Abstract In this paper a coupled mono-inductance transmission line is consideredand the envelope modulation is reduced to the generalized 2D-Ginzburg-Landausystem (2D-GL system). The case of scalar equation is also considered and we deducefrom the obtained 2D-GL system a 2D-cubic Ginzburg-Landau equation (2D-GLequation) containing the derivatives with respect to one spatial variable in the cubicterms. The obtained 2D-GL system and 2D-GL equation admit spatial wave solu-tions. The modulational instability of these spatial wave solutions is investigated. Inthe case of the 2D-GL system we have restricted ourselves in the zero wavenumber ofthe perturbations, and the obtained modulational instability conditions depends onlyon the system’s coefficients and the wavenumber of the carriers. The modulationalinstability criterion is established for non zero wavenumber of the perturbation, anddepends on both the wavenumber of the perturbation and the carrier. For the 2D-GLsystem we also study the coherent structures and present some numerical studies.
Nonlinear discrete electrical transmission lines (NLTLs) that consist of a largenumber of identical sections have been used to experimentally study the propagation
E. Kengne (B) · R. VaillancourtDepartment of Mathematics and Statistic, University of Ottawa, 585 King Edward Avenue,Ottawa, ON K1N 6N5, Canadae-mail: [email protected]
E. KengneDepartment of Mathematics & Computer Science, Faculty of Science, University of Dschang,P.O. Box 4509, Douala, Republic of Cameroon
680 J Infrared Milli Terahz Waves (2009) 30:679–699
of solitons obeying the Korteweg deVries (KdV) equation. Since the pioneeringworks by Hirota and Suzuki [1, 2] and Nagashima and Amagishi [3] on a singleelectrical line simulating the Toda lattice [4], a growing interest has been devotedto the use of NLTLs [5–9].
Extending the transmission line model to higher dimensions has been provendifficult both theoretically and experimentally. There have been particular applica-tions of the KdV equations [10, 11] and nonlinear Schrodinger equations [12, 13].Experiments have been performed on cylindrical solitons for a two-dimensionaltransmission line [14]. Resonances of a two-dimensional transmission line have alsobeen examined [15]. However, with rapid developments in the integrated-circuittechnology, we believe that wave propagation in higher-dimensional transmissionlines is a topic that will continue to receive attention. In the present paper, we use asystem of Ginzburg-Landau type to study the nonlinear transmission line model ofhigher dimension.
The paper is organized as follows. In Section 2 we write down the circuit equa-tions governing small-amplitude pulses on systems of NTL’s coupled via constantcapacitors. After scaling coordinates and taking a continuum limit, the method ofmultiple-scale perturbation is used in Section 3 to reduce the circuit equations toa new nonlinear system of PDE that we call 2D Ginzburg-Landau system. Weundertake an analytic study of the stability of the particular solutions of the obtained2D generalized Ginzburg-Landau system in Section 4. The 2D coherent structure isstudied in Section 5, and the main results are summarized in Section 6.
2 Electrical models
Before examining coupled systems, we introduce our notation by means of a singlenonlinear transmission line illustrated in Fig. 1. Here the nonlinear transmissionlines are studied as circuit models in terms of circuit theory where voltages andcurrents, instead of fields, are the variables. Figure 1 shows an infinitesimal segmentof the physical transmission line. In this figure, R, L, C, and G are the distributed
Fig. 1 Part of a single dissipative nonlinear transmission line.
J Infrared Milli Terahz Waves (2009) 30:679–699 681
parameters of the line. They are the resistance per unit length, inductance per unitlength, capacitance per unit length, and conductance per unit length, respectively.
For the circuit in Fig. 1, we no longer have the familiar linear charge-voltagerelation CQn = Q, but rather the nonlinear differential relationship
C(Vn) = dQn
dVn. (1)
To solve for the voltage as a function of time, we use the familiar relations at thenodes
Vn−1 − Vn = d�n
dt+ RIn, In − In+1 = dQn
dt+ GVn. (2)
To obtain an equation in terms of V only, one notes that the magnetic flux can beexpressed in terms of the current using �n = LIn, while the charge can be eliminatedusing (1). Using these relations in (2) leads to the following formulae:
Vn−1 − Vn = LdIn
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.
Eliminating In and In+1, we find the following second order ordinary differencedifferential equation:
ddt
[C(Vn)
dVn
dt
]+ G
dVn
dt+ R
LC(Vn)
dVn
dt= 1
L(Vn−1 − 2Vn + Vn+1) .
This is the circuit equation which describes the voltage Vn(t) on a single line.Now we imagine coupling many identical lines such as this by means of capacitors
C2 at each node. Such a configuration is shown in Fig. 2. The nodes in the systemare labelled with two discrete coordinates: n specifies the nodes in the direction ofpropagation of the pulse, and m labels the lines in the transverse direction. We applyKirchhoff’s laws in orthogonal loops and obtain
LdI1
n,m
dt+ RI1
n,m = Vn−1,m − Vn,m; I2n,m = C2
ddt
(Vn,m − Vn,m+1
) ;dQn,m
dt= I1
n,m − I1n+1,m + I2
n,m−1 − I2n,m − GVn,m;
d2 Qn,m
dt2= dI1
n,m
dt− dI1
n+1,m
dt+ dI2
n,m−1
dt− dI2
n,m
dt− G
dVn,m
dt.
The circuit equation for this system is therefore
d2 Qn,m
dt2− 1
L
(Vn−1,m−2Vn,m+Vn+1,m
)−C2d2
dt2
(Vn,m−1−2Vn,m+Vn,m+1
)
+ GdVn,m
dt+ R
LdQn
dt+ RG
LVn,m− RC2
Lddt
(Vn,m−1−2Vn,m+Vn,m+1
)=0.
(3)
We shall take the continuum limit of (3) in both the n and m directions, alongwith the additional provisos that the amplitude of the voltage pulse is small and its
682 J Infrared Milli Terahz Waves (2009) 30:679–699
Fig. 2 Part of a system of dissipative nonlinear transmission line coupled by a capacitor C2.
wavelength is much greater than the lattice spacings. In this analysis, we shall take thenonlinear capacitance C(Vn,m) to be of the form C(V) = C0/
[1 + (V/V0)
p] , whereC0 and V0 are arbitrary capacitance and voltage scales, respectively, and p > 0.Similar forms for C(V) have been used in the past to model the capacitance of certainvaractor diodes as part of comparisons with experimental measurements of solitarywaves on NTL’s (Hicks and al., 1996).
If we treat n and m as continuous variables and V as a function of x = n, y = mand t, and assume that V � V0, then (3) becomes
C0∂2
∂t2
(V− V p+1
(p+1) V p0
)− 1
L∂2
∂x2
(V+ 1
12
∂2V∂x2
+ ...
)−C2
∂4
∂t2∂y2
(V+ 1
12
∂2V∂y2
+ ...
)
+ G∂V∂t
+ RC0
L∂
∂t
(V− V p+1
(p+1) V p0
)+ RG
LV
− RC2
L∂3
∂t∂y2
(V+ 1
12
∂2V∂y2
+ ...
)=0. (4)
In what follows, we study the case where p = 2 and truncate equation (4). We thenobtain the following nonlinear partial differential equation for the voltage:
C0∂2
∂t2
(V + b V3
) − 1
L∂2V∂x2
− C2∂4V
∂t2∂y2+(
G + RC0
L
)∂V∂t
+ RC0
L∂V3
∂t+ RG
LV
− RC2
L∂3V∂t∂y2
= 0, (5)
where b = −1/(3V2
0
).
J Infrared Milli Terahz Waves (2009) 30:679–699 683
3 Derivation of the 2D-like generalized Ginzburg-Landau system
In this section we use equation (5) of the voltage to derive a second order partialdifferential system that we will call a 2D-Ginzburg-Landau system. To construct thesaid system we use the method of multiple-scale by introducing two slow time scalesT1 = εt and T2 = ε2t in addition to the initial time T0 = t and one large length scaleX1 = εx and Y1 = εy in addition to the spatial variables X0 = x and Y0 = y. Here ε
is a small parameter. We then seek a solution of equation (5) in the form
where c.c stands for the complex conjugate of preceding terms and
θ1 =kX0−ω1T0, θ2 =qY0−ω2T0, u jk =u jk(X1,Y1,T1,T2), vjk =vjk(X1,Y1,T1,T2).
In Eq. (5), we follow Kengne et al. [16] and order the damping coefficient so thatthe effect of damping and nonlinearity appear in the same perturbation equation.In other words, we assume G + RC0/L to be of order O (ε) . Coefficient RG/L inEq. (5) is considered to be of order O (1) .
Inserting the perturbation expansion (6) into the nonlinear equation (5) we obtaina series of nonhomogeneous equations at different orders of (ε, exp [iθ1] , exp [iθ2]) .
At order(ε1/2, exp [iθ1] , exp [iθ2]
)we obtain
(−C0ω
21 + k2
L+ RG
L
)u11 exp [iθ1] +
(−C0ω
22 − C2q2ω2
2 + RGL
)v11 exp [iθ2] = 0,
and the condition of non triviality (non-nullity) of u11 and v11 gives the followinglinear dispersion relations:
− C0ω21 + k2
L+ RG
L= 0; − (C0 + C2q2
)ω2
2 + RGL
= 0. (7)
At order (ε, exp [2iθ1] , exp [2iθ2]) we have
(−4C0ω
21 +4
k2
L+ RG
L
)u22 exp [2iθ1]+
(−4C0ω
22 −4C2q2ω2
2 + RGL
)v22 exp [2iθ2]=0.
For simplicity, we can take u22 = v22 = 0.
The obtained equations at order(ε3/2, exp [iθ1] , exp [iθ2]
)reads
[−2iC0ω1
∂u11
∂T1− 2ik
L∂u11
∂ X1− 3C0bω1
(ω1 + i
RL
) (|u11|2 + 2 |v11|2)
u11
]exp (iθ1)
+[−2iω2
(C0 + C2q2
) ∂v11
∂T1+ 2iC2qω2
2∂v11
∂Y1
−3C0bω2
(ω2 + i
RL
) (|v11|2 + 2 |u11|2)v11
]exp (iθ2) = 0.
684 J Infrared Milli Terahz Waves (2009) 30:679–699
From this equation we obtain the system
∂u11
∂T1= −∂ω1
∂k∂u11
∂ X1+ A3
(|u11|2 + 2 |v11|2)
u11;∂v11
∂T1= −∂ω2
∂q∂v11
∂Y1+ B3
(|v11|2 + 2 |u11|2)v11, (8)
where
A3 = −3b2
(RL
− iω1
); B3 = 3C0b
2(C0 + C2q2
)(
− RL
+ iω2
); ∂ω1
∂k= 1
C0 Lkω1
;
∂ω2
∂q= − C2qω2
C0 + C2q2. (9)
At order(ε3/2, exp [3iθ1] , exp [3iθ2]
)we have
[(−9C0ω
21 + 9
k2
L+ RG
L
)u33 − 3C0bω1
(3ω1 + R
L
)u3
11
]exp (3iθ1)
+[(
−9C0ω22 − 81C2q2ω2
2 + RGL
)v33 − 3C0bω2
(3ω2 + R
L
)v3
11
]exp (3iθ2) = 0.
Using the dispersion relations (7), the last equation becomes
u33 = A4u311; v33 = B4v
311, (10)
with
A4 = 3C0bω1
8(
k2
L − C0ω21
)(
3ω1 + RL
); B4 = −3C0bω2
8(C0 + 10C2q2
)ω2
2
(3ω2 + R
L
). (11)
Equation of order(ε2, exp [2iθ1] , exp [2iθ2]
)reads
[(−4C0ω
21 + 4
k2
L+ RG
L
)u42
]exp (2iθ1)
+[(
−4C0ω22 − 16C2q2ω2
2 + RGL
)v42
]exp (2iθ2) = 0
from where we take
u42 = v42 = 0.
J Infrared Milli Terahz Waves (2009) 30:679–699 685
At order(ε5/2, exp [iθ1] , exp [iθ2]
)we have the system
[C0
(∂2u11
∂T21
− 2iω1∂u11
∂T2
)− 1
L∂2u11
∂ X21
+ C2ω21∂2u11
∂Y21
− 4C0bω1
(ω1 + i
RL
)u∗2
11u33
−iμ1ω1u11 + 3C0b(
RL
− 2iω1
)∂
∂T1
((|u11|2 + 2 |v11|2)
u11)]
exp (iθ1) ,
[(C0 + C2q2
) ∂2v11
∂T21
− 2iω2(C0 + C2q2
) ∂v11
∂T2+ C2ω
22∂2v11
∂Y21
− 4C2qω2∂2v11
∂T1∂Y1
−i(μ1 − q2μ2
)ω2v11 − 1
L∂2v11
∂ X21
− 4C0bω2
(ω2 + i
RL
)v∗2
11v33
+3C0b(
RL
− 2iω2
)∂
∂T1
((|v11|2 + 2 |u11|2)v11)]
exp (iθ1) = 0.
Restricting ourselves to the case where
C0 = 2C2; k = −√
8C2 LRG1 − 8C2 L
; ω1 =√
RG2C2 L (1 − 8C2 L)
; ω2 = 2(C2 + q2
)q
;
4C2 L(2 + q2
)3 − RGq2 = 0,
and taking equations (8) and (10) into account, the last equation in terms of originalcoordinates x, y, and t becomes,
∂u∂t
+ i(
P11∂2u∂x2
+ P12∂2u∂y2
)+ γ1u + Q1
∂
∂x
((|u|2 + 2 |v|2) u) = 0, (12)
∂v
∂t+ i(
P21∂2v
∂x2+ P2
2∂2v
∂y2
)+ γ2v + Q2
∂
∂y
((|v|2 + 2 |u|2) v) = 0, (13)
where
u = ε1/2u11; v=ε1/2v11; P11 = C0
(∂ω1∂k
)2− 1L
2C0ω1; P1
2 = C2ω1
2C0; P2
1 = −1
2L(C0+C2q2
)ω2
;
P22 =
4C2qω2∂ω2∂q +C2ω
22 + (C0 + C2q2
) (∂ω2∂q
)2
2(C0 + C2q2
)ω2
; Q1 =−iA3
2ω1
∂ω1
∂k; γ1 = LG+RC0
2C0 L;
Q2 = iB3
4C2qω2 − (C0 + C2q2)
∂ω2∂q
2(C0 + C2q2
)ω2
; γ2 = LG + R(C0 − C2q2
)2L(C0 + C2q2
) . (14)
If we write system (8) in terms of the original coordinates x, y, and t, multiply eachequation of this system by ε1/2 and then add the obtained system to system (12)–(13)side by side, after using the transformations
t = τ = t/2, u = u exp (iαx) , v = v exp (iβy) , with α = 1
2P11
∂ω1
∂k, β = 1
2P22
∂ω2
∂q,
686 J Infrared Milli Terahz Waves (2009) 30:679–699
we obtain the final system
∂u∂t
+ i(
P11∂2u∂x2
+ P12∂2u∂y2
)+ γ1u + D1
(|u|2 + 2 |v|2) u + Q1∂
∂x
((|u|2+2 |v|2) u) = 0,
(15)
∂v
∂t+ i(
P21∂2v
∂x2+ P2
2∂2v
∂y2
)+ γ2v + D2
(|v|2 + 2 |u|2) v + Q2∂
∂y
((|v|2+2 |u|2) v) = 0,
(16)
where
γ1 = γ1 + i(
α∂ω1
∂k− P1
1α2
); γ2 = γ2 + i
(β
∂ω2
∂q− P2
2β2
); D1 = iαQ1 − A3;
D2 = iβQ2 − B3.
System (15)–(16) is the required 2D Ginzburg-Landau system. We have preferredto call this system ”2D” Ginzburg-Landau system (see [18–24]) even if it does notcontain the expression � = ∂2
∂x2 + ∂2
∂y2 in both equations of the system. Moreover
our system contains terms like ∂∂x
((|u|2 + 2 |v|2)u)
and ∂∂y
((|v|2 + 2 |u|2) v) , and
consequently contains the following terms: ∂u∂x , ∂u
∂y , ∂v∂x , ∂v
∂y , ∂u∂x , ∂u
∂y , ∂v∂x , and ∂v
∂y . It is
possible that the cubic terms involving ∂u∂x , ∂u
∂y , ∂v∂x , ∂v
∂y , ∂u∂x , ∂u
∂y , ∂v∂x , and ∂v
∂y significantlyslow down the propagation speed of moving fronts and pulses. This is interesting inview of the slowly drifting pulse states. It is seen from (11), (14) and (9) that P1
1,
P12, P2
1, and P22 are real numbers, while Q1, Q2, D1, D2, γ1, and γ2 are complex
parameters. In what follows we denote by f r and f i the real and imaginary partsof a complex number f, respectively. Then if f is a complex number we write it asf = f r + i f i.
if we set either v(x, y, t) = 0 or u(x, y, t) = 0, we obtain a generalized 2D–cubicGinzburg-Landau equation. If, for example, we set v(x, y, t) = 0, we obtain
∂u∂t
+ i(
P11∂2u∂x2
+ P12∂2u∂y2
)+ γ1u + D1 |u|2 u + Q1
∂
∂x|u|2 u = 0,
which we can be written in the form
∂u∂t
+ i(
P11∂2u∂x2
+ P12∂2u∂y2
)+ γ1u + D1 |u|2 u + Q1
(2 |u|2 ∂u
∂x+ u2 ∂u∗
∂x
)= 0.
(17)
4 2D-Modulational instability
In this section we study the modulational instability of the spatial wave solutions ofthe 2D-system (15)–(16). We also establish the modulational instability criterion ofthe spatial wave solutions of the generalized 2D-Ginzburg-Landau equation (17).
J Infrared Milli Terahz Waves (2009) 30:679–699 687
4.1 2D-modulational instability of the 2D-Ginzburg-Landau system (15)–(16):The case of the perturbation zero wavenumber
By considering a first order perturbation of the amplitudes of plane waves, weexamine in this subsection the conditions of instability of harmonic solutions. First,the 2D Ginzburg-Landau system (15)–(16) has spatial wave solutions with constantamplitudes of the form
u(x, y, t) = u0 exp i (k0x + l0 y − ω0t) , v(x, y, t) = v0 exp i (k0x + l0 y − ω0t) , (18)
in which u0, v0, and ω0 are real constants defined by
ω0 = − (P11k2
0 + P12l2
0
)− γ r1
(Di
1 + k0 Qi1
)Dr
1 − k0 Qi1
, u20 = γ r
2
3(Dr
2 − l0 Qi2
) − 2γ r1
3(Dr
1 − k0 Qi1
) ,
v20 = γ r
1
3(Dr
1 − k0 Qi1
) − 2γ r2
3(Dr
2 − l0 Qi2
) . (19)
Here k0 and l0 are two parameters satisfying((
P21−P1
1
)k2
0+(P2
2−P12
)l20
) (Dr
1−k0 Qi1
) (Dr
2−l0 Qi2
)− γ r1
(Di
1+k0 Qr1
)+γ r
2
(Di
2+l0 Qr2
)=0. (20)
Next, we perturb u and v from the spatial wave solutions (18)–(20). That is, we let
u = (u0 + u1(x, y, t)) exp i (k0x + l0 y − ω0t) ;v = (v0 + v1(x, y, t)) exp i (k0x + l0 y − ω0t) , (21)
where u1 and v1 are assumed to be infinitesimal. Substituting (21) into (15)–(16) andkeeping only linear terms in the perturbation quantities, we obtain
∂u1
∂t+ P1
1
(i∂2u1
∂x2− 2k0
∂u1
∂x
)+ P1
2
(i∂2u1
∂y2− 2l0
∂u1
∂y
)
+ Q1
[u2
0
(∂u∗
1
∂x+ 2
∂u1
∂x
)+ 2u0v0
(∂v∗
1
∂x+ ∂v1
∂x
)+ 2v2
0∂u1
∂x
]
+ (D1 + ik0 Q1)[u2
0
(u1 + u∗
1
)+ 2u0v0(v∗
1 + v1)] = 0, (22)
∂v1
∂t+ P2
1
(i∂2v1
∂x2− 2k0
∂v1
∂x
)+ P2
2
(i∂2v1
∂y2− 2l0
∂v1
∂y
)
+ Q2
[v2
0
(∂v∗
1
∂y+ 2
∂v1
∂y
)+ 2u0v0
(∂u∗
1
∂y+ ∂u1
∂y
)+ 2v2
0∂v1
∂y
]
+ (D2 + il0 Q2)[v2
0
(v1 + v∗
1
)+ 2u0v0(u∗
1 + u1)] = 0. (23)
In equations (22) and (23), (∗) denotes the complex conjugate. Since (22)–(23) haveconstant coefficients, one can represent their solutions in the form
u1 = a0 exp (i�t) + b ∗0 exp
(−i�∗t) ; v1 = c0 exp (i�t) + d∗
0 exp(−i�∗t
), (24)
688 J Infrared Milli Terahz Waves (2009) 30:679–699
where a0, b 0, c0, d0, and � are constants. Here, we have considered the wavenumberof the perturbations to be null. Substituting (24) into (22)–(23) yields
b 0 = D∗1 − ik0 Q∗
1
D1 + ik0 Q1a0; c0 = D2 + il0 Q2
D∗2 − il0 Q∗
2
d0;
⎧⎨⎩[i� + 2
(Dr
1 − k0 Qi1
)u2
0
]a0 + 4u0v0
(Dr
2 − l0 Qi2
) D1+ik0 Q1
D∗2−il0 Q∗
2d0 = 0,
4u0v0(Dr
1 − k0 Qi1
) D∗2−il0 Q∗
2D1+ik0 Q1
a0 + [i� + 2(Dr
2 − l0 Qi2
)v2
0
]d0 = 0.
(25)
For a nontrivial solution the determinant of the coefficient matrix of (25) must vanish.That is,
�2−2i[(
Dr1−k0 Qi
1
)u2
0+(Dr
2−l0 Qi2
)v2
0
]�+12
(Dr
1−k0 Qi1
) (Dr
2−l0 Qi2
)u2
0v20 = 0.
Solving this second order equation we obtain
� = [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]i ±
√δ
′,
with
δ′ = − [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]2 − 12(Dr
1 − k0 Qi1
) (Dr
2 − l0 Qi2
)u2
0v20 .
Using (19), we write δ′ as
δ′ = −[(
γ r2
(Dr
1 − k0 Qi1
)3(Dr
2 − l0 Qi2
) − 2γ r1
3
)+(
γ r1
(Dr
2 − l0 Qi2
)3(Dr
1 − k0 Qi1
) − 2γ r2
3
)]2
−12
(γ r
2
(Dr
1 − k0 Qi1
)3(Dr
2 − l0 Qi2
) − 2γ r1
3
)(γ r
1
(Dr
2 − l0 Qi2
)3(Dr
1 − k0 Qi1
) − 2γ r2
3
). (26)
It follows from (26) that δ′ is always non positive if(Dr
1 − k0 Qi1
) (Dr
2 − l0 Qi2
) ≥ 0.
Thus in order that δ′ be nonnegative it is necessary that(Dr
1 − k0 Qi1
) (Dr
2 − l0 Qi2
)< 0. (27)
To analyze the modulational instability of the spatial wave solutions (18)–(20), wedistinguish two cases:
1. If δ′ ≥ 0 (in this case we necessarily have (27)) then
i� = − [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]± i√
δ′
and
−i�∗ = − [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]∓ i√
δ′.
That is,
Re (i�) = Re(−i�∗) = − [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
],
and hence u1 and v1 are bounded according to (24) if, and only if,(Dr
1 − k0 Qi1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0 > 0.
J Infrared Milli Terahz Waves (2009) 30:679–699 689
Otherwise u1 and v1 will be unbounded as t → +∞. In other words, for themodulational instability of the spatial wave solutions (18)–(20) in the case whereδ′ ≥ 0, it is necessary and sufficient that
(Dr
1 − k0 Qi1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0 < 0.
Using now (19) and taking into account (27) we obtain the following conditionof the modulational instability of the spatial wave solutions (18)–(20) the casewhere δ′ ≥ 0
2γ r2
(Dr
1−k0 Qi1
)2 + 2γ r1
(Dr
2 − l0 Qi2
)2−(γ1 + γ2)(Dr
1 − k0 Qi1
) (Dr
2 − l0 Qi2
)< 0.
2. If δ′ < 0 then
i� = − [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]∓√−δ′,
and
−i�∗ = − [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]±√−δ′.
Thus i� and −i�∗ are real numbers and hence u1 and v1 are unboundedaccording to (24) if, and only if,
− [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]±√−δ′> 0,
that is,
− [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]+√−δ′> 0 (28)
and
− [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]−√−δ′> 0. (29)
Because −√−δ′< 0, in order to have (29) it is necessary that
− [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]> 0.
In this case, Eq. (28) is satisfied. Hence for the spatial wave solutions (18)–(20)to be modulationally unstable, it is necessary that
(Dr
1 − k0 Qi1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0 < 0
and sufficient that
− [(Dr1 − k0 Qi
1
)u2
0 + (Dr2 − l0 Qi
2
)v2
0
]−√−δ′> 0,
that is,(γ r
2
(Dr
1−k0 Qi1
)3(Dr
2−l0 Qi2
) − 2γ r1
3
)(γ r
1
(Dr
2−l0 Qi2
)3(Dr
1−k0 Qi1
) − 2γ r2
3
)<0⇔(Dr
1−k0 Qi1
)(Dr
2−l0 Qi2
)<0.
690 J Infrared Milli Terahz Waves (2009) 30:679–699
According to (19) the necessary condition reads
2γ r2
(Dr
1 − k0 Qi1
)2 + 2γ r1
(Dr
2 − l0 Qi2
)2 − (γ1 + γ2)(Dr
1 − k0 Qi1
) (Dr
2 − l0 Qi2
)< 0.
From 1. and 2. above we obtain the following theorem:
Theorem
I. If the wavenumber k0 and l0 satisfying the (20) and defining u20 and v2
0 in (19) aresuch that
[(γ r
2
(Dr
1 − k0 Qi1
)3(Dr
2 − l0 Qi2
) − 2γ r1
3
)+(
γ r1
(Dr
2 − l0 Qi2
)3(Dr
1 − k0 Qi1
) − 2γ r2
3
)]2
+ 12
(γ r
2
(Dr
1 − k0 Qi1
)3(Dr
2 − l0 Qi2
) − 2γ r1
3
)(γ r
1
(Dr
2 − l0 Qi2
)3(Dr
1 − k0 Qi1
) − 2γ r2
3
)≤ 0
then for the spatial wave solutions (18)–(20) of the Ginzburg-Landau system(15)–(16) to be modulationally unstable it is necessary and sufficient that
2γ r2
(Dr
1−k0 Qi1
)2+2γ r1
(Dr
2−l0 Qi2
)2−(γ1+γ2)(Dr
1−k0 Qi1
) (Dr
2−l0 Qi2
)<0;
(30)
II. If the wavenumber k0 and l0 satisfying (20) and defining u20 and v2
0 in (19) aresuch that
[(γ r
2
(Dr
1 − k0 Qi1
)3(Dr
2 − l0 Qi2
) − 2γ r1
3
)+(
γ r1
(Dr
2 − l0 Qi2
)3(Dr
1 − k0 Qi1
) − 2γ r2
3
)]2
+ 12
(γ r
2
(Dr
1 − k0 Qi1
)3(Dr
2 − l0 Qi2
) − 2γ r1
3
)(γ r
1
(Dr
2 − l0 Qi2
)3(Dr
1 − k0 Qi1
) − 2γ r2
3
)> 0
then for the spatial wave solutions (18)–(20) of the Ginzburg-Landau system(15)–(16) to be modulationally unstable it is necessary that
2γ r2
(Dr
1 − k0 Qi1
)2+2γ r1
(Dr
2 − l0 Qi2
)2 − (γ1 + γ2)(Dr
1 − k0 Qi1
) (Dr
2 − l0 Qi2
)<0
(31)
and sufficient that
(Dr
1 − k0 Qi1
) (Dr
2 − l0 Qi2
)< 0. (32)
We note that conditions (30) or (31) and (32) of the modulational instabilitydepend on the wavenumbers k0 and l0 of the spatial wave solutions (18)–(20). If(20) defines k0 (or l0) as an implicit function of l0 (or of k0), then the conditions of themodulational instability of the spatial wave solutions (18)–(20) will depend on onlyone parameter l0 (or k0).
J Infrared Milli Terahz Waves (2009) 30:679–699 691
4.2 2D-modulational instability of the 2D-Ginzburg-Landau equation (17)
We now pass to the study of the modulational instability of the spatial wave solutionsof equation (17). We first note that equations (17) admits spatial wave solutions ofthe form
u(x, y, t) = a0 exp[i (k1x + q1 y − ω1t)
], a0 =
√γ r
1
Qi1k1 − Dr
1
,
ω1 = − (P11k2
1 + P12q2
1
)+ γ i1 + Di
1a21 + Qr
1a21k1. (33)
Note that the local wavenumber of the spatial wave solutions (33) are constant:k(x, y, t) = k1 and l(x, y, t) = l1.
To investigate the modulational instability of the carrier, we consider a smallperturbation of (33) by setting
u(x, y, t) = (a0 + u1(x, y, t)) exp i (k1x + q1 y − ω1t) , (34)
where u1 is assumed to be infinitesimal. Substituting (34) into (17) and keeping onlylinear terms in the perturbation quantities, we obtain
∂u1
∂t+ P1
1
(i∂2u1
∂x2− 2k1
∂u1
∂x
)+ P1
2
(i∂2u1
∂y2− 2q1
∂u1
∂y
)+ Q1a2
0
(∂u∗
1
∂x+ 2
∂u1
∂x
)
+ (D1 + ik1 Q1) a20
(u1 + u∗
1
). (35)
Here (∗) denotes the complex conjugate. Since (35) has constant coefficients, one canrepresent its solutions in the form
u1 = α0 exp[i (Kx + qy + �t)
]+ β∗0 exp
[−i(Kx + qy + �∗t
)], (36)
where α0 and β0 are constants. Substituting (36) into (35) yields:[i� − iP1
1
(K2 + 2k1 K
)− iP12
(q2 + 2q1q
)+ 2ia20 Q1 K + (D1 + ik1 Q1) a2
0
]α0
+ a20
[iQ1 K + (D1 + ik1 Q1)
]β0 = 0, (37)
[i� + iP1
1
(K2 − 2k1 K
)+ iP12
(q2 − 2q1q
)+ 2ia2∗0 Q1 K + (D∗
1 − ik1 Q∗1
)a2
0
]β0
+ a20
[iQ∗
1 K + (D∗1 − ik1 Q∗
1
)]α0 = 0. (38)
For a nontrivial solution (α0, β0) , the determinant of the matrix of coefficients ofsystem (37)–(38) must vanish. This condition gives an equation for determining of �.
The said equation has the form
(� − ib)2 =⎛⎝√
α +√α2 + β2
2+ i
√−α +√α2 + β2
2
⎞⎠
2
,
where α and β are real numbers and b a complex number. This last equation gives
� = ±√
α +√α2 + β2
2− Im(b) + i
⎛⎝Re(b) ±
√−α +√α2 + β2
2
⎞⎠ ,
692 J Infrared Milli Terahz Waves (2009) 30:679–699
from where we have
i� = ±i
√α +√α2 + β2
2+⎛⎝−Re b ∓
√−α +√α2 + β2
2
⎞⎠ . (39)
Here
b = (Dr1 − k1 Qi
1
)a2
0 + 2i[a2
0 Qr1 K − P1
1k1 K − P12q1q] ; α = Re c; β = Im c,
with
c = −b 2 + a40
(Q∗
1 K − (k1 Q∗1 + iD∗
1
))(Q1 K + (k1 Q1 − iD1))
+ [P11
(K2 + 2k1 K
)+ P12
(q2 + 2q1q
)− (2Q1 K + k1 Q1 − iD1) a20
]× [P1
1
(K2 − 2k1 K
)+ P12
(q2 − 2q1q
)+ (2Q∗1 K − k1 Q∗
1 − iD∗1
)a2
0
].
Hence
Re b = (Dr1 − k1 Qi
1
)a2
0;α = − (Dr
1 − k1 Qi1
)2a4
0 + 4[a2
0 Qr1 K − P1
1k1 K − P12q1q]2 + a4
0 |Q1|2 K2
+ (P11
(K2 + 2k1 K
)+ P12
(q2 + 2q1q
)− (2Qr1 K + k1 Qr
1 + Di1
)a2
0
)× (P1
1
(K2 − 2k1 K
)+ P12
(q2 − 2q1q
)+ (2Qr1 K − k1 Qr
1 − Di1
)a2
0
)+ (2Qi
1 K + k1 Qi1 − Dr
1
) (−2Qi1 K + k1 Qi
1 − Dr1
)a4
0.
Substituting (39) into (36) helps understanding the behaviour of (36). This behav-
iour depends on the sign of the quantities − Re b ∓√
−α+√
α2+β2
2 , which correspond
to the imaginary part of �. The asymptotic behaviour of (36) is related to the sign of
the constant −Re b ∓√
−α+√
α2+β2
2 .
If −Re b ∓√
−α+√
α2+β2
2 > 0, then the solution (36) increases exponentially ast → +∞. The system remains unstable under modulation, and the spatial wavesolutions (33) are said to be modulationally unstable. The condition −Re b ∓√
−α+√
α2+β2
2 > 0 means that
−Re b +√
−α +√α2 + β2
2> 0 and − Re b −
√−α +√α2 + β2
2> 0.
In order to have the last inequality it is necessary that b should be negative. If thisnecessary condition is satisfied, then the first inequality will be valid. Then we can saythat for the modulational instability of the spatial wave solutions (33) it is necessarythat Re b > 0 and sufficient that
−Re b −√
−α +√α2 + β2
2> 0.
J Infrared Milli Terahz Waves (2009) 30:679–699 693
The last inequality is equivalent to
α + 2 (Re b)2 >√
α2 + β2. (40)
Then
α + 2 (Re b)2 > 0. (41)
Solving (41) for Re b we have
Re b ∈⎛⎝−∞,−
√−α +√α2 + β2
2
⎞⎠ ∪⎛⎝√
−α +√α2 + β2
2, +∞
⎞⎠ .
Using the necessary condition Re b > 0, we finally obtain
Re b = (Dr1 − k1 Qi
1
)a2
0 ∈⎛⎝√
−α +√α2 + β2
2,+∞
⎞⎠ . (42)
Using the expressions for α and Re b , (41) takes the form
2(Dr
1 − k1 Qi1
)2a4
0 + 4[a2
0 Qr1 K − P1
1k1 K − P12q1q]2 + a4
0 |Q1|2 K2 + P121 K4
+ P122 q4 + k2
1a40 Qr2
1 + Di21 a4
0
+ 2P11 P1
2
(q2 K2 − 4k1q1 Kq
)+ 2k1a40 Qr
1 Di1
+ a20
(6k1 P1
1 Qr1 K2 + 8q1 P1
2 Qr1 Kq − 2k1 P1
2 Di1q2)
− 2a20
(P1
1 + P12
)Di
1 K2 > 4Qi21 K2a4
0 + 4P121 k2
1 K2 + 4P122 q2
1q2
+ 4a40 Qr2
1 K2 +√
α2 + β2 > 0,
and necessarily
1
2a20 K2
{2(Dr
1 − k1 Qi1
)2a4
0 + 4[a2
0 Qr1 K − P1
1k1 K − P12q1q]2 + a4
0 |Q1|2 K2 + P121 K4
+P122 q4 + 2k1a4
0 Qr1 Di
1 + a20
(6k1 P1
1 Qr1 K2 + 8q1 P1
2 Qr1 Kq − 2k1 P1
2 Di1q2)
+k21a4
0 Qr21 + Di2
1 a40 + 2P1
1 P12
(q2 K2 − 4k1q1 Kq
)}− (P11 + P1
2
)Di
1 > 0. (43)
We finally obtain the following modulational instability criterion:
Theorem In order that the spatial wave solutions (33) of the generalized 2D-Ginzburg-Landau equation (17) be modulationally unstable it is necessary that thewavenumbers of the perturbation, K and q, and the wavenumbers k1 and q1 of thecarrier satisfy inequality (43), and sufficient that condition (42) be valid.
5 2D coherent structures
In this section we discuss the coherent structure framework in the Ginzburg-Landausystem. The important coherent structures that organize much of the dynamical
694 J Infrared Milli Terahz Waves (2009) 30:679–699
properties of traveling wave system are sources and sinks [25]. As in the case of asingle complex Ginzburg-Landau (see for example Ref. [26]), the temporal evolu-tion of coherent structures in the Ginzburg-Landau system amounts to a uniformpropagation with velocity vcoh and an overall phase-oscillation with frequencies ω1coh
and ω2coh :
u(x, y, t) = a(z) exp
[i∫
F(z)dz − iω1coht]
, v(x, y, t)=b(z) exp
[i∫
G(z)dz − iω2coht]
,
z = kx + qy − vcoht. (44)
Here F(z) and G(z) are the local wavenumber and are functions of z. As we can seefrom Eq. (44), coherent structures have a fixed spatial profile, and their dynamics is acombination of propagation and oscillation. When the ansatz (44) is substituted intothe Ginzburg-Landau system (15)–(16), one obtains a set of six coupled first orderreal ordinary equations
a′ = x, (45)
b ′ = y,
x′ = aF2 + 1
k2 P11 + q2 P1
2
{vcohaF − Di
1
(a3 + 2ab 2
)+ (ω1 coh − γ i1
)a
−kQi1
(3a2x + 2b 2x + 4ab y
)− kQr1
(a3 F + 2ab 2 F
)},
y′ = b G2 + 1
k2 P21 + q2 P2
2
{vcohb G − Di
2
(b 3 + 2ba2)+ (ω2 coh − γ i
2
)b
−qQi2
(3b 2 y + 2a2 y + 4ab x
)− qQr2
(b 3G + 2ba2G
)},
F ′ = −2Fxa
+ 1
k2 P11 + q2 P1
2
{−vcoh
xa
+ γ r1 + Dr
1
(a2 + 2b 2
)
+kQr1
(3ax + 2b 2 x
a+ 4b y
)− kQi
1
(a2 F + 2b 2 F
)},
G′ = −2Gyb
+ 1
k2 P21 + q2 P2
2
{−vcoh
yb
+ γ r2 + Dr
2
(b 2 + 2a2
)
+qQr2
(3b y + 2a2 y
b+ 4ax
)− qQi
2
(b 2 F + 2a2 F
)}. (46)
The (a, b , x, y, G, F) system (45)–(46) has singularities at a = 0 and b = 0. Inorder to overcome this difficulty we introduce the ”blow up” transform or σ -process[27]. Letting
xa
= X; yb
= Y,
J Infrared Milli Terahz Waves (2009) 30:679–699 695
we compute x′a = x2 + X ′, y′
b = Y2 + Y ′ and (45)–(46) becomes the desingularized(a, b , X, Y, G, F) system
a′ = aX, (47)
b ′ = bY,
X ′ = −X2 + F2 + 1
k2 P11 + q2 P1
2
{vcoh F − Di
1
(a2 + 2b 2)+ ω1coh − γ i
1
−kQi1
(3a2 X + 2b 2 X + 4b 2Y
)− kQr1
(a2 F + 2b 2 F
)},
Y ′ = −Y2 + G2 + 1
k2 P21 + q2 P2
2
{vcohG − Di
2
(b 2 + 2a2
)+ ω2coh − γ i2
−qQi2
(3b 2Y + 2aY + 4a2 X
)− qQr2
(b 2G + 2a2G
)},
F ′ = −2F X + 1
k2 P11 + q2 P1
2
{−vcoh X + γ r1 + Dr
1
(a2 + 2b 2)
+kQr1
(3a2 X + 2b 2 X + 4b 2Y
)− kQi1
(a2 F + 2b 2 F
)},
G′ = −2GY + 1
k2 P21 + q2 P2
2
{−vcohY + γ r2 + Dr
2
(b 2 + 2a2
)
+qQr2
(3b 2Y + 2a2Y + 4a2 X
)− qQi2
(b 2 F + 2a2 F
)}. (48)
On the invariant space a = 0, b = 0, the (a, b , X, Y, G, F) system reduces to thefollowing (X, F) and (Y, G)
⎧⎨⎩
X ′ = −X2 + 1(k2 P1
1+q2 P12)
(vcoh F + ω1coh − γ i
1
)+ F2,
F ′ = 1(k2 P1
1+q2 P12)
(−vcoh X + γ r1
)− 2F X.(49)
⎧⎨⎩
Y ′ = −Y2 + 1k2 P2
1+q2 P22
(vcoh G + ω2coh − γ i
2
)+ G2,
G′ = 1k2 P2
1+q2 P22
(−vcohY + γ r2
)− 2GY.(50)
The (X, F) and (Y, G) systems (49) and (50) characterize the behavior of thesolutions of (a, b , X, Y, G, F) system (47)–(48) as (a, b) → (0, 0) .
The (X, F) and (Y, G) systems (49) and (50) admit a number of fixed points. Itis evident that
(0, 0, X, F, Y, G
)will be a fixed point of system (47)–(48) if, and
only if(X, F
)and(Y, G
)are fixed points of (49) and (50), respectively. Hence, for
the stability of the fixed point(0, 0, X, F, Y, G
)of system (47)–(48), it is necessary
and sufficient that the fixed points(X, F
)and
(Y, G
)of system (49) and (50),
respectively, be stable. The eigenvalues of the linearized systems associated with thefixed points
(X, F
)and(Y, G
)read
λ1,2 = −2X ± i
(2F + vcoh(
k2 P11+q2 P1
2
))
and λ3,4 = −2Y ± i
(2G + vcoh(
k2 P21+q2 P2
2
))
,
696 J Infrared Milli Terahz Waves (2009) 30:679–699
respectively. Therefore the eigenvalues of the linearized system associated with thefixed point
(0, 0, X, F, Y, G
)of system (47)–(48) are
λ1,2 = −2X ± i
(2F + vcoh(
k2 P11 + q2 P1
2
))
, λ3,4 = −2Y ± i
(2G + vcoh(
k2 P21 + q2 P2
2
))
,
λ5,6 = 0.
Thus if X > 0 and Y > 0 (respectively X < 0 or Y < 0), then the fixed point(0, 0, X, F, Y, G
)of system (47)–(48) will be stable (respectively, unstable).
As an example let us consider the stationary case:
vcoh = ω1coh = ω2coh = 0.
In this case, we have
X = ±√√√√1
2
(|γ1|∣∣k2 P1
1 + q2 P12
∣∣ − γ i1
k2 P11 + q2 P1
2
);
F = ±√√√√1
2
(|γ1|∣∣k2 P1
1 + q2 P12
∣∣ + γ i1
k2 P11 + q2 P1
2
);
Y = ±√√√√1
2
(|γ2|∣∣k2 P2
1 + q2 P22
∣∣ − γ i2
k2 P21 + q2 P2
2
);
G = ±√√√√1
2
(|γ2|∣∣k2 P2
1 + q2 P22
∣∣ + γ i2
k2 P21 + q2 P2
2
).
We then obtain the following results:
⎛⎝0, 0,
√√√√1
2
(|γ1|∣∣k2 P1
1 + q2 P12
∣∣ − γ i1
k2 P11 + q2 P1
2
), ±√√√√1
2
(|γ1|∣∣k2 P1
1 + q2 P12
∣∣ + γ i1
k2 P11 + q2 P1
2
),
√√√√1
2
(|γ2|∣∣k2 P2
1 + q2 P22
∣∣ − γ i2
k2 P21 + q2 P2
2
), ±√√√√1
2
(|γ2|∣∣k2 P2
1 + q2 P22
∣∣ + γ i2
k2 P21 + q2 P2
2
)⎞⎠
and⎛⎝0, 0,−
√√√√1
2
(|γ1|∣∣k2 P11+q2 P1
2
∣∣ − γ i1
k2 P11+q2 P1
2
),±√√√√1
2
(|γ1|∣∣k2 P11+q2 P1
2
∣∣ + γ i1
k2 P11+q2 P1
2
),
−√√√√1
2
(|γ2|∣∣k2 P2
1 + q2 P22
∣∣ − γ i2
k2 P21 + q2 P2
2
),±√√√√1
2
(|γ2|∣∣k2 P2
1 + q2 P22
∣∣ + γ i2
k2 P21 + q2 P2
2
)⎞⎠
are stable fixed points and unstable fixed points of system (47)–(48), respectively.
J Infrared Milli Terahz Waves (2009) 30:679–699 697
Fig. 3 Figures obtained for the stable fixed point (0, 0, 29462, 29462, .968 25, .968 25) with initialconditions a(0) = b(0) = .0001, F(0) = G(0) = X(0) = Y(0) = .001. (a,d): amplitudes of u and v asfunctions of z, , respectively. (b, e): Local wavenumber as functions of z. (c, f): F and G curves in thephase.
For the line parameters
C0 = 540 pF, C2 = 270 pF; L = 28 μH; R = 105�, G = 10−4�−1; b = .16V−1,
we plot in the stationary case the amplitudes |u(z)| = a(z) and |v(z)| = b(z), thelocal wavenumber F = F(z) and G = G(z), and the local wavenumber F = F(a),
and G = G(b) as functions of the amplitudes a and b . Here the plots a(z), b(z),
F(z), and G(z) are all solutions of Cauchy problems associated with the ODE’s(47)–(48) near the fixed points when k2 = q2 = 10−7. (a, F) and (b , G) are the phase
Fig. 4 Figures obtained for the unstable fixed point (0, 0,−29462, 29462,−.968 25, .968 25) withinitial conditions a(0) = b(0) = .0001, F(0) = G(0) = X(0) = Y(0) = .001. (a,d): amplitudes of uand v as functions of z, , respectively. (b, e): Local wavenumber as functions of z. (c, f): F and Gcurves in the phase plane (a, F) and (b , G) , respectively.
698 J Infrared Milli Terahz Waves (2009) 30:679–699
planes corresponding to system (49) and (50), respectively. In the case of the stablefixed point, Fig. 3 (b) and (e) show that F(z), G(z) → 0 as z → +∞. The situationis not the same if
(0, 0, X, F, Y, G
)is an unstable point of system (47)–(48) when(
X, F)
and(Y, G
)are unstable fixed points of (49) and (50), respectively. As one
can observe from Fig. 4 (b) and (e), F(z), G(z) → 0 as z → −∞.
6 Conclusion
The present paper presents a study of modulated wave trains in the discrete couplednonlinear dissipative transmission line. Using the method of multiple scales weestablished that the evolution of nonlinear excitations in the network is governed by2D-Ginzburg-Landau system with derivatives in the cubic terms. As far as we know,this is the first time is derived such a system using coupled nonlinear transmissionline. The conditions of the modulational instability of the spatial wave solutionsof both the 2D-Ginzburg-Landau system and the generalized 2D-Ginzburg-Landauequation are expressed in terms of coefficients of the obtained equations. In the caseof the 2D-Ginzburg-Landau system, we set the wavenumber of the perturbation to bezero. Next, the coherent structures of the 2D-Ginzburg-Landau system are studied.
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