where our nonlinear solver failed to converge
entirely. This observation raises an important problem that merits a further yet separate investigation in
developing suitable computational methods for identifying such periodic orbits. Regardless, and for the
single KM solutions we present herein, we found that the use of 123 collocation points in time together
with a lattice of 100 sites accurately resolves the time-translation mode 1 + 0i within 10−7 − 10−8 error.
Finally, we provide relevant dynamics of the KM solutions that are numerically obtained in order to
explore the corresponding evolution of such states. For these direct numerical computations, we use the
Burlisch-Stoer method [19] with 10−14 relative and absolute errors for all time integrations (including
those that are needed for the Floquet computations, i.e., variational problem). As a diagnostic in order
to assess the accuracy of the dynamical results, we leverage the integrable structure of the AL model
and, in particular, its bearing of an infinite number of associated conservation laws. The conserved
quantities in the case of waveforms whose tails asymptotically vanish can be derived recursively using
formulas (3.4.155) and (3.4.157) in [4], and simplified using the summation by parts formula (A.1.2).
The conserved quantities for non-zero boundary conditions can be obtained by subtracting the boundary
conditions to ensure all the series are convergent. Specifically, we have
+∞
X +∞
X
Qn Q∗n−1 − Q2o , Qn−1 Q∗n − Q2o ,
 
Γ1 = Γ̄1 = (49)
n=−∞ n=−∞

+∞  
X
1 3 4
Γ2 = Qn Q∗n−2 1 − |Qn−1 |2 − (Qn Q∗n−1 )2 − Q2o + Qo , (50)
n=−∞
2 2
+∞  
X
1 3 4
Γ̄2 = Qn−2 Q∗n 1 − |Qn−1 |2 − (Qn−1 Q∗n )2 − Q2o + Qo . (51)
n=−∞
2 2

The IST also shows the following infinite product is conserved:

+∞
Y |Qn |2 − 1
c∞ = . (52)
n=−∞
Q2o − 1

13
Finally, from the expression of the Hamiltonian in the case of zero boundary conditions, once the non-
trivial boundary conditions are subtracted out we obtain (cf. also [36]):
+∞ +∞
X  ∗ X |Qn |2 − 1
H= Qn (Qn+1 + Qn−1 ) − 2Q2o + 2 log . (53)
n=−∞ n=−∞
Q2o − 1

with
H = Γ1 + Γ̄1 + 2 log c∞ . (54)
It is worth pointing out that both the infinite product (52) and the related log series in (53) satisfy the
necessary condition for convergence, since |Qn | → Qo as n → ±∞ for both Qo < 1 and P Qo > 1. In the
∞
case of zero boundary conditions, the defocusing AL lattice conserves the functional P = n=−∞ log(1−
2
|Qn | ), and global existence is ensured if the initial condition satisfies ||Qn (0)||∞ < 1, but to the best
of our knowledge the analog of this result has not yet been established in the case of nonzero boundary
conditions even when 0 < Qo < 1. For our exact solutions, we observe that when Qo > 1, even though
|Qn | can dip below Qo at some lattice sites, one has |Qn | > 1 for all n ∈ Z. In this case, the terms in the
infinite product (52) and the arguments of the logarithm in (53) are positive, and requiring Qn − Qo ∈ ℓ1
is sufficient to guarantee the convergence of both (52) and (53). It is interesting to add here that we have
observed that in the course of the numerical evolution dynamics of such unstable solutions, round-off
error may lead, through the resulting destabilization, to the indefinite growth of the modulus of particular
nodes. As this happens, we have noticed that (pairwise) some sites drop below Qo and indeed approach
unit modulus, in a way that can be seen to conserve both the infinite product of (52) and the energy of
(53). How this collapse dynamics manifests itself and its bearing on the AL lattice conservation laws are
clearly topics meriting further investigation.
We now turn our focus on Fig. 8 which presents numerical results on the existence, stability and
spatio-temporal dynamics in panels (a)-(d), (e)-(h), and (i)-(l), respectively. Those are complemented
by the panels (m)-(p) which depict the relative error (with respect to t = 0) of the conserved quantities
given by Eqs. (49), (51), (52), and (54), respectively (see the legend in panel (m)) all as a function of
time. Moreover, the first, second, third, and fourth rows of Fig. 8 connect with the KM breathers of
Figs. 2(e), (b), (c), and (d), respectively. The panels (a)-(d) showcase the spatial dependence of the
amplitude Qn of the numerically obtained solution (with blue open circles) against the respective exact
solution (with red stars), see the legend in panel (a). Besides the expected matching between numerically
exact and analytical solutions, we report that the ones shown in panels (a) and (c) of Fig. 8 bear two-
site excitations, i.e., bond centered waveforms, whereas the ones of panels (b) and (d) correspond to
site-centered solutions.
The respective results of the Floquet spectra of the solutions shown in (a)-(d) are presented in panels
(e)-(h) where the Floquet multipliers λ = λr + iλi are depicted with blue filled circles. To understand
the spectrum of the KM breathers, we complement the relevant spectra by including in each panel the
Floquet multipliers (see, the legend in panel (e)) of the respective constant background (CW) solution
in panels (e)-(h) with red open circles (see, the MI dispersion relation in [33], cf. Eqs. (10) and (14)
therein with g = 1 and C = −1). In all cases that we briefly report herein, all KM breathers are
deemed spectrally unstable due to the presence of a band of real Floquet multipliers (i.e., with λi = 0)
where |λ| > 1. However, the instabilities identified in these panels emanate from the instability of the
background, that is, the CW is modulationally unstable here since the background amplitude in all cases
we consider is greater than 1, i.e., Q0 > 1 [cf. [15]]. Indeed, executing the relevant computations in larger
lattices, we have verified that the bands of associated instabilities become more dense as is expected
due to the excitation of additional unstable wavenumbers therein. We find no additional unstable modes
getting generated by the mounting of KM breathers atop the CW waveform at least for the cases we have
studied numerically. This is in line with earlier results in both continuum [16] and discrete [33] problems
of the discrete nonlinear Schrödinger type. It should be mentioned in passing that the breather itself
slightly perturbs the Floquet multipliers of the CW solution, see, indicatively, panels (g) and (h). More
importantly, we find that the site-centered solutions of panels (b) and (d) as well as the bond-centered
one of panel (a) are less unstable compared

to the bond-centered solution of panel (c) which itself involves
a maximal growth rate of O 107 and which is accordingly expected to be manifested over very short
time scales. The stability trait of the KM breathers is reflected in the spatio-temporal evolution of the
amplitude |Qn (τ )| in panels in (i)-(l). Here, we use the numerically exact solution as an initial conditions,
and advance Eq. (2) forward in time until the integrator breaks down due to the (numerically induced)
blow-up of the solutions.

14
 18 1 0 10 -8
(a) (i) (m)
15 0.2 15
0.5 10 -10
12 0.4
0 10 10 -12
9 0.6
6 -0.5 10 -14
0.8 5
3 (e)
-1
10 -16
-20 -10 0 10 20 0 2 4 -10 -5 0 5 10 0 0.5 1

25 1 0
(b) (j) (n)
20 2 25 10 -12
0.5
20
15 4
0 15
10 6 10 -14
-0.5 10
5
(f ) 8 5
-1
0 10 -16
-20 -10 0 10 20 0 2 4 -10 -5 0 5 10 0 0.5 1 1.5

8 1 0
(c) (k) (o)
40 10 -12
6 0.5
0.5
30
0
4 1 20 10 -14
-0.5
10
2 (g)
-1 1.5
0 5 10 10 -16
-20 -10 0 10 20 -10 -5 0 5 10 0 0.5 1 1.5
10 6
50 1 0 10 -8
(d) (l) (p)
40
40
0.5 1 10 -10
30 30
0 10 -12
20 20
2
-0.5 10 -14
10 10
-1 (h) 3
0 10 -16
-20 -10 0 10 20 0 5 10 -10 -5 0 5 10 0 1 2 3

Figure 8: Summary of numerical results associated with the KM solutions of Fig. 2(e), (b), (c), and (d)
shown here in the first, second, third, and fourth rows, respectively. The panels (a)-(d) show the spatial
distribution of numerically exact KM breathers with blue open circles (connected by solid line as a guide
to the eye). For comparison, the exact solution is shown too with red stars (see, the legend in panel (a)).
As expected, in all cases, the two are practically identical. The spectral stability results are summarized
in panels (e)-(h) where the Floquet multipliers λ = λr + iλi are shown with blue filled circles against the
MI prediction associated with the uniform background state shown with red open circles. Moreover, the
spatio-temporal evolution of Qn (τ ) is depicted in (i)-(l) where the respective conserved quantities (in a
semi-logarithmic scale) are shown in panels (m)-(p); see the legend and Eqs. (49), (51), (52), and (54),
as well as the associated discussion in the text.

In line with our spectral stability calculations shown in panels (e)-(h), we observe that the waveforms
of panels (i), (j), and (l) do breathe in time, and in a staggered or beating fashion without changing their
structural characteristics until modulational instability (MI) gets triggered due to perturbations. For
example, see the waveform shown in Fig. 8(i) close to the end of the time window, but also the one
of Fig. 8(k) where MI gets manifested itself more quickly given the very large instability growth rate,
causing the solution to not even return back after one period of time integration.
The perturbations mentioned above stem from not only round-off and local truncation errors of
the integrator, but also from the error that gets introduced by the truncation of Eq. (48), i.e., in the
computation of the initial condition as a fixed-point solution. For reference, we report the integration
times [0, tf ] of the respective cases with (i) tf = 22.143Tb ≈ 1.0322, (j) tf = 18.4Tb ≈ 9.7596, (k)
tf = Tb ≈ 1.6265, and (l) tf = 13Tb ≈ 3.39720, respectively. It should be noted that the spatio-temporal
dynamics of |Qn (τ )| shown in these panels are up to times where the solution has not blown up yet, i.e., up
to times slightly before blow-up where the structures can still be visible. Despite the evolution of the KM
breathers towards blow-up, we report that the conserved quantities introduced in Eqs. (49), (51), (52),
and (54) are very well preserved with relative errors O(10−9 ) − O(10−16 ).This happens even very close to
blow-up times as this shown in panels (m)-(p) where the terminal times therein are up to and including
those when the integrator breaks down. This is an additional indication of the fidelity of our numerical
computations discussed in this section.
We speculate that what is happening here is that the relevant solutions are very proximal to ones

15
featuring collapse dynamics, as discussed in the previous sections, given the relevant variation of pa-
rameters such as c. The dynamical manifestation of the waveforms’ MI (often with a very large growth
rate) due to its background rapidly leads to growth and opens a pathway towards exploring these prox-
imal collapse-prone dynamical features of the model. Understanding these pathways towards collapse
is naturally an interesting direction for further exploration. This is especially so since it’s worthwhile
to bear in mind that the exact analytical solutions constructed herein have been designed to feature no
collapse, and it is only the quite small perturbations of roundoff, integration local truncation error, and
inexactness due to a finite mode truncation that lead to the observed blowup.

7 Concluding remarks
In the present work we have examined the existence (including the analytical form), stability and dy-
namics of an important, periodic-in-time class of solutions of the defocusing Ablowitz-Ladik lattice on
an arbitrarily large background which are discrete analogs of the KM breathers of the focusing NLS
equation. Specifically, we derived conditions on the background and on the spectral parameters that
guarantee the exact KM solution to be non-singular on the lattice for all times. Furthermore, a novel
KM-type breather solution is presented which is also regular on the lattice under suitable conditions.
Furthermore, we employed Darboux transformations to obtain a multi-KM breather solution, and showed
that parameters choices exist for which the newly obtained double KM breather to be regular on the
lattice. Finally, we presented numerical results on the stability and spatio-temporal dynamics of the
single KM breathers. The stability analysis manifested that these states were modulationally unstable
due to the modulational instability of the background on top of which they existed, with their Floquet
multipliers suitably modified (due to the nonlinear nature of the waveform), yet without novel instabili-
ties introduced (similarly to what was found for the continuum NLS problem in [16]). Another important
feature of the numerical computations was that small perturbations (introduced by roundoff and local
truncation errors, or by the approximate nature of our numerical solution) led to the finite time blowup
of the KM structures. The preservation of the model’s conservation laws strongly suggested that this
feature was not a numerical artifact, but rather a real feature which merits further examination.
In a forthcoming work, a more systematic stability analysis will be carried out together with ho-
motopic continuations of solutions presented herein towards the discrete nonlinear Schrödinger (DNLS)
equation, and in the same spirit as in [33]. This may provide significant insights towards the existence
of corresponding KM waveforms in the more experimentally relevant (within nonlinear optics) model of
the discrete NLS equation [22]. Another interesting direction is to continue the waveforms (especially
the novel ones discovered herein) towards smaller frequencies in order to acquire structures closer to the
rogue waves that are known to exist in the AL model (see, e.g., [9]). Further such studies of rogue waves
in both defocusing and focusing AL lattices and their non-integrable analogs are currently in progress
and will be reported in future publications.

Acknowledgments
This work has been supported by the U.S. National Science Foundation under Grants No. DMS-2204782
(EGC), and DMS-2110030 and DMS-2204702 (PGK), and DMS-2406626 (BP). The authors also acknowl-
edge the Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, for support and hospitality
during the satellite programme “Emergent phenomena in nonlinear dispersive waves” (supported by
EPSRC grant EP/R014604/1), where work on this paper was undertaken. Finally, we would also like
to thank Gino Biondini for his suggestions with one of the figures, and the anonymous referee for their
useful comments which helped us improve the work.

A Construction of the Darboux matrix

The DM in [15] is assumed to be of the form
PJ (J−2l) J−2l PJ (J−2l+1) J−2l+1 !
zJ + l=1 an  z l=1 bn  z 
Dn[0,J] (z, τ ) = PJ  (J−2l+1)
∗
−J+2l−1
PJ (J−2l)
∗ (55)
l=1 bn z z −J + l=1 an z −J+2l

16
where the symmetry has already been taken into account. The unknown quantities in the DM are then
determined by requiring that the DM is singular at the discrete eigenvalues ±zj , ±1/zj∗ , with

Dn[0,J] (zj , τ )ϕ[0] [0,J]

n (zj , τ ) = γj Dn (zj , τ )ψn[0] (zj , τ ) , (56)
 
[0] [0] [0]
where Ψn (z, τ ) = ϕn (z, τ ), ψn (z, τ ) is a fundamental analytic matrix solution of the Lax pair corre-
[0]
sponding to the potential Qn (τ ), and γj are the proportionality constants that relate the column vectors
at the discrete eigenvalue. Inserting the ansatz (55) into (56), one obtains
J
! J
!
X X
zjJ + a(J−2l)
n zjJ−2l + zjJ + bn(J−2l+1) zjJ−2l+1 rj = 0, (57)
l=1 l=1
J
! J
!
(J−2l) (J−2l+1)
1 X an X bn 1
∗ + = 0, (58)
(zj )J (zj∗ )J−2l (zj∗ )J−2l+1 rj∗
l=1 l=1

where
[0],2 [0],2
ϕn (zj ) − γj ψn (zj )
rj = [0],1 [0],1
ϕn (zj ) − γj ψn (zj )
[0] [0]
and the superscript denotes the corresponding component of the vectors ϕn (zj , τ ), ψn (zj , τ ). If the
number of singular points zj is equal to the order J of the Darboux transformation, the above equations
(l) (l)
for j = 1, . . . , J define an linear system of 2J equations for the 2J unknowns an , bn for l = 1, . . . , J.
[J] [0]
Finally, the solution Qn (τ ) can be obtained from Qn (τ ) using the first of (32), yielding
(−J) (−J+1)
Q[0,J]
n (τ ) = Q[0]
n (τ )an+1 (τ ) + bn+1 (τ ). (59)

The solutions of the above equations for the simplest cases, J = 1 and J = 2, are provided in [15].
Specifically, if J = 1:

z1 − |r1 |2 /z1∗ r1∗

a(−1) b(0) |z1 |2 − 1/|z1 |2 ,


n (τ ) = , n (τ ) = − (60)
|r1 |2 z1∗ − 1/z1 |r1 |2 z1∗ − 1/z1

yielding
(−1) (0)
Q[0,1] [0]
n (τ ) = Qn (τ )an+1 (τ ) + bn+1 (τ ) . (61)
Similarly, when J = 2 the DM elements are given by:
(2) (2)
∆a ∆b
a(−2)
n (τ ) = , b(−1)
n (τ ) = , (62)
∆(2) ∆(2)
where
1 1/z12
 
r1 z1 r1 /z1
1 1/z22 r2 z2 r2 /z2 
∆(2) = det 
1 (z1∗ )2 1/(r1 z1 )∗ z1∗ /r1∗ 


1 (z2∗ )2 1/(r2 z2 )∗ z2∗ /r2∗

(2) (2)
∆a , ∆ b are obtained replacing the second and forth column, respectively, with the vector

(−z12 , −z22 , −(z1∗ )−2 , −(z2∗ )−2 )T ,

and the potential is reconstructed by means of Eq. (59), i.e.

(−2) (−1)
Q[0,2] [0]
n (τ ) = Qn (τ )an+1 (τ ) + bn+1 (τ ) . (63)

B Errata for Ref. [29]

1. The first equation without number in Remark 1 should read

|λ|2 ≤ 1 ⇔ (|ζ|2 + 1) |ζ|2 + ir(ζ − ζ ∗ ) − 1 ≤ 0 |ζ − ir|2 ≤ Q2o

 
⇔

17
 2. The expression between Eq. (32b) and Eq. (33a) should read
(Mn (τ, ζ) M̄n (τ, ζ)) = (N̄n (τ, ζ) Nn (τ, ζ))Λn eiσ3 Ω(ζ)τ T̃ (ζ)e−iσ3 Ω(ζ)τ Λ−n (64)
−1
where T̃ = A(λ(ζ))T (z(ζ))A (λ(ζ)) is defined as in [29] as
3. As a result, Eqs. (38c) and (38d) should respectively be
n
Q20 ∆n (τ )

ζ(ζ − ir)
β(ζ) = − e2iΩ(ζ)τ W r(Mn (τ, ζ), N̄n (τ, ζ)) (65)
ir(ζ + 1/ζ − 2ir) irζ − 1
−n
Q20 ∆n (τ )

ζ(ζ − ir)
β̄(ζ) = e−2iΩ(ζ)τ W r(M̄n (τ, ζ), Nn (τ, ζ)) (66)
ir(ζ + 1/ζ − 2ir) irζ − 1
4. There is a typo in a sign in the exponents in Eqs. (63a) and (63b). The correct expression for (63a)
and (63b) is respectively shown below
M̄n (τ, ζ) b̄k
Resζ=ζ̄k = C̄k λ2n (ζ̄k )e2iΩ(ζ̄k )τ N̄n (τ, ζ̄k ), C̄k = (67)
ā(ζ) λ(ζ̄k )ā′ (ζ̄k )
Mn (τ, ζ) bk λ(ζk )
Resζ=ζk = Ck λ−2n (ζk )e−2iΩ(ζk )τ Nn (τ, ζk ), C̄k = ′ (68)
a(ζ) a (ζk )

5. Equation (63b) is used to obtain equations (68a) and so it contains a typo in a sign in the exponent.
The correct expression for (68a) is
  X 2J
1/∆n (τ ) (ζ − ir)
N̄n (τ, ζ) = + Ck λ−2n (ζk )e−2iΩ(ζk )τ Nn (τ, ζk )
(ζ − ir)/Q+ (ζk − ir)(ζ − ζk )
k=1
(ζ − ir)
Z
1
− Ck λ−2n (w)e−2iΩ(w)τ Nn (τ, w)ρ(w)dw (69)
2πi w∈Σ (w − ζ)(w − ir)

6. Similarly, Equation (63a) is used to obtain equations (68b), whose correct expression is
2J
(1/ζ − ir)/Q∗+
  X
(ζ + i/r)
Nn (τ, ζ) = + C̄k λ2n (ζ̄k )e2iΩ(ζ̄k )τ N̄n (τ, ζ̄k )
1/∆n (τ ) ( ζ̄ k + i/r)(ζ − ζ̄ k )
k=1
Z
1 (ζ + i/r)
+ λ2n (w)e2iΩ(w)τ N̄n (τ, w)ρ̄(w)dw (70)
2πi w∈Σ (w − ζ)(w + i/r)

7. Following the correction in (68b), Equations (69a) and (69b) are respectively corrected as
2J
1 X 1
=1 + Q∗+ C̄k λ2n (ζ̄k )e2iΩ(ζ̄k )τ N̄n(1) (τ, ζ̄k )
∆n (τ )
k=1
1 − ir ζ̄ k

Q∗+
Z
1
− λ2n (w)e2iΩ(w)τ N̄n(1) (τ, w)ρ̄(w)dw (71)
2πi w∈Σ 1 − irw
2J
Q∗n (τ ) Q∗ (τ ) X 1
= + Q∗+ C̄k λ2n (ζ̄k )e2iΩ(ζ̄k )τ N̄n(2) (τ, ζ̄k )
∆n (τ ) ∆n (τ ) ζ̄ k + i/r
k=1
Q∗+
Z
1
− λ2n (w)e2iΩ(w)τ N̄n(2) (τ, w)ρ̄(w)dw (72)
2πi w∈Σ w + i/r

8. Consequently, equation (70) is corrected as

!
2iΩ(ζ̄k )τ (2)
P2J 2n
∗ ∗ k=1 (C̄k λ (ζ̄k )e N̄n (τ, ζ̄k ))/(ζ̄k + i/r)
Qn (τ ) = Q+ 1 + P2J (1)
(73)
1 + Q∗+ k=1 (C̄k λ2n (ζ̄k )e2iΩ(ζ̄k )τ N̄n (τ, ζ̄k ))/(1 − irζ̄k )

9. Eq. (87) should read

ξ = n log ρ + 2(Im Ω̄1 )τ , η = nϕ + 2(Re Ω̄1 )τ . (74)

10. Equation (92) should read

(Q0 + r sin α) cos α
Ω̄1 = iω, ω = 2rQ20 . (75)
2Q0 (Q0 + r sin α) − 1

18
References
[1] MJ Ablowitz, G Biondini and B Prinari, “Inverse scattering transform for the integrable discrete
nonlinear Schrödinger equation with non-vanishing boundary conditions”, Inv. Probl. 23, 1711–1758
(2007)
[2] MJ Ablowitz and JF Ladik, “Nonlinear differential-difference equations”, J. Math Phys. 16, 598–
603 (1975)
[3] MJ Ablowitz and JF Ladik, “Nonlinear differential-difference equations and Fourier analysis”, J.
Math Phys. 17, 1011–1018 (1976)
[4] MJ Ablowitz, B Prinari and AD Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems,
LMS Lecture Notes Series 302, Cambridge University Press (2004)
[5] NN Akhmediev, A Ankiewicz and JM Soto-Crespo, “Rogue waves and rational solutions of the
nonlinear Schrödinger equation”, Phys. Rev. E 80, 026601 (2009)
[6] NN Akhmediev, A Ankiewicz and M Taki, “Waves that appear from nowhere and disappear without
a trace”, Phys. Lett. A 373, 675–678 (2009)
[7] NN Akhmediev, VM Eleonskii and NE Kulagin, “Generation of a periodic sequence of picosecond
pulses in an optical fiber. Exact solutions.”, Sov Phys JETP. 89, 1542–1551 (1985)
[8] NN Akhmediev and VI Korneev, “Modulational instability and periodic solutions of the nonlinear
Schrödinger equation”, Theor. Math. Phys. 69, 1089–1093 (1987)
[9] A Ankiewicz, AN Akhmediev and JM Soto-Crespo, “Discrete rogue waves of the Ablowitz-Ladik
and Hirota equations”, Phys Rev. E. 82, 026602 (2010)
[10] A Ankiewicz, N Devine, M Unal, A Chowdury and N Akhmediev, “Rogue waves and other solutions
of single and coupled Ablowitz-Ladik and nonlinear Schrödinger equations”, J. Optics 15, 064008
(2013)
[11] D Cai, AR Bishop, and N Grønbech-Jensen, Perturbation theories of a discrete, integrable nonlinear
Schrödinger equation, Phys. Rev. E 53 4131–4136 (1996)
[12] EG Charalampidis, G James, J Cuevas-Maraver, D Hennig, N Karachalios and PG Kevrekidis,
“Existence, stability and spatio-temporal dynamics of time-quasiperiodic solutions on a finite back-
ground in discrete nonlinear Schrödinger models”, Wave Motion 128, 103324 (2024)
[13] J Chen and DE Pelinovsky, “Rogue waves arising on the standing periodic waves in the Ablowitz-
Ladik equation”, Stud. Appl. Math. 152, 147–173 (2024)
[14] OA Chubykalo, VV Konotop, L Vazquez and VE Vekslerchik, “Some features of the repulsive
discrete nonlinear Schrödinger equation”, Phys. Lett. A 169, 359–363 (1992)
[15] F Coppini and PM Santini, “Modulation instability, periodic anomalous wave recurrence, and blow
up in the Ablowitz–Ladik lattices”, J. Phys. A: Math. Theor. 57, 015202 (2024)
[16] J Cuevas-Maraver, PG Kevrekidis, DJ Frantzeskakis, N I Karachalios, M Haragus, and G James,
“Floquet analysis of Kuznetsov-Ma breathers: A path towards spectral stability of rogue waves”,
Phys. Rev. E 96, 012202 (2017)
[17] A Doliwa and PM Santini, “Integrable dynamics of a discrete curve and the Ablowitz-Ladik hier-
archy”, J. Math. Phys. 36, 1259–1273 (1995)
[18] X Geng, “Darboux transformation of the discrete Ablowitz-Ladik eigenvalue problem”, Acta Math.
Sci. 9, 21–26 (1989)
[19] E Hairer, SP Nørsett and G Wanner, “Solving Ordinary Differential Equations I: Nonstiff Prob-
lems”, Springer-Verlag (2008)
[20] Y Ishimori, “An integrable classical spin chain”, J. Phys. Soc. Japan 51, 3417–3418 (1982)
[21] AR Its, AG Izergin, VE Korepin and NA Slavnov, “Temperature correlations of quantum spins”,
Phys. Rev. Lett. 70 1704–1706 (1991)
[22] PG Kevrekidis, The Discrete Nonlinear Schrödinger Equation, Springer (Heidelberg, 2009).

19
[23] EA Kuznetsov, “Solitons in a parametrically unstable plasma”, Sov. Phys. Dokl. (Engl. Transl).
22, 507–508 (1977)
[24] E Lieb, T Schultz and D Mattis, “Two soluble models of an antiferromagnetic chain”, Ann. Phys.
16 407–446 (1961)
[25] X Liu and Y Zeng, “On the Ablowitz–Ladik equations with self-consistent sources”, J. Phys. A:
Math. and Theor., 30, 8765–8790 (2007)
[26] Y-C Ma, “The perturbed plane-wave solutions of the cubic Schrödinger equation”, Stud. Appl.
Math. 60, 43–58 (1979)
[27] P Marquié, JM Bilbault and M Remoissenet, “Observation of nonlinear localized modes in an
electrical lattice”, Phys. Rev. E 51, 6127–6133 (1995)
[28] Y Ohta and J Yang, “General rogues waves in the focusing and defocusing Ablowitz-Ladik equa-
tions”, J. Phys. A. 47, 255201 (2014)
[29] AK Ortiz and B Prinari, “Inverse scattering transform for the defocusing Ablowitz-Ladik equation
with arbitrarily large background”, Stud. App. Math. 143, 337–448 (2019)
[30] DH Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions”, J. Aust. Math.
Soc. Series B 25 16–43 (1983)
[31] B Prinari, “Discrete solitons of the Ablowitz-Ladik equation with nonzero boundary conditions via
inverse scattering”, J. Math. Phys. 57, 083510 (2016)
[32] B Prinari and F Vitale, “Inverse scattering transform for the focusing Ablowitz-Ladik system with
nonzero boundary conditions”, Stud. App. Math. 137, 28–52 (2016)
[33] J Sullivan, EG Charalampidis, J Cuevas-Maraver, PG Kevrekidis and N Karachalios,
“Kuznetsov–Ma breather-like solutions in the Salerno model”, Eur. Phys. J. Plus 135, 607 (2020)
[34] M Tajiri and Y Watanabe, “Breather solutions to the focusing nonlinear Schrödinger equation”,
Phys. Rev. E. 57, 3510–3519 (1998)
[35] S Takeno and K Hori, “A propagating self-localized mode in a one-dimensional lattice with quartic
anharmonicity”, J. Phys. Soc. Japan 59, 3037–3040 (1990)
[36] VE Vekslerchik and VV Konotop, “Discrete nonlinear Schrödinger equation under non-vanishing
boundary conditions”, Inv. Probl. 8, 889–909 (1992)

20