Ablowitz M.A. Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons
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9.10 Outline of the IST for a general evolution system 255
corresponding norming constant C
1
). In the relevant physical case, when the
symmetry r = −q
∗
holds, using (9.75) and (9.76) in the above system, we get
N
(1)
1
(x) = −
C
1
k
1
− k
∗
1
e
−2ik
∗
1
x
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 −
|
C
1
|
2
e
2i
(
k
1
−k
∗
1
)
x
k
1
− k
∗
1
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
−1
,
N
(2)
1
(x) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 −
|
C
1
|
2
e
2i
(
k
1
−k
∗
1
)
x
k
1
− k
∗
1
2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
−1
,
where, as before, N
1
(x) =
N
(1)
1
(x), N
(2)
1
(x)
T
. Then, if we set
k
1
= ξ + iη, e
2δ
=
|
C
1
|
2η
,
it follows from (9.79) that
q(x) = −2iη
C
∗
1
|
C
1
|
e
−2iξx
sech
(
2ηx − 2δ
)
.
Taking into account the time dependence of C
1
as given by (9.89), we find
q(x) = 2ηe
−2iξx+2i Im A
0
(k
1
)t−iψ
0
sech
%
2
(
η(x − x
0
) + Re A
0
(k
1
)t
)
&
,
where C
1
(0) = 2ηe
2ηx
0
+i(ψ
0
+π/2)
.
Thus when we take
(a) A
0
(k
1
) = 2ik
2
1
= −4ξη + 2i(ξ
2
− η
2
) where k
1
= ξ + iη, r = −q
∗
we get the
well-known bright soliton solution of the NLS equation
q(x, t) = 2ηe
−2iξx+4i
(
ξ
2
−η
2
)
t−iψ
0
sech
%
2η
(
x −4ξt − x
0
)
&
.
(b) A
0
(k
1
) = −4ik
3
1
= −4η
3
where k
1
= iη, r = −q, real; recall that due to
symmetry all eigenvalues must come in pairs
k
j
, −k
∗
j
hence with only
one eigenvalue Re k
1
= ξ = 0. Then we get the bright soliton solution of
the mKdV equation (9.92):
q(x, t) = 2η sech
2η
x −4η
2
t − x
0
.
(c) A
0
(k
1
) =
i
4k
=
1
4η
where k
1
= iη, r = −q =
u
x
2
, real; again due to symmetry
Re k
1
= ξ = 0. Then we get the soliton-kink solution of the sine–Gordon
equation (9.93):
q(x, t) = −
u
x
2
= −2η sech
2η
x +
1
4η
t − x
0
,
or in terms of u
u(x, t) = 4tan
−1
exp
2η
x +
1
4η
t − x
0
.
256 Inverse scattering transform for the KdV equation
Exercises
9.1 Use (9.15) and (9.16) to find the Neumann series for M(x; k) and N(x; k).
Show these series converge uniformly when u(x) decays appropriately
(e.g., u ∈ L
1
2
).
9.2 Using the results in Exercise 9.1 establish that a(k) is analytic for Im k >
0 and continuous for Im k = 0.
9.3 Find a one-soliton solution to an equation associated with the time-
independent Schr
¨
odinger equation (9.4) when
b(k, t) = b(k, 0)e
−32ik
5
t
, a(k, t) = a(k, 0),
c
j
(t) = c
j
(0)e
32k
5
j
t
, k
j
= iκ
j
.
Use the concepts of Chapter 8 to determine the nonlinear evolution
equation this soliton solution solves. (Hint: use (8.51)–(8.52).)
9.4 Use (8.51)–(8.52) to deduce the time-dependence of the scattering data
for a nonlinear evolution equation associated with (9.4) whose linear
dispersion relation is ω(k).
9.5 From (9.53)–(9.54) find the next two conserved quantities C
7
, C
9
associated with the KdV equation. Determine the conservation laws.
9.6 Use (9.67) to establish that M(x; k), N(x; k) are analytic for Im k > 0,
M(x; k), N(x; k)forImk < 0 where q, r ∈ L
1
(R).
9.7 Suppose A
0
(k) = 8ik
4
, associated with (9.85) and (9.90) with r = −q
∗
.
Find the time dependence of the scattering data and the one-soliton solu-
tion. Use the results of Chapter 8 to find the nonlinear evolution equation
that this soliton solution satisfies.
9.8 Consider the equation
(u
t
+ 6uu
x
+ u
xxx
)
x
− 3u
yy
= 0.
(a) Show that the equation has the rational solution
u(x, t) = 4
p
2
y
2
− X
2
+ 1/p
2
p
2
y
2
+ X
2
+ 1/p
2
where X = x + 1/p − 3p
2
t and p is a real constant.
(b) Make the transformation y → iy to formally derive the following KP
equation
(u
t
+ 6uu
x
+ u
xxx
)
x
+ 3u
yy
= 0
and thus show that the above solution is a singular solution of the
KP.
Exercises 257
9.9 Suppose K(x, z; t) satisfies the Marchenko equation
K(x, z; t) + F(x, z; t) +
∞
X
K(x, y; t)F(y, z; t) dy = 0,
where F is a solution of the pair of equations
F
xx
− F
zz
= (x − z)F
3tF
t
− F + F
xxx
+ F
zzz
= xF
x
+ zF
z
.
(a) Show that
u
t
+
u
2t
+ 6uu
x
+ u
xxx
= 0,
where u(x, t) =
2
(12t)
2/3
∂
∂X
K(X, X; t) with X =
x
(12t)
1/3
.
(b) Show that a solution for F is
F(x, z; t) =
∞
−∞
f (yt
1/3
)Ai(x + y)Ai(y + z) dy
where f is an arbitrary function and Ai(x) is the Airy function.
Hint: See Ablowitz and Segur (1981) on how to operate on K.
9.10 Show that the KP equation
(u
t
+ 6uu
x
+ u
xxx
)
x
+ 3σ
2
u
yy
= 0
with σ = ±1 can be derived from the compatibility of the system
∂
2
v
∂x
2
+ σ
∂v
∂y
+ uv = 0
v
t
+ 4
∂
3
v
∂x
3
+ 6u
∂v
∂x
+
3u
x
− 3σ
∞
∞
u
y
dx + α
v = 0
where α is constant. Note: there is no “eigenvalue” in this equation. The
scattering parameter is inserted when carrying out the inverse scattering
(see Ablowitz and Clarkson, 1991).
9.11 Consider the following time-independent Schr
¨
odinger equation
v
xx
+ (k
2
+ Q sech
2
x)v = 0
where Q is constant.
(a) Make the transformation v(x) =Ψ(ξ), with ξ = tanh x (hence −1 <
ξ<1 corresponds to −∞ < x < ∞) to find
(1 − ξ
2
)
d
2
Ψ
dξ
2
− 2ξ
dΨ
dξ
+
Q +
k
2
1 − ξ
2
Ψ=0
which is the associated Legendre equation,cf.Abramowitz and
Stegun (1972).
258 Inverse scattering transform for the KdV equation
(b) Show that there are eigenvalues k = iκ when
κ =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
Q +
1
4
1/2
− m −
1
2
⎤
⎥
⎥
⎥
⎥
⎥
⎦
> 0,
where m = 0, 1, 2,... and therefore a necessary condition for
eigenvalues is Q +
1
4
>
1
4
.
(c) Show that when Q = N(N + 1), N = 1, 2, 3,..., the reflection coef-
ficient vanishes, the discrete eigenvalues are given by κ
n
= n, n =
1, 2, 3,...,N and the discrete eigenfunctions are proportional to the
associated Legendre function P
n
N
(tanh x).
9.12 Show that the solution to the Gel’fand–Levitan–Marchenko (GLM)
equation (9.42)–(9.43) with pure continuous spectra is unique. Hint: one
method is to show the homogeneous equation has only the zero solution,
hence by the Fredholm alternative, the solution to the GLM equation is
unique. Then in the homogeneous equation let K(x, y) = 0, y < x and
take the Fourier transform.
PART III
APPLICATIONS OF NONLINEAR WAVES IN
OPTICS
10
Communications
Nonlinear optics is the branch of optics that describes the behavior of light
in nonlinear media; such as, media in which the induced dielectric polariza-
tion responds nonlinearly to the electric field of the light. This nonlinearity
is typically observed at very high light intensities such as those provided by
pulsed lasers. In this chapter, we focus on the application of high bit-rate
communications. We will see that the nonlinear Schr
¨
odinger (NLS) equation
and the dispersion-managed nonlinear Schr
¨
odinger (DMNLS) equation play a
central role.
10.1 Communications
In 1973 Hasegawa and Tappert (Hasegawa and Tappert, 1973a; Hasegawa
and Kodama, 1995) showed that the nonlinear Schr
¨
odinger equation derived
in Chapter 7 [see (7.26), and the subsequent discussion] described the prop-
agation of quasi-monochromatic pulses in optical fibers. Motivated by the
fact that the NLS equation supports special stable, localized, soliton solu-
tions, Mollenauer et al. (1980) demonstrated experimentally that solitons
can propagate in a real fiber. However, it was soon apparent that due to
unavoidable damping in optical fibers, solitons lose most of their energy
over relatively short distances. In the mid-1980s all-optical amplifiers (called
erbium doped fiber amplifiers: EDFAs) were developed. However with such
amplifiers there is always some additional small amount of noise. Gordon and
Haus (1986)(seealsoElgin, 1985) showed that solitons suffered seriously from
these noise effects. The frequency and temporal position of the soliton was
significantly shifted over long distances, thereby limiting the available trans-
mission distance and speed of soliton-based systems. Subsequently researchers
began to seriously consider so-called wavelength division multiplexed (WDM)
261
262 Communications
communication systems in which many optical pulses are transmitted simul-
taneously. The inevitable pulse interactions caused other serious problems,
called four-wave mixing instabilities and collision-induced time shifts. In order
to alleviate these penalties, researchers began to study dispersion-managed
(DM) transmission systems. A DM transmission line consists of fibers with
different types of dispersion fused together. The success of DM technology
has already led to its use in commercial systems. In this chapter we discuss
some of the important issues in the context of nonlinear waves and asymp-
totic analysis. There are a number of texts that cover this topic, which the
interested reader can consult. These include Agrawal (2002, 2001), Hasegawa
and Kodama (1995), Hasegawa and Matsumoto (2002), and Molleneauer and
Gordon (2006).
10.1.1 The normalized NLS equation
In Chapter 7 the NLS equation was derived for weakly nonlinear, quasi-
monochromatic electromagnetic waves. It was also discussed how the NLS
equation should be modified to apply to optical fibers. Using the assumption
for electromagnetic fields in the x-direction
E
x
= ε(A(Z, T )e
i(kz−ωt)
+ cc) + O(ε
2
),
where Z = z, T = t, the NLS equation in optical fibers was found, in Chapter
7 [see (7.27) and the equations following], to satisfy
i
∂A
∂˜z
+
−
k
(ω)
2
∂
2
A
∂T
2
+ ν|A|
2
A = 0, (10.1)
with the nonlinear coefficient ν = ν
eff
=
n
2
ω
4cA
eff
where the retarded coordinate
frame is given by T = t −
Z
v
g
= t − k
Z,˜z = Z with k =
ω
c
n
0
, and the
nonlinear index of refraction is given by n = n
0
+ n
2
|E
x
|
2
= n
0
+ 4n
2
|A|
2
;
v
g
= 1/k
(ω) is the group velocity and A
eff
is the effective area of the fiber. It
is standard to introduce the following non-dimensional coordinates: ˜z = z
z
∗
,
T = t
t
∗
, and A =
√
P
∗
u, where a subscript ∗ denotes a characteristic scale.
Usually, z
∗
is taken as being proportional to the nonlinear length, the length
over which a nonlinear phase change of one radian occurs, t
∗
is taken as being
proportional to the pulse full width at half-maximum (FWHM), and P
∗
is taken
as the peak pulse power. With the use of this non-dimensionalization, (10.1)
becomes
10.1 Communications 263
i
∂u
∂z
+
z
∗
t
2
∗
(−k
)
1
2
∂
2
u
∂t
2
+ z
∗
P
∗
ν|u|
2
u = 0.
For solitons, z
∗
P
∗
ν = 1 gives the nonlinear length as z
∗
= 1/νP
∗
.Onthe
other hand the linear dispersive length is given by z
∗L
= t
2
∗
/|k
|. Further
assuming the nonlinear length to be equal to the linear dispersive length,
yields the relation P
∗
= |k
|/
νt
2
∗
, and the normalized NLS equation
follows:
i
∂u
∂z
+
1
2
∂
2
u
∂t
2
+ |u|
2
u = 0. (10.2)
An exact solution of (10.2), called here the classical soliton solution, is given
by u = η sech(ηt
) e
iη
2
z
/2
(see also Chapter 6). As mentioned earlier, solitons
in optical fibers were predicted theoretically in 1973 (Hasegawa and Tappert,
1973a) and then demonstrated experimentally a few years later (Mollenauer
et al., 1980; see also Hasegawa and Kodama, 1995, for additional references
and historical background).
10.1.2 FWHM: The full width at half-maximum
The FWHM corresponds to the temporal distance between the two points
where the pulse is at half its peak power. This is schematically shown in
Figure 10.1. For a sech pulse this gives
|u|
2
= η
2
sech
2
ηT
t
∗
=
η
2
2
.
Thus T is such that sech
2
(ηT/t
∗
) = 1/2. If we assume η = 1, then
T/t
∗
≈1.763/2 = 0.8815. The FWHM for classical solitons is notationally
taken to be τ with τ = 2T ≈ 1.763t
∗
. Recall that t
∗
is the normalizing value of
−T
t
*
T
t
*
Half power
Figure 10.1 The full width at half-maximum.
264 Communications
5τ 5τ 5τ
Figure 10.2 Pulses are frequently spaced five FWHM apart in a single-
channel optical fiber communication system.
t.Ifτ = 20 ps (ps denotes picoseconds) with pulse spacing of 5τ = 100 ps, this
implies one bit is transmitted every 100 ps, as indicated in Figure 10.2.The
bit-rate is therefore found to be:
Bit-Rate =
1
100 × 10
−12
= 10 ×10
9
bits/sec = 10 gb/s
where gb/s denotes gigabits per second. Typical numbers for classical solitons
are: τ = 20 ps which gives t
∗
= 20/1.763 = 11.3ps.
On the other hand if the pulse were of Gaussian shape (which more
closely approximates dispersion-managed solitons, described later in this
chapter), then
u = u
0
e
−t
2
/2t
2
∗
,
|u|
2
= |u
0
|
2
e
−t
2
/t
2
∗
= |u
0
|
2
/2,
and
t = (log 2)
1/2
t
∗
,
hence in this case the FWHM is given by τ = 2
log 2t
∗
= 1.665t
∗
.
10.1.3 Loss
In reality there is a damping effect, or loss, in an optical fiber. Otherwise, using
typical fibers, after 100 km pulses would lose most of their energy. Usually
linear damping is assumed, and the NLS equation is given by
i
∂A
∂˜z
+
−
k
(ω)
2
∂
2
A
∂T
2
+ ν|A|
2
A = −iγA, (10.3)
where γ is the physical damping rate. Non-dimensionalizing the equation as
before, see (10.2), we find equation (10.3) takes the form
i
∂u
∂z
+
1
2
∂
2
u
∂t
2
+ |u|
2
u = −iΓu, (10.4)