# Explicit breather solution of the nonlinear Schrödinger equation

Robert Conte
Université Paris-Saclay, ENS Paris-Saclay, CNRS,
Centre Borelli, F-91190 Gif-sur-Yvette, France
Department of Mathematics, The University of Hong Kong,
Pokfulam Road, Hong Kong
E-mail<EMAIL_ADDRESS>
ORCID https://orcid.org/0000-0002-1840-5095

(April 12, 2021)

###### Abstract

We present a one-line closed form expression for the three-parameter breather
of the nonlinear Schrödinger equation. This provides an analytic proof of the
time period doubling observed in experiments. The experimental check that some
pulses generated in optical fibers are indeed such generalized breathers will
be drastically simplified.

Keywords: modulational instability, nonlinear Schrödinger equation, nonlinear
optics, breather, exact solutions.

PACS 02.30.Hq, 02.30.Jr, 03.75.Kk, 03.75.Lm, 42.65.J, 42.65.-k

###### Contents

  1. 1 Introduction
  2. 2 The generic solution
  3. 3 Nongeneric solutions
    1. 3.1 Degeneracies of the generic solution
    2. 3.2 Nongeneric solutions $Q_{x}z^{\prime}=0$
    3. 3.3 Nongeneric solutions $Q_{x}z^{\prime}\not=0$ and $k_{1}=0$
  4. 4 On constraints of higher degree
  5. 5 Conclusion and discussion

##  1 Introduction

The complex amplitude of many nonlinear media displays two generic features.
The first one is to obey an evolution equation (first order in the time
variable $t$) with a (linear) dispersion term (second order in the space
variable $x$) and the simplest nonlinearity preserving the phase invariance of
$A$,

$\displaystyle{\hskip-11.38109pt}iA_{t}+pA_{xx}+q|A|^{2}A=0,\ pq\not=0,\
p,q\hbox{ real}.$ (1)

The second feature, observed in the “focusing” régime ($pq>0$) of this
nonlinear Schrödinger equation (NLS), is the “modulational instability” (MI)
[1], also known as Benjamin-Feir instability: an initial plane wave grows
exponentially, then saturates and decreases to its original state, with only a
shift of its phase. This MI has enormous applications, which we now recall.

In the ocean, deep water waves are suitably described by the focusing NLS [2],
where one observes “bright” solitons. Sailors have also reported the sudden
occurence of huge waves (“freak” or “rogue” waves) which disappear as quickly
as they appeared, and these solutions of very high amplitude and energy can
also be described by NLS [3]. However, experimental setups able to reproduce
this rare observation are quite difficult. As to the “defocusing” régime
($pq<0$), it is more adapted to shallow water waves, where only “dark”
solitons occur.

The situation is quite different in Bose-Einstein condensation (BEC), where
the wave function of the condensate obeys the Gross-Pitaevskii equation, a
three-dimensional analogue of NLS. It has been proven analytically [4] that MI
is the mechanism which generates wave functions of soliton type in a Bose-
Einstein condensate, a prediction confirmed by the experimental observation
[5] of MI in a cigar-shaped BEC.

But nowadays the main playground of MI no more water waves nor even BEC but
nonlinear optics, for two reasons. The first one is the huge recent progress
in manufacturing optical fibers with prescribed physical properties
(refractive index, etc), making experiments easier, cheaper and easily
reproducible. The second reason is more fundamental: as opposed to a three-
dimensional BEC, a fiber is quasi one-dimensional and thus well described by
the NLS, $t$ being the propagation distance and $x$ the transverse coordinate.
For instance, one has succeeded [6, 7] to generate rogue waves in optical
fibers, an achievement with potentially important industrial applications.
Nonlinear optics has become an excellent field to perform an experimental
check of the beautiful analytic description of MI, which we first recall.

Indeed, the later stages of MI can be computed exactly, resulting in a two-
parameter111The scaling invariance $(x,t,A)\to(kx,k^{2}t,kA)$ of NLS reduces
this number by one. bright soliton localized in space and periodic in time,
whose asymptotic behaviour as $|x|\to+\infty$ is the plane wave
$e^{-i\omega_{0}t}$, see (74) below. This achievement of Kuznetsov [8] was
obtained in plasma physics where the Langmuir waves are appropriately
described by the focusing NLS. Changing the sign of one parameter converts
this soliton to another quite important physical solution, localized in time
and periodic in space, known as the Akhmediev breather [9], see (78) below.
Finally, using a quite simple Ansatz, Akhmediev, Eleonskii and Kulagin [10]
extrapolated the Kuznetsov soliton to a three-parameter breather solution, in
which $Ae^{i\omega_{0}t}$ is elliptic222 We never use the ambiguous term
“periodic” for elliptic solutions, but always either “doubly periodic” alias
“elliptic” (example: Jacobi $\mathop{\rm dn}\nolimits$, Weierstrass $\wp$), or
“quasi-doubly periodic” alias “quasi-elliptic” alias “elliptic of the second
kind” in Hermite’s terminology [11, tome I p. 227, tome II p. 506] (example:
the solution $\mathop{\raise 0.15pt\hbox{\rm H}}\nolimits(t,a)$ of Lamé
equation (25)). in $x$ and quasi-elliptic in $t$.

In a recent experiment [12] with an optical fiber, this breather was observed
by matching the three arbitrary parameters with experimental data, showing a
“good” agreement, however during only two quasi-periods of $t$. The difficulty
did not arise from the sophisticated experimental setup but from “the
complexity of this class of solutions” [13]. Indeed, its current analytic
representation [10, (3), (22), (24)–(25)] does not clearly separates the
elliptic dependence on $x$ and quasi-elliptic dependence on $t$, despite
several later attempts [14] [15] [16] [17], forcing the authors to expand the
amplitude in Fourier series of $x$ and to retain only the first two
coefficients.

In this article, we provide a one-line closed form expression for this three-
parameter solution, see Eq. (43), and perform a full classification of the
solutions of the Ansatz of Ref. [10], thus uncovering a new solution, Eq.
(97), elliptic in $x$ and trigonometric in $t$, together with its degeneracy.
The present three-parameter closed form makes it possible to check the
agreement on a much larger number of quasi-periods of $t$, and therefore to
determine more accurately the nonlinear range of validity of MI as sketched in
[13]. Another puzzling phenomenon observed in Ref. [12], namely a time period
twice the one expected, is naturally explained by our three-parameter
solution.

##  2 The generic solution

Ref. [9] assumes a constraint between $A$ and ${\bar{A}}$, defined by three
real functions $\varphi(t)$, $\delta(t)$ and $Q(x,t)$,

$\displaystyle{\hskip-25.60747pt}\left\\{\begin{array}[]{l}\displaystyle{\sin\varphi(t)\mathop{\rm
Re}\nolimits(A)-\cos\varphi(t)\mathop{\rm Im}\nolimits(A)+\delta(t)=0,}\\\
\displaystyle{\cos\varphi(t)\mathop{\rm
Re}\nolimits(A)+\sin\varphi(t)\mathop{\rm
Im}\nolimits(A)-Q(x,t)=0.}\end{array}\right.$ (4)

Since $A=(Q/\delta+i)(\delta e^{i\varphi})$ is single-valued [18, 19], both
terms $Q/\delta$ and $\delta e^{i\varphi}$ are single-valued, while $Q$,
$\delta$ and $e^{i\varphi}$ may be multivalued. Because of the absence of
methods to handle multivaluedness, the strategy is therefore to only consider
$\delta e^{i\varphi}$, its complex conjugate and $Q/\delta$.

Remark. The real and imaginary parts of $A$ are,

$\displaystyle
A=[Q-\delta\psi+i(Q\psi+\delta)]/\sqrt{1+\psi^{2}},\psi=\tan\varphi.$ (5)

Let us first recall the result of [10], then proceed to the explicit
dependence on $x$ and $t$. By elimination of $A$, the system to be solved is
made of two coupled real PDEs for $Q(x,t)$ [10, Eqs. (4)–(5)],

$\displaystyle{\hskip-25.60747pt}\left\\{\begin{array}[]{l}\displaystyle{Q_{t}+q\delta
Q^{2}-\varphi^{\prime}\delta+q\delta^{3}=0,}\\\
\displaystyle{pQ_{xx}+qQ^{3}+(q\delta^{2}-\varphi^{\prime})Q-\delta^{\prime}=0,}\end{array}\right.$
(8)

and the second equation admits the first integral $h(t)$,

$\displaystyle(Q_{x}\not=0)\
:h=p{Q_{x}}^{2}+qQ^{4}/2+(q\delta^{2}-\varphi^{\prime})Q^{2}-2\delta^{\prime}Q.$
(9)

The integrability of (8)1 and (9) defines the ODEs,

$\displaystyle{\hskip-25.60747pt}\left\\{\begin{array}[]{l}\displaystyle{\varphi^{\prime\prime}+4q\delta\delta^{\prime}=0,\
h^{\prime}+2\delta\delta^{\prime}\varphi^{\prime}-2q\delta^{3}\delta^{\prime}=0,}\\\
\displaystyle{\delta^{\prime\prime}+\delta{\varphi^{\prime}}^{2}-2q\delta^{3}\varphi^{\prime}+2q\delta
h+q^{2}\delta^{5}=0.}\end{array}\right.$ (12)

This system admits three real first integrals $\omega_{0},k_{1},k_{2}$,

$\displaystyle{\hskip-5.69054pt}\left\\{\begin{array}[]{l}\displaystyle{q\delta^{2}=2z,\
\varphi^{\prime}=-4z-\omega_{0},\ qh=2(3z^{2}+\omega_{0}z+k_{2}),}\\\
\displaystyle{(z^{\prime}\not=0):\
{z^{\prime}}^{2}=-4(4z+\omega_{0})^{2}z^{2}-16k_{2}z^{2}+4k_{1}z,}\end{array}\right.$
(15)

characterized by the three nonzero roots of $z^{\prime}$,

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle{\omega_{0}=-2(z_{1}+z_{2}+z_{3}),\
k_{1}=16z_{1}z_{2}z_{3},}\\\
\displaystyle{k_{2}=(z_{1}+z_{2}+z_{3})^{2}-2(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}).}\end{array}\right.$
(18)

In the generic case $Q_{x}zk_{1}\not=0$ (nongeneric cases are detailed in
section 3), the product $\delta^{2}$ is an elliptic function [10, Eq. (13)]
which in the notation of Weierstrass333To convert to the notation of Jacobi,
see [21, §18.9.11, 18.10.8]. takes the quite simple form ($ia$ is real),

$\displaystyle(k_{1}\not=0)\left\\{\begin{array}[]{l}\displaystyle{z=\frac{k_{1}}{\wp(t)-\wp(a)},\
\wp(a)=-\frac{\omega_{0}^{2}+4k_{2}}{3}\raise 2.0pt\hbox{,}\
\wp^{\prime}(a)=-8ik_{1},}\\\
\displaystyle{g_{2}=(4/3)\left[(\omega_{0}^{2}+4k_{2})^{2}+24k_{1}\omega_{0}\right],}\\\
\displaystyle{g_{3}=(8/27)\left[(\omega_{0}^{2}+4k_{2})^{3}+36k_{1}(\omega_{0}(\omega_{0}^{2}+4k_{2})+6k_{1})\right],}\\\
\displaystyle{\Delta^{(t)}\equiv g_{2}^{3}-27g_{3}^{2}=-2^{12}k_{1}^{2}}\\\
\displaystyle{\phantom{1}\times\left[16k_{2}^{3}+8\omega_{0}^{2}k_{2}^{2}+\omega_{0}^{4}k_{2}+36\omega_{0}k_{2}k_{1}+\omega_{0}^{3}k_{1}+27k_{1}^{2}\right].}\end{array}\right.$
(24)

Let us next determine simultaneously $\delta e^{i\varphi}$ and $\delta
e^{-i\varphi}$, not by the multivalued quadrature
$\int\varphi^{\prime}\hbox{d}t$ as usually done, but as the two complex
conjugate solutions of a real second order ODE. The phase invariance of NLS
only allowing the contribution of $\varphi^{\prime}$, not of $\varphi$, by
elimination of $z$ one easily obtains the Lamé equation of index $n=1$,

$\displaystyle\left(\frac{\hbox{d}^{2}}{\hbox{d}t^{2}}-(2\wp(t)+\wp(a))\right)\left(\delta^{-1}e^{\mp
i(\varphi+\omega_{0}t)}\right)=0.$ (25)

Its two independent solutions are generically,

$\displaystyle{\hskip-14.22636pt}\delta^{-1}e^{\mp
i\varphi}=\sqrt{-q/k_{1}}e^{\displaystyle\pm i\omega_{0}t}\mathop{\raise
0.15pt\hbox{\rm H}}\nolimits(t,\pm a),\ $ (26)

with the definition [11, tome II p. 506],

$\displaystyle\mathop{\raise 0.15pt\hbox{\rm
H}}\nolimits(t,a)=e^{\displaystyle-\zeta(a)t}\sigma(t+a)/(\sigma(a)\sigma(t)).$
(27)

At this point, Ref. [10] chooses to integrate the $x$-elliptic ODE (9) with
$t$-dependent coefficients. It is more efficient to integrate the $t$-Riccati
ODE (8)1 with $x$-independent coefficients, and this will allow us to uncover
a new solution, Eq. (97). Indeed, an affine transformation on $Q(x,t)$ maps
the equation (8)1 to a canonical Riccati equation,

$\displaystyle{\hskip-5.69054pt}\left\\{\begin{array}[]{l}\displaystyle{(z\not=0):\
Q(x,t)/\delta(t)=y(x,t)/(2z)+z^{\prime}/(8z^{2})\raise 2.0pt\hbox{,}}\\\
\displaystyle{\partial_{t}y+y^{2}-(3/4)\wp(t)=0,}\end{array}\right.$ (30)

equivalent to a particular Lamé equation of index $n=1/2$, whose solution is
[20, §20 p. 104] [11, tome II p. 482],

$\displaystyle
y=\partial_{t}\log\left[{\wp^{\prime}(t/2)^{-1/2}}\left(4\sqrt{k_{1}}F(x)+\wp(t/2)-\wp(a)\right)\right].$

The real-valued function $\sqrt{k_{1}}F(x)$ is defined by,

$\displaystyle{\hskip-19.91692pt}\begin{array}[]{l}\displaystyle{p{F^{\prime}}^{2}+P(F)=0,\
P(F)\equiv F^{4}+\omega_{0}F^{2}-2\sqrt{k_{1}}F-k_{2},}\end{array}$ (32)

and evaluates to (all $\sqrt{}$ signs are allowed),

$\displaystyle{\hskip-5.69054pt}\left\\{\begin{array}[]{l}\displaystyle{F=\sqrt{z_{1}}+\sqrt{z_{2}}+\sqrt{z_{3}}}\\\
\displaystyle{\phantom{F=}-\frac{2}{p}\frac{(\sqrt{z_{2}}+\sqrt{z_{3}})(\sqrt{z_{3}}+\sqrt{z_{1}})(\sqrt{z_{1}}+\sqrt{z_{2}})}{\wp(x,G_{2},G_{3})-\wp(b,G_{2},G_{3})}\raise
2.0pt\hbox{,}}\\\
\displaystyle{\wp(b)=-\frac{z_{1}+z_{2}+z_{3}+3(\sqrt{z_{2}}\sqrt{z_{3}}+\sqrt{z_{3}}\sqrt{z_{1}}+\sqrt{z_{1}}\sqrt{z_{2}})}{3p}\raise
2.0pt\hbox{,}}\\\
\displaystyle{\sqrt{k_{1}}=4\sqrt{z_{1}}\sqrt{z_{2}}\sqrt{z_{3}},}\\\
\displaystyle{G_{2}=(\omega_{0}^{2}-12k_{2})/(12p^{2}),G_{3}=(\omega_{0}^{3}+36\omega_{0}k_{2}+54k_{1})/(6p)^{3},}\\\
\displaystyle{\Delta^{(x)}\equiv
G_{2}^{3}-27G_{3}^{2}=2^{-16}p^{-6}k_{1}^{-2}\Delta^{(t)}.}\end{array}\right.$
(39)

To summarize, the complex amplitude is,

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle{A=\left\\{\frac{16\sqrt{k_{1}}}{\wp^{\prime}(t/2,g_{2},g_{3})}\left[\frac{P(V)}{F(x)-V(t)}+\frac{\hbox{d}P(V)}{4\hbox{d}V}\right]+i\right\\}}\\\
\displaystyle{\phantom{1234}\times\sqrt{-k_{1}/q}e^{\displaystyle-i\omega_{0}t}/\mathop{\raise
0.15pt\hbox{\rm H}}\nolimits(t,a),}\\\
\displaystyle{V(t)=(\wp(a,g_{2},g_{3})-\wp(t/2,g_{2},g_{3}))/(4\sqrt{k_{1}}),}\end{array}\right.$
(43)

with $P(V(t))$ and $F(x)$ defined in (32) and (39), $a$ in (24), and
$\mathop{\raise 0.15pt\hbox{\rm H}}\nolimits(t,a)$ in (27), and its complex
conjugate results from the change $(i,a)\to(-i,-a)$.

This amplitude (43) depends on three arbitrary real constants
$\omega_{0},k_{1},k_{2}$ and is elliptic in $x$. The ratio two between the
$t$’s in $\wp(t/2)$ and in $\mathop{\raise 0.15pt\hbox{\rm H}}\nolimits(t,a)$
makes the quasi-$t$-periods of $A(x,t)e^{i\omega_{0}t}$ twice the periods of
$\wp(*,g_{2},g_{3})$, thus proving the period doubling observation [12].

Remarks.

  1. 1.

The generality of (43) is worth being emphasized. This unique formula (the
advantage of Weierstrass notation) covers both signs of the discriminant:
$\Delta^{(t)}<0$ (“B-type” solutions, one nonzero real $z_{j}$),
$\Delta^{(t)}>0$ (“A-type” solutions, three nonzero real $z_{j}$), it involves
no multivalued expression and even applies to both NLS régimes (focusing,
defocusing).

  2. 2.

The argument doubling formula $\sigma(2y)=-\wp^{\prime}(y)\sigma^{4}(y)$ [21,
18.4.8] allows one to express (43) with the unique argument $t/2$, i.e. $t$
because of the homogeneity of $\wp$.

  3. 3.

To be physically admissible, the amplitude (43) must obey two constraints. The
first one $\delta^{2}(t)>0$ is: $pz_{3}>0$ [10, p. 811] and $pz_{1},pz_{2}$
positive or complex conjugate, with bounds $0<pz(t)<\hbox{the smallest
positive }pz_{j}$. The second one $Q/\delta$ real, which was painful to
implement [10, p. 811], is equivalent to $\sqrt{k_{1}}F(x)$ real, i.e. the
transposition to the four zeroes $\sqrt{z_{1}}+\sqrt{z_{2}}+\sqrt{z_{3}}$ of
$F^{\prime}(x)$ of the constraints on the zeroes $0,z_{1},z_{2},z_{3}$ of
$z^{\prime}(t)$. The bounded solutions of this focusing régime result from
(43) by applying to the origins of $x$ and $t$ a shift of either zero or a
nonreal half-period, depending on the common sign of the two discriminants
$\Delta^{(x)}$, $\Delta^{(t)}$, see formulae [21, 16.8,18.4.1].

##  3 Nongeneric solutions

They are defined by either $Q_{x}=0$ (inexistence of $h(t)$) or
$z^{\prime}(t)=0$ (inexistence of $k_{1}$) or $z(t)=0$ (undefined link (30)
between $Q$ and $y$) or $k_{1}=0$ (independence of (30) on $z$) or
$\wp^{\prime}(a)=0$ (linear dependence of the two solutions (26) of (25)) or
$\Delta^{(t)}=0$ (degeneracy of elliptic functions to either trigonometric
functions or rational functions). Because Eq. (30) was not considered in [10],
the nongeneric case $k_{1}=0$ will yield the new solution Eq. (97).

###  3.1 Degeneracies of the generic solution

They are characterized by $Q_{x}\delta^{\prime}k_{1}\not=0$,
$\wp^{\prime}(a)\Delta^{(t)}=0$.

When $\wp^{\prime}(a)=0$, then $a$ is a purely imaginary half-period
$\omega^{\prime}$, the multipliers444 Under addition of anyone of the two
periods, a quasi-elliptic function is multiplied by a constant factor, the
multiplier. of $\mathop{\raise 0.15pt\hbox{\rm
H}}\nolimits(t,\omega^{\prime})$ are $(-1,1)$ but $k_{1}$ is zero, which is
forbidden. Fortunately, the form invariance of the ODE for $\wp$ by halving
one period changes $\wp^{\prime}(a)$ to $\wp^{\prime}(2a)\equiv
8i(k_{1}+\omega_{0}k_{2})$, now allowed to vanish. This “Landen
transformation” [22, p. 39] [21, 16.14.2]

$\displaystyle{\hskip-11.38109pt}\left\\{\begin{array}[]{l}\displaystyle{\wp(t,g_{2},g_{3})\equiv\wp(t|\omega,\omega^{\prime})\to\mathcal{P}(t,\gamma_{2},\gamma_{3})\equiv\mathcal{P}(t|\omega,2\omega^{\prime}),}\\\
\displaystyle{\wp(t)=\mathcal{P}(t)+(e_{2}-e_{1})(e_{3}-e_{1})/(\mathcal{P}(t)-e_{1}),}\\\
\displaystyle{{\wp^{\prime}}^{2}=4(\wp-e_{1})(\wp-e_{2})(\wp-e_{3}),\
e_{1}=(8k_{2}-\omega_{0}^{2})/3,}\\\
\displaystyle{{\mathcal{P}^{\prime}}^{2}=4(\mathcal{P}+2e_{1})(\mathcal{P}-\varepsilon_{2})(\mathcal{P}-\varepsilon_{3}),}\\\
\displaystyle{g_{2}=-4\gamma_{2}+60e_{1}^{2},\
g_{3}=8\gamma_{3}+56e_{1}^{3},}\end{array}\right.$ (49)

makes both multipliers unity (i.e. $\mathop{\raise 0.15pt\hbox{\rm
H}}\nolimits$ elliptic), yielding Jacobi functions as solutions to the Lamé
ODE (25),

$\displaystyle{\hskip-17.07164pt}\sqrt{\mathcal{P}(t)-\varepsilon_{2}},\
\sqrt{\mathcal{P}(t)-\varepsilon_{3}}.$ (50)

This leads to the two elliptic breathers in an algorithmic way, instead of the
kind of magic derivation of Ref. [10], and the notation of Halphen

$\displaystyle{\hskip-5.69054pt}\begin{array}[]{l}\displaystyle{\mathop{\raise
0.15pt\hbox{\rm
h}}\nolimits_{a}(x)=\sqrt{\wp(x,G_{2},G_{3})-E_{a}},\mathop{\raise
0.15pt\hbox{\rm
h}}\nolimits_{\alpha}(t)=\sqrt{\mathcal{P}(t,\gamma_{2},\gamma_{3})-\varepsilon_{\alpha}},}\\\
\displaystyle{{\wp^{\prime}(x)}^{2}=4\wp^{3}-G_{2}\wp-G_{3}=4(\wp-E_{a})(\wp-
E_{b})(\wp-E_{c}),}\end{array}$ (53)

allows one to unify them in a very symmetric expression. Characterized by the
relation $pz_{1}+pz_{2}=pz_{3}>0$ between the three roots of
${z^{\prime}}^{2}$ in (15)3, their two singlevalued parts,

$\displaystyle{\hskip-17.07164pt}\left\\{\begin{array}[]{l}\displaystyle{\delta
e^{i\varphi}=\left(-\frac{\omega_{0}}{q}\right)^{1/2}\omega_{0}\frac{\mathop{\raise
0.15pt\hbox{\rm h}}\nolimits_{1}(t)e^{-i\omega_{0}t}}{\mathop{\raise
0.15pt\hbox{\rm h}}\nolimits_{1}(t)\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{3}(t)+i\omega_{0}\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{2}(t)}\raise 2.0pt\hbox{,}}\\\
\displaystyle{\frac{Q}{\delta}=\frac{-\omega_{0}^{2}\mu\mathop{\raise
0.15pt\hbox{\rm h}}\nolimits_{2}(t)\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{2}(x)+\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{1}^{2}(t)\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{3}(t)\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{3}(x)}{\omega_{0}\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{1}(t)[\mu\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{3}(t)\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{2}(x)+\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{2}(t)\mathop{\raise 0.15pt\hbox{\rm h}}\nolimits_{3}(x)]}\raise
2.0pt\hbox{,}}\\\ \displaystyle{k_{2}=(\mu^{2}-1)^{2}\omega_{0}^{2}/2,\
k_{1}=-\omega_{0}k_{2},}\end{array}\right.$ (57)

yield the amplitude [9, Eq. (18)] [10, Eqs. (45), (59)]

$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle{A=\left(-\frac{\omega_{0}}{q}\right)^{1/2}(\mu^{2}-1)\frac{\mathop{\raise
0.15pt\hbox{\rm h}}\nolimits_{\alpha}(t)\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{c}(x)+i\mu\omega_{0}\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{b}(x)}{\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{\beta}(t)\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{c}(x)+\mu\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{\gamma}(t)\mathop{\raise 0.15pt\hbox{\rm
h}}\nolimits_{b}(x)}e^{-i\omega_{0}t}}\\\
\displaystyle{\frac{\varepsilon_{\beta}-\varepsilon_{\alpha}}{(\mu^{2}-1)^{2}}=\frac{\varepsilon_{\gamma}-\varepsilon_{\beta}}{-1}=\frac{\varepsilon_{\alpha}-\varepsilon_{\gamma}}{\mu^{2}(2-\mu^{2})}=\omega_{0}^{2},}\\\
\displaystyle{\frac{E_{b}-E_{c}}{2(\mu^{2}-1)}=\frac{E_{c}-E_{b}}{-\mu^{2}}=\frac{E_{a}-E_{c}}{2-\mu^{2}}=\frac{\omega_{0}^{2}}{2p}\raise
2.0pt\hbox{,}}\end{array}\right.$ (61)

with $(\alpha,\beta,\gamma)$ and $(a,b,c)$ two independent permutations of
$(1,2,3)$. Its two arbitrary constants are $(\omega_{0},\mu)$. The conversion
to Jacobi notation [23, Appendix B] yields the two types A ($\Delta^{(t)}>0$)
and B ($\Delta^{(t)}<0$).

Next, $\Delta^{(t)}=0$ can be represented in terms of $\Omega$ as,

$\displaystyle\begin{array}[]{l}\displaystyle{k_{1}=-\Omega(\Omega-\omega_{0})^{2}/2,\
k_{2}=-\Omega(3\Omega-\omega_{0})/4,}\\\
\displaystyle{{z^{\prime}}^{2}=-64\left(z-(\Omega-\omega_{0})/4\right)^{2}\left(z+\Omega/2\right)z.}\end{array}$
(64)

The first degeneracy ($k\not=0$),

$\displaystyle{\hskip-34.1433pt}\left\\{\begin{array}[]{l}\displaystyle{k^{2}=4(\Omega-\omega_{0})(3\Omega-\omega_{0}),\
pK^{2}=3\Omega-\omega_{0},}\\\ \displaystyle{\delta
e^{i\varphi}=\left(\frac{\Omega-\omega_{0}}{2q}\right)^{1/2}e^{\displaystyle{-i\Omega
t}}\frac{\sin(kt/2)}{\sin(k(t-t_{3})/2)}\raise 2.0pt\hbox{,}}\\\
\displaystyle{\frac{Q}{\delta}=\frac{k}{2(\Omega-\omega_{0})}\left(\mathop{\rm
cotg}\nolimits(kt/2)\right.}\\\
\displaystyle{\left.\phantom{12}+\frac{6\Omega-3\omega_{0}+3\Omega\cos(kt)}{3\Omega[\alpha\cosh(Kx)+\cos(kt/2)]\sin(kt/2)}\right),}\\\
\displaystyle{\cos(kt_{3})=-(2\Omega-\omega_{0})/\Omega,\
\sin(kt_{3})=ik/(2\Omega)\raise 2.0pt\hbox{,}}\end{array}\right.$ (70)

is the first iterate of the plane wave (80) by the Bäcklund transformation, it
depends on two arbitrary real constants $\omega_{0},\Omega$ restricted to
$0<\Omega/\omega_{0}<1$ by the reality of $y(x,t)$. Depending on the signs of
$(K^{2},k^{2})$, this mathematical solution defines four physical solutions:
two unbounded in the defocusing régime, and two in the focusing régime: the
Kuznetsov bright soliton solution [8] [24, (6.10)] [25, (41a)], localized in
space and periodic in time,

$\displaystyle{\hskip-5.69054pt}\left\\{\begin{array}[]{l}\displaystyle{A=\sqrt{-\Omega/q}e^{\displaystyle{-i\Omega
t}}}\\\
\displaystyle{\phantom{1}\times\left[1-\frac{2(1-\alpha^{2})\Omega\cos(kt/2)+i(k/2)\sin(kt/2)}{\Omega[\alpha\cosh(Kx)+\cos(kt/2)]}\right]}\\\
\displaystyle{K^{2}=2\Omega(1-\alpha^{2}),\
k^{2}=-16\Omega^{2}\alpha^{2}(1-\alpha^{2}),\
1<\alpha^{2},}\end{array}\right.$ (74)

and the breather solution of Akhmediev [9, Eq. (11)] localized in time and
periodic in space,

$\displaystyle{\hskip-5.69054pt}\left\\{\begin{array}[]{l}\displaystyle{A=\sqrt{-\Omega/q}e^{\displaystyle{-i\Omega
t}}}\\\
\displaystyle{\phantom{1}\times\left[1+\frac{2(\alpha^{2}-1))\Omega\cosh(\kappa
t/2)+i(\kappa/2)\sinh(kt/2)}{\Omega[\alpha\cos({K^{\prime}}x)+\cosh(\kappa
t/2)]}\right]}\\\ \displaystyle{{K^{\prime}}^{2}=-2\Omega(1-\alpha^{2}),\
\kappa^{2}=16\Omega^{2}\alpha^{2}(1-\alpha^{2}),\
0<\alpha^{2}<1.}\end{array}\right.$ (78)

A rigorous proof of their instability under small perturbations can be found
in [26].

The second degeneracy ($k=0,\omega_{0}=3\Omega\not=0$) yields the Peregrine
soliton [27], whose complex amplitude is rational in $x$ and $t$,

$\displaystyle{\hskip-22.76219pt}A=\left(-\frac{\Omega}{q}\right)^{1/2}\left[1+4p\frac{1-2i\Omega
t}{2\Omega x^{2}-p(1+(2\Omega t)^{2})}\right]e^{\displaystyle-i\Omega t}.$
(79)

whose large maximum amplitude $3$ above its background makes it a simple
prototype of rogue wave.

###  3.2 Nongeneric solutions $Q_{x}z^{\prime}=0$

If $Q_{x}=0$, the solution is a particular plane wave,

$\displaystyle{\hskip-34.1433pt}A=\sqrt{-\omega_{0}/q}e^{\displaystyle-i\omega_{0}t},\
$ (80)

which is also the limit $\Omega\to\omega_{0}$ of both (74) and (78).

If $Q_{x}\not=0$ and $z=z_{0}\not=0$, one obtains a two-parameter particular
“dark” one-soliton solution [28, (28)],

$\displaystyle\begin{array}[]{l}\displaystyle{A=\left(-\frac{2p}{q}\right)^{1/2}\left(\lambda\tanh(\lambda(x-ct))+i\frac{c}{2p}\right)e^{\displaystyle-i\Omega_{0}t},}\\\
\displaystyle{\lambda^{2}=\Omega_{0}/(2p)-c^{2}/(4p^{2}),\
\Omega_{0}=\omega_{0}+2z_{0},}\end{array}$ (83)

and its one-parameter rational degeneracy $\lambda=0$,

$\displaystyle{\hskip-19.91692pt}A=\left(-\frac{2p}{q}\right)^{1/2}\left(\frac{1}{x-ct}+i\frac{c}{2p}\right)e^{\displaystyle-i\Omega_{0}t},\
c^{2}=2p\Omega_{0}.$ (84)

$Q_{x}\not=0$ and $z=0$ defines the envelope solution,

$\displaystyle{\hskip-34.1433pt}\left\\{\begin{array}[]{l}\displaystyle{A=\sqrt{2p/q}\mathop{\rm
dn}\nolimits(\lambda x,m_{x})e^{\displaystyle-i\omega_{0}t}}\\\
\displaystyle{\omega_{0}=p\lambda^{2}(m_{x}-2),\
k_{2}=p^{2}\lambda^{4}(m_{x}-1),}\end{array}\right.$ (87)

and its degeneracy “bright” one-soliton solution [29],

$\displaystyle{\hskip-14.22636pt}k_{2}=0:\
A=\left(\frac{2p}{q}\right)^{1/2}\frac{\lambda}{\cosh(\lambda
x)}e^{\displaystyle-i\omega_{0}t},\ \lambda^{2}=-\frac{\omega_{0}}{p}.$ (88)

The other trigonometric degeneracy $k_{2}=-\omega_{0}^{2}/4$ is identical to
the limit $z_{0}=0$ of (83), and their common rational degeneracy is also the
limit $\Omega_{0}\to 0$ of (84).

###  3.3 Nongeneric solutions $Q_{x}z^{\prime}\not=0$ and $k_{1}=0$

One must distinguish $k_{2}(\omega_{0}^{2}+4k_{2})$ zero or nonzero.

For $k_{2}(\omega_{0}^{2}+4k_{2})\not=0$, one obtains,

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle{z^{-1}=2a\left(\cos(kt)-\cos(k{t_{1}})\right),\
\sin(k{t_{1}})=-4i/(ak),}\\\
\displaystyle{y=\partial_{t}\log\left[F(x)\sin(kt/4)+\cos(kt/4)\right],}\\\
\displaystyle{8p{F^{\prime}}^{2}=2\omega_{0}(F^{2}+1)^{2}-a(k^{2}/4)(F^{4}-6F^{2}+1),}\\\
\displaystyle{a^{2}=-\frac{16k_{2}}{(\omega_{0}^{2}+4k_{2})^{2}}\raise
2.0pt\hbox{,}\ k^{2}=4(\omega_{0}^{2}+4k_{2}),\
\cos(k{t_{1}})=\frac{8\omega_{0}}{ak^{2}}\raise
2.0pt\hbox{,}}\end{array}\right.$ (93)

and the reality of $z(t)$ restricts $k_{2}$ to be negative.

To our knowledge, this is a new solution, depending on two constants
$\omega_{0},k_{2}$. The reason why it was not found earlier is the choice of
all authors to integrate the $x$-elliptic ODE (9) instead of the $t$-Riccati
ODE (8)1, preventing $k_{1}=0$ to be singled out. The physically admissible
solutions, elliptic in $x$, exist in focusing and defocusing régimes but are
not bounded. When $-\omega_{0}^{2}/4<k_{2}<0$, the amplitude
$Ae^{i\omega_{0}t}$ is periodic in time,

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle{A=\left(-\frac{a}{qk_{2}}\right)^{1/2}\frac{k^{2}}{16\sin(k{t_{1}}/2)}e^{\displaystyle{-i\omega_{0}t}}}\\\
\displaystyle{\phantom{12}\times\frac{\frac{ak}{4}(\cos(k{t_{1}})-1)[1+F(x)c]+i[F(x)-c]}{F(x)+c}\raise
2.0pt\hbox{,}}\\\ \displaystyle{F(x)=c_{0}\mathop{\rm cs}\nolimits(\lambda
x,m)\hbox{ real},\ c=\mathop{\rm cotg}\nolimits(kt/4),}\end{array}\right.$
(97)

and, when $k_{2}<-\omega_{0}^{2}/4$, only periodic in $x$,

$\displaystyle\left\\{\begin{array}[]{l}\displaystyle{A=\left(\frac{a}{qk_{2}}\right)^{1/2}\frac{k^{2}}{16\sinh(\kappa{t_{1}}/2)}e^{\displaystyle{-i\omega_{0}t}}}\\\
\displaystyle{\phantom{12}\times\frac{\frac{a\kappa}{4}(\cosh(\kappa{t_{1}})-1)[1+G(x)c]+i[G(x)+c]}{-G(x)+c}\raise
2.0pt\hbox{,}}\\\ \displaystyle{G(x)=iF(x)\hbox{ real},\ c=\coth(\kappa t/4),\
\kappa^{2}=-k^{2}>0.}\end{array}\right.$ (101)

The degeneracy $k_{1}=0$, $k_{2}=-\omega_{0}^{2}/4\not=0$ of (97),

$\displaystyle{\hskip-14.22636pt}A=\sqrt{-\frac{2p}{q}}\frac{K}{2}\left[1-\frac{2(2\omega_{0}t-i)}{\sinh(\sqrt{2\omega_{0}/p}x)+2\omega_{0}t}\right]e^{\displaystyle-i\omega_{0}t},\
$ (102)

is the limit $\Omega\to\omega_{0}$ of the degeneracy (74) of (43), obtained by

$\displaystyle\begin{array}[]{l}\displaystyle{\Omega=\omega_{0}(1-2\varepsilon^{2}),\
\alpha=\varepsilon,\ k=-4i\omega_{0}\varepsilon,}\\\
\displaystyle{\cosh(Kx+i\pi/2)=i\sinh(\lambda x),\ \varepsilon\to
0,}\end{array}$ (105)

and expanding $\sin$ and $\cos$ near $kt=0$. Although we could not find (102)
explicitly written somewhere, it is certainly not new, see for instance [30].

Last, the degeneracy $k_{2}=0$ has a nonreal value of $z(t)$.

Table 1 displays all solutions generated by (4).

Table 1:  All solutions of the constraint (4). Each solution is separated by a single line from its degeneracies. Columns display: $x$ and $t$-dependences of $Ae^{i\omega_{0}t}$ (quasi-elliptic Q, elliptic E, trigonometric T, rational R, none $0$), the arbitrary constants, the complex amplitude, the initial reference.  | $Q_{x}$ | $z^{\prime}(t)$ | $z$ | $k_{1}$ | $x$ | $t$ | arb | Eq | reference
---|---|---|---|---|---|---|---|---|---
$A$ | $\not=0$ | $\not=0$ | $\not=0$ | $\not=0$ | E | Q | $\omega_{0}k_{1}k_{2}$ | (43) | [10, (3), (22), (24)–(25)]
$B$ | $\not=0$ | $\not=0$ | $\not=0$ | $\not=0$ | E | E | $\omega_{0}k_{1}$ | (61) | [9, (18)]
$C$ | $\not=0$ | $\not=0$ | $\not=0$ | $\not=0$ | T | T | $\Omega\alpha$ | (74) | [8]
$D$ | $\not=0$ | $\not=0$ | $\not=0$ | $\not=0$ | T | T | $\Omega\alpha$ | (78) | [9, (11)]
$E$ | $\not=0$ | $\not=0$ | $\not=0$ | $\not=0$ | R | R | $\omega_{0}$ | (79) | [27, (6.7)]
$1$ | $0$ |  |  |  | 0 | 0 | $\omega_{0}$ | (80) | [10, (37),(51)]
$2$ | $\not=0$ | $0$ | $\not=0$ |  | T | T | $\Omega_{0}c$ | (83) | [28, (3)]
$3$ | $\not=0$ | $0$ | $\not=0$ |  | R | R | $\omega_{0}$ | (84) | [28]
$4$ | $\not=0$ | $0$ | $0$ |  | E | 0 | $\omega_{0}k_{2}$ | (87) | [10, (54), (60)]
$5$ | $\not=0$ | $0$ | $0$ |  | T | 0 | $\omega_{0}$ | (88) |  [29] [10, (46)]
$6$ | $\not=0$ | $\not=0$ | $\not=0$ | $0$ | E | T | $\omega_{0}k_{2}$ | (97) | New
$7$ | $\not=0$ | $\not=0$ | $\not=0$ | $0$ | T | R | $\omega_{0}$ | (102) |

##  4 On constraints of higher degree

Since those singularities of $A$ and ${\bar{A}}$ which depend on the initial
conditions are simple poles [18, 19], the next constraint after (4) should be,

$\displaystyle\begin{array}[]{l}\displaystyle{(g_{2,1}R^{2}+2g_{2,2}RI+g_{2,3}I^{2}+g_{2,4}R_{x}+g_{2,5}I_{x})}\\\
\displaystyle{+(g_{1,1}R+g_{1,2}I)+g_{0}=0,\ R=\mathop{\rm
Re}\nolimits(A),I=\mathop{\rm Im}\nolimits(A),}\end{array}$ (108)

in which the real coefficients $g_{Nij\dots}$ depend on $t$ (and maybe on
$x$). Indeed, the relevant degree is the singularity degree (two in (108)),
not the polynomial degree, which is why the restrictive assumption [10, Eq.
(61)] ($g_{2,4}=g_{2,5}=0$) finds nothing new. The larger freedom of (108)
should generate more solutions, this will be the subject of future work.

##  5 Conclusion and discussion

The present work, which makes explicit the three-parameter extrapolation of
the NLS breather, explains the $t$-period doubling experimentally observed
[12]. It should provide a much better precision in all the experiments on the
phenomenon of modulational instability.

The Lamé equation is fundamental in the solution of the constraint (4): (i) it
leads to the compact expression (43), (ii) it provides a natural derivation of
the breather (61), initially obtained by expert manipulations [10, 14].

Since the Kuznetsov solution (74), identical in the complex plane to the
Akhmediev breather (78), is generated by the plane wave (80) via the Bäcklund
transformation (BT), it is natural to ask which seed generates the three-
parameter solution (43), an extrapolation of (74). We conjecture that this
could be the general traveling wave

$\displaystyle{\hskip-5.69054pt}A=\left(-\frac{2p}{q}\right)^{1/2}\frac{\sigma(\xi+d)}{\sigma(\xi)}e^{\displaystyle{-i\omega
t-\zeta(d)\xi+i\frac{c}{2p}\xi}},\xi=x-ct,$ (109)

with $id$ real (again Lamé!) for two reasons: (i) Since the BT involves the
integration of a linear differential system (the Lax pair) depending on the
seed, this seed must be elliptic in $x$ and $t$; (ii) The elliptic
discriminants $\Delta^{(x)}$, $\Delta^{(t)}$ of (43) have a never zero ratio,
just like the elliptic discriminants of (109) have for ratio a power of $c$.

## Acknowledgments

The author is pleased to thank Micheline Musette for a critical reading of the
manuscript. This work was initiated in Centre international de rencontres
mathématiques, Marseille (grant 2311, year 2019), whose hospitality is
gratefully acknowledged.

## Conflict of interest

The author declares that he has no conflicts of interest.

## References

  * [1] V.I. Bespalov and V.I. Talanov, Filamentary structure of light beams in nonlinear liquids, Pis’ ma Zh. Eksp. Teor. Fiz. 3 (1966) 471–476 [English translation: JETP Lett. 3 (1966) 307–310.] `http://www.jetpletters.ac.ru/ps/1621/article_24803.pdf`
  * [2] V.E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, Zh. Prikl. Mekh. Tekh. Fiz. 9 (2) (1968) 86–94. [English J. Appl. Mech. Tech. Phys. 9 (2) (1968) 190–194] http://zakharov75.itp.ac.ru/static/local/zve75/zakharov/1968/1968-03-art3A10.10072FBF00913182.pdf
  * [3] N. Akhmediev, A. Ankiewicz and M. Taki, Waves that appear from nowhere and disappear without a trace, Phys. Lett. A 373 (2009) 675–678 https://doi.org/10.1016/j.physleta.2008.12.036
  * [4] V.V. Konotop and M. Salerno, Modulational instability in Bose-Einstein condensates in optical lattices, Phys. Rev. A 65 (2002) 021602(R) https://doi.org/10.1103/PhysRevA.65.021602
  * [5] P.J. Everitt, M.A. Sooriyabandara, M. Guasoni, P.B. Wigley, C.H. Wei, G.D. McDonald, K.S. Hardman, P. Manju, J.D. Close, C.C.N. Kuhn, S.S. Szigeti, Y.S. Kivshar and N.P. Robins, Observation of a modulational instability in Bose-Einstein condensates, Phys. Rev. A 96 (2017) 041601(R) http://dx.doi.org/10.1103/PhysRevA.96.041601
  * [6] D.R. Solli, C. Ropers, P. Koonath and B. Jalali, Optical rogue waves, Nature 450 (2007) 1054–1057. https://doi.org/10.1038/nature06402
  * [7] D.I. Yeom and B.J. Eggleton, Rogue waves surface in light, Nature 450 (2007) 953–954. https://doi.org/10.1038/450953a
  * [8] E.A. Kuznetsov, Solitons in a parametrically unstable plasma, Dokl. Akad. Nauk SSSR 236 (1977) 575–577 [English : Sov. Phys. Dokl. 22 (1977) 507–508]. http://mi.mathnet.ru/dan41246
  * [9] N.N. Akhmediev and V.I. Korneev, Modulation instability and periodic solutions of the nonlinear Schrödinger equation, Teoreticheskaya i Matematicheskaya Fizika 69 (1986) 189–194 [English : Theor. and Math. Phys. 69 (1987) 1089–1093]. https://doi.org/10.1007/BF01037866http://mi.mathnet.ru/tmf5217
  * [10] N.N. Akhmediev, V.M. Eleonskii and N.E. Kulagin, Exact first-order solutions of the nonlinear Schrödinger equation, Teoreticheskaya i Matematicheskaya Fizika 72 (1987) 183–196 [English : Theor. and Math. Phys. 72 (1987) 809–818]. https://doi.org/10.1007/BF01017105
  * [11] G.-H. Halphen, Traité des fonctions elliptiques et de leurs applications (Gauthier-Villars, Paris, 1886, 1888, 1891). https://gallica.bnf.fr/ark:/12148/bpt6k7348q https://gallica.bnf.fr/ark:/12148/bpt6k73491
  * [12] Guillaume Vanderhaegen, Pascal Szriftgiser, Corentin Naveau, Alexandre Kudlinski, Matteo Conforti, Stefano Trillo, Nail Akhmediev, and Arnaud Mussot, Observation of doubly periodic solutions of the nonlinear Schrödinger equation in optical fibers, Optics letters 45, 13 (2020) 3757–3760. https://doi.org/10.1364/OL.394604
  * [13] Matteo Conforti, Arnaud Mussot, Alexandre Kudlinski, Stefano Trillo and Nail Akhmediev, Doubly periodic solutions of the focusing nonlinear Schrödinger equation: recurrence, period doubling, and amplification outside the conventional modulation-instability band, Phys. Rev. A 101 (2020) 023843. https://doi.org/10.1103/PhysRevA.101.023843
  * [14] N.N. Akhmediev and A. Ankiewicz, First order exact solutions of the nonlinear Schrödinger equation in the normal dispersion regime, Phys. Rev. A 47 (1993) 3213–3221. https://doi.org/10.1103/PhysRevA.47.3213
  * [15] D. Mihalache and N.C. Panoiu, Exact solutions of nonlinear Schrödinger equation for positive group velocity dispersion, J. Math. Phys. 33 (1992) 2323–2328. https://doi.org/10.1063/1.529603
  * [16] D. Mihalache and N.C. Panoiu, Exact solutions of the nonlinear Schrödinger equation for the normal-dispersion regime in optical fibers, Phys. Rev. A 45 (1992) 6730–6734. https://doi.org/10.1103/PhysRevA.45.6730
  * [17] Kwok W. Chow, A class of doubly periodic waves for nonlinear evolution equations, Wave motion 35 (2002) 71–90. https://doi.org/10.1016/S0165-2125(01)00078-6
  * [18] D.V. Chudnovsky, G.V. Chudnovsky and M. Tabor, Painlevé property and multicomponent isospectral deformation equations, Phys. Lett. A 97 (1983) 268–274. https://doi.org/10.1016/0375-9601(83)90686-2
  * [19] R. Conte and M. Musette, The Painlevé handbook, 2nd ed., 391 pages (Springer Nature, Switzerland, 2020). https://doi.org/10.1007/978-3-030-53340-3
  * [20] G.-H. Halphen, Mémoire sur la réduction des équations différentielles, Recueil des savants étrangers 28 (1884) 1–301. https://gallica.bnf.fr/ark:/12148/bpt6k33332?rk=21459
  * [21] M. Abramowitz, I. Stegun, Handbook of mathematical functions, Tenth printing (Dover, New York, 1972). https://kfk.pw/182101-uploads.pdf
  * [22] L. Kiepert, Auflösung der Transformationsgleichungen und Division der elliptischen Functionen, J. für die reine und angewandte Math. 76 (1873) 34–44. https://eudml.org/doc/148207
  * [23] Chow K.-w., R. Conte and Neil Xu, Analytic doubly periodic wave patterns for the integrable discrete nonlinear Schrödinger (Ablowitz-Ladik) model, Phys. Lett. A 349 (6) (2006) 422–429. http://dx.doi.org/10.1016/j.physleta.2005.09.053 http://arXiv.org/abs/nlin.PS/0509005
  * [24] T. Kawata and H. Inoue, Inverse scattering method for the nonlinear evolution equations under nonvanishing conditions, J. Phys. Soc. Japan 44 (1978) 1722–1729. https://doi.org/10.1143/JPSJ.44.1722
  * [25] Ma Yan-chow, The perturbed plane-wave solutions of the cubic Schrödinger equation, Stud. Appl. Math. 60 (1979) 43–58. https://doi.org/10.1002/sapm197960143
  * [26] M.A. Alejo, L. Fanelli and C. Muñoz, The Akhmediev breather is unstable, São Paulo J. Math. Sci. 13 391–401 (2019). https://doi.org/10.1007/s40863-019-00145-4https://arxiv.org/abs/1308.5565
  * [27] D.H. Peregrine, Water waves, nonlinear Schrödinger equations and their solutions, J. Austral. Math. Soc. B 25 (1983) 16–43. https://doi.org/10.1017/S0334270000003891
  * [28] V.E. Zakharov and A.B. Shabat, Interaction between solitons in a stable medium, Zh. Eksp. Teor. Fiz. 64 (1973) 1627–1639 [English : Soviet Physics JETP 37 (1973) 823–828]. $http://jetp.ac.ru/cgi-bin/dn/e_{0}37_{0}5_{0}823.pdf$
  * [29] V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Zh. Eksp. Teor. Fiz. 61 (1971) 118–134 [English : Soviet physics JETP 34 (1972) 62–69]. `http://jetp.ac.ru/cgi-bin/dn/e_034_01_0062.pdf`
  * [30] Kwok W. Chow, Solitary waves on a continuous background, J. Phys. Soc. Jpn 64 (1995) 1524–1528. https://doi.org/10.1143/JPSJ.64.1524