Journal ArticleDOI

# Parametric resonance and radiative decay of dispersion-managed solitons

01 Jan 2004-Siam Journal on Applied Mathematics (Society for Industrial and Applied Mathematics)-Vol. 64, Iss: 4, pp 1360-1382
TL;DR: The nonlinear Fermi golden rule for radiative decay of dispersion-managed solitons is derived from the solvability condition for the perturbation series expansions.
Abstract: We study propagation of dispersion-managed solitons in optical fibers which are modeled by the nonlinear Schrodinger equation with a periodic dispersion coefficient. When the dispersion variations are weak compared to the average dispersion, we develop perturbation series expansions and construct asymptotic solutions at the first and second orders of approximation. Due to a parametric resonance between the dispersion map and the dispersion-managed soliton, the soliton generates continuous-wave radiation leading to its radiative decay. The nonlinear Fermi golden rule for radiative decay of dispersion-managed solitons is derived from the solvability condition for the perturbation series expansions. Analytical results are compared to direct numerical simulations, and good agreement is obtained.

### 1. Introduction.

• The authors analysis starts with the standard NLS equation (1.3) for m = 0, such that the right-hand side of (1.3) is treated as a small perturbation.
• Methods of their analysis are similar to the soliton perturbation theory in [13] , but their calculations are more systematic.
• Rigorous analysis of decay rates in the linear Schrödinger equation was recently considered in [17, 18] , where the bound states were supported by a time-dependent periodic potential in [17] and by a time-independent potential in [18] .
• Section 4 describes a comparison between the analytical and numerical results.
• Appendices A and B describe technical details of the first-order solution in the perturbation series expansions.

### 4. Numerical simulations of DM solitons.

• When the initial value of µ is large, the asymptotic analysis predicts that the soliton decays exponentially according to the bounds in (3.18) .
• Thus, the accuracy of the analytical prediction needs to be examined.
• Further evolution of the DM soliton shows that the DM soliton character is lost after the fifth critical resonance at the average amplitude about 9.38.
• Figures 6(b) and (c ) indicate that when µ(0) is close to or above the lowest critical resonance value µ 1 = 2π, the pulse deviates further from the DM soliton than in the case of m = 0.1, and the pulse amplitude oscillates with a period further away from the unit period of the dispersion map.

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PARAMETRIC RESONANCE AND RADIATIVE DECAY OF
DISPERSION-MANAGED SOLITONS
DMITRY E. PELINOVSKY
AND JIANKE YANG
SIAM J. A
PPL. MAT H .
c
2004 Society for Industrial and Applied Mathematics
Vol. 64, No. 4, pp. 1360–1382
Abstract. We study propagation of dispersion-managed solitons in optical ﬁbers which are
modeled by the nonlinear Schr¨odinger equation with a periodic dispersion coeﬃcient. When the
dispersion variations are weak compared to the average dispersion, we develop perturbation series
expansions and construct asymptotic solutions at the ﬁrst and second orders of approximation. Due
to a parametric resonance between the dispersion map and the dispersion-managed soliton, the soliton
for radiative decay of dispersion-managed solitons is derived from the solvability condition for the
perturbation series expansions. Analytical results are compared to direct numerical simulations, and
good agreement is obtained.
Key words. dispersion management, optical solitons, perturbation series, parametric resonance,
AMS subject classiﬁcations. 35Q55, 78M30, 78M35
DOI. 10.1137/S0036139903422358
1. Introduction. This paper addresses the dispersion-periodic nonlinear Schr¨o-
dinger (NLS) equation,
i
∂u
∂z
+
m
2
D
(z)
2
u
∂t
2
+
1
2
D
0
2
u
∂t
2
+ |u|
2
u =0,(1.1)
which models optical pulse propagation in dispersion-managed communication sys-
tems. Here u C is the wave envelope of the electromagnetic ﬁeld, z ( 0) is the
distance along the optical ﬁber, t R is the retarded time of the optical pulse, D
0
is the average dispersion, D
(z)isan-periodic mean-zero dispersion map, and m is
the strength of the map variations. Lump ampliﬁcation and losses are not included
in the model (1.1) for the sake of simplicity.
Special solutions of the dispersion-periodic NLS equation (1.1) are called disper-
sion-managed (DM) solitons. They have been the subject of growing interest in recent
literature [1, 2, 3]. DM solitons are periodic solutions of (1.1) in the form
u(z,t)=Φ(z, t) e
iµz
,(1.2)
where Φ(z + , t)=Φ(z, t) and µ R. Existence of periodic solutions of (1.1) is
studied with the normal-form transformations in the limit 0 [4]. The normal-
form transformations average the fast periodic variations of
1
D
(z) and reduce the
dispersion-periodic NLS equation (1.1) to an integral NLS equation [5, 6]. Bound
states of the integral NLS equation exist in the case of D
0
> 0 [7] and in the case
Received by the editors February 6, 2003; accepted for publication (in revised form) November
1, 2003; published electronically May 20, 2004.
http://www.siam.org/journals/siap/64-4/42235.html
Department of Mathematics, McMaster University, 1280 Main Street West, Hamilton, Ontario,
Canada, L8S 4K1 (dmpeli@math.mcmaster.ca). The work of this author was supported by NSERC
grant 5-36694.
Department of Mathematics, University of Vermont, Burlington, VT 05401 (jyang@emba.uvm.
edu). The work of this author was supported by NSF grant DMS-9971712 and by a NASA EPSCoR
minigrant.
1360

RESONANCE AND DECAY OF DISPERSION-MANAGED SOLITONS 1361
D
0
= 0 [8]. Numerical results indicate nonexistence of bound states of the integral
NLS equation in the case D
0
< 0 [9].
In what follows we consider the case D
0
> 0 only. Early papers by Nijhof et
al. [11] reported numerically the existence of “exactly” periodic bound states in the
dispersion-periodic NLS equation (1.1), which do not radiate any energy. Later, more
careful numerics [3] showed that such bound states actually had nonvanishing radi-
ation tails. Recent results of Yang and Kath [10] showed that exactly-periodic DM
solitons do not exist in the dispersion-periodic NLS equation (1.1) because resonances
in the perturbation series generate nonvanishing radiation tails. These tails can be ex-
tremely small in certain parameter regimes [10], but they do not vanish when D
0
> 0.
Radiation tails of DM solitons occur due to parametric resonance between the
DM soliton and the periodic variation of the dispersion. This parametric resonance
drains energy out of the DM soliton and leads to its radiative damping. Parametric
resonances can be predicted by viewing the periodic term of (1.1) as an external
forcing term:
i
∂u
∂z
+
1
2
D
0
2
u
∂t
2
+ |u|
2
u =
m
2
D
(z)
2
u
∂t
2
.(1.3)
We expand D
(z) into a Fourier series,
D
(z)=
n=−∞
d
n
e
2πinz
,d
0
=0,d
n
=
¯
d
n
,(1.4)
where
¯
d
n
is the complex conjugate of d
n
. When the nonlinear term in (1.3) is neglected
and the averaged DM soliton u(z, t)=Φ(t) e
iµz
is substituted into the right-hand
side of (1.3), we ﬁnd a solution of the linear inhomogeneous problem in the form of
the Fourier series in z,
u(z,t)=
n=−∞
u
n
(t) e
2πinz
e
iµz
.(1.5)
The correction terms u
n
(t) take the form of Fourier integrals in t,
u
n
(t)=
md
n
4π
−∞
ω
2
ˆ
Φ(ω)e
iωt
1
2
D
0
ω
2
+
2πn
+ µ
,(1.6)
where
ˆ
Φ(ω) is the Fourier transform of Φ(t). The inhomogeneous solution has resonant
denominators at
ω
2
= ω
2
n
=
2
D
0
µ +
2πn
> 0.(1.7)
Resonances are absent if D
0
= 0 and µ = 2πn/ for any integer n. This is the
only case when DM soliton solutions (1.2) may exist in the dispersion-periodic NLS
equation (1.1). In this case, the asymptotic representation of Φ(z, t) in (1.2) was
found recently in [12] in the limit =O(m) 1 with the use of the inverse scattering
transform methods.
If D
0
> 0, suﬃciently large negative terms of the Fourier series (1.5)–(1.6) are in
resonance (1.7) for n ≤−N
µ
, where N
µ
=
µ
2π
is the integer ceiling of
µ
2π
> 0. The
periodic variations of the dispersion map D
(z) lead to a coupling of a bound state

1362
DMITRY E. PELINOVSKY AND JIANKE YANG
and linear waves of the averaged dispersion map and to the energy transfer from the
bound state to radiative waves. As a result, the pulse solution has resonant peaks in
the spectrum ˆu(z,ω)atω = ±ω
n
, and nonzero values of u(z, t) in the far-ﬁeld |t|1,
as reported numerically in [3, 10].
Radiation damping of solitons in the presence of a weak sinusoidal dispersion
variation was considered analytically in [13]. The radiative wave amplitudes and
decay rates of solitons were computed by means of the soliton perturbation theory
for the standard NLS equation. Dynamics of DM solitons was studied in [14, 15, 16]
by variational and numerical methods. Recently, analytical and numerical studies of
the same problem were undertaken in [10] by asymptotic beyond-all-orders methods
in the limit =O(m) 1. Radiation-tail amplitudes and decay rates of DM solitons
were found to be exponentially small in this limit. It was also shown in [10] that
radiation-tail amplitudes drop to near-zero values in certain windows on the m-axis.
We study here nonlinear parametric resonance of DM solitons for average-anoma-
lous dispersion (D
0
> 0) in the limit m 1, while we keep = O(1). This is a
diﬀerent limit from the one studied in [10]. In this limit, the DM soliton decays
much faster because radiation-tail amplitudes are only algebraically small in terms of
O(m). The new feature of our analysis is that the periodic dispersion map D
(z)is
allowed to be arbitrary in (1.4) as compared to a single sine function in [13]. Thus,
our dispersion maps include the piecewise-constant dispersion map which is widely
used in ﬁber communication systems.
Our analysis starts with the standard NLS equation (1.3) for m = 0, such that the
right-hand side of (1.3) is treated as a small perturbation. The ﬁrst-order perturbation
theory describes generation of linear waves due to parametric resonances (1.7), and the
second-order perturbation theory leads to the decay rate of DM solitons. Methods of
our analysis are similar to the soliton perturbation theory in [13], but our calculations
are more systematic. We ﬁnd that the DM soliton decays according to a nonlinear
Fermi golden rule, which generalizes the Fermi golden rule for radiative decay of bound
states in the linear Schr¨odinger equation with a time-periodic potential. Rigorous
analysis of decay rates in the linear Schr¨odinger equation was recently considered
in [17, 18], where the bound states were supported by a time-dependent periodic
potential in [17] and by a time-independent potential in [18].
This paper is structured as follows. Section 2 contains perturbation series expan-
sions and derivations of the Fermi golden rule for DM solitons. Section 3 is devoted
to analytical approximations of radiative decay of DM solitons. Section 4 describes
a comparison between the analytical and numerical results. Section 5 concludes the
paper. Appendices A and B describe technical details of the ﬁrst-order solution in
the perturbation series expansions.
NLS equation in the form (1.3), where is ﬁnite and m is small. If D
0
> 0, we
employ the following rescaling of variables:
z = ˆz, u =
ˆu
,t=
D
0
ˆ
t, m = D
0
ˆm.(2.1)
When the hats are dropped, (1.3) becomes
iu
z
+
1
2
u
tt
+ |u|
2
u =
m
2
D
1
(z)u
tt
,(2.2)
where the dispersion map D
1
(z) has unit period. In other words, we have normalized
and D
0
in (1.3) so that = 1 and D
0
=1.

RESONANCE AND DECAY OF DISPERSION-MANAGED SOLITONS 1363
When m = 0, the standard NLS equation (2.2) has a bound state:
u(z,t)=Φ(t; µ)e
iµz
,(2.3)
where µ>0 and Φ(t; µ)=
2µ sech
2µt
. When m = 0, the NLS soliton (2.3)
would generate radiative tails and decay accordingly. Parameter µ of the NLS soli-
ton (2.3) changes in z, such that the z-dependence of µ(z) serves as a condition for
Poincar´e continuation of the perturbation series for u(z, t) in powers of m. The Fermi
golden rule of radiative decay of NLS solitons follows from the dynamical equation for
µ = µ(z). In order to formalize this qualitative picture, we employ the transformation
u(z,t)=U(z,t; µ(z))e
i
z
0
µ(z
)dz
,(2.4)
where U(z,t; µ) solves the problem
i
∂U
∂z
+ i ˙µ
∂U
∂µ
µU +
1
2
2
U
∂t
2
+ |U|
2
U =
m
2
D
1
(z)
2
U
∂t
2
(2.5)
with the initial data U (0,t; µ
0
)=Φ(t; µ
0
) and µ(0) = µ
0
. The transformation (2.4)
describes the adiabatically varying orbit of the NLS soliton (2.3). We present the
asymptotic solution of (2.5) as a perturbation series for U(z, t; µ) and µ(z)inpowers
of m:
U(z,t; µ)=
k=0
m
k
U
(k)
(z,t; µ)(2.6)
and
˙µ =
k=1
m
2k
Γ
(2k)
(µ),(2.7)
where Γ
(2k)
(µ) are corrections of the Fermi golden rule for radiative decay of NLS
solitons. Substitution of (2.6)–(2.7) into (2.5) produces a chain of equations for cor-
rections of the perturbation series. At the leading, ﬁrst and second orders, the chain
of perturbative equations takes the form
i
∂U
(0)
∂z
µU
(0)
+
1
2
2
U
(0)
∂t
2
+ |U
(0)
|
2
U
(0)
=0,(2.8)
i
∂U
(1)
∂z
µU
(1)
+
1
2
2
U
(1)
∂t
2
+2|U
(0)
|
2
U
(1)
+ U
(0)2
¯
U
(1)
=
1
2
D
1
(z)
2
U
(0)
∂t
2
,(2.9)
and
i
∂U
(2)
∂z
µU
(2)
+
1
2
2
U
(2)
∂t
2
+2|U
(0)
|
2
U
(2)
+ U
(0)2
¯
U
(2)
(2.10)
= iΓ
(2)
(µ)
∂U
(0)
∂µ
1
2
D
1
(z)
2
U
(1)
∂t
2
2|U
(1)
|
2
U
(0)
U
(1)2
¯
U
(0)
.
Initial conditions for these equations are
U
(0)
(0,t; µ
0
)=Φ(t; µ
0
)(0) = µ
0
,(2.11)

1364
DMITRY E. PELINOVSKY AND JIANKE YANG
and
U
(k)
(0,t; µ
0
)=0,k 1.(2.12)
Order O(1). The nonlinear equation (2.8) at order O(1) with initial data (2.11) has
a unique solution, U
(0)
(z,t; µ)=Φ(t; µ), which is the NLS soliton with the adiabatic
change of µ = µ(z).
Order O(m). The linear inhomogeneous equation (2.9) at order O(m) has the
Fourier series solution
U
(1)
(z,t; µ)=
n=−∞
U
(1)
n
(z,t; µ) e
2π inz
,(2.13)
where U
(1)
0
= 0 and (U
(1)
n
,
¯
U
(1)
n
)atn 1 solve the coupled equations
i
∂U
(1)
n
∂z
(µ +2πn) U
(1)
n
+
1
2
2
U
(1)
n
∂t
2
2
(t; µ)
2U
(1)
n
+
¯
U
(1)
n
(2.14)
=
d
n
2
Φ

(t; µ),
i
¯
U
(1)
n
∂z
(µ 2πn)
¯
U
(1)
n
+
1
2
2
¯
U
(1)
n
∂t
2
2
(t; µ)
2
¯
U
(1)
n
+ U
(1)
n
(2.15)
=
d
n
2
Φ

(t; µ).
It follows from (2.12) that the system (2.14)–(2.15) is supplemented with zero initial
conditions: U
(1)
n
(0,t; µ
0
) = 0 for any |n|≥1. Solutions of the system (2.14)–(2.15) are
constructed in Appendix A with the use of the spectral decomposition for a linearized
NLS operator [19, 20]. Asymptotic limits of the correction terms U
(1)
n
(z,t; µ) are
obtained in Appendix B with the use of generalized functions. These calculations
show that the continuous-wave radiation in the solution U
(1)
n
(z,t; µ) at large distance
z and time t is given by the following expression [see (A.1) and (B.9)]:
lim
|t|→∞,z→∞
U
(1)
n
=
πi
2µd
n
(k
n
+ i)
2
4k
n
sech
πk
n
2
e
i
2µk
n
|t|
,n N
µ
,(2.16)
and
lim
|t|→∞,z→∞
U
(1)
n
=0, n<N
µ
,(2.17)
where
k
n
=
2πn
µ
1 > 0,N
µ
=
µ
2π
.(2.18)
This result will be used at order O(m
2
) to calculate the decay rate Γ
(2)
(µ)ofDM
solitons.
Order O(m
2
). Solution of the linear inhomogeneous equation (2.11) at order
O(m
2
) can also be represented by the Fourier series:
U
(2)
(z,t; µ)=
n=−∞
U
(2)
n
(z,t; µ) e
2π inz
.(2.19)

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115 citations

### "Parametric resonance and radiative ..." refers background in this paper

• ...For this dispersion map, the DM soliton is chirp-free at mod(z, 1) = 0 and mod(z, 1) = 1 2 (see [21], for instance)....

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