Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line
The envelope modulation of a monoinductance transmission line is reduced to generalized coupled Ginzburg – Landau equations from which is deduced a single cubic-quintic Ginzburg – Landau equation containing derivatives with respect to the spatial variable in the cubic terms. We investigate the modul...
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irk-123456789-1781732021-02-19T01:26:48Z Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line Kengne, E. Vaillancourt, R. The envelope modulation of a monoinductance transmission line is reduced to generalized coupled Ginzburg – Landau equations from which is deduced a single cubic-quintic Ginzburg – Landau equation containing derivatives with respect to the spatial variable in the cubic terms. We investigate the modulational instability of the spatial wave solutions of both the system and the single equation. For the generalized coupled Ginzburg – Landau system we consider only the zero wavenumbers of the perturbations whose modulational instability conditions depend only on the system’s coefficients and the wavenumbers of the carriers. In this case, a modulational instability criterion is established which depends both on the perturbation wavenumbers and the carrier. We also study the coherent structures of the generalized coupled Ginzburg – Landau system and present some numerical studies. Огортуючу модуляцiю моноiндуктивної лiнiї передач зведено до узагальнених пов’язаних мiж собою рiвнянь Гiнзбурга – Ландау, звiдки отримано одне рiвняння Гiнзбурга – Ландау третього – п’ятого порядку, яке мiстить похiднi вiдносно просторової змiнної в кубiчних членах. Для системи та рiвняння дослiджено модуляцiйну нестiйкiсть розв’язкiв у формi просторової хвилi. Для системи Гiнзбурга – Ландау розглянуто лише збурення з нульовими хвильовими числами, для яких умови модуляцiйної нестiйкостi залежать тiльки вiд коефiцiєнтiв системи та хвильових чисел носiїв. У цьому випадку отримано критерiй для модуляцiйної нестiйкостi, який залежить як вiд хвильових чисел збурень, так i вiд носiя. Також вивчаються когерентнi структури системи Гiнзбурга – Ландау та проведено деякий числовий аналiз. 2006 Article Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line / E. Kengne, R. Vaillancourt // Нелінійні коливання. — 2006. — Т. 9, № 4. — С. 451-489. — Бібліогр.: 31 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178173 517.9 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
The envelope modulation of a monoinductance transmission line is reduced to generalized coupled Ginzburg – Landau equations from which is deduced a single cubic-quintic Ginzburg – Landau equation containing derivatives with respect to the spatial variable in the cubic terms. We investigate the modulational
instability of the spatial wave solutions of both the system and the single equation. For the generalized
coupled Ginzburg – Landau system we consider only the zero wavenumbers of the perturbations whose
modulational instability conditions depend only on the system’s coefficients and the wavenumbers of
the carriers. In this case, a modulational instability criterion is established which depends both on the
perturbation wavenumbers and the carrier. We also study the coherent structures of the generalized coupled
Ginzburg – Landau system and present some numerical studies. |
| format |
Article |
| author |
Kengne, E. Vaillancourt, R. |
| spellingShingle |
Kengne, E. Vaillancourt, R. Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line Нелінійні коливання |
| author_facet |
Kengne, E. Vaillancourt, R. |
| author_sort |
Kengne, E. |
| title |
Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line |
| title_short |
Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line |
| title_full |
Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line |
| title_fullStr |
Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line |
| title_full_unstemmed |
Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line |
| title_sort |
ginzburg-landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line |
| publisher |
Інститут математики НАН України |
| publishDate |
2006 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/178173 |
| citation_txt |
Ginzburg-Landau system of complex modulation equations for a distributed nonlinear-dispersive transmission line / E. Kengne, R. Vaillancourt // Нелінійні коливання. — 2006. — Т. 9, № 4. — С. 451-489. — Бібліогр.: 31 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT kengnee ginzburglandausystemofcomplexmodulationequationsforadistributednonlineardispersivetransmissionline AT vaillancourtr ginzburglandausystemofcomplexmodulationequationsforadistributednonlineardispersivetransmissionline |
| first_indexed |
2025-07-15T16:32:27Z |
| last_indexed |
2025-07-15T16:32:27Z |
| _version_ |
1837731307149328384 |
| fulltext |
UDC 517 . 9
GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION
EQUATIONS FOR A DISTRIBUTED NONLINEAR-DISSIPATIVE
TRANSMISSION LINES
СИСТЕМА РIВНЯНЬ ГIНЗБУРГА – ЛАНДАУ НА КОМПЛЕКСНУ
МОДУЛЯЦIЮ ДЛЯ РОЗПОДIЛЕНОЇ НЕЛIНIЙНО ДИСИПАТИВНОЇ
ЛIНIЇ ПЕРЕДАЧ
E. Kengne
Univ. Dschang
PO Box 4509 Douala, Republic of Cameroon
and Univ. Ottawa
Ottawa, Ontario, Canada K1N 6N5
e-mail: ekengne6@yahoo.fra
R. Vaillancourt
Univ. Ottawa
Ottawa, Ontario, Canada K1N 6N5
e-mail: remi@uottawa.ca
The envelope modulation of a monoinductance transmission line is reduced to generalized coupled Ginz-
burg – Landau equations from which is deduced a single cubic-quintic Ginzburg – Landau equation contai-
ning derivatives with respect to the spatial variable in the cubic terms. We investigate the modulational
instability of the spatial wave solutions of both the system and the single equation. For the generalized
coupled Ginzburg – Landau system we consider only the zero wavenumbers of the perturbations whose
modulational instability conditions depend only on the system’s coefficients and the wavenumbers of
the carriers. In this case, a modulational instability criterion is established which depends both on the
perturbation wavenumbers and the carrier. We also study the coherent structures of the generalized coupled
Ginzburg – Landau system and present some numerical studies.
Огортуючу модуляцiю моноiндуктивної лiнiї передач зведено до узагальнених пов’язаних мiж
собою рiвнянь Гiнзбурга – Ландау, звiдки отримано одне рiвняння Гiнзбурга – Ландау третьо-
го – п’ятого порядку, яке мiстить похiднi вiдносно просторової змiнної в кубiчних членах. Для
системи та рiвняння дослiджено модуляцiйну нестiйкiсть розв’язкiв у формi просторової хви-
лi. Для системи Гiнзбурга – Ландау розглянуто лише збурення з нульовими хвильовими числами,
для яких умови модуляцiйної нестiйкостi залежать тiльки вiд коефiцiєнтiв системи та хви-
льових чисел носiїв. У цьому випадку отримано критерiй для модуляцiйної нестiйкостi, який
залежить як вiд хвильових чисел збурень, так i вiд носiя. Також вивчаються когерентнi струк-
тури системи Гiнзбурга – Ландау та проведено деякий числовий аналiз.
1. Introduction. Nonlinear transmission lines (NTL’s) are networks which consist of nonlinear
capacitors periodically placed between sections of the transmission line to guide and contain
electromagnetic radiation. Recently, the terahertz region of the electromagnetic spectrum loose-
ly defined as the range between 100 GHz and 10 THz has become very attractive for future
wide-band systems. In this regard, electronic engineers are now exploring different possibilities
to develop signal sources operating at these frequencies [1]. There are various kinds of transmi-
c© E. Kengne, R. Vaillancourt, 2006
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т .9 , N◦ 4 451
452 E. KENGNE, R. VAILLANCOURT
ssion lines, such as twin lead, coaxial cable and wave guides. Transmission line theory is quite
relevant to engineering physics, especially in the area of optics and wave theory. Propagating
waves in transmission lines are reflected from abrupt discontinuities just as optical waves are
partially reflected off boundaries between two materials of different refraction indexes. Also,
standing waves can occur in transmission lines if there are two boundaries present, just as on a
vibrating string with two fixed points.
Different physico-chemical systems driven out of equilibrium may undergo Hopf bifurcati-
ons leading to rich spatio-temporal behavior. When these bifurcations occur with broken spatial
symmetries, they induce the formation of wave patterns described by order parameters of the
form:
Ψ = A ei(kcx−ωct) + B ei(−kcx−ωct) + c.c, (1.1)
where c.c stands for the complex conjugate of the preceding terms. The slow dynamics of the
complex-valued wave amplitudes A and B obey complex Ginzburg – Landau (GL) equations,
and kc and ωc are the critical wavenumber and the critical frequency, respectively. This is the
case, for example, for Rayleigh – Bénard convection in binary fluids, Taylor – Couette instabiliti-
es between co-rotating cylinders, electro-convection in nematic liquid crystals, or the transverse
field of high Fresnel number lasers. The main purpose of this paper is to study the dynamics of
modulated wave trains in a distributed nonlinear electrical transmission line. As primary modes,
we consider traveling waves. When these primary modes are essentially one-dimensional and
the system possesses left-right reflection symmetry, weakly nonlinear patterns are of the form
of Eq. (1.1) where A and B are the complex-valued amplitudes of the right- and left-traveling
waves. Using a perturbation method and passing to the continuum limit, we show that A and B
obey the generalized Ginzburg – Landau (GGL) systems.
In the next section we write down the circuit equations governing small-amplitude pulses
on systems of dissipative NTL’s. After scaling the coordinates and taking a continuum limit,
the multiple-scales method and the perturbation method are used in Section 3 to reduce the
circuit equations to a new nonlinear system of partial differential equations (PDEs) that we call
generalized coupled Ginzburg – Landau (GCGL) equations. We undertake an analytic study of
the stability of the particular solutions of these equations in Section 4. The coherent structures
are studied in Section 5. The results are summarized in Section 6.
2. Electrical models. Let us consider a distributed transmission line with the simplest peri-
odical structure consisting of the elements shown in Fig. 1, where the capacitance C is a function
of voltage [2]. Here, the transmission lines are studied as circuit models in terms of circuit theory
where voltages and currents, instead of fields, are the variables. Figure 1 shows an infinitesimal
segment of a physical dissipative transmission line. In the figure, the distributed parameters
of the line, R, L, C, and G, are the per-unit-length resistance, inductance, capacitance, and
conductance, respectively.
For the circuit in Fig. 1, we no longer have the familiar linear charge-voltage relation
CQn = Q, but rather the nonlinear differential relationship
C(Vn) =
dQn
dVn
. (2.1)
To solve (2.1) for the voltage as a function of time, we use the familiar relations at the nodes
Vn−1 − Vn =
dΦn
dt
+ RIn, In − In+1 =
dQn
dt
+ GVn. (2.2)
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 4
GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 453
Fig. 1. A typical section of a nonlinear transmission
line.
To obtain an equation in terms of V only, one notes that the magnetic flux can be expressed
in terms of the current using Φn = LIn, while the charge can be eliminated using (2.1). Using
these relations in (2.2) leads to the following formulae:
Vn−1 − Vn = L
dIn
dt
+ RIn, Vn − Vn+1 = L
dIn+1
dt
+ RIn+1,
In − In+1 = C(Vn)
dVn
dt
+ GVn,
dIn
dt
− dIn+1
dt
=
d
dt
[
C(Vn)
dVn
dt
]
+ G
dVn
dt
.
(2.3)
Eliminating In and In+1, we obtain the following second-order difference-differential equation:
d
dt
[
C(Vn)
dVn
dt
]
+ G
dVn
dt
+
R
L
C(Vn)
dVn
dt
=
1
L
(Vn−1 − 2Vn + Vn+1)−
RG
L
Vn. (2.4)
This is the circuit equation describing the voltage Vn(t) on a single line.
In this analysis, the nonlinear capacitance C(Vn,m) is of the form
C(V ) =
C0
1 + (V/V0)p
, (2.5)
where C0 and V0 are arbitrary capacitance and voltage scales, respectively, and p > 0. Simi-
lar forms for C(V ) have been used in the past to model the capacitance of certain varactor
diodes as part of comparisons with experimental measurements of solitary waves in NTL’s [3].
Substituting (2.5) into (2.4), we find
d2Qn
dt2
+ G
dVn
dt
+
R
L
dQn
dt
=
1
L
(Vn−1 − 2Vn + Vn+1)−
RG
L
Vn. (2.6)
To obtain an approximate solution of discretized Eq. (2.6), we can invoke the continuum limit
by a Taylor expansion of the voltage at Vn±1 as follows:
Vn±1 = Vn ±
∂Vn
∂n
+
1
2!
∂2Vn
∂n2
± 1
3!
∂3Vn
∂n3
+ . . . . (2.7)
By defining the quantity δ ≡ x′/n, eliminating terms of order higher than δ2 from (2.7) and
assuming that V � V0, we obtain the following wave equation valid in a weakly dispersive and
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 4
454 E. KENGNE, R. VAILLANCOURT
nonlinear regime:
C0
∂2
∂t2
(
V − V p+1
(p + 1) V p
0
)
+G
∂V
∂t
+
RC0
L
∂
∂t
(
V − V p+1
(p + 1) V p
0
)
− δ2
L
∂2V
∂x′2
+
RG
L
Vn = 0. (2.6′)
In what follows, we study the case where p = 2 and use the transformation x = x′/δ. Then
Eq. (2.6′) becomes
C0
∂2
∂t2
(
V + bV 3
)
− 1
L
∂2V
∂x2
+
(
G +
RC0
L
)
∂V
∂t
+
bRC0
L
∂V 3
∂t
+
RG
L
V = 0, (2.8)
where b = −1/ (3V p
0 ).
3. Derivation of the GCGL equations. In this section we use the voltage Eq. (2.8) to derive a
second-order partial differential system that will be called GCGL equations. To construct these
equations we use the method of multiple-scale by introducing two slow time scales T1 = εt and
T2 = ε2t in addition to the initial time T0 = t and one large length scale X1 = εx in addition
to the initial spatial variable X0 = x:
V =
3∑
j=1
εj/2
(
ujj ejiθ1 + vjj ejiθ2
)
+ ε5/2
(
u42e
2iθ1 + v42 e2iθ2
)
+ c.c + . . . , (3.1)
where
θ1 = kX0 − ωT0, θ2 = −kX0 − ωT0, ujk = ujk(X1, T1, T2), vjk = vjk(X1, T1, T2),
and ujk and vjk are complex-valued amplitudes of the right- and left-traveling waves. We then
order the damping coefficient in (2.8) so that the damping and nonlinearity effects appear in
the same perturbation equations. Thus, we set G + RC0/L = ε2µ1.
Inserting the perturbation expansion (3.1) into the nonlinear Eq. (2.8) we obtain a series of
nonhomogeneous equations at different orders of
(
ε, eiθ1 , eiθ2
)
.
At order
(
ε1/2, eiθ1 , eiθ2
)
we have(
−C0ω
2 +
k2
L
+
RG
L
)(
u11 eiθ1 + v11 eiθ2
)
= 0,
and the nontriviality condition of u11 and v11 gives the following linear dispersion relation,
which, for further use, we solve for k and differentiate with respect to k:
−C0ω
2 +
k2
L
+
RG
L
= 0, ω =
√
k2 + RG
C0L
,
∂ω
∂k
=
k√
C0L (k2 + RG)
=
1
C0L
k
ω
. (3.2)
At order
(
ε, eiθ1 , eiθ2
)
we have(
−4C0ω
2 + 4
k2
L
+
RG
L
)(
u22 e2iθ1 + v22 e2iθ2
)
= 0,
thus, we can take u22 = v22 = 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 4
GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 455
At order
(
ε3/2, eiθ1 , eiθ2
)
the equation is
[
−2iC0ω
∂u11
∂T1
− 2ik
L
∂u11
∂X1
− 3C0bω
(
ω + i
R
L
)(
|u11|2 + 2 |v11|2
)
u11
]
eiθ1+
+
[
−2iC0ω
∂v11
∂T1
+
2ik
L
∂v11
∂Y1
− 3C0bω
(
ω + i
R
L
)(
|v11|2 + 2 |u11|2
)
v11
]
eiθ2 = 0.
From this equation we obtain the system
∂u11
∂T1
+
∂ω
∂k
∂u11
∂X1
+ A3
(
|u11|2 + 2|v11|2
)
u11 = 0,
∂v11
∂T1
− ∂ω
∂k
∂v11
∂X1
+ A3
(
|v11|2 + 2|u11|2
)
v11 = 0,
(3.3)
where
A3 =
3b
2
(
R
L
− iω
)
,
∂ω
∂k
=
1
C0L
k
ω
. (3.4)
At order
(
ε3/2, eiθ1 , eiθ2
)
we have
[(
−9C0ω
2 + 9
k2
L
+
RG
L
)
u33 − 3C0bω
(
3ω +
R
L
)
u3
11
]
e3iθ1+
+
[(
−9C0ω
2 + 9
k2
L
+
RG
L
)
v33 − 3C0bω
(
3ω +
R
L
)
v3
11
]
e3iθ2 = 0.
By the dispersion relation (3.2), the last equation becomes
u33 = A4u
3
11, v33 = A4v
3
11, (3.5)
with
A4 =
3C0bω
8 (k2/L− C0ω2)
(
3ω +
R
L
)
. (3.6)
The equation of order
(
ε2, e2iθ1 , e2iθ2
)
is
[(
−4C0ω
2 + 4
k2
L
+
RG
L
)
u42
]
e2iθ1 +
[(
−4C0ω
2 + 4
k2
L
+
RG
L
)
v42
]
e2iθ2 = 0
from which we take
u42 = v42 = 0. (3.7)
ISSN 1562-3076. Нелiнiйнi коливання, 2006, т . 9, N◦ 4
456 E. KENGNE, R. VAILLANCOURT
At order
(
ε5/2, eiθ1 , eiθ2
)
we have{
C0
(
∂2u11
∂T 2
1
− 2iω
∂u11
∂T2
)
− 1
L
∂2u11
∂X2
1
+ 4C0bω
(
ω + i
R
L
)
u∗211u33 −
−iµ1ωu11 + 3C0b
(
R
L
− 2iω
)
∂
∂T1
[(
|u11|2 + 2 |v11|2
)
u11
]}
eiθ1+
+
{
C0
(
∂2v11
∂T 2
1
− 2iω
∂v11
∂T2
)
− 1
L
∂2v11
∂X2
1
+ 4C0bω
(
ω + i
R
L
)
v∗211v33 −
−iµ1ωv11 + 3C0b
(
R
L
− 2iω
)
∂
∂T1
[(
|v11|2 + 2 |u11|2
)
v11
]}
eiθ2 = 0.
From this equation, we obtain, after using (3.5), the following system:
−C0
ω1
∂2u11
∂T 2
1
+ 2iC0
∂u11
∂T2
+
1
Lω
∂2u11
∂X2
1
+ 4C0A4b
(
ω + i
R
L
)
|u11|4 u11+
+ iµ1u11 −
3C0b
ω
(
R
L
− 2iω
)
∂
∂T1
[(
|u11|2 + 2 |v11|2
)
u11
]
= 0,
−C0
ω
∂2v11
∂T 2
1
+ 2iC0
∂v11
∂T2
+
1
Lω2
∂2v11
∂X2
1
+ 4C0A4b
(
ω + i
R
L
)
|v11|4 v11+
+ iµ1v11 −
3C0b
ω
(
R
L
− 2iω
)
∂
∂T1
[(
|v11|2 + 2 |u11|2
)
v11
]
= 0.
(3.8)
If we write the systems (3.3) and (3.8) in terms of the original coordinates x, y, t, multiply
each equation of system (3.3) by ε1/2 (1 + 2iC0) and each equation of system (3.8) by ε1/2 and
then sum each side of these systems, we obtain, after using the transformations u = ε1/2u11 and
v = ε1/2v11,
(1 + 2iC0)
[
∂u
∂t′
+
∂ω
∂k
∂u
∂x
]
− C0
ω
∂2u
∂t2
+
1
Lω
∂2u
∂x2
+ 4C0A4b
(
ω + i
R
L
)
|u|4 u +
+ A3 (1 + 2iC0)
(
|u|2 + 2 |v|2
)
u + i
(
G +
RC0
L
)
u−
− 3C0b
ω
(
R
L
− 2iω
)
∂
∂t
[(
|u|2 + 2 |v|2
)
u
]
= 0,
(1 + 2iC0)
[
∂v
∂t′
− ∂ω
∂k
∂v
∂x
]
− C0
ω
∂2v
∂t2
+
1
Lω
∂2v
∂x2
+ 4C0A4b
(
ω + i
R
L
)
|v|4 v +
+ A3 (1 + 2iC0)
(
|v|2 + 2 |u|2
)
v + i
(
G +
RC0
L
)
v−
− 3C0b
ω
(
R
L
− 2iω
)
∂
∂t
[(
|v|2 + 2 |u|2
)
v
]
= 0,
(3.9)
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 457
where t′ = t/2. Now we can use (3.3) to eliminate the terms
∂u
∂t2
,
∂v
∂t2
,
∂
∂t
[(
|v|2 + 2|u|2
)
v
]
,
∂
∂t
[(
|u|2 + 2|v|2
)
u
]
from (3.9) and obtain
∂u
∂t
+ S0
∂u
∂x
+ P
∂2u
∂x2
+ γu + Q
(
|u|2 + 2 |v|2
)
u+
+
(
D |u|4 + F |u|2 |v|2 + K |v|4
)
u + E
∂
∂x
[(
|u|2 + 2 |v|2
)
u
]
+ Hu
∂
∂x
|v|2 = 0, (3.10)
∂v
∂t
− S0
∂v
∂x
+ P
∂2v
∂x2
+ γv + Q
(
|v|2 + 2 |u|2
)
v+
+
(
D |v|4 + F |v|2 |u|2 + K |u|4
)
v − E
∂
∂x
[(
|v|2 + 2 |u|2
)
v
]
−Hv
∂
∂x
|u|2 = 0, (3.11)
where
t = t′, P =
1− 2iC0
1 + 4C2
0
[
1
Lω
− C0
ω
(
∂ω
∂k
)2
]
, γ =
i + 2C0
1 + 4C2
0
(
G +
RC0
L
)
,
Q = A3 =
3b
2
(
R
L
− iω
)
, E =
1− 2iC0
1 + 4C2
0
[
3C0b
ω
(
R
L
− 2iω
)
− 2A3C0
ω
]
∂ω
∂k
,
D =
1− 2iC0
1 + 4C2
0
[
3C0b
ω
(
R
L
− 2iω
)
(A∗
3 + 2A3) + 4C0A4b
(
ω + i
R
L
)]
,
H = 4
1− 2iC0
1 + 4C2
0
[
A3
C0
ω
− 3C0b
ω
(
R
L
− 2iω
)]
∂ω
∂k
,
F =
1− 2iC0
1 + 4C2
0
3C0b
ω
(
R
L
− 2iω
)
(6A∗
3 + 10A3) ,
K =
1− 2iC0
1 + 4C2
0
3C0b
ω
(
R
L
− 2iω
)
(6A3 + 2A∗
3) , S0 =
∂ω
∂k
=
1
C0L
k
ω
.
(3.12)
Equations (3.10), (3.11) are the required GCGL equations: they form a cubic-quintic GL
system with derivatives in the cubic terms. These equations are amplitude equations descri-
bing the slow modulations of a right-traveling mode with (x, t) dependence exp [i (kx− ωt)]
and amplitude u, coupled with a left-traveling mode exp [i (−kx− ωt)] with amplitude v. In the
special case where D = F = K = E = H = 0, system (3.10), (3.11) was studied in [4 – 9].
The existence and stability of the modulated amplitude waves in the complex plane were studi-
ed in [4] and the existence of soliton-like solutions has been shown.
For wave propagation occurring in one direction (v = 0 or u = 0), system (3.10), (3.11)
give the following GGL equation:
∂U
∂t
± S0
∂U
∂x
+ P
∂2U
∂x2
+ γU + Q|U |2U + D|U |4U + E
∂
∂x
|U |2U = 0. (3.13)
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458 E. KENGNE, R. VAILLANCOURT
The term
∂
∂x
|U |2U = U2 ∂U∗
∂x
+ 2|U |2 ∂U
∂x
appears in the asymptotic derivation. Deissler et al.
[10] showed numerically that this term can significantly slow down the propagating speed of
pulses and also cause nonsymmetric pulses. For equation (3.13), Saarloos and Hohenberg [11]
present a framework for the discussion of front, pulse and domain wall dynamics. Doelman and
Eckhaus [12] (Theorem 3.4) find some homoclenic domain walls with wave speed c = 0, via Poi-
ncaré maps and Melnikov integrals, extending and correcting the work of Holmes [13]. Kengne
[14] studied the Benjamin – Feir instability of the monochromatic wave solutions of Eq. (3.13).
In the case where S0 = 0, Kengne and Liu [15] found some exact solutions of Eq. (3.13).
In the special case where E = 0 (for example when k = 0), Eq. (3.13) becomes the so-called
quintic GL equation
∂U
∂t
± S0
∂U
∂x
+ P
∂2U
∂x2
+ γU + Q|U |2U + D|U |4U = 0. (3.14)
Thual and Fauve [16] and Fauve and Thual [17] discuss pulses for this equation. Kolyshkin et al.
[18, 19] investigated Eq. (3.14) in the special case where D = 0 for a suddenly blocked unsteady
channel.
It is seen from (3.12) that all the coefficients of system (3.10), (3.11), except for the linear
group velocity S0, are complex-valued functions of the wavenumber k. In what follows, we shall
denote by f r and f i the real and imaginary parts of the complex f , respectively.
In the CGL Eqs. (3.10), (3.11), S0 is the linear group velocity, i.e., the group velocity of the
fast modes. It is important to realize that the group velocity s is different from S0. To see this,
note that the CGL equations admit single mode traveling waves of the form
u(x, t) = a ei(qx−ωut), v(x, t) = 0, or v(x, t) = aei(qx−ωvt), u(x, t) = 0. (3.15)
Substituting these wave solutions in the amplitude equations (3.10), (3.11) we obtain the nonli-
near dispersion relation
ωu,v = ±
(
S0 + Eia2
)
q − P iq2 + γi + Qia2 + Dia4, (3.16)
where the real amplitude a is a solution of the equation
Dra4 +
(
Qr − Eiq
)
a2 − P rq2 + γr = 0. (3.17)
Solving (3.17) for a we get
a2 =
1
2Dr
[
Eiq−Qr ±
√(
(Ei)2+4DrP r
)
q2−2EiQrq+(Qr)2−4Drγr
]
, if Dr 6= 0,
(P rq2 − γr)/(Qr − Eqi), if Dr = 0.
(3.18)
Therefore, if Dr 6= 0, the group velocity s = ∂ω/∂k of these traveling waves, as functions of
the wavenumber q, becomes
su = S0 + Eia2 − 2P iq, sv = −S0 − Eia2 − 2P iq. (3.19)
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 459
a b
Fig. 2. Behavior of the group velocities su (a) and sv (b) as function of the wavenumber q, correspond-
ing to the line parameters (Lp1) for q ∈ [0, q− = 89580], respectively.
We see that, if either Dr = 0 or
(
Ei
)2 + 4DrP r ≥ 0, the amplitude a goes to infinity as q
increases. Therefore we frequently assume the global existence condition
(
Ei
)2 + 4DrP r < 0.
For the sign of the square root in (3.18) to make sense, we assume that Dr < (Qr)2 /(4γr) (it is
seen from Appendix A that γr > 0). In this case, plane waves exist for q in a bounded interval
[q+, q−], where
q± =
EiQr ±
√
4Drγr (Ei)2 + 16 (Dr)2 P rγr − 4 (Qr)2 DrP r
(Ei)2 + 4DrP r
.
For the line parameters [4]
C0 = 540 pF, L = 28 µH, R = 105 Ω, G = 10−4 Ω−1, b = 0.16 V −1, (Lp1)
C0 = 540 pF, L = 28 µH, R = 0.0525 Ω, G = 190.48 Ω−1, b = 0.16 V −1, (Lp2)
and wavenumber k=0.2, the analytical expressions for the coefficients of the GCGL Eqs. (3.10),
(3.11) are given in Appendix B. For the line parameters (Lp1) the above conditions for the
existence of wave solutions (3.15) are not satisfied. These conditions are satisfied for (Lp2),
and the curves of the group velocities su and sv are shown in Fig. 2. As one can see from
Figs. 2(a) and 2(b), su is monotone increasing and sv is monotone decreasing for wavenumbers
q ∈ [0, q− = 89580].
4. Amplitude dynamics of phase winding solutions. Having computed the analytical expressi-
ons for the coefficients of the GCGL equations (3.10), (3.11) (see Appendix A), we are now
able to study their stability. We shall analyze the so-called phase winding solutions which possess
a spatial periodic structure. A phase winding solution to (3.10), (3.11) is a pair of functions (u, v)
of the form
u(x, t) = a(t) ei(kux+Ωu(t)), v(x, t) = b(t) ei(kvx+Ωv(t)), (4.1)
for (x, t) ∈ R×R+, where a, b, and Ωu, Ωv are real amplitudes and phases, respectively,
depending only on time t ∈ R+, and ku, kv ∈ R are phase winding numbers. Note that under
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460 E. KENGNE, R. VAILLANCOURT
assumption (4.1) only amplitude instabilities can be analyzed. The phase functions do not affect
the stability properties in this case. If we insert Ansatz (4.1) into system (3.10), (3.11), we obtain
the following planar system of ordinary differential equations (ODEs) for the real amplitudes
a and b:
a′ =
(
P rk2
u − γr
)
a−
(
Qr + kuEi
) (
a2 + 2b2
)
a−
(
Dra4 + F ra2b2 + Krb4
)
a, (4.2)
b′ =
(
P rk2
v − γr
)
b−
(
Qr − kvE
i
) (
b2 + 2a2
)
b−
(
Drb4 + F ra2b2 + Kra4
)
b. (4.3)
The phase functions Ωu and Ωv are given by
Ωu(t) = Ω0
u +
(
P ik2
u − S0ku − γi
)
t−
−
t∫
0
[(
Qi + kuEr
) (
a2(τ) + 2b2(τ)
)
+
(
Dia4(τ) + F ia2(τ)b2(τ) + Kib4(τ)
)]
dτ,
Ωv(t) = Ω0
v +
(
P ik2
v + S0kv − γi
)
t−
−
t∫
0
[(
Qi − kvE
r
) (
b2(τ) + 2a2(τ)
)
+
(
Dib4(τ) + F ia2(τ)b2(τ) + Kia4(τ)
)]
dτ,
where Ω0
u and Ω0
v are the initial phases.
Let (u, v) be a solution of system (4.2), (4.3) corresponding to the phase winding numbers
ku, kv ∈ R. A straightforward phase-plane analysis enables us to conclude that the first quadrant
R+
0 ×R+
0 is invariant with respect to solutions of (4.2), (4.3). Furthermore, by introducing the
logarithmic transformation of the variables A = log a, B = log b and taking into account the
Poincaré – Bendixon criterion applied to the transformed planar system of ODEs, we conclude
that there are no periodic orbits and no heteroclinic cycles in system (4.2), (4.3).
The system of ODEs (4.2), (4.3) can have a number of fixed points. If (a0, b0) is a fixed point
of this system, then the linear flow near the stationary solution (a0, b0) is
A′ =
[
P rk2
u − γr − 2a2
0
(
Qr + kuEi
)
− 4Dra4
0 − 2F ra2
0b
2
0
]
A−
−
[
4a0b0
(
Qr + kuEi
)
+ 2F rb0a
3
0 + 4Kra0b
3
0
]
B, (4.4)
B′ =
[
4a0b0
(
kvE
i −Qr
)
− 2F ra0b
3
0 − 4Krb0a
3
0
]
A+
+
[
P rk2
v − γr − 2b2
0
(
Qr − kvE
i
)
− 4Drb4
0 − 2F ra2
0b
2
0
]
B. (4.5)
As in Jones, Kapitula and Powell [20], we can show that system (4.4), (4.5) characterizes the
behavior of the solutions of system (4.2), (4.3) as a → a0 and b → b0.
Since there are two flow equations, there are two eigenvalues of the linear flow near each
fixed point. When performing the counting analysis for these fixed points we will only need the
signs of the real parts of the two eigenvalues, since these determine whether the flow along the
corresponding eigendirection is inwards (−) or outwards (+).
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 461
Let
M =
P rk2
u − γr − 2a2
0
(
Qr + kuEi
)
− −4a0b0
(
Qr + kuEi
)
−
−4Dra4
0 − 2F ra2
0b
2
0 − 2F rb0a
3
0 − 4Kra0b
3
0
4a0b0
(
kvE
i −Qr
)
− P rk2
v − γr − 2b2
0
(
Qr − kvE
i
)
−
−2F ra0b
3
0 − 4Krb0a
3
0 − 4Drb4
0 − 2F ra2
0b
2
0
be the matrix of the flow (4.4), (4.5) near the stationary solution (a0, b0) of system (4.2), (4.3)
with eigenvalues λ1 and λ2. To have a stable stationary solution (a0, b0) of system (3.2), (3.3),
it is necessary and sufficient that Re λj > 0, j ∈ {1, 2}. For example, for the stability of the
zero amplitude wave critical point (0, 0) of system (4.2), (4,3), it is necessary and sufficient that
P rk2
u−γr < 0 and P rk2
v−γr < 0. In what follows, we give some numerical solutions of system
(4.4), (4.5) for (Lp1) and (Lp2).
For given coefficients of the GCGL Eqs. (3.10), (3.11), the matrix M contains two parame-
ters, the phase winding numbers ku and kv. For some arbitrary values of ku and kv we find some
stationary solutions to system (4.2), (4.3) and in Figs. 3 and 4 (near the obtained stationary
solutions) we show the phase graph for (a, b), |u(x, t)| = |a(t)| and |v(x, t)| = |b(t)|, according
to (4.1), and for a(t) and b(t). Figures 5(a) and 5(b) show the behavior of the real parts of u(x, t)
and v(x, t) corresponding to Figs. 3 and 4, respectively.
4.1. The line parameters (Lp1). In this subsection, for notational simplicity, we set α =
= 1.4653 × 10−7. For the line parameters (Lp1) and the phase winding numbers ku = 0.2 and
kv = −0.2, we find that
(−α,−α), (α, α), (−α, α), (α,−α),
are stationary solutions of system (4.2), (4.3) with a = b or a = −b. For the stationary solutions
(α, α) the matrix M and eigenvalues are
M =
(
0.000018407 −0.000073615
0.000073615 0.000018407
)
,
λ1 = 1.8407× 10−5 + 7.3615× 10−5i,
λ2 = 1.8407× 10−5 − 7.3615× 10−5i.
For the stationary solutions (α,−α) the matrix M and eigenvalues are
M =
(
1.8407× 10−5 7.3615× 10−5
7.3615× 10−5 1.8407× 10−5
)
,
λ′1 = −5.5208× 10−5,
λ′2 = 9.2022× 10−5.
Because Re λ1 > 0 and Re λ2 > 0, the stationary solutions (α, α) and (−α,−α) are unstable
foci. For the stationary solutions (−α, α) and (α,−α), the eigenvalues λ′1 and λ′2 of the matrix
M are real and λ′1 < 0 and λ′2 > 0. Therefore (−α, α) and (α,−α) are two saddle points.
Figure 3 corresponds to the stationary solution (α, α) of system (4.2), (4.3). Figure 3(a)
shows the solution in the (a, b) phase plane. Figures 3(b) and 3(c) show the plots of |u(x, t)| =
= |a(t)| and |v(x, t)| = |b(t)|, respectively Figures 3(d) and 3(e) show that the waves travel from
left to right and are unbounded as t → +∞.
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462 E. KENGNE, R. VAILLANCOURT
a b
c d
e
Fig. 3. (a): Plot of the stationary solution (α, α) of system (4.2), (4.3) in the (a, b) phase plane;
(b) and (c): plots of |u(x, t)| = |a(t)| and |v(x, t)| = |b(t)|, respectively; (d) and (e): graphs
of a(t) and b(t), respectively.
4.2. The line parameters (Lp2). For the line parameters (Lp2) and the phase winding
numbers ku = 0.57664 and kv = 0.80947 we find that
(
10−3, 10−4
)
is a stationary solution
to system (4.2), (4.3), and the matrix M and eigenvalues corresponding to this solution are
M =
(
−6.21× 10−4 −1.8× 10−4
−1.8× 10−4 −1.755× 10−4
)
,
λ1 = −6.8464× 10−4,
λ2 = −1.1186× 10−4.
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 463
a b
c d
Fig. 4. Graphs corresponding to the stationary solution (10−3, 10−4) to system (4.2), (4.3). (a): Plot of the
solution in the (a, b) phase plane; (b) and (c): plots of |u(x, t)| = |a(t)| and |v(x, t)| = |b(t)|, respecti-
vely; (b) and (d): plots of a(t) and b(t), respectively.
Because λ1 < 0 and λ2 < 0,
(
10−3, 10−4
)
is a stable node. Figure 4(a) shows the plot of
the solution in the (a, b) phase plane. Figure 4(b) shows the curves a(t) and |u(x, t)| = |a(t)|,
because a(t) > 0 for all t. As one can see from Figs. 4(b), (c), and (d), the curves a(t), |u(x, t)|,
|v(x, t)|, and b(t) are bounded as t → +∞.
5. Coherent structures. Many patterns that occur in experiments on traveling wave systems
or numerical simulations of the single and CGL equations exhibit local structures that have
an essentially time-independent shape and propagate with a constant velocity v. For these so-
called coherent structures, the spatial and temporal degrees of freedom are not independent:
apart from a phase factor, they are stationary in the co-moving frame z = x − vcoht. Since
the appropriate functions that describe the profiles of these coherent structures depend only
on the single variable z, these functions can be determined by ODEs. These are obtained by
substituting the appropriate Ansatz in the original CGL equations, which of course are parti-
al differential equations. Since the ODEs can themselves be written as a set of first-order
flow equations in a simple phase space, the coherent structures of the amplitude equations
correspond to certain orbits of these ODEs. Note that plane waves, since they have constant
profiles, are trivial examples of coherent structures (in the flow equations they correspond to fi-
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464 E. KENGNE, R. VAILLANCOURT
xed points stationary solutions). Sources and sinks connect, asymptotically, plane waves, and so
the corresponding orbits in the ODEs connect fixed points. Many different coherent structures
have been identified within this framework [21 – 26].
The counting arguments that give the multiplicity of such solutions are essentially based on
determining the dimensions of the stable and unstable manifolds near the fixed points. These
dimensions, together with the parameters of the Ansatz such as vcoh, determine, for some orbit,
the number of constraints and the number of free parameters that can be varied to fulfill these
constraints. We may illustrate the theoretical importance of counting arguments by recalling
that for the single CGL equation a continuous family of hole solutions has been known to exist
for some time [24]. Later, however, counting arguments showed that these source type solutions
were on general grounds expected to come as discrete sets, not as a continuous one-parameter
family [22, 23]. This suggested that there is some accidental degeneracy or hidden symmetry in
the single CGL equation, so that by adding a seemingly innocuous perturbation to the CGL
equation, the family of hole solutions should collapse to a discrete set. This was indeed found
to be the case [27, 28]. For further details of the results and implications of these counting
arguments for coherent structures in the single CGL equation, we refer to [22, 23, 29].
It should be stressed that counting arguments cannot prove the existence of certain coherent
structures nor can they establish the dynamical relevance of the solutions. They can only establi-
sh the multiplicity of the solutions, assuming that the equations have no hidden symmetries.
Imagine that we know either by an explicit construction or from numerical experiments that a
certain type of coherent structure solution does exist. The counting arguments then establish
whether this should be an isolated or discrete solution (at most a member of a discrete set of
them), or a member of a one-parameter family of solutions, etc. In the case of an isolated soluti-
on, there are no nearby solutions if we slightly change one of the parameters (like the velocity
vcoh). For a one-parameter family, the counting argument implies that when we start from a
known solution and change the velocity, we have enough other free parameters available to
make sure that there is a perturbed trajectory that flows into the proper fixed point as z → ∞.
For the two CGL Eqs. (3.10), (3.11) the counting can be performed by a straightforward
extension of the counting for the single CGL equation [22, 23, 30]. The Ansatz for coherent
structures of the CGL equations (3.10), (3.11) is the following generalization of the Ansatz for
the single CGL equation
u(x, t) = a(z = x− vcoht) exp
[
i
∫
Φ(z)dz − iωut
]
,
v(x, t) = b(z = x− vcoht) exp
[
i
∫
Ψ(z)dz − iωvt
]
.
(5.1)
Note that we take the velocities of the structures in the left and right modes equal, while
the frequencies ω are allowed to be different. This is due to the form of the coupling of the left-
and right-traveling modes, which is through the moduli of the amplitudes. It obviously does
not make sense to choose the velocities of u and v differently: for large times the cores of the
structures in u and v would then get arbitrarily far apart, and at the technical level, this would
be reflected by the fact that with different velocities we would not obtain simple ODEs for a
and b. Since the phases of u and v are not directly coupled, there is no a priori reason to take
the frequencies ωu and ωv equal. In (5.1) a(z) and b(z) are real amplitudes while Φ(z) and Ψ(z)
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 465
are the local wavenumbers and are functions of z = x − vcoht. In other words,
∫
Φ(z)dz and∫
Ψ(z)dz are real phases and are functions of z = x− vcoht.
Substituting Ansatz (5.1) into the CGL Eqs. (3.10), (3.11), we obtain the following set of
ODEs:
a′ = X, (5.2a)
b′ = Y, (5.3a)
X ′ = Φ2a +
P r (vcoh − S0)
|P |2
X − ErP r + EiP i
|P |2
(
3a2 + 2b2
)
X−
− 2
(
HrP r + H iP i
|P |2
+ 2
ErP r + EiP i
|P |2
)
abY − γrP r + γiP i − ωuP i
|P |2
a−
− QrP r + QiP i
|P |2
(
a2 + 2b2
)
a−
(
DrP r + DiP i
|P |2
a4 +
F rP r + F iP i
|P |2
a2b2+
+
KrP r + KiP i
|P |2
b4
)
a− ErP i − EiP r
|P |2
(
a3 + 2ab2
)
Φ +
(
vcohP
i − S0P
i
|P |2
)
aΦ, (5.4a)
Y ′ = Ψ2b +
P r (vcoh + S0)
|P |2
Y +
ErP r + EiP i
|P |2
(
3b2 + 2a2
)
Y +
+ 2
(
HrP r + H iP i
|P |2
+ 2
ErP r + EiP i
|P |2
)
abX − γrP r + γiP i − ωvP
i
|P |2
b−
− QrP r + QiP i
|P |2
(
b2 + 2a2
)
b−
(
DrP r + DiP i
|P |2
b4 +
F rP r + F iP i
|P |2
a2b2+
+
KrP r + KiP i
|P |2
a4
)
b +
ErP i − EiP r
|P |2
(
b3 + 2ba2
)
Ψ +
(
vcohP
i + S0P
i
|P |2
)
bΨ, (5.5a)
Φ′ = −2Φ
X
a
+
P i (S0 − vcoh)
|P |2
X
a
− EiP r − ErP i
|P |2
(
3aX + 2b2 X
a
)
−
− 2
(
H iP r −HrP i
|P |2
+ 2
EiP r − ErP i
|P |2
)
bY +
γrP i − γiP r − ωuP r
|P |2
−
− QiP r −QrP i
|P |2
(
a2 + 2b2
)
−
(
DiP r −DrP i
|P |2
a4 +
F iP r − F rP i
|P |2
a2b2 +
+
KiP r −KrP i
|P |2
b4
)
− ErP r + EiP i
|P |2
Φ
(
a2 + 2b2
)
+
(
vcohP
i − S0P
r
|P |2
)
Φ, (5.6a)
Ψ′ = −2Ψ
Y
b
− P i (S0 + vcoh)
|P |2
Y
b
+
EiP r − ErP i
|P |2
(
3bY + 2a2 Y
b
)
+
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466 E. KENGNE, R. VAILLANCOURT
+ 2
(
H iP r −HrP i
|P |2
+ 2
EiP r − ErP i
|P |2
)
aX +
γrP i − γiP r − ωvP
r
|P |2
−
− QiP r −QrP i
|P |2
(
b2 + 2a2
)
−
(
DiP r −DrP i
|P |2
b4 +
F iP r − F rP i
|P |2
a2b2+
+
KiP r −KrP i
|P |2
a4
)
+
ErP r + EiP i
|P |2
Ψ
(
b2 + 2a2
)
+
(
vcohP
i + S0P
r
|P |2
)
Ψ. (5.7a)
The solutions of these ODEs correspond to coherent structures of the GCGL equations (3.10),
(3.11) and vice-versa. From Appendix A, we have ErP r + EiP i = 0.
System (5.2a) – (5.7a) for (a,X, Φ, b, Y, Ψ) has singularities at a = 0 and b = 0. To overcome
this difficulty we introduce the “blow up” transform or σ-process [31]. Letting
X
a
= x,
Y
b
= y, (5.8)
we compute
X ′
a
= x2 + x′ and
Y ′
b
= y2 + y′ so that (5.2a) – (5.7a) become the following
regularized system for (a, x,Φ, b, y, Ψ):
a′ = ax, (5.2)
b′ = by, (5.3)
x′ = −x2 + Φ2 +
P r (vcoh − S0)
|P |2
x−
2
(
HrP r + H iP i
)
|P |2
b2y − γrP r + γiP i − ωuP i
|P |2
−
− QrP r + QiP i
|P |2
(
a2 + 2b2
)
−
(
DrP r + DiP i
|P |2
a4 +
F rP r + F iP i
|P |2
a2b2+
+
KrP r + KiP i
|P |2
b4
)
− ErP i − EiP r
|P |2
(
a2 + 2b2
)
Φ +
(
vcohP
i − S0P
i
|P |2
)
Φ, (5.4)
y′ = −y2 + Ψ2 +
P r (vcoh + S0)
|P |2
y +
2
(
HrP r + H iP i
)
|P |2
a2x− γrP r + γiP i − ωvP
i
|P |2
−
− QrP r + QiP i
|P |2
(
b2 + 2a2
)
−
(
DrP r + DiP i
|P |2
b4 +
F rP r + F iP i
|P |2
a2b2+
+
KrP r + KiP i
|P |2
a4
)
+
ErP i − EiP r
|P |2
(
b2 + 2a2
)
Ψ +
(
vcohP
i + S0P
i
|P |2
)
Ψ, (5.5)
Φ′ = −2Φx +
P i (S0 − vcoh)
|P |2
x− EiP r − ErP i
|P |2
(
3a2 + 2b2
)
x−
− 2
(
H iP r −HrP i + 2
(
EiP r − ErP i
)
|P |2
)
b2y +
γrP i − γiP r − ωuP r
|P |2
−
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 467
− QiP r −QrP i
|P |2
(
a2 + 2b2
)
−
(
DiP r −DrP i
|P |2
a4 +
F iP r − F rP i
|P |2
a2b2+
+
KiP r −KrP i
|P |2
b4
)
+
(
vcohP
i − S0P
r
|P |2
)
Φ, (5.6)
Ψ′ = −2Ψy − P i (S0 + vcoh)
|P |2
y +
EiP r − ErP i
|P |2
(
3b2 + 2a2
)
y+
+ 2
(
H iP r −HrP i + 2
(
EiP r − ErP i
)
|P |2
)
a2x +
(
γrP i − γiP r − ωvP
i
|P |2
)
−
− QiP r −QrP i
|P |2
(
b2 + 2a2
)
−
(
DiP r −DrP i
|P |2
b4 +
F iP r − F rP i
|P |2
a2b2+
+
KiP r −KrP i
|P |2
a4
)
+
(
vcohP
i + S0P
r
|P |2
)
Ψ. (5.7)
Compared to the flow equations for the single CGL equation [22, 23], there are two important
differences that should be noted: (i) Instead of the velocity vcoh we now have velocities vcoh±S0.
This is simply due to the fact that the linear group velocity terms cannot be transformed away.
(ii) The nonlinear coupling term in the CGL equations shows up only in the flow equations for
x and y.
The fixed points of these flow equations, the points in phase space at which the right-hand
sides of Eqs. (5.2) – (5.7) vanish, describe the asymptotic states for z → ±∞ of the coherent
structures. What are these fixed points? From Eq. (5.2) we find that either x or a is equal to
zero at a fixed point, and similarly, from Eq. (5.3) it follows that either y or b vanishes. For the
sources and sinks of (3.10) and (3.11) that we wish to study, the asymptotic states are left- and
right-traveling waves. Therefore the fixed points of interest to us have either both x and b or
both y and a equal to zero, and we search for heteroclinic orbits connecting these two fixed
points.
5.1. Coherent structures in systems described by the GGL equation (3.13). As it is noted
above the GGL Eqs. (3.13) is obtained from the GCGL Eqs. (3.10), (3.11) by setting either
u = 0 or v = 0. Suppose that the GGL Eq. (3.13) is obtained from (3.10), (3.11) by setting
v(x, t) = 0. Then by setting b = 0, Y = 0, F = K = H = 0, Ψ(z) = 0 in system (5.2a) –
(5.7a) and using (5.8) for Y and a, we obtain Eqs. (5.1), (5.3), and (5.5) in which we set b = 0:
a′ = ax, (5.9)
x′ = −x2 + Φ2 +
P r (vcoh − S0)
|P |2
x− γrP r + γiP i − ωuP i
|P |2
− QrP r + QiP i
|P |2
a2−
− DrP r + DiP i
|P |2
a4 − ErP i − EiP r
|P |2
a2Φ +
P i (vcoh − S0)
|P |2
Φ, (5.10)
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468 E. KENGNE, R. VAILLANCOURT
Φ′ = −2Φx +
P i (S0 − vcoh)
|P |2
x− 3
EiP r − ErP i
|P |2
a2x +
γrP i − γiP r − ωuP r
|P |2
−
− QiP r −QrP i
|P |2
a2 − DiP r −DrP i
|P |2
a4 +
vcohP
i − S0P
r
|P |2
Φ. (5.11)
The solutions of these ODEs correspond to coherent structures of the GGL equation (3.13)
and vice-versa.
The fixed points of the ODEs have, according to (5.9), either a = 0 or x = 0. The values
of x and Φ for the fixed points with a = 0 are related through the dispersion relation of the
linearized equation, or, what amounts to the same, by the equation obtained by setting the
right-hand side of (5.10) and (5.11) equal to zero and taking a = 0. We refer to these fixed
points as linear fixed points [22, 23] and we denote them by L±, where the subscript indicates
the sign of x. This means that the behavior near some L+ corresponds to a situation in which the
amplitude is growing away from zero to the right, while the behavior near some L− describes
the situation in which the amplitude a decays to zero.
Since a fixed point with a 6= 0 and x = 0 corresponds to nonlinear traveling waves, the
corresponding fixed points are referred to as nonlinear fixed points [22, 23] which we denote
by N±, where the subscript now indicates the sign of the nonlinear group velocity su = s of
the corresponding traveling wave. Thus, the amplitude near an N+ can either grow (x > 0) or
decay (x < 0) with increasing z.
Coherent structures correspond to orbits which go from one of the fixed point to another
one or back to the original one, and the counting analysis amounts to establishing the dimensi-
ons of the in- and out-going manifolds of these fixed points. In combination with the number of
free parameters (in this case vcoh and ωu), this yields the multiplicity of orbits connecting these
fixed points, and, therefore, of the multiplicity of the corresponding coherent structures.
Since there are three flow Eqs. (5.9) – (5.11), there are three eigenvalues of the linear flow
near each fixed point. When we perform the counting analysis for these fixed points we will only
need the signs of the real parts of the three eigenvalues, since they determine whether the flow
along the corresponding eigendirection is inwards (−) or outwards (+). We will denote the signs
by pluses and minuses, so that L−(+,+,−) denotes an L− fixed point with two eigenvalues with
positive real parts, and one with a negative real part.
From Eqs. (5.9) – (5.11), we obtain the fixed point equations:
ax = 0, (5.12)
−|P |2x2 + |P |2Φ2 + P r (vcoh − S0) x− γrP r − γiP i + ωuP i −
(
QrP r + QiP i
)
a2−
−
(
DrP r + DiP i
)
a4 +
(
EiP r − ErP i
)
a2Φ + P i (vcoh − S0) Φ = 0, (5.13)
−2|P |2Φx + P i (S0 − vcoh) x +
(
QrP i −QiP r
)
a2 + γrP i − γiP r−
− ωuP r +
(
DrP i −DiP r
)
a4 −
(
ErP r + EiP i
)
Φa2 +
(
vcohP
i − S0P
r
)
Φ = 0.
(5.14)
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 469
From (5.12) we immediately obtain that fixed points either have a = 0 (linear fixed points
denoted as L) or a 6= 0 and x = 0 (nonlinear fixed points denoted by N). Let (a0, x0,Φ0) be a
fixed point of the ODEs (5.9) – (5.11). Then the linear flow equations in its neighborhood is
a′ = x0a + a0x, (5.15)
x′ =
−2a0
(
QrP r + QiP i
)
− 4a3
0
(
DrP r + DiP i
)
− 2Φ0a0
(
EiP r − ErP i
)
|P |2
a+
+
P r (vcoh − S0)− 2|P |2x0
|P |2
x +
vcohP
i − S0P
i + 2 |P |2 Φ0Φ + a2
0
|P |2
Φ, (5.16)
Φ′ =
2a0
(
QrP i −QiP r
)
+ 6a0x0
(
ErP i − EiP r
)
+ a3
0
(
DrP i −DiP r
)
|P |2
a+
+
P i (S0 − vcoh)− 2|P |2Φ0 + 3a2
0
(
ErP i − EiP r
)
|P |2
x +
vcohP
i − S0P
r − 2|P |2x0
|P |2
Φ,
(5.17)
with matrix
ML =
x0 a0 0
ã1 x̃1 Φ̃1˜̂a2 x̃2 Φ̃2
where
ã1 =
−2a0
(
QrP r + QiP i
)
− 4a3
0
(
DrP r + DiP i
)
− 2Φ0a0
(
EiP r − ErP i
)
|P |2
,
ã2 =
2a0
(
QrP i −QiP r
)
+ 6a0x0
(
ErP i − EiP r
)
+ a3
0
(
DrP i −DiP r
)
|P |2
,
x̃1 =
P r (vcoh − S0)− 2 |P |2 x0
|P |2
,
x̃2 =
P i (S0 − vcoh)− 2 |P |2 Φ0 + 3a2
0
(
ErP i − EiP r
)
|P |2
,
Φ̃1 =
vcohP
i − S0P
i + 2 |P |2 Φ0 + a2
0
|P |2
,
Φ̃2 =
vcohP
i − S0P
r − 2 |P |2 x0
|P |2
.
Solving the fixed point Eqs. (5.12) – (5.14) and calculating the eigenvalues of ML yield the di-
mensions of the incoming and outgoing manifolds of these fixed points. Note that according
to our convention, a fixed point with a two-dimensional outgoing and one-dimensional ingoi-
ng manifold is denoted as (+,+,−). We can restrict calculations to the case of positive vcoh,
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470 E. KENGNE, R. VAILLANCOURT
since the case of negative vcoh can be found by the left-right symmetry operation z → −z,
vcoh → −vcoh, x → −x, and Φ → −Φ (for some examples we may take vcoh negative).
The eigenvalues of ML are solutions of the cubic equation
λ3 −
(
x0 + x̃1 + Φ̃2
)
λ2 +
(
x0x̃1 − ã1a0 + x0Φ̃2 − x̃1Φ̃2 − x̃2Φ̃1
)
λ−
−
(
x0x̃1Φ̃2 − x0x̃2Φ̃1 − a0ã1Φ̃2 + a0ã2Φ̃1
)
= 0,
with coefficients
P2 = −
(
x0 + x̃1 + Φ̃2
)
, P1 = x0x̃1 − ã1a0 + x0Φ̃2 − x̃1Φ̃2 − x̃2Φ̃1,
P0 = −x0x̃1Φ̃2 + x0x̃2Φ̃1 + a0ã1Φ̃2 − a0ã2Φ̃1.
We may read the signs of the real parts of the solution of these three equations from the follow-
ing [22, 23]:
P0 > 0
{
P2 > 0, P1P2 > P0 : (−,−,−) case (i),
else : (+,+,−) case (ii),
P0 < 0
{
P2 < 0, P1P2 < P0 : (+,+,+) case (iii),
else : (+,−,−) case (iv).
According to these rules, we need to know the sign of the real parts of the eigenvalues of ML
for the three combinations of the coefficients, namely, P0, P2 and
P1P2 − P0 =(a0x0 + a0x̃1) ã1 +
(
x̃− x2
0
)
Φ̃2 + (x̃1x̃2 + a0ã2) Φ̃1−
− x0x̃
2
1 − x2
0x̃1 + (x̃1 − x0) Φ̃2
2 + x̃2Φ̃1Φ̃2.
For a0 = 0 (a linear fixed point) and x0 = 0 and a0 6= 0 (a nonlinear fixed point), we have the
respectively relations:
P1P2 − P0 =
(
x̃− x2
0
)
Φ̃2 + x̃1x̃2Φ̃1 − x0x̃
2
1 − x2
0x̃1 + (x̃1 − x0) Φ̃2
2 + x̃2Φ̃1Φ̃2,
P1P2 − P0 = a0x̃1ã1 + x̃Φ̃2 + (x̃1x̃2 + a0ã2) Φ̃1 + x̃1Φ̃2
2 + x̃2Φ̃1Φ̃2.
5.1.1. Linear fixed points. In the case of a linear fixed point, we have a = 0, and from
(5.12) – (5.14) we obtain the fixed-point equations
|P |2Φ2 + P i (vcoh − S0) Φ + P r (vcoh − S0) x− |P |2x2 − γrP r − γiP i + ωuP i = 0, (5.18)(
vcohP
i − S0P
r
)
Φ− 2|P |2Φx + P i (S0 − vcoh) x + γrP i − γiP r − ωuP r = 0. (5.19)
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 471
Since we may choose ωu and vcoh freely, we can take x such that P i (S0 − vcoh) x+γrP i−γiP r−
−ωuP r = 0. Equation (5.19) then gives
x0 =
vcohP
i − S0P
r
2|P |2
,
and from (5.18) we obtain
Φ0 =
P iS0−vcoh ±
√
[P i (vcoh−S0)]
2+4|P |2
(
P r (S0−vcoh) x0+|P |2 x2
0+γrP r+γiP i−ωuP i
)
2|P |2
with
ωu =
2|P |2
(
γrP i − γiP r
)
+ P i (S0 − vcoh)
(
vcohP
i − S0P
r
)
2|P |2P r
.
Under these conditions, we obtain three linear fixed points(
0,
vcohP
i − S0P
r
2|P |2
, Φ0
)
. (5.20)
To obtain all the solutions of system (5.18), (5.19), we can proceed as follows: we solve (5.19)
for x and obtain
x =
(
vcohP
i − S0P
r
)
Φ + γrP i − γiP r − ωuP r
2|P |2Φ + P i(vcoh − S0)
. (5.21a)
Replacing this expression for x in (5.18) we obtain the following quartic equation for Φ:
(
|P |2Φ2 + P i(vcoh − S0)Φ
) [
2|P |2Φ + P i(vcoh − S0)
]2 +
+ P r(vcoh − S0)
[(
vcohP
i − S0P
r
)
Φ + γrP i − γiP r − ωuP r
] [
2|P |2Φ + P i (vcoh − S0)
]
−
− |P |2
[(
vcohP
i − S0P
r
)
Φ + γrP i − γiP r − ωuP r
]2−
−
[
2|P |2Φ + P i (vcoh − S0)
]2 (
γrP r + γiP i − ωuP i
)
= 0. (5.21)
At the fixed points, the eigenvalues are given by
x0,
x̃1 + Φ̃1 ±
√(
x̃1 + Φ̃1
)2
− 4x̃1Φ̃2 + 4x̃2Φ̃1
2
:=
x̃1 + Φ̃1 ±
√
∆
2
.
To establish the signs of the real parts of the eigenvalues, if ∆ < 0 we need to determine the
signs of x0 and x̃1 + Φ̃1. But if ∆ > 0 we need to determine the signs of x0 and x̃1 + Φ̃1 ±
√
∆.
Let us first establish the signs of x0. This is important in establishing whether the evanescent
wave decays to the left (L+) or to the right (L−). For the fixed points (5.20) we have
x0 =
vcohP
i − S0P
r
2|P |2
.
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472 E. KENGNE, R. VAILLANCOURT
Because P r > 0 and P i < 0 and we are in the case where vcoh > 0, we conclude that x0 < 0,
and this means that the evanescent wave decays to the right (L−).
For the line parameters (Lp1) and vcoh = 6×105 and ωu = 106, (5.20) gives us the two fixed
points
p1 = (0,−1.8617× 108,−1.5765× 1011), p2 = (0,−1.8617× 108,−1.5722× 1011).
For p1, the eigenvalues of ML are
−1.8617× 108, 2.1757× 108 ± 3.153× 1011i
which means that p1 is an L−(+,+,−) fixed point. Now, p2 is an L−(+,+,−) fixed point because
the corresponding matrix ML has two eigenvalues with positive real parts and one eigenvalue
with negative real part.
For the above values of vcoh and ωu and line parameters (Lp1), (5.21) gives
Φ0 ∈
{
−0.11601, 3.1256× 109, 3.155× 109, 5.477× 109
}
.
For these values of Φ we obtain the following values of x (see (5.21a)):
x ∈
{
10045, −1.8617× 108, −1.8617× 108, −1.8617× 108
}
.
We then have four linear fixed points:
(0, 10045,−0.11601), (0,−1.8617× 108, 3.1256× 109),
(0,−1.8617× 108, 3.155× 109), (0,−1.8617× 108, 5.477× 109),
that is, one L+ fixed point and three L− fixed points. The L+ fixed point is an L+(+,+,−) fixed
point while the three L− fixed points are L+(+,+,−) fixed points. For the L+(+,+,−) fixed
point (0, 10045,−0.11601) and the L−(+,+,−) fixed point
(
0,−1.8617× 108, 3.1256× 109
)
, Fi-
gs. 5 and 6 plot the real and imaginary parts of u, respectively, as functions of x at fixed time t.
In these figures we observe a left-right symmetry of the waves that is broken in domains where
they are traveling to the left and domains where they are traveling to the right. In Fig. 5 the
waves are emitted from a point and in Fig. 6, they are emitted from a wall. This point or wall is
a source.
If we take Φ = 0 in (5.21a) – (5.21), we have the fixed point(
0,
γrP i − γiP r − ωuP r
P i (vcoh − S0)
, 0
)
,
where vcoh and ωu are such that
P rP i (vcoh − S0)
2 (γrP i − γiP r − ωuP r
)
− |P |2
(
γrP i − γiP r − ωuP r
)2−
−
[
P i (vcoh − S0)
]2 (
γrP r + γiP i − ωuP i
)
= 0.
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 473
a b
c d
Fig. 5. Plots of (a) Re u(x, t) and (b) Re v(x, t) as functions of t at fixed spatial variable x. Both waves travel
to the right changing form as x increases.
a b
Fig. 6. Plots of Re u(x, t) as a function of z, (a) at t = 10−5 and (b) at t = 10−6.
If these values of vcoh and ωu verify the condition
γrP i − γiP r − ωuP r
(vcoh − S0)
< 0,
then x0 > 0. Reversing the inequality, we have x0 < 0.
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474 E. KENGNE, R. VAILLANCOURT
If we choose vcoh so that vcoh = S0, then (5.21a) – (5.21) gives the linear fixed points(
0,
S0
(
P i − P r
)
Φ0 + γrP i − γiP r − ωuP r
2|P |2Φ0
,Φ0
)
,
where Φ0 is any real solution of the quartic equation
4|P |6Φ4 −
(
4|P |4
(
γrP r + γiP i − ωuP i
)
+ |P |2S2
0
(
P i − P r
)2 Φ2
)
Φ2−
− 2|P |2S0
(
P i − P r
) (
γrP i − γiP r − ωuP r
)
Φ− |P |2
(
γrP i − γiP r − ωuP r
)2 = 0.
If, for a given Φ0, any real solution of this last equation satisfies one of the conditions
γrP i − γiP r − ωuP r
2|P |2 + S0 (P r − P i)
> Φ0 > 0 or
γrP i − γiP r − ωuP r
2|P |2 + S0 (P r − P i)
< Φ0 < 0,
then x0 > 0; else x0 < 0.
5.1.2. Nonlinear fixed points. The analysis of the nonlinear fixed points goes along the same
lines. Since the nonlinear fixed point has x = 0 and a 6= 0, the fixed point becomes
|P |2Φ2 +
(
EiP r − ErP i
)
a2Φ + P i (vcoh − S0) Φ−
(
DrP r + DiP i
)
a4−
−
(
QrP r + QiP i
)
a2 − γrP r − γiP i + ωuP i = 0, (5.22)(
DrP i −DiP r
)
a4 +
(
QrP i −QiP r
)
a2 + γrP i − γiP r − ωuP r+
+
(
vcohP
i − S0P
r
)
Φ = 0. (5.23)
System (5.22), (5.23) gives
Φ =
(
DrP i −DiP r
)
a4 +
(
QrP i −QiP r
)
a2 + γrP i − γiP r − ωuP r
S0P r − vcohP i
,
where a is any real solution of the equation of degree eight:
0 = |P |2
((
DrP i −DiP r
)
a4 +
(
QrP i −QiP r
)
a2 + γrP i − γiP r − ωuP r
)2 +
+
(
S0P
r − vcohP
i
) ((
EiP r − ErP i
)
a2 + P i (vcoh − S0)
) ((
DrP i −DiP r
)
a4+
+
(
QrP i −QiP r
)
a2 + γrP i − γiP r − ωuP r
)
−
(
S0P
r − vcohP
i
)2 (
DrP r + DiP i
)
a4−
−
((
QrP r + QiP i
)
a2 + γrP r + γiP i − ωuP i
) (
S0P
r − vcohP
i
)2
.
For the line parameters (Lp1), we find the following N+(+,+,−):
(3301.8, 0, 2.7937× 109) and (−3301.6, 0, 2.7937× 109),
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 475
a b
Fig. 7. Plot of Im u(z, t) as function of z, (a) at t = 0 and (b) at t = 10−14.
and corresponding to ωu = 106, vcoh = 6× 105, and the following N+(+,+,−):
(3301.6, 0, 2.7935× 109) and (−3301.8, 0, 2.7937× 109),
corresponding to vcoh = S0 = 5.1332 × 105 and ωu = 106. For the nonlinear fixed point(
3301.8, 0, 2.7937× 109
)
we show the evolution of (the real part of) u(x, t) = u(z, t) in Fig. 7.
This figure shows the dependence of u on a fixed time t. From Fig. 7, we observe the existence
of a sink, a point that absorbs waves.
5.2. Coherent structures in systems described by the GCGL equations (3.10), (3.11). While
the counting for the CGL equations follows unambiguously from that for the single CGL, there
are various nontrivial subtleties in the extension of those results to the CGL equations that
require careful discussion.
Suppose we want to perform the fixed points with x = b = 0, which corresponds to the
case in which only a right-traveling wave is present. The fixed point equations that follow from
(5.5) – (5.7) are, up to a change of vcoh → vcoh + S0, equal to the fixed point equations for the
nonlinear fixed points of the single CGL Eq. (3.13) with U = u and ±S0 = S0. To solve the
fixed point equations that follow from (5.4) – (5.6), note that a is a constant at the fixed point
and so term −γrP r − γiP i − a2
(
QrP r + QiP i
)
− a4
(
DrP r + DiP i
)
can be absorbed in the
P iωu term. Since we may choose ωu freely, for the counting analysis we can forget about the
−γrP r − γiP i − a2
(
QrP r + QiP i
)
− a4
(
DrP r + DiP i
)
as we may think of it as having been
absorbed into the frequency. The fixed point equations that follow from (5.4) and (5.6) give
a = ±
√
(vcoh − S0) P i
ErP i − EiP r
, (5.24)
and
Φ =
(
ErP i − EiP r
)−1
[(
γrP i − γiP r − ωuP r
) (
ErP i − EiP r
)2 +
+
(
vcohP
i − S0P
i
) (
QrP i −QiP r
) (
ErP i − EiP r
)
+
(
vcohP
i − S0P
i
)2 (
DrP i −DiP r
)]
×
×
[(
ErP i − EiP r
) (
S0P
r − vcohP
i
)]−1
(5.25)
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476 E. KENGNE, R. VAILLANCOURT
and we obtain the expression for ωu:
ωu =
γrP r + γiP i
P i
+
(
vcohP
i − S0P
i
) (
QrP r + QiP i
)
(ErP i − EiP r) P i
+
(
vcohP
i − S0P
i
)2 (
DrP r + DiP i
)
P i (ErP i − EiP r)2
.
(5.26)
Here vcoh is a parameter and will be chosen so that vcoh−S0 < 0, because P i
(
ErP i−EiP r
)
< 0.
The fixed point equations that follow from (5.5) – (5.7) are two equations with two unknowns,
y and Ψ. These two equations contain two parameters, vcoh and ωv. Because a is a constant at
the fixed point term γrP i − γiP r + 2a2
(
QrP i −QiP r
)
+
(
KrP i −KiP r
)
a4 can be absorbed
in the −ωvP
i term. We then choose ωv from condition
γrP i − γiP r − ωvP
i + 2a2
(
QrP i −QiP r
)
+
(
KrP i −KiP r
)
a4 = 0
and obtain
ωv =
γrP i − γiP r
P i
+
2
(
vcohP
i − S0P
i
) (
QrP i −QiP r
)
(ErP i − EiP r) P i
+
(
KrP i −KiP r
) (
vcohP
i − S0P
i
)2
(ErP i − EiP r)2 P i
.
(5.27)
One of the fixed point equations that follow from (5.5) – (5.7) then becomes
−
(
P i (S0 + vcoh) + 2a2
(
ErP i − EiP r
))
y +
(
vcohP
i + S0P
r − 2|P |2y
)
Ψ = 0.
Solving this last equation in Ψ, we obtain
Ψ =
[
P i (S0 + vcoh) + 2a2
(
ErP i − EiP r
)]
y
vcohP i + S0P r − 2|P |2y
. (5.28)
If we replace this expression for Ψ in the second equation of the fixed points equations, we
obtain for determination of y the following fourth degree equation:
− |P |2
(
vcohP
i + S0P
r − 2 |P |2 y
)2
y2 + |P |2
[
P i (S0 + vcoh) + 2a2
(
ErP i − EiP r
)]2
y2+
+
(
vcohP
i + S0P
r − 2|P |2y
)2 [P r (vcoh + S0)] y +
(
vcohP
i + S0P
r − 2|P |2y
)
×
×
(
2a2
(
ErP i − EiP r
)
+ vcohP
i + S0P
i
) [
P i (S0 + vcoh) + 2a2
(
ErP i − EiP r
)]
y+
+
{
a4
(
KrP r + KiP i
)
−
[
γrP r + γiP i − ωvP
i − 2a2
(
QrP r + QiP i
)]}
×
×
(
vcohP
i + S0P
r − 2|P |2y
)2 = 0. (5.29)
Since the fixed points of interest in for sources and sinks always have either b = 0 or a = 0,
the linearization around them largely parallels the analysis of the single GCGL equation. For,
when we linearize about the fixed point b = 0, we do not have to take the variation of a
into account in the coupling term and this allows us, for the counting argument, to absorb
these terms into a frequency and redefined ω as discussed above. Once this is done, the linear
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 477
Fig. 8. Plot of Re u(z, t) as a function of z at
fixed time t. One wave travels to the
right and the other to the left.
equations for the mode whose amplitude vanishes at the fixed point do not involve the other
mode variables at all.
For the line parameters (Lp1) and for vcoh = 6× 105, equations (5.24) – (5.29) give
ωu =
(
−8.681× 1016
)
, ωv =
(
−2.5841× 1014
)
,
a = ±328.83, Φ = 19.842,
y = −1.5382× 108, Ψ = 1.7486× 105,
or y = 8.879× 108, Ψ = 4.881× 105,
whence the following fixed points of system (5.2) – (5.7) for (a, x,Φ, b, y, Ψ):
p±(±328.83, 0, 19.842, 0,−1.5382× 108, 1.7486× 105),
q±(±328.83, 0, 19.842, 0, 8.879× 108, 4.881× 105).
After some investigation we find that p+ is an (N+(−,−,−), L−(+,+,−)) fixed point, that is,
the matrix of the flow equations corresponding to this fixed point possesses four eigenvalues
with negative real parts and two eigenvalues with positive real parts. For this fixed point we
show the evolution of (the real parts of) u(z, t) and v(z, t) in Figs. 8 and 9, respectively. Figure 8
show the dependence of (the real part of) u on z (or, what is the same, on x) for a given time t.
In this figure, we notice the existence of a source (see Figs. 9(a) and 9(b)). Figure 10 shows
the dependence of v on z (that is, on x) for a given time t. In this figure, one can see a wave
traveling form −∞ to the right, and vanishing somewhere before z = −26.5 (see Fig. 10(a)).
From z = −25.5 to −3.6, there is not any movement (see Fig. 10(b)). There exists a source at
some point after z = −2.2 and before z = 2000 (see Fig. 10(c, d)). The wave that propagates
from this source to the left vanish before z = −3.6, while the wave that travels from the source
to the right vanishes before z = 6200 (see Fig. 10(e)).
System (5.2) – (5.7) has the invariant planes a = 0 and b = 0. Thus, by the σ-process, a
singular point is transformed to a (singular) invariant plane. On the invariant planes a = 0 and
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478 E. KENGNE, R. VAILLANCOURT
a b
Fig. 9. Plot of Re u(z, t) as a function of z at fixed time t; (a) wave traveling to the right and (b) wave
traveling to the left.
b = 0, the system for (a, x,Φ, b, y, Ψ) reduces to
x′ = −x2 + Φ2 +
P r (vcoh − S0)
|P |2
x−
(
vcohωu
P r
|P |2
+
γrP r + γiP i
|P |2
)
+
(
vcohP
i − S0P
i
|P |2
)
Φ,
(5.4b)
y′ = −y2 + Ψ2 +
P r (vcoh + S0)
|P |2
y +
(
vcohωv
P r
|P |2
+
γrP r + γiP i
|P |2
)
+
(
vcohP
i + S0P
i
|P |2
)
Ψ,
(5.5b)
Φ′ = −2Φx +
P i (S0 − vcoh)
|P |2
x +
(
vcohωu
P i
|P |2
− γiP r − γrP i
|P |2
)
+
(
vcohP
i − S0P
r
|P |2
)
Φ,
(5.6b)
Ψ′ = −2Ψy − P i (S0 + vcoh)
|P |2
y +
(
vcohωv
P i
|P |2
− γiP r − γrP i
|P |2
)
+
(
vcohP
i + S0P
r
|P |2
)
Ψ.
(5.7b)
Because (5.4b) and (5.6b) depend only on x and Φ, and (5.5b) and (5.7b) depend only on y
and Ψ, we obtain the following two systems of ODEs:
x′ = −x2 + Φ2 +
P r (vcoh − S0)
|P |2
x−
(
vcohωu
P r
|P |2
+
γrP r + γiP i
|P |2
)
+
(
vcohP
i − S0P
i
|P |2
)
Φ,
(5.8a)
Φ′ = −2Φx +
P i (S0 − vcoh)
|P |2
x +
(
vcohωu
P i
|P |2
− γiP r − γrP i
|P |2
)
+
(
vcohP
i − S0P
r
|P |2
)
Φ
(5.9a)
and
y′ = −y2 + Ψ2 +
P r (vcoh + S0)
|P |2
y +
(
vcohωv
P r
|P |2
+
γrP r + γiP i
|P |2
)
+
(
vcohP
i + S0P
i
|P |2
)
Ψ,
(5.10a)
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 479
a b
c d
e
Fig. 10. Plot of Re v(z, t) as a function of z at fixed time t. (a) z ∈ [−∞,−26.33], (b) z ∈ [−26.33,−3.71],
(c) z ∈ [−3.71, 0], (d) z ∈ [0, 2120], (e) z ∈ [2120, 6200].
Ψ′ = −2Ψy − P i (S0 + vcoh)
|P |2
y +
(
vcohωv
P i
|P |2
− γiP r − γrP i
|P |2
)
+
(
vcohP
i + S0P
r
|P |2
)
Ψ.
(5.11a)
Note that, if (x0,Φ0) and (y0,Ψ0) are fixed points of systems (5.8a), (5.9a) and (5.10a), (5.11a),
respectively, then (0, x0,Φ0, b0, y0,Ψ0) is a fixed point of system (5.2) – (5.7). Because we wish
to study the sources and sinks of (3.10) and (3.11), the fixed points of interest to us have either
x or y equal to zero.
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480 E. KENGNE, R. VAILLANCOURT
If we consider the six variables ã, x̃, Φ̃, b̃, ỹ, and Ψ̃ as the elements of a vector w and linearize
the flow Eqs. (5.2) – (5.7) about the fixed point (0, x0,Φ0, 0, y0,Ψ0), we can write the linearized
equations in the form w′
i =
∑
j Mijwj , where the 6× 6 matrix M has the following structure:
M =
x0 0 0 0 0 0
0
−2x0+
+
P r (vcoh − S0)
|P |2
2Φ0+
+
vcohP
i − S0P
i
|P |2
0 0 0
0
−2Φ0+
+
P i (S0 − vcoh)
|P |2
−2x0+
+
vcohP
i − S0P
r
|P |2
0 0 0
0 0 0 y0 0 0
0 0 0 0
−2y0+
+
P r (vcoh + S0)
|P |2
2Ψ0+
+
vcohP
i + S0P
i
|P |2
0 0 0 0
−2Ψ0−
−P i (S0 + vcoh)
|P |2
−2y0+
+
vcohP
i + S0P
r
|P |2
.
The eigenvalue of M are simply given by the eigenvalues of the upper-left and lower-right block
matrices. The secular equation for the eigenvalues of each of the two blocks has the form
P3λ
3 + P2λ
2 + P1λ + P0 = 0.
We need only know the number of solutions of the secular equation that have positive real part,
and instead of solving the equation explicitly, we can proceed as follows.
For the cubic equation of the above form where P3 > 0, we may read off the signs of the
real parts of the solution to this equation from the following [22, 23]:
P0 > 0
{
P2 > 0, P1P2 > P0P3 : (−,−,−) case (i),
else : (+,+,−) case (ii),
P0 < 0
{
P2 < 0, P1P2 < P0P3 : (+,+,+) case (iii),
else : (+,−,−) case (iv).
According to these rules, there are six combinations of the coefficients whose sign we need to
know, namely, P0, P2, P1P2 − P0P3 for each of the two blocks (see (5.30) – (5.35)).
For the upper-left block matrices, we have
λ3 +
(
3x0 −
(
P i + P r
)
vcoh − 2S0P
r
|P |2
)
λ2 +
(
4Φ2
0−
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 481
−
(
P i
)2 (vcoh − S0)
2 − 2x0P
r (vcoh − S0)
(
vcohP
i − S0P
r
)
+ x0|P |2
[(
P i + P r
)
vcoh − 2S0P
r
]
|P |4
)
λ+
+ x0
((
P i
)2 (vcoh − S0)
2 − 2x0P
r (vcoh − S0)
(
vcohP
i − S0P
r
)
+ 2x0|P |2
[(
P i + P r
)
vcoh − 2S0P
r
]
|P |4
−
−4Φ2
0 − 4x2
0
)
= 0,
and
P0 = x0×
×
((
P i
)2 (vcoh−S0)
2−2x0P
r (vcoh−S0)
(
vcohP
i−S0P
r
)
+2x0|P |2
[(
P i+P r
)
vcoh−2S0P
r
]
|P |4
−
−4Φ2
0 − 4x2
0
)
, (5.30)
P2 = 3x0 −
(
P i + P r
)
vcoh − 2S0P
r
|P |2
, (5.31)
P1P2 − P0P3 =
((
P i
)2 (vcoh − S0)
2 − 2x0P
r (vcoh − S0)
(
vcohP
i − S0P
r
)
+
+x0|P |2
[(
P i + P r
)
vcoh − 2S0P
r
]) (P i + P r
)
vcoh − 2S0P
r
|P |6
−
−
4Φ2
0
[(
P i + P r
)
vcoh − 2S0P
r
]
|P |2
+ 4x3
0 + 16x0Φ2
0 − x0×
×
3
(
P i
)2 (vcoh−S0)
2−8x0P
r (vcoh−S0)
(
vcohP
i−S0P
r
)
+5x0|P |2
[(
P i+P r
)
vcoh−2S0P
r
]
|P |4
.
(5.32)
For the lower-right block, we have
λ3 +
(
3y0 −
(
P i + P r
)
vcoh + 2S0P
r
|P |2
)
λ2−
−
((
P i
)2 (vcoh+S0)
2−2x0P
r (vcoh+S0)
(
vcohP
i+S0P
r
)
+y0|P |2
[(
P i+P r
)
vcoh+2S0P
r
]
|P |4
−
−4Ψ2
0
)
λ + y0×
×
((
P i
)2 (vcoh+S0)
2−2y0P
r (vcoh+S0)
(
vcohP
i+S0P
r
)
+2y0|P |2
[(
P i+P r
)
vcoh+2S0P
r
]
|P |4
−
−4Ψ2
0 − 4y2
0
)
= 0
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482 E. KENGNE, R. VAILLANCOURT
and
P0 = y0×
×
((
P i
)2 (vcoh+S0)
2−2y0P
r (vcoh+S0)
(
vcohP
i+S0P
r
)
+2y0|P |2
[(
P i+P r
)
vcoh+2S0P
r
]
|P |4
−
−4Ψ2
0 − 4y2
0
)
, (5.33)
P2 = 3y0 −
(
P i + P r
)
vcoh + 2S0P
r
|P |2
, (5.34)
P1P2 − P0P3 =
((
P i
)2 (vcoh + S0)
2 − 2y0P
r (vcoh + S0)
(
vcohP
i + S0P
r
)
+
+x0|P |2
[(
P i+P r
)
vcoh+2S0P
r
]) (P i+P r
)
vcoh+2S0P
r
|P |6
−
4Ψ2
0
[(
P i+P r
)
vcoh+2S0P
r
]
|P |2
+
+ 16y0Ψ2
0 + 4y3
0 − y0×
×
3
(
P i
)2 (vcoh+S0)
2−8y0P
r (vcoh+S0)
(
vcohP
i+S0P
r
)
+5y0|P |2
[(
P i+P r
)
vcoh+2S0P
r
]
|P |4
.
(5.35)
It is seen from (5.30) and (5.33) that for P0 to be different from zero it is necessary that
x0 6= 0 and y0 6= 0. Let
F (ω, β) = |P |2
(
γiP r − γrP i − vcohωP i
)2 +
+ vcoh
(
vcohP
i + (−1)βS0P
r
)(
P i + (−1)βS0P
i
) (
γiP r − γrP i − vcohωP i
)
−
−
(
vcohP
i + (−1)βS0P
r
)2 (
vcohωP r + γrP r + γiP i
)
= 0.
For x = 0, system (5.8a), (5.9a) admits the fixed point(
0,
γiP r − γrP i − vcohωuP i
vcohP i − S0P r
)
,
if vcoh and ωu verify the condition
F (ωu, 1) = 0. (5.12a)
For y = 0, system (5.10a), (5.11a) admits the fixed point(
0,
γiP r − γrP i − vcohωvP
i
(vcohP i + S0P r)
)
,
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 483
if vcoh and ωv verify the condition
F (ωv, 2) = 0. (5.13a)
If conditions (5.12a) and (5.13a) are satisfied, then(
0, 0, 0, 0,
γiP r − (γr + vcohωu) P i
vcohP i − S0P r
,
γiP r − (γr + vcohωv) P i
vcohP i + S0P r
)
(5.36)
is a fixed point of system (5.2) – (5.7).
Now let
G(ω, β) = −|P |2
(
vcohωuP i − γiP r + γrP i
)2 +
+ P iP r
(
vcoh + (−1)βS0
)2 (
vcohωuP i − γiP r + γrP i
)
−
− P i2
(
vcoh + (−1)βS0
)2 (
vcohωuP r + γrP r + γiP i
)
.
If vcoh and ωu are such that
G(ωu, 1) = 0, (5.14a)
then (
(vcohωu + γr) P i − γiP r
P i (vcoh − S0)
, 0
)
is a fixed point of system (5.8a), (5.9a), and if vcoh and ωv are such that
G(ωv, 2) = 0, (5.15a)
then (
(vcohωv + γr) P i − γiP r
P i (S0 + vcoh)
, 0
)
is a fixed point of system (5.10a), (5.11a). Thus if conditions (5.14a) and (5.13a) or (5.15a) and
(5.12a) are satisfied, then(
0, 0,
vcohωuP i − γiP r + γrP i
P i (vcoh − S0)
, 0, 0,
γiP r − γrP i − vcohωvP
i
vcohP i + S0P r
)
(5.37)
or (
0, 0, 0,
vcohωvP
i − γiP r + γrP i
P i (S0 + vcoh)
,
γiP r − γrP i − vcohωuP i
vcohP i − S0P r
, 0
)
(5.38)
is a fixed point of system (5.2) – (5.7).
If
vcoh = −S0 and ωv =
γrP i − γiP r
S0P i
,
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484 E. KENGNE, R. VAILLANCOURT
then (
S0
(
P r − P i
)
2|P |2
,
±1
2|P |2
√
S2
0 (P r − P i)2 − 4|P |4γi
P i
)
are fixed points of system (5.10a), (5.11a). Moreover, if vcoh and ωu verify (5.12a) (with vcoh =
= −S0), then(
0, 0, 0,
S0
(
P r − P i
)
2|P |2
,
(γr − S0ωu) P i − γiP r
S0 (P i + P r)
,
±1
2|P |2
√
S2
0 (P r − P i)2 − 4|P |4γi
P i
)
(5.39)
are fixed points of system (5.2) – (5.7). We should note that the quantity S2
0
(
P r − P i
)2−4|P |4γi
P i
is always positive, because P iγi < 0 (see Appendix A).
If
vcoh = S0 and ωu =
γiP r − γrP i
S0P i
,
then S0
(
P i − P r
)
2|P |2
,
±1
2|P |2
√
−P iS2
0 (P i − P r)2 + 4|P |4γi
P i
are fixed points of system (5.8a), (5.9a), if the line parameters and the wavenumber k in coeffi-
cients of the GCGL Eqs. (3.10), (3.11) satisfy the inequality
k4
(
k2 + RG
)
< 4
C4
0
(
R2G2 + 4C2
0R2G2
)4(
1 + 4C2
0
)4
L4
(GL + RC0)
2 C0L
C4
0R2G2 (RG + 2C0RG)4
.
Moreover, if condition (5.13a) is satisfied with vcoh = S0, then0, 0,
S0
(
P i − P r
)
2|P |2
, 0,
±1
2|P |2
√
−P iS2
0 (P i − P r)2 + 4|P |4γi
P i
,
γiP r − (γr + S0ωv) P i
S0 (P i + P r)
(5.40)
are fixed points of system (5.2) – (5.7).
Thus, we have obtained the five particular fixed points, (5.36) – (5.40). We now study the
stability of these fixed points. Linearizing (5.2) – (5.7) at the obtained fixed points we obtain the
following five sets of eigenvalues. To save space and make reading easier, we let λ
[+]
j ≡ λj and
λ
[−]
j+1 ≡ λj+1 be the eigenvale with the “+” sign and “−” sign, respecively for j = 3 and 5, in
the following formulae.
1. For (5.36), x0 = y0 = 0 and we have
λ1 = λ2 = 0,
λ
[+]
3 , λ
[−]
4 =
1
2|P |2
{
−2S0P
r
(
P r + P i
)
vcoh ±
[
16|P |4Φ2
0 + v2
cohP
r2+
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 485
+
(
8S0vcoh − 3v2
coh − 6S2
0
)
P i2 − 2v2
cohP
rP i
]1/2
}
,
λ
[+]
5 , λ
[−]
6 =
1
2|P |2
{
2S0P
r +
(
P r + P i
)
vcoh ±
[
v2
cohP
r2 − 2v2
cohP
rP i−
−
(
4S2
0 + 3v2
coh + 8S0vcoh
)
P i2 − 16|P |4Ψ2
0 − 16P i|P |2 (S0 + vcoh) Ψ0
]1/2
}
.
2. For (5.37), y0 = Φ0 = 0 and we find
λ1 =
vcohωuP i − γiP r + γrP i
P i (vcoh − S0)
, λ2 = 0,
λ
[+]
3 , λ
[−]
4 =
1
2|P |2
{
−4x0|P |2 − 2S0P
r +
(
P r + P i
)
vcoh±
±
[
v2
cohP
r2 −
(
3v2
coh + 4S2
0 − 8S0vcoh
)
P i2 − 2v2
cohP
iP r
]1/2
}
,
λ
[+]
5 , λ
[−]
6 same as λ
[+]
5 , λ
[−]
6 in case 1.
3. For (5.38), x0 = Ψ0 = 0 and we obtain
λ1 = 0, λ2 =
vcohωvP
i − γiP r + γrP i
P i (S0 + vcoh)
,
λ
[+]
3 , λ
[−]
4 same as λ
[+]
3 , λ
[−]
4 in case 1,
λ
[+]
5 , λ
[−]
6 =
1
2|P |2
{
−4x0|P |2 − 2S0P
r +
(
P r + P i
)
vcoh±
±
[
v2
cohP
r2 −
(
3v2
coh + 4S2
0 + 8S0vcoh
)
P i2 − 2v2
cohP
iP r
]1/2
}
.
4. For (5.39), x0 = 0, vcoh = −S0, and ωv = (γrP i − γiP r)/(S0P
i) and we obtain
λ1 = 0, λ2 =
S0
(
P r − P i
)
2|P |2
,
λ
[+]
3 , λ
[−]
4 =
1
2|P |2S0
{
−S2
0
(
3P r + P i
)
±
±
[
16 |P |4
(
γiP r − γrP i + S0ωuP i
P i + P r
)2
+ S4
0
(
P r2 − 2P rP i − 15P i2
)]1/2
,
λ
[+]
5 , λ
[−]
6 =
1
2|P |2
√
−P i
{[
S0
(
P r−P i
)
−4|P |2y0
]√
−P i±
[
3P iS2
0
(
P r−P i
)2−64|P |8γi
]1/2
}
.
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486 E. KENGNE, R. VAILLANCOURT
5. For (5.40), y0 = 0, vcoh = S0, and ωu = (γiP r − γrP i)/(S0P
i) we have
λ1 =
S0
(
P i − P r
)
2|P |2
, λ2 = 0,
λ
[+]
3 , λ
[−]
4 =
1
2|P |2P i
{
−P i
[
S0
(
P i − P r
)
− 4x0|P |2
]
±
±
[
4P iS2
0
(
P i − P r
)2 + 16|P |4γi − P iS2
0
(
P i − P r
)2]1/2
}
,
λ
[+]
5 , λ
[−]
6 =
(
3P r+P i
)
S0
2|P |2
± 1
2S0|P |2|P i+P r|
[
S4
0
(
P r2−2P rP i−15P i2
) (
P i+P r
)2−16|P |4×
×
(
γiP r−γrP i−S0ωvP
i
)2−32S2
0P i|P |2
(
γiP r−γrP i−S0ωvP
i
) (
P i+P r
)]1/2
.
6. Conclusion. The dynamics of modulated wave trains in a distributed nonlinear mono-
inductance electrical line was analyzed in this paper. The GGL system of complex modulati-
on equations was derived. From this system we have derived the cubic-quintic and the GGL
equations. The coefficients of these equations are analytically given in terms of the line parame-
ters. The stability properties of the GCGL equations were studied for a class of phase winding
solutions. The coherent structures have been studied for the GGL equation and system. Whi-
le, in general, it is not clear how to go from the tools developed for low-dimensional dynamic
systems to an effective description of systems with many degrees of freedom, the coherent
structure framework sketched in this paper may be such a bridge in the case of CGL. Possibly
the greatest advantage of studying 1D systems is that their time evolution can be captured in
two-dimensional space-time plots, which may help one understand these systems. Without such
plots, the discovery of many dynamical properties would have been much more difficult.
Acknowledgments. This research was partially supported by the Natural Sciences and Engi-
neering Research Council of Canada and the Centre de recherches mathématiques of the Uni-
versité de Montréal.
7. Appendix. 7.1. Appendix A. We use the dispersion relation (3.2)
ω =
√
k2 + RG
C0L
,
∂ω
∂k
=
k√
C0L (k2 + RG)
and (3.12) to compute the coefficients of system (3.10), (3.11):
P = P r + iP i =
RG(
1 + 4C2
0
)
L
√
C0L
(k2 + RG)3
− i
2C0RG(
1 + 4C2
0
)
L
√
C0L
(k2 + RG)3
,
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GINZBURG – LANDAU SYSTEM OF COMPLEX MODULATION EQUATIONS . . . 487
γ = γr + iγi =
2C0 (GL + RC0)
L
(
1 + 4C2
0
) + i
GL + RC0
L(1 + 4C2
0 )
,
Q = Qr + iQi =
3bR
2L
− i
3b
2
√
k2 + RG
C0L
,
E = Er + iEi = − 6C2
0bk(
1 + 4C2
0
)√
C0L (k2 + RG)
− i
3C0bk(
1 + 4C2
0
)√
C0L (k2 + RG)
,
D = Dr + iDi =
3C0b
1+4C2
0
9bR2
2L2
√
C0L
k2+RG
− 3b
√
k2+RG
C0L
− LC0b
2RG
3
√(
(k2+RG)
C0L
)3
+
+
R
L
k2 + RG
C0L
)
− 21bRC0
L
− bC0
GL
(
3
(
k2 + RG
)
+
R
L
√
C0L (k2 + RG)
)]
+
+ i
3C0b
1 + 4C2
0
−9bC0R
2
L2
√
C0L
k2 + RG
+ 6C0b
√
k2 + RG
C0L
+
LC2
0b
RG
3
√(
(k2 + RG)
C0L
)3
+
+
R
L
k2 + RG
C0L
)
− 21bR
2L
− b
2GL
(
3
(
k2 + RG
)
+
R
L
√
C0L (k2 + RG)
)]
,
H = Hr + iH i =
(
36C2
0b− 6C0bR
L
√
C0L
k2+RG
)
k(
1 + 4C2
0
)√
C0L (k2 + RG)
+ i
(
12C0bR
L
√
C0L
k2+RG
+ 18C0b
)
k(
1 + 4C2
0
)√
C0L (k2 + RG)
,
F = F r + F i =
3C0b
2
1 + 4C2
0
24R2
L2
√
C0L
k2 + RG
− 12
√
k2 + RG
C0L
− 108bC0R
L
−
− i
3C0b
2
1 + 4C2
0
48C0R
2
L2
√
C0L
k2 + RG
− 24C0
√
k2 + RG
C0L
+
54R
L
,
K = Kr + iKi =
3C0b
2
1 + 4C2
0
12R2
L2
√
C0L
k2 + RG
− 12
√
k2 + RG
C0L
− 60C0R
L
−
− i
3C0b
2
1 + 4C2
0
12C0R
2
L2
√
C0L
k2 + RG
− 24C0
√
k2 + RG
C0L
+
30bR
L
,
S0 =
∂ω
∂k
=
k√
C0L (k2 + RG)
.
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488 E. KENGNE, R. VAILLANCOURT
7.2. Appendix B (Analytical expressions for the coefficients of the GCGL equations (3.10),
(3.11) corresponding to the line parameters (Lp)). Using Appendix A, we compute the coeffi-
cients of the GCGL equations (3.10), (3.11) corresponding to the line parameters (Lp).
(Lp) (Lp1) (Lp2)
P 1.3804× 10−3 − 1.4909× 10−6i 1.3804× 10−3 − 7.827× 10−14i
γ 2.083× 10−9 + 1.9287i 2.0681× 10−13 + 1.9149× 10−4i
Q 8.5714× 108 − 1.9905× 106i 450.0− 6.1845× 106i
E −1.437× 10−13 − 1.330 5× 10−4i −1.437× 10−13 − 1.3305× 10−4i
H −3.6881× 10−2 + 0.07456i −1.9361× 10−8 + 7.9834× 10−4i
D 92.297− 12.083i −4.8155× 10−3 − 9.3358× 10−7i
F 492.66− 7.9982i −1.2824× 10−2 − 4.199× 10−6i
K 246.32− 4.4434i −1.2824× 10−2 − 2.3328× 10−6i
S0 5.1332× 105 5.1331× 105
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Received 12.03.2004
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