On Time-Bandwidth Product of Multi-Soliton Pulses
Alexander Span∗, Vahid Aref† , Henning Bülow† , and Stephan ten Brink∗

arXiv:1705.09468v1 [] 26 May 2017

∗

Institute of Telecommunications, University of Stuttgart, Stuttgart, Germany
†
Nokia Bell Labs, Stuttgart, Germany

Abstract—Multi-soliton pulses are potential candidates for
fiber optical transmission where the information is modulated
and recovered in the so-called nonlinear Fourier domain. While
this is an elegant technique to account for the channel nonlinearity, the obtained spectral efficiency, so far, is not competitive
with the classic Nyquist-based schemes. In this paper, we study
the evolution of the time-bandwidth product of multi-solitons
as they propagate along the optical fiber. For second and third
order soliton pulses, we numerically optimize the pulse shapes
to achieve the smallest time-bandwidth product when the phase
of the spectral amplitudes is used for modulation. Moreover, we
analytically estimate the pulse-duration and bandwidth of multisolitons in some practically important cases. Those estimations
enable us to approximate the time-bandwidth product for higher
order solitons.

I. I NTRODUCTION
Advances made over the past decade in coherent optical
technology have significantly improved transmission capacities
to a point where Kerr nonlinearity once again becomes the
limiting factor. The equalization of nonlinear effects is usually
very complex and has a limited gain due to the mixing of
signal and noise on the channel. The optical channel is usually modeled by the Nonlinear Schrödinger Equation (NLSE)
which describes the interplay between Kerr nonlinearity and
chromatic dispersion along the fiber.
The Nonlinear Fourier Transform (NFT) is a potential way
of generating pulses matched to a channel governed by the
NLSE. It maps a pulse to the nonlinear Fourier spectrum with
some beneficial properties. This elegant technique, known also
as inverse scattering method [1], has found applications in
fiber optics when the on-off keying of first order solitons was
developed in the 1970s [2]. Following [3], [4], it has regained
attention as coherent technology allows to exploit all degrees
of freedom offered by the nonlinear spectrum.
Multi-soliton pulses are specific solutions of the NLSE.
Using the NFT, an N −th order soliton, denoted here by
N −soliton, is mapped to a set of N distinct nonlinear frequencies, called eigenvalues, and the corresponding spectral
amplitudes. The key advantage of this representation is that
the complex pulse evolution along the fiber can be expressed
in terms of spectral amplitudes which evolve linearly in the
nonlinear spectrum. Moreover, the transformation is independent of the other spectral amplitudes and eigenvalues. These
properties motivate to modulate data using spectral amplitudes.
On-off keying of 1-soliton pulses, also called fundamental
solitons, has been intensively studied two decades ago for
different optical applications (see [2] and reference therein). To
increase spectral efficiency, it has been proposed to modulate
multi-solitons [3]. One possibility is the independent on-off

keying of N predefined eigenvalues. The concept has been
experimentally shown up to using 10 eigenvalues in [5], [6].
The other possibility is to modulate the spectral amplitudes of
N eigenvalues. The QPSK modulation of spectral amplitudes
has been verified experimentally up to 7 eigenvalues in [7],
[8], [9]. All of these works have a small spectral efficiency.
Characterizing the spectral efficiency of multi-soliton pulses
is still an open problem. First, the statistics of noisy received
pulses in the nonlinear spectrum have not yet been fully
understood, even though there are insightful studies for some
special cases and under some assumptions [10], [11], [12].
Second, the bandwidth and the pulse-duration change as a
multi-soliton propagates along a fiber or as spectral amplitudes
are modulated. The nonlinear evolution makes it hard to
estimate the time-bandwidth product of a multi-soliton.
In this paper, we study the evolution of pulse-duration
and bandwidth of multi-soliton pulses along an optical fiber
link. We numerically optimize the time-bandwidth product
of N −soliton pulses for N = 2 and 3. The results provide
some guidelines for N > 3. We focus on scenarios where the
phases of N spectral amplitudes are modulated independently.
However, our results can also be applied to on-off keying
modulation schemes. We assume that the link is long enough
so that the pulse-duration and bandwidth can reach their respective maximum. We also neglect inter-symbol interference.
Our results show that the optimization of [13] is suboptimal
when the evolution along the fiber is taken into account.
We further introduce a class of N −solitons which are
provably symmetric. A subset of these pulses are already
used in [7], [8], [13]. We derive an analytic approximation
of their pulse-duration. Numerical observations exhibit that
the approximation is tight and can serve as a lower-bound for
other N −solitons. To the best of our knowledge, this is the
first result on the pulse-duration of multi-solitons. We also
approximate the time-bandwidth product by lower-bounding
the maximal bandwidth.
II. P RELIMINARIES ON M ULTI -S OLITON P ULSES
In this section, we briefly explain the nonlinear Fourier
transform (NFT), the characterization of multi-soliton pulses
in the corresponding nonlinear spectrum and how they can be
generated via the inverse NFT.
A. Nonlinear Fourier Transform
The pulse propagation along an ideally lossless and noiseless fiber is characterized using the standard Nonlinear
Schrödinger Equation (NLSE)
∂2
∂
q(t, z) + j 2 q(t, z) + 2j|q(t, z)|2 q(t, z) = 0.
∂z
∂t

(1)

The physical pulse Q(τ, ℓ) at location ℓ along the fiber is then
described by


p
|β2 |
|β2 |
τ
, ℓ 2 with P0 · T02 =
,
Q (τ, ℓ) = P0 q
T0 2T0
γ

where β2 < 0 is the chromatic dispersion and γ is the Kerr
nonlinearity of the fiber, and T0 determines the symbol rate.
The closed-form solutions of the NLSE (1) can be described
in a nonlinear spectrum defined by the following so-called
Zakharov-Shabat system [1]

 


∂ ϑ1 (t; z)
−jλ
q (t, z)
ϑ1 (t; z)
=
, (2)
−q ∗ (t, z)
jλ
ϑ2 (t; z)
∂t ϑ2 (t; z)
with the boundary condition


 
ϑ1 (t; z)
1
→
exp (−jλt) for t → −∞
ϑ2 (t; z)
0

each recursion. The main advantage of this algorithm is that
it is exact with a low computational complexity and it can be
used to derive some properties of multi-soliton pulses.
Algorithm 1: INFT from Darboux Transform [16]
Input : Discrete Spectrum {λk , Qd (λk )}; k = 1, . . . , N
Output: N −soliton waveform q(t)
begin
for k ← 1 to Ndo
QN
(0)
d (λk )
ρk (t) ←− Q
m=1,m6=k
λk −λ∗
k

λk −λm
λk −λ∗
m

q (0) ←− 0;
for k ← 1 to N do
(k−1)
ρ(t) ←− ρk
(t);

e2jλk t ;

∗

under the assumption that q(t; z) → 0 decays sufficiently
fast as |t| → ∞ (faster than any polynomial). The nonlinear
Fourier coefficients (Jost coefficients) are defined as

ρ (t)
q (k) (t) ←− q (k−1) (t) + 2j(λk − λ∗k ) 1+|ρ(t)|
2;
(4)
for m ← k + 1 to N do
(k)
ρm (t) ←−
λ −λ∗

(k−1)
k
k (ρ(k−1) (t)−ρ(t))
(λm −λk )ρm
(t)+ 1+|ρ(t)|
2
m


;
λk −λ∗
(k−1)
∗
k
∗
(t)
λm −λk − 1+|ρ(t)|2 1+ρ (t)ρm

a (λ; z) = lim ϑ1 (t; z) exp (jλt)
t→∞

b (λ; z) = lim ϑ2 (t; z) exp (−jλt) .

(5)

t→∞

The set Ω denotes the set of simple roots of a(λ; z) with
positive imaginary part, which are called eigenvalues as they
do not change in terms of z, i.e. λk (z) = λk . The nonlinear
spectrum is usually described by the following two parts:
(i) Continuous Part: the spectral amplitude Qc (λ; z) =
b(λ; z)/a(λ; z) for real frequencies λ ∈ R.
(ii) Discrete Part: {λk , Qd (λk ; z)} where λk ∈ Ω, i.e.
a(λk ; z) = 0, and Qd (λk ; z) = b(λk ; z)/ ∂a(λ;z)
∂λ |λ=λk .
An N −soliton pulse is described by the discrete part only and
the continuous part is equal to zero (for any z). The discrete
part contains N pairs of eigenvalue and the corresponding
spectral amplitude, i.e. {λk , Qd (λk ; z)}, 1 ≤ k ≤ N .
An important property of the nonlinear spectrum is its
simple linear evolution given by [3]
Qd (λk ; z) = Qd (λk ) exp(−4jλ2k z),



(3)

where we define Qd (λk ) = Qd (λk ; z = 0). The transformation is linear and depends only on its own eigenvalue λk . This
property motivates for modulation of data over independently
evolving spectral amplitudes.
Note that there are several methods to compute the nonlinear
spectrum by numerically solving the Zakharov-Shabat system.
Some of these methods are summarized in [3],[14].
B. Inverse NFT
The Inverse NFT (INFT) maps the given nonlinear spectrum
to the corresponding pulse in time-domain. For the special
case of the spectrum without the continuous part, the Darboux
Transformation can be applied to generate the corresponding
multi-soliton pulse [15]. Algorithm 1 shows the pseudo-code
of the inverse transform, as described in [16]. It generates an
N −soliton q (t) recursively by adding a pair {λk , Qd (λk )} in

(λ∗ denotes the complex conjugate of λ)

C. Definition of Pulse Duration and Bandwidth
In this paper, we consider an N −soliton with the eigenval+
ues on the imaginary axis, i.e. {λk = jσk }N
k=1 and σk ∈ R .
Without loss of generality, we assume that σk < σk+1 . As
such an N −soliton propagates along the fiber, the pulse does
not disperse and the pulse shape can be repeated periodically.
An N −soliton pulse has unbounded support and exponentially decreasing tails in time and (linear) frequency domain.
As this pulse is transformed according to the NLSE, e.g. propagation along the ideal optical fiber, its shape can drastically
change as all Qd (λk ; z) are evolved in z. Despite of nontrivial
pulse variation and various peak powers, the
PNenergy of the
pulse remains fixed and equal to Etotal = 4 k=1 Im{λk }.
As a result, the pulse-duration and the bandwidth of a
multi-soliton pulse are well-defined if they are characterized
in terms of energy: the pulse duration Tw (and bandwidth
Bw , respectively) is defined as the smallest interval (frequency
band) containing Etrunc = (1 − ε)Etotal of the soliton energy.
Note that truncation causes small perturbations of eigenvalues.
In practical applications, the perturbations become even larger
due to inter-symbol-interference (ISI) when a train of truncated
soliton pulses is used for fiber optical communication. Thus,
there is a trade-off: ε must be kept small such that the
truncation causes only small perturbations, but large enough
to have a relatively small time-bandwidth product.
Note that truncating a signal in time-domain may slightly
change its linear Fourier spectrum in practice. For simplicity,
we however computed Tw and Bw with respect to the original
pulse as the difference is negligible for ε ≪ 1.

III. S YMMETRIC M ULTI -S OLITON P ULSES
In this section, we address the special family of multi-soliton
pulses which are symmetric in time domain. An application
of such solitons for optical fiber transmission is studied in [7]
where the symmetric 2-solitons are used for data modulation.
Theorem 1. Let Ω = {jσ1 , jσ2 , . . . , jσN } be the set of
eigenvalues on the imaginary axis where σk ∈ R+ , for
1 ≤ k ≤ N . The corresponding N −soliton q(t) is a symmetric
pulse, i.e. q(t) = q(−t), and keeps this property during the
propagation in z, if and only if the spectral amplitudes are
chosen as
N
Y
σk + σm
.
(6)
|Qd,sym (jσk )| = 2σk
σk − σm
m=1;m6=k

Sketch of Proof. The proof is based
on Algorithm 1 with
ρ∗ (t)
the following steps: (i) g(t) = 1+|ρ(t)|
2 is symmetric, if
ρ∗ (−t)ρ(t) = 1.

(7)

(ii) The update rule (5) preserves the property (7): if ρ(t) and
(k−1)
(k)
ρm (t) satisfy (7), then ρm (t) will satisfy (7) as well.
(0)
(iii) Because of (6), ρk (t) satisfies (7) for all k.
(k)
(iv) Using induction, ρm satisfies (7) for all m and k.
(v) According to (4) and step (i), q(t) is symmetric.
It is already mentioned in [17] that (6) leads to a symmetric
multi-soliton in amplitude. Theorem 1 implies that (6) is not
only sufficient but also necessary to have q(t) = q(−t).
As it is shown in the next section, we numerically observe
that these symmetric pulses have the smallest pulse duration1
among all solitons with the same set of eigenvalues Ω (but
different |Qd (λk )|). Assuming σ1 = mink {σk }, this minimum
pulse-duration can be well approximated by


N
X
1
σm + σ1
Tsym (ε) ≈
2
ln
2σ1
σm − σ1
m=2
!!
PN
 
2
m=1 σm
− ln
+ ln
, (8)
ε
σ1
where ε is defined earlier as the energy threshold. The approximation
becomes tight as ε → 0 and is only valid if
PN
ε ≪ σ1 / m=1 σm . Verification of (8) follows readily by
describing an N −soliton by the sum of N terms according to
(4), and showing that in the limit |t| → ∞, the dominant term
behaves as sech(2σ1 (|t| − t0 )) for some t0 and all other terms
decay exponentially faster.
IV. T IME -BANDWIDTH P RODUCT
Consider the transmission of an N −soliton with eigenvalues
{jσk }N
k=1 over an ideal fiber of length zL . Each spectral amplitude Qd (jσk ; z) = |Qd (jσk ; z)| exp(jφk (z)) is transformed
along the fiber according to (3). Equivalently,
|Qd (jσk ; z)| = |Qd (jσk ; z = 0)|
φk (z) = φk (0) + 4σk2 z
1 It

is correct when ε is small enough.

for z ≤ zL . It means that φk (z) changes with a distinct speed
proportional to σk2 . Different phase combinations correspond
to different soliton pulse shapes with generally different pulseduration and bandwidth. It implies that Tw and Bw of a pulse
are changing along the transmission. Furthermore, if the φk (0)
are independently modulated for each eigenvalue with a constellation of size M , e.g. M −PSK, this results in M N initial
phase combinations (N log2 (M ) bits per soliton) associated
with different initial pulse shapes. Such transmission scenarios
are demonstrated experimentally for M = 4, N = 2 [7] and
N = 7 [8]. To avoid a considerable ISI between neighboring
pulses in a train of N -solitons for transmission in time or
frequency, we should consider Tw and Bw larger than their
respective maximum along the link.
For a given set of eigenvalues and fixed |Qd (jσk ; z = 0)|,
the maxima depend on M N initial phase combinations and
the fiber length zL . To avoid these constraints, we maximize
Tw and Bw over all possible phase combinations:
Tmax =

max

φk ,1≤k≤N

Tw and Bmax =

max

φk ,1≤k≤N

Bw .

These quantities occur in the worst case but can be reached
in their vicinity when N is small, e.g. 2 or 3, or when M is
very large, or the transmission length zL is large enough.
In the rest of this section, we address the following fundamental questions:
(i) How do Tmax and Bmax change in terms of {jσk }N
k=1
and {|Qd (jσk )|}N
k=1 ?
(ii) What is the smallest time-bandwidth product for a given
N , i.e.
(Tmax Bmax )⋆ =

min

min

σk ,1≤k≤N |Qd (jσk )|,1≤k≤N

Tmax Bmax

and which is the optimal choice for {jσk⋆ }N
k=1 and
.
{|Q⋆d (jσk⋆ )|}N
k=1
The following properties preserving the time-bandwidth
product decrease the number of parameters to optimize:
(i) If q(t) has eigenvalues {jσk }N
k=1 , then 1/σ1 · q(t/σ1 ) will
with
the same time-bandwidth
have eigenvalues {j σσk1 }N
k=1
product. It implies that Tmax Bmax only depends on the N − 1
eigenvalue ratios σk /σ1 .
N
(ii) If {φk }N
k=1 corresponds to q(t), then {φk − φ1 }k=1
corresponds to q(t) exp(jφ1 ). Thus, we assume φ1 = 0.
(iii) Instead of directly optimizing {|Qd (jσk )|}N
k=1 , it is
equivalent to optimize ηk > 0 defined by
|Qd (jσk )| = ηk |Qd,sym (jσk )|.
Using {ηk }N
k=1 has two advantages. The first one is the
generalization of Theorem 1. If {ηk }N
k=1 corresponds to q(t),
then {1/ηk }N
corresponds
to
q(−t).
The proof is similar to
k=1
the one of Theorem 1. Moreover, {e−2σk t0 ηk }N
k=1 corresponds
to q(t+t0 ). Thus, it suffices to assume η1 = 1 and η2 ∈ (0, 1].
A. Optimization of Spectral Amplitudes
Consider a given set of eigenvalues Ω = {jσk }N
k=1 . We
want to optimize {ηk }N
k=2 to minimize Tmax Bmax . Recall that
N
{|Qd (jσk )|}N
k=1 , and thus {ηk }k=1 do not change along z.

with σN = maxk {σk }. The bound becomes tight when an
N −soliton is the linear superposition of N separate 1-solitons.
We performed such a numerical optimization for N = 2
and N = 3 and for different { σσk1 }N
k=2 . For each ε, we found
the optimal {ηk⋆ }N
k=2 with the smallest Tmax (ε)Bmax (ε).
B. Optimization of Eigenvalues
In general, an N −soliton has a larger Tmax Bmax than
a 1−soliton but it has also N times, e.g. Qd (jσk ), more
dimensions for encoding data. To have a fair comparison,
we use a notion of “time-bandwidth product per eigenvalue”
defined as
σk
1
σk
T ·B N ({ }N
) = Tmax Bmax ({ }N
, {ηk⋆ }N
k=2 )
σ1 k=2
N
σ1 k=2
where Tmax Bmax is already optimized in terms of {ηk }, seperately for each eigenvalue combination. This is an important
parameter as the spectral efficiency will be O(1/T ·B N ). For
a 1-soliton with “sech” shape in time and frequency domain,
we have
T ·B 1 = Tw (ε)Bw (ε) = π −2 ln2 (2/ε),

Tmax

30

20

10

(a)
-5

-4

-3

log (η2 )

-2

-1

0

-2

-1

0

8

Bmax , Bmin

We present the optimization method for N = 2. In this
case, there are two parameters to optimize: φ2 and η2 ∈ (0, 1].
Consider a given energy threshold ε. For each chosen η2 , we
find Tmax (ε) and Bmax (ε) by exhaustive search. The phase
φ2 ∈ [0, 2π) is first quantized uniformly by 64 phases. At
each phase, a 2-soliton is generated using Algorithm 1 and
then Tw (ε) and Bw (ε) are computed. To estimate Tmax (ε),
another round of search is performed with a finer resolution
around the quantized phase with the largest Tw (ε). Similarly,
Bmax (ε) is estimated.
Fig. 1 illustrates Tmax (ε) and Bmax (ε) in terms of log(η2 )
for different energy thresholds ε when Ω = { 21 j, 1j}. We also
depict Bmin (ε), the minimum bandwidth of 2-soliton pulses
with a given η2 and various φ2 . Fig. 1 indicates the following
features that we observed for any pairs of {jσ1 , jσ2 }.
We can see that for any ε, the smallest Tmax is attained at
η2 = 1 (log(η2 ) = 0) which corresponds to the symmetric 2soliton defined in Sec. III. We also observe that Bmax reaches
the largest value at η2 = 1 while Bmin reaches its minimum.
As log(η2 ) decreases, Tmax increases gradually up to some
point and then it linearly increases in | log(η2 )|. The behaviour
of Bmax is the opposite. It decreases very fast in | log(η2 )| up
to some η2 and then converges slowly to the bandwidth defined
by the 1-soliton spectrum with λ = jσ2 . In fact, we have two
separate 1-solitons without any interaction when η2 = 0. As
η2 increases to 1, the distance between these two 1-solitons
decreases, resulting in more nonlinear interaction but smaller
Tmax . The largest Bmax − Bmin at η2 = 1 indicates the largest
amount of interaction.
The above features seem general for N −solitons. In particular, Tmax becomes minimum if the N −soliton is symmetric.
Moreover, Bmax can be lower-bounded by
!!
PN
 
2
2σN
k=1 σk
− ln
ln
Bsep (ε) = 2
π
ε
σN

Bmax
Bmin

6
4
2

(b)

0
-5

-4
ε = 10−3

-3

log (η2 )

ε = 10−4

ε = 10−6

ε = 10−10

Fig. 1. (a) Pulse duration Tmax and (b) bandwidth Bmax/min for 2-soliton
pulse (λ1 = 0.5j, λ2 = 1j) when maximized (minimized) over all phase
combinations of spectral amplitudes

where ε is the energy threshold defined in Section II-C.
For N = 2 and N = 3, we numerically optimized Tmax (ε)Bmax (ε) for different values of { σσk1 }N
k=2 and
{ηk }N
.
Fig.
2-(a)
shows
the
numerical
optimization
of
k=2
T ·B 2 in terms of σ2 /σ1 for different choices of ε where
the best {ηk⋆ }N
k=2 were chosen for each eigenvalue ratio. We
normalized T ·B 2 by T ·B 1 to see how much the “timebandwidth product per eigenvalue” can be decreased. Fig. 2(b) shows a similar numerical optimization for N = 3 and
ε = 10−4 . We have the following observations:
(i) T ·B N is sensitive to the choice of eigenvalues. For
instance, equidistant eigenvalues, i.e. σk = kσ1 , are a bad
choice in terms of spectral efficiency.
(ii) The ratio T ·B N /T ·B 1 gets smaller as ε vanishes. The
intuitive reason is that as ε → 0, we get Tmax ≈ 2σ11 ln( 2ε )
(see (8)) which is the pulse-duration of the 1-soliton.
(iii) For a practical value of ε ∼ 10−4 −10−3 , T ·B N decreases
very slowly in N . Moreover, the optimal σk⋆ are close. This
can make the detection challenging in presence of noise. For
ε = 10−4 ,
T ·B 2 /T ·B 1 = 0.87 for σ2⋆ /σ1⋆ = 1.11
T ·B 3 /T ·B 1 = 0.83 for σ2⋆ /σ1⋆ = 1.28, σ3⋆ /σ1⋆ = 1.35
(iv) Choosing the above optimal {σk⋆ /σ1⋆ }, and the optimal
{|Q⋆d (σk⋆ )}, the resulting solitons for N = 2, 3 are shown
in Fig. 3 for different phase combinations and two energy
thresholds ε. This figure gives some guidelines for a larger N :
the optimal N −soliton has eigenvalues close to each other and
significantly seperated pulse centers, why the optimum pulse
looks similar to a train of 1-solitons with eigenvalues close
to each other. The pulse centers should be close to minimize

T ·B 2 /T ·B 1

1.2
ε = 10−3

1

ε = 10−6

ε = 10−4
0.8

simulation
approximation (9)

ε = 10−10
0.6
1

1.2

1.4

1.6

1.8

2

σ2 /σ1

(a)
2

1.2

T ·B 3 /T ·B 1
1.1

σ2 /σ1

1.7

1
1.3

0.9

1

0.8
1.2

1.4

1.6

1.8

2

σ3 /σ1

(b)

Fig. 2. Gain of time-bandwidth product per eigenvalue of (a) second and (b)
third order solitons with eigenvalues jσk in relation to first order pulses

Tmax but not too close to avoid a large interaction which comes
along with a growth of Bmax .
1.5

|q (t)|

ε = 10−4

1

ε = 10−10

0.5
0
−10

−5

0

5

10

5

10

t

|q (t)|

1.5
1
0.5
0
−10

−5

0

t
Fig. 3. Time domain signal of optimum second and third order soliton pulse
for different phase combinations of the spectral amplitudes (same color)

For ε ≪ 1, an estimate on T ·B N at optimal {ηk∗ } can be
given by (9), where Tmax and Bmax are estimated by Tsym (ε)
and Bsep (ε), respectively (see Fig. 1).
Tsym (ε)Bsep (ε)
,
(9)
N
For the second order case, these approximations for various
ε are plotted in Fig. 2-(a) by dashed lines. We see that the
approximation becomes better for small ε. This approximation
can be used to predict T ·B N for a large N .
T ·B N ≈

V. C ONCLUSION
We studied the evolution of the pulse-duration and the
bandwidth of N −soliton pulses along the optical fiber. We
focused on solitons with eigenvalues located on the imaginary

axis. The class of symmetric soliton pulses was introduced
and an analytical approximation of their pulse-duration was
derived.
The phase of the spectral amplitudes was assumed to be
used for modulation while their magnitudes were kept fixed.
We numerically optimized the location of eigenvalues and
the magnitudes of spectral amplitudes for 2− and 3−solitons
in order to minimize the time-bandwidth product. It can
be observed that the time-bandwidth product per eigenvalue
improves in the soliton order N , but very slowly. Another
observation is, that the optimal N −soliton pulse looks similar
to a train of first-order pulses.
There are some remarks about our optimization. As an
N −soliton propagates, the phases of the spectral amplitudes
change with different speeds. We assumed that all possible
combinations of phases occur during transmission. This is the
worst case scenario which is likely to happen for N = 2 and
N = 3 but becomes less probable for large N . Moreover, the
same magnitudes of spectral amplitudes are used for any phase
combination while they can be tuned according to the phases.
Without these assumptions, the time-bandwidth product will
decrease. However, it becomes harder to estimate as there are
many more parameters to optimize.
R EFERENCES
[1] A. Shabat and V. Zakharov, “Exact theory of two-dimensional selffocusing and one-dimensional self-modulation of waves in nonlinear
media,” Soviet physics JETP, vol. 34, no. 1, p. 62, 1972.
[2] L. F. Mollenauer and J. P. Gordon, Solitons in optical fibers: fundamentals and applications. Academic Press, 2006.
[3] M. Yousefi, F. Kschischang, “Information transmission using the nonlinaer fourier transform: I-III,” IEEE Trans. Inf. Theory, vol. 60, 2014.
[4] J. E. Prilepsky, S. A. Derevyanko, K. J. Blow, I. Gabitov, and S. K. Turitsyn, “Nonlinear inverse synthesis and eigenvalue division multiplexing
in optical fiber channels,” Phys. review lett., vol. 113, no. 1, 2014.
[5] Z. Dong, S. Hari, et al., “Nonlinear frequency division multiplexed
transmissions based on nft,” IEEE Photon. Technol. Lett., vol. 27, 2015.
[6] V. Aref, Z. Dong, and H. Buelow, “Design aspects of multi-soliton pulses
for optical fiber transmission,” in IEEE Photon. Conf. (IPC), 2016.
[7] V. Aref, H. Buelow, K. Schuh, and W. Idler, “Experimental demonstration of nonlinear frequency division multiplexed transmission,” in 41st
Europ. Conf. on Optical Comm. (ECOC), 2015.
[8] H. Buelow, V. Aref, and W. Idler, “Transmission of waveforms determined by 7 eigenvalues with psk-modulated spectral amplitudes,” in
42nd Europ. Conf. on Optical Comm. (ECOC), 2016.
[9] A. Geisler and C. Schaeffer, “Experimental nonlinear frequency division
multiplexed transmission using eigenvalues with symmetric real part,”
in 42nd Europ. Conf. on Optical Comm. (ECOC), 2016.
[10] S. A. Derevyanko, S. K. Turitsyn, and D. A. Yakushev, “Fokker-planck
equation approach to the description of soliton statistics in optical fiber
transmission systems,” JOSA B, vol. 22, no. 4, 2005.
[11] Q. Zhang and T. H. Chan, “A spectral domain noise model for optical
fibre channels,” in IEEE Int. Symp. on Inf. Theory (ISIT), 2015, 2015.
[12] S. Wahls, “Second order statistics of the scattering vector defining the d-t nonlinear fourier transform,” in Int. ITG Conf. on Systems, Comm. and Coding (ITG SCC), 2017.
[13] S. Hari, M. I. Yousefi, and F. R. Kschischang, “Multieigenvalue communication,” J. Lightw. Technol., vol. 34, no. 13, 2016.
[14] S. Wahls and H. V. Poor, “Fast numerical nonlinear fourier transforms,”
IEEE Trans. Inf. Theory, vol. 61, no. 12, 2015.
[15] V. B. Matveev and V. Matveev, Darboux transformations and solitons.
Springer-Verlag, 1991.
[16] V. Aref, “Control and detection of discrete spectral amplitudes in
nonlinear fourier spectrum,” arXiv preprint arXiv:1605.06328, 2016.
[17] H. A. Haus and M. N. Islam, “Theory of the soliton laser,”
IEEE J. Quantum Electron., vol. QE-21, no. 8, 1985.

