# Elliptic solitons in optical fiber media

Défi Jr. Jubgang Fandio<EMAIL_ADDRESS>Alain M. Dikandé
<EMAIL_ADDRESS>Laboratory of Research on Advanced Materials and
Nonlinear Sciences, Department of Physics, Faculty of Science, University of
Buea P.O. Box 63, Buea Cameroon  A. Sunda-Meya<EMAIL_ADDRESS>Department
of Physics, Xavier University of Louisiana, 1 Drexel Drive New Orleans, LA
70125

###### Abstract

We examine the evolution of a time-varying perturbation signal pumped into a
mono-mode fiber in the anomalous dispersion regime. We analytically establish
that the perturbation evolves into a conservative pattern of periodic pulses
which structures and profiles share close similarity with the so-called
soliton-crystal states recently observed in fiber media [see e.g. A. Haboucha
et al., Phys. Rev. A78, 043806 (2008); D. Y. Tang et al., Phys. Rev. Lett.
101, 153904 (2008); F. Amrani et al., Opt. Express 19, 13134 (2011)]. We
derive mathematically and generate numerically a crystal of solitons using
time division multiplexing of identical pulses. We suggest that at very fast
pumping rates, the pulse signals overlap and create an unstable signal that is
modulated by the fiber nonlinearity to become a periodic lattice of pulse
solitons which can be described by elliptic functions. We carry out a linear
stability analysis of the soliton-crystal structure and establish that the
correlation of centers of mass of interacting pulses broadens their internal-
mode spectrum, some modes of which are mutually degenerate. While it has long
been known that high-intensity periodic pulse trains in optical fibers are
generated from the phenomenon of modulational instability of continuous waves,
the present study provides evidence that they can also be generated via
temporal multiplexing of an infinitely large number of equal-intensity single
pulses to give rise to stable elliptic solitons.

###### pacs:

42.55.Wd, 42.65.Tg, 42.70.Mp

††preprint: APS/123-QED

##  I  Introduction

Optical sources able to generate wavelength tunable ultra-short pulse trains
with high quality and high-repetition rates have attracted a great deal of
attention in the recent past agr1 ; agr2 . These specific structures find
important application in optical networks combining wavelength division
multiplexing and optical time domain multiplexing transmission techniques agr1
; agr2 . Traditionally soliton operations in optical media have been
associated with a net negative dispersion, given that the balance between a
negative dispersion and the nonlinear optical responses (Kerr and non-Kerr) of
the media is more likely to promote nonlinear transform-limited pulses mol .
Although a full description of most real optical fiber networks as well as
fiber laser media generally requires the account of several complex processes
interfering in the system dynamics (high-order dispersions processes, loss and
gain dynamics and so on), the fundamental optical soliton (pulse-shape signal)
observed in these systems is generic from the nonlinear Schrödinger equation
menyuk . Among other possible optical soliton structures that can also be
derived from the nonlinear Schrödinger equation multi-soliton complexes are
particular, due to their suitability for high-bit and high-repetition rate
data transfer technology. These include harmonically mode-locked vector
solitons, soliton crystals and bound soliton states representing a periodic
lattice of pulses distributed in a one-dimensional chain and that have
recently been the subject of intense theoretical investigations, in connection
with a broad range of optical-wave transmission phenomena tang ; kart1 ; kart2
; kart3 ; dik1 ; dik2 ; zhao .
Our main point of focus in this work are the periodic trains of high-intensity
signals amra1 ; amra2 ; abouch1 ; abouch2 ; abouch3  that have been reported
recently in passively mode-locked double cladd fiber lasers, where each pulse
in the trains appears to be a double time scale comb-like signal consisting of
a pulse soliton hosting a quasi-periodic pattern of solitons at its apex and
whose self-ordering may stem from the phase-locking of wave components
modulated by gain variation abouch1 ; abouch2 . To this last point theoretical
investigations of the stability of steady wave spread in optical fibers have
long predicted agr1 ; agr2  that the nonlinear Schrödinger equation could
admit a family of periodic soliton solutions represented by Jacobian elliptic
functions. On the other hand it is well established that in the anomalous
regime of dispersion, modulational instability can generate a web of bound
pulses whose inter-pulse length is controlled by an external mechanism chern .
In general high-power input pulses are necessary to trigger pulse interaction
but the effect is relatively unstable and the bound pulse train eventually
splits into independent pulses after few kilometers of propagation. Methods to
compensate for this inconvenience might be enclosing the fiber into a Fabry-
Perot resonator, or by using dispersion-managed systems. The gain acquired by
modulational instability in the first method or by timing-jitter reduction, as
illustrated in mu  in the second case, provides enough energy for pulse
interaction. Pulse repetition control can be achieved by means of self-
starting laser pumps menyuk1 ; chen  able to generate ultra-short high
energetic pulses into the fiber.
In this paper we examine the intimate structure (width, tails and mutual
separation between pulses) of the periodic trains of pulses reported recently
in fiber lasers and called soliton-crystal signals amra1 ; amra2 ; abouch1 ;
abouch2 ; abouch3 . In particular we wish to establish unambiguously the
connection between the elliptic-wave solution of the cubic nonlinear
Schrödinger equation, and the pattern formed by means of time division
multiplexing of pumped pulses menyuk  thus emphasizing the minor role of the
gain dynamics in the formation mechanism of soliton crystals while the gain is
expected to hold a more major role in the pulse interaction stengths and the
soliton-crystal stability in general.
In section II we present the model consisting of a cubic nonlinear Schrödinger
equation for a single-mode fiber, and find exact nonlinear periodic-wave
solution representing elliptic soliton structures. In section III we revisit a
familiar reconstruction scheme (see e.g. menyuk ; abouch3 ) in order to
identify the pulse distribution in the soliton crystal. In this respect we
consider a standard ansatz menyuk ; abouch3  describing a multiplex state of
periodically spaced solitons, that can be obtained by repetitively pumping an
infinitely large number of pulses into the fiber. We expand the ansatz and
find an analytical wave function that permits a direct comparision with the
elliptic-soliton solution of the nonlinear propagation equation. Next a
numerical implementation of the time division multiplexing of identical pulses
is carried out now with a finite number of pulses, to illustrate the
consistency of the proposed theoretical reconstruction scheme with
experimental results amra1 ; amra2 . In section IV we carry out a linear
stability analysis to check the robustness of the soliton-crystal structure
against small-amplitude noises, in this context we point out the implication
of the multiplexing of equal-intensity pulses on the stability of individual
pulses as well as the multiplex state. Concluding remarks follow in section V.

##  II  The model and elliptic-soliton structures

The propagation of slowly varying wave envelopes in optical fibers exhibiting
weak dispersion and weak nonlinearity, is governed by the cubic nonlinear
Schrödinger equation:

$i\frac{\partial q(z,t)}{\partial
z}-\frac{1}{2}\beta_{2}\frac{\partial^{2}q(z,t)}{\partial
t^{2}}+\gamma|q(z,t)|^{2}q(z,t)=0,$ (1)

where $z$ is the spatial coordinate of the envelope, $t$ is the propagation
time, $\beta_{2}$ is the group velocity dispersion of the fiber material and
$\gamma$ is the nonlinear Kerr coefficient accounting for self-phase
modulation. The slowly varying wave envelope $q(z,t)$ is normalized such that
the square of its magnitude represents the power transmitted throughout the
fiber. For weak fields, nonlinear effects are neglected and the solution to
eq(1) is a steady-state wave that disperses along the fiber core and vanishes
in the cladding.
We assume that the slowly varying amplitude $q(z,t)$ injected by the pump is a
strong time-varying perturbation that temporally awakes nonlinear effects in
the fiber, in this respect we set $q(z,t)=a(t)e^{i\beta z}$ where $\beta$ is
the wave parameter. The amplitude $a(t)$ of the signal governing the temporal
evolution of the envelope then obeys the equation:

$\left(\frac{da}{dt}\right)^{2}=-\frac{2\beta}{\beta_{2}}a^{2}+\frac{\gamma}{\beta_{2}}a^{4}+C,$
(2)

where $C$ is the constant of energy integral that determines profiles of the
amplitude $a(t)$. For a localized profile we expect a rapid evanescence of the
wave outside its time-bandwidth such that the constant $C$ tends to zero. In
the anomalous dispersion regime ($\beta_{2}<0$), the localized-wave solution
is a single hyperbolic-secant pulse given by:

$a(t)=\sqrt{\frac{2\beta}{\gamma}}\
sech\left[\sqrt{\frac{-2\beta}{\beta_{2}}}(t-t_{0})\right],$ (3)

the shape of which is independent on the autocorrelation trace and propagation
distance such that its ”soliton” identity is always preserved. When the
constant $C$ is non-zero and negative, energetic conditions become detrimental
to the hyperbolic-secant pulse. However we can still find nonlinear solutions
to the amplitude equation as:

$a(t)=\sqrt{\frac{2\beta}{\gamma\
(2-k^{2})}}\,dn\left[\sqrt{\frac{-2\beta}{\beta_{2}(2-k^{2})}}(t-t_{0}),k\right].$
(4)

In the last formula $dn$ is the Jacobi elliptic delta function of modulus $k$
($0<k\leq 1$), while $t_{0}$ is an arbitrary initial time.
The Jacobi elliptic $dn$ function is periodic in its time argument $t$ with a
time period:

$\tau=2K(k)\sqrt{\frac{\beta_{2}(2-k^{2})}{-2\beta}},$ (5)

with $K(k)$ the Elliptic Integral of first kind. In fig. 1, we sketched the
autocorrelation trace of the Jacobi elliptic delta function signal (4).

(a)

(b)

Figure 1:  Elliptic-soliton solution to the nonlinear amplitude equation (2),
for two different values of the parameter of pulse dispersion $k$: (a)
Periodic network of sharp pulse signals for large dispersion parameter
($k=0.98$), (b) Elliptic-wave decay into a quasi-continuous signal for small
dispersion parameter ($k=0.6$).

Possible numerical values for characteristic parameters are $\beta=1.55\mu m$,
$\beta_{2}=-15ps^{2}/km$ and $\gamma=25W^{-1}/km$ and correspond to typical
data common in most fiber laser setups amra1 ; amra2 ; abouch1 ; abouch2 . As
fig. 1 shows, the $dn$ function describes bound states of periodically spaced
pulses forming a soliton crystal kart1 ; kart2 ; kart3 ; dik1 ; dik2 .
Remarkably small values of the modulus $k$ tend to increase the pulse density
in the crystal whereas large values of this parameter isolate each pulse, as a
matter of fact when $k=1$ the soliton crystal breaks down and the single-pulse
regime is favored. Its turns out that the modulus $k$ can be readily regarded
as the parameter of pulse dispersion.
The presence of the autocorrelation propagation parameter of the envelope, as
well as its conservative nature manifested in the distance-independent power
transmitted, are relevant characteristics providing the envelope eq. (4) with
a ”soliton” feature. We can therefore conclude that the Jacobi $dn$ function
stands for a good mathematical representation of a soliton crystal. Since the
generic wave equation (1) is integrable, all physical features arising from
the propagation of waves will be intrinsic to such system such that the
soliton crystal structure is clearly an essential profile of light envelopes
in Kerr nonlinear optical media.

##  III  Soliton-crystal reconstruction

###  III.1 Analytical reconstruction

It has long been established agr2 ; chern ; mu  that nonlinear-wave generation
in optical fibers results from the modulational instability of continuous
waves. The modulational instability in this case can lead either to a single
pulse menyuk1 , as for instance a strong hyperbolic secant pulse produced by
the superposition of continuous waves oscillating at a constant frequency
shift, or to a train of pulses milon ; akhmed . It would however be
interesting to examine the structure of a periodic train of pulses constructed
by time division multiplexing menyuk  of an infinitely large number of pulses,
which in fact would be the output signal resulting from the interaction of
pulses pumped into the fiber by a pulse-mode laser at a given period. In this
last goal we consider a laser pump injecting pulses into a standard mono-mode
fiber at finite wavelengths, in the weak anomalous dispersion regime
$\beta_{2}$. Let the hypothetical time multiplexed output signal A(t) be
expressed as menyuk ; abouch3 :

$A(t)=\sum\limits_{n=-\infty}^{\infty}\sqrt{\frac{2\beta_{m}}{\gamma}}\,sech\left[\sqrt{\frac{-2\beta_{m}}{\beta_{2}}}(t-t_{0}-n\tau_{A})\right],$
(6)

where the subscript $m$ refers to the propagation mode and $\tau_{A}$ is the
period at which the pulses are pumped into the fiber. In the case of multi-
mode transmission the sum in eq. (6) is not integrable dik1  and thus does not
yield an exact solution. However if we consider the case of identical pulses
($\beta_{m}=\beta$) this sum becomes exact forming the bound states:

$A(t)=2\sqrt{\frac{\beta_{2}}{\gamma}}\frac{K(k)}{\tau_{A}}\,dn\left[2K(k)\frac{t}{\tau_{A}},k\right].$
(7)

According to eq (7) as well as fig.2, pulses periodically pumped into the
optical fiber interact instantaneously to form a periodic pattern of identical
pulse solitons under the anormalous regime of dispersion.

Figure 2: Autocorrelation trace of the periodic train of bound solitons,
obtained from analytical reconstruction by time division multiplexing of
identical pulses (eq. 7).

Because of the similarity between formula (7) and the elliptic-wave solution
to the nonlinear amplitude equation obtained in the previous section, we can
state unambiguously that the soliton-crystal state created by time division
multiplexing has the same structural feature as the elliptic-wave solution to
the propagation equation. Namely we also expect the time division multiplexed
pulses state to decay into a quasi-continuous wave signal when the dispersion
parameter $k$ decreases from its maximum $k=1$, while pulses in the soliton
crystal will be sharp and well separated as $k$ tends to one. It is relevant
at this step to stress that from the standpoint of exact mathematical
treatment the sum of an infinite number of secant hyperbolic pulses as given
by formula (6), to obtain the elliptic-wave structure formula (7), must be
supplemented with the existence condition:

$\tau_{A}=\frac{\pi K(k^{\prime})}{\lambda K(k)},$ (8)

where

$\lambda=\sqrt{\frac{-2\beta}{\beta_{2}}},\quad\hskip
2.84544ptk^{\prime}=\sqrt{1-k^{2}},$ (9)

that links the pumping period $\tau_{A}$ to the width at half height of
individual pulses $\lambda$ in the pulse multiplex. This existence condition
also determines the stability of the soliton-crystal state, indeed when $k$
tends to one $\tau_{A}$ tends to infinity which is consistent with the fact
that $dn$ tends to $sech$ when $k\rightarrow 1$. On the other hand, as $k$
decreases $\tau_{A}$ also decreases such that the separation between pulses is
not large enough for the pulses to preserve their full identity in the
soliton-crytal state, causing a decay of the soliton crystal into quasi-
continuous wave signals.
Experimental evidences on soliton-crystal structures abouch1 ; abouch2 ;
abouch3 ; amra1 ; amra2  have established that high pumping rates favor
solitons interaction and increase the intensity of the soliton crystal. When
the group velocity dispersion is sufficiently high, the input pulses overlap
and become a bound state that is modulated by the fiber nonlinearity into a
chain of periodic pulses liu . It is remarkable from formula (7) that the
amplitude of the time division multiplexed signal is inversely proportional to
the puming period $\tau_{A}$ which depends solely upon the pump. In particular
this formula suggests that to generate a regular well-shaped pulse train
practically it is desirable to pump high energetic pulses with the shortest
widths possible, and passively mode-locked lasers are suitable for such pump
because they can repetitively generate pulses of the order of femtosecond
menyuk1 ; chen . Titanium-doped sapphire lasers passively mode-locked by use
of slow saturable absorbers, for instance, are able to generate 10 to 30 fs
pulses. Furthermore self-starting passively mode-locked lasers that
continuously generate trains of pulses once turned on appear to be good
candidates for pumping periodically a large number of pulses in the process of
time division multiplexing menyuk ; chen , in this last context the separation
between pulses would theoretically correspond to the round trip time of the
laser cavity as observed in the harmonic mode-locking-induced periodic
bunching of pulse signals amra1 ; niang1 .

###  III.2 Numerical reconstruction

In experiments on multi-pulse processing it can be necessary to reconstruct
the observed periodic pattern of pulses analytically as well as numerically,
for a better understanding of their fundamental properties. Indeed, unlike the
analytical reconstruction scheme where we needed to sum an infinite number of
pulses to be able to find a single function comparable to the elliptic-wave
solution to the propagation equation, in the numerical reconstruction we can
consider a large but finite number of pulses and comparing the structure
obtained from this numerical sum over a finite number of pulses with the
elliptic-wave signal, is also relevant.
For the numerical reconstruction we use the ansatz:

$A(t)=\sum\limits_{n=-N/2}^{N/2}\sqrt{\frac{2\beta_{m}}{\gamma}}sech\left[\sqrt{\frac{-2\beta_{m}}{\beta_{2}}}(t-t_{0}-n\tau_{A})\right],$
(10)

where N is the number of pumped pulses. By considering a hundred of identical
pulses ($\beta_{m}=\beta$) a two-time-scale signal is produced as seen in fig.
3: a slow-time scale large pulse, and a fast-time scale soliton crystal.
Suppose each pulse in the crystal is $44ps$ of width and is separated from its
neighbour by $\Delta t=40fs$. Evaluating the number of pulses in the crystal
whose total length is $5ns$, taking into consideration the pulse width and
pulse seperation, yields 113 pulses in the soliton crystal.

(a)

(b)

Figure 3:  Numerically reconstructed signal at the pumping rate
$\tau_{A}=23.248\ 1/ns$: (a) Double time scale comb-like pulse, (b) Apex of
the large pulse: Soliton crystal of finite length.

The excess in the number of pulses comes from the non-linear gain induced by
the modulational instability of the interacting input pulses. The pulses are
not as sharp as those experimentally observed in rare earth doped fiber lasers
abouch1 ; abouch2 ; abouch3 , because in the later context the immediate
response of the medium to the laser gain dynamics modifies the propagation and
the interaction of pulses in the crystal. This is not the case for our optical
fiber model where we neglected the laser gain dynamics, so pulses are less
intense and the autocorrelation compels them to a broader profile. In fact the
formation of the analytically obtained time-division multiplexed signal,
confirmed by the numerical reconstruction, perfectly agrees with the theory
that pulse interaction caused by the overlap of pulses at the fiber input end
generates an unstable signal that is reformed into a one-dimensional lattice
of pulses. The pattern owes its periodic nature to the constant rate at which
identical pulses are pumped into the fiber. A close observation of the
periodic pattern behaviour, however, reveals a peculiar behaviour: pulse
interaction is inversely proportional to the pumping period and we expect it
to be minimal when the pumping period falls below the temporal width of each
pulse, such that a pulse entering the optical fiber collides with its
predecessor. However the numerical reconstruction illustrated by fig. 4 shows
a strong modulational instability when the pumping period is of the order of
the pumped pulse width.

(a)

(b)

Figure 4: Autocorrelation trace of the reconstructed signal: (a) Strong
modulation illustrated by the amplitude gap between the soliton crystal and
the large pulse, (b) Minimal modulation at the threshold frequency.

The strength of the modulation can be qualitatively evaluated by the amplitude
difference between the floor of the giant pulse and the lowest point of the
soliton crystal. From a theoritical stand point, pulses start to interact when
their tails overlap and we numerically determined that for our system this
interaction starts when the separation time between pulses is a dozen times
the width of each pulse as shown in fig. 4(b). This observation is manifest of
the wide bandwidth pulse solitons display in optical fibres.

##  IV  Linear stability analysis of the Soliton-crystal structure

A not less relevant issue related to the study of nonlinear structures in
optical media is their stability, once their existence is proven. This issue
has been investigated at length for single pulse and dark soliton signals,
particularly within the framework of linear stability analysis which provides
an interesting way of testing their robustness against small-amplitude noises
(see e.g. pelynov1 ). For single-soliton signals the linear stability analysis
of the wave equation (1) leads to an eignvalue problem for which the discrete
spectrum consists of three localized modes with non-zero spatial modulations.
From the standpoint of Physics these localized modes describe internal
oscillations in the structure of the soliton signals that propagate together
with the signals, and because of their non-zero energies they can be
associated with ”radiation-carrying” excitations in the propagating pulse
background.
In the case of time-division multiplexed pulse states the interaction between
pulses holds a key role in the stability of the structure as emphasized for
instance in ref. bul ). To this last point, for the soliton-crystal structure
obtained in the previous section the separation between individual pulses must
be equal and therefore it is necessary to minimize their mutual interactions
as well as the Gauss-Haus effect, two phenomena that tend to promote strong
correlations between centers of mass of the interacting pulses. To explain the
shape preservation (referred to as non-coalescence in ref. bul ) of individual
pulses as well as their propagation in a fixed time slot within the soliton-
crystal structure, Buryak et al. bul  suggested the possibility of oscillating
tails on pulses that would compensate for the energy cost in pulse
interactions with neighbour pulses through some effective potential.
In our investigations of stability of the soliton-crystal structure we shall
proceed through the linear stability analysis, in follozing this approach we
expect the emergence of additional internal modes, besides the boundstates
proper to each individual pulse, that would be consistent with new internal
degees of freedom due exclusively to the correlation of centers of mass of the
interacting pulses. Proceeding with, consider a small disturbance $a_{1}(t)$
moving bound to the pulse-lattice envelope (4) of the soliton crystal, which
we now denote $a_{0}(t)$. With this let

$a(t)=a_{0}(t)+a_{1}(t)$ (11)

be the soliton-crystal envelope dressed with the noise $a_{1}(t)$, replacing
$a(t)$ in the envelope equation derived from (1), to the linear order we find
the following eigenvalue equation:

$\left[\frac{\partial^{2}}{\partial\tau^{2}}-6k^{2}\,sn^{2}(\tau)\right]a_{1}(\tau)=E(k)a_{1}(\tau),\hskip
7.11317ptE(k)=2\frac{3\beta_{2}\lambda^{2}+\beta}{\beta_{2}\lambda^{2}}.$ (12)

This is Lamé’s eigenvalue problem of second order dik1 ; dik2  in which
$sn(\tau)$ is the Jacobi sine function of argument $\tau=\lambda\,t$. The
eigenvalue problem (12) possesses both discrete and continuous modes all
describing radiative modes of the soliton-crystal profile. Discrete modes are
of particular interest since they represent localized excitations in the
background of the soliton crystal.
For eq. (12) there are five distinct localized modes dikan  namely:

$a_{11}(\tau)=a_{11}(k)cn(\tau)dn(\tau),\hskip
7.11317pt\beta_{11}=(5-k^{2})|\beta_{2}|\lambda^{2}/2,$ (13)
$a_{12}(\tau)=a_{12}(k)cn(\tau)sn(\tau),\hskip
7.11317pt\beta_{12}=(2-k^{2})|\beta_{2}|\lambda^{2}/2,$ (14)
$a_{13}(\tau)=a_{13}(k)sn(\tau)dn(\tau),\hskip
7.11317pt\beta_{13}=(5-4k^{2})|\beta_{2}|\lambda^{2}/2,$ (15)

$\displaystyle a_{14,15}(\tau)$ $\displaystyle=$ $\displaystyle
a_{14}(k)\left[sn^{2}(\tau)-\frac{1+k^{2}}{3k^{2}}\mp\frac{\sqrt{1-k^{2}(1-k^{2})}}{3k^{2}}\right],$
$\displaystyle\beta_{14,15}$ $\displaystyle=$
$\displaystyle\left[2-k^{2}\pm\frac{\sqrt{1-k^{2}(1-k^{2})}}{2}\right]|\beta_{2}|\lambda^{2},$
(16)

where $a_{1j=1,2,3,4,5}(k)$ are their (constant) amplitudes. In fig. 5 we
sketched the five internal modes for $k=0.97$ (left column) and $k=1$ (right
column), respectively.

Figure 5: Wave profiles of the five boundstates $a_{1j}$ ($j=1,2,3,4,5$) given
by eqs. (13)-(16), for $k=0.98$ (left column) and $k=1$ (right column).

Graphs in the right column of fig. 5 are particularly instructive given that
when $k=1$, the separation between pulses becomes infinite and the soliton
crystal decays into a singl pulse. Thus, according to shape profiles in the
right column of fig. 5, the single-pulse signal possesses three distinct
internal modes including a single-pulse shaped, a symmetric two-pulse and an
asymmetric two-pulse boundstates all with positive spatial modulations
$\beta_{1j}$. Actually this feature of the internal-mode spectrum of the
single pulse suggests that the two additional modes appearing when $k$ is
smaller that one, as seen in the right column of fig. 5, are related to the
correlation of centers of mass of the interacting pulses.
As a final remark, we would like to stress that the linear stability analysis
provides a very efficient way of probing the soliton stability and
particularly its shape invariance under translation. To this last point, for
the single-pulse signal one of the three localized modes is precisely the
translation mode while the two others are related to localized excitations in
the internal structure of the pulse interacting with the noise. We can
determine which among the three boundstates is the translation mode by simply
slighly translating the pulse by $\tau_{0}$ from some temporal position $t$,
which yields:

$a_{1s}(\tau+\tau_{0})\propto
a_{1s}(\tau)+\tau_{0}\,a^{\prime}_{1s}(\tau)+0(\tau_{0}^{2}),$ (17)

where $a^{\prime}_{1s}(t)$ refers to time derivative of the field. For the
single pulse the second term in the right hand side of formula (17) is
proportional to $-sech(\tau)tanh(\tau)$, this is actually the function formula
(14) and the first graph in the right column of fig. 5. Similarly, an
infinitesimal translation of the soliton crystal as a hole by $\tau_{0}$ leads
to the same expression (17) with a translation mode proportional to
$sn(\tau)cn(\tau)$. However, if we translate individual pulses prior to
temporal multiplexing we find an additional interaction-bearing excitation
proportional to $sn(\tau)dn(\tau)$, coinciding with the second boundstate
(15). Formulas (14) and (15) show that the two modes tend to the same limit
(same wave profile and same eigen energy) as $k\rightarrow 1$, so we can
conclude that the correlation of centers of mass of individual pulses due to
their interactions, broadens their internal-mode spectrum while inducing mode
degeneracies.

##  V Conclusion

In summary we have established that in the regime of anomalous dispersion
soliton crystal patterns can emerge in a nonlinear optical medium through a
time division multiplexing process, and we proceeded with its reconstruction
by considering an experimental setup consisting of periodically pumping
identical pulses at high repetition rates. The Jacobi elliptic delta function
was found to provide a good representation of the temporal envelope signal in
the soliton-crystal state. The numerical reconstruction of the signal by time
divsion multiplexing of a large but finite number of pulses, has led to a
comb-like signal similar to those observed in passively mode-locked fiber
lasers and confirmed that modulational instability of pulses fuels the
formation of soliton crystals. At some threshold pumping limit a pulse train
is formed and evolves into a more stable double-time scale signal as the
pumping frequency increases. The self-preserving features of the pulse-soliton
lattice offer a suitable means of long distance signal convoy in fiber
waveguides, as widely illustrated in several past studies.
To end it would be instructive stressing that the issue of theoretical
reconstruction of bound-soliton states observed experimentally in fiber laser
media has already been addressed in few past works. Among these past studies
the account given in ref. tang  on the formation of pulses and their
subsequent evolution into muti-pulse soliton states, is consistent with our
theoretical results. Indeed the authors observed that while the formation of
fundamental pulses is governed mainly by the balance of nonlinearity and the
anomalous group velocity dispersion, higher-energy pulses will split into
identical lower-energy multisolitons with exactly the same physical
properties. They found that the separation of neighboring pulses in the bound-
soliton states was variational in the temporal domain, which suggests a time
division multiplexing of identical pulses to form a periodic multi-soliton
structure with temporal and spectral characteristics having a sizable
dependence on the laser cavity roundtrip.

###### Acknowledgements.

Jubgang Fandio thanks Atoneche Fred for enriching discussions on fiber lasers
technology. A. M. Dikandé wishes to acknowledge supports of the Alexander von
Humboldt foundationm and The World Academy of Science (TWAS).

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