Available at: http://www.ictp.it/~pub−off IC/2006/035

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

SUPPRESSION OF THE FAST AND SLOW MODULATED WAVES MIXINGIN THE COUPLED NONLINEAR DISCRETE LC TRANSMISSION LINES

David Yemele1

Laboratoire de Mecanique et de Modelisation des Systemes Physiques,

Faculte des Sciences, Universite de Dschang, B.P. 067, Dschang, Cameroun

and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

Timoleon C. KofaneLaboratoire de Mecanique, Faculte des Sciences, Universite de Yaounde I,

B.P. 812, Yaounde, Cameroun.

Abstract

The conditions of propagation of fast and slow-modes of modulated waves on the two coupled

discrete nonlinear LC transmission lines are examined, each line of the network containing a finite

number of cells. It is found that the use of an appropriate unit-cell, a band-pass filter, associated

to a convenient choice of the intermediate coupling capacitor between the two lines allows to

avoid the crucial problem of mixing of waves of different modes in the network. Good qualitative

and quantitative agreements are found between analytical predictions and numerical results.

MIRAMARE – TRIESTE

May 2006

1Junior Associate of ICTP.

1 Introduction

In nonlinear dispersive media, the propagation of modulated waves, such as envelope (bright)

solitons or hole (dark) solitons, has been the subject of considerable interest for many decades [1].

In nonlinear optics [2, 3, 4, 5, 6], the permanent progress about fiber loss, birefringent fibers,

Raman Effect and other undesirable effects allow important improvement of practical results

concerning the distortion-less signal transmission in ultrahigh-speed communication and are

potentially promising for a number of other applications. In general, the propagation of modu-

lated waves in nonlinear dispersive media is characterized by a series of complex changes in its

structure. A theoretical interpretation of these phenomena is crucial in order to understand the

nonlinear properties of the medium and the complex dynamics of such nonlinear systems. As

an example, the unstable propagation of slowly modulated plane waves (known as modulational

instability (MI)) in the nonlinear dispersive medium exhibits the capacity of this medium to

support the propagation of envelope soliton in this parameter regime [7].

Similarly, nonlinear discrete electrical transmission lines are convenient tools to study wave

propagation in nonlinear dispersive media (see [7, 8] and references therein). In particular, they

provide a useful way to check how the nonlinear excitations behave inside the nonlinear medium

and to model the exotic properties of new systems [9]. These are the reasons why, since the

pioneering works by Hirota and Suzuki [10] and Nagashima and Amagishi [11] on a single elec-

trical line simulating the Toda lattice [12], a growing interest has been devoted to the use of

the nonlinear discrete electrical transmission lines [13, 14, 15, 16, 17, 18]. However, the above

cited studies are limited to a single mode soliton that the single LC transmission line adequately

describes in certain parameter regimes, whereas there are many physical phenomena which can

be investigated by the use of more than a single electrical transmission line. Indeed, to cite

just two examples, the single electrical transmission line gives a good description of nonlinear

deep water waves while it fails when considering nonlinear wave in the long gravity wave re-

gion [19, 20]. Similarly, if the single electrical line shares many phenomena with certain optical

fibers, it cannot model the birefringent fiber which allows two modes of propagation, i.e. the

fast and slow modes [6].

There are only a few works, as far as we know, which report the study of soliton in the

coupled nonlinear transmission lines. Yoshinaga and Kakutani [19, 20, 21] have investigated

theoretically and experimentally the Korteweg de Vries (KdV) solitons on a coupled LC trans-

mission line consisting of two nonlinear LC ladder lines connected by identical intermediary

capacitors, and have shown that the network admits two different modes (a fast-mode and a

slow-mode), in each direction of wave propagation. Similarity of this line with a two-layer fluid

in the long gravity wave region has also been discussed. Next, the extension of these studies

2

to envelope solitons has been made by Essimbi et al. [22, 23] who concentrated efforts to ex-

citation modes, for the wave vector close to the limit k = π/2 [22] and in the vicinity of the

gap of the linear spectrum [23]. The soliton propagation and interaction on two-dimensional

nonlinear transmission lines have also been studied (see [24] and references therein). However,

experiments and analytical studies on the basic coupled NLTL show that whenever the network

is excited by an electrical wave, two modes of propagation (slow and fast-modes) are generated

in each line and enters unavoidably into play with the wave-coupling behavior causing qualita-

tively different phenomena compared with the ordinary process of MI, such as the annihilation

of both modes. This slow and fast-mode mixing effect is undesirable for the nonlinear modulated

wave’s experiments in the coupled NLTL as well as for its practical applications. Therefore it

is useful to find a way to suppress this undesirable effect. This is the main objective of the

paper. For this purpose, we first present carefully the analytical and numerical investigations

concerning the nonlinear modulated waves and the possible propagation of envelope solitons on

the two coupled LC transmission lines, depending on the appropriate choice of the carrier wave

frequency (or angular frequency) and the capacitance of the intermediary coupling capacitor.

The paper is organized as follows: In Sec.2 we present the basic characteristics of the coupled

nonlinear discrete electrical transmission line under consideration. In Sec.3, in the low-amplitude

limit, we derive the coupled nonlinear Schrodinger (NLS) equations governing the propagation

of slowly modulated waves in the network and show that they can reduce to the single NLS

equation. The frequency domain where the network allows the propagation of envelope solitons

is also determined. In Sec.4, by introducing a linear inductor in parallel with each nonlinear

capacitor of the two lines, we show that it is possible to avoid the mixing of the fast- and

slow-modes of propagation in the network. Numerical experiments are considered in Sec.5 in

order to verify the validity of the theoretical predictions, namely the MI phenomenon and the

propagation of fast- and slow-envelope solitons. Finally, concluding remarks are devoted to Sec.6.

2 Basic coupled nonlinear discrete LC transmission lines

The basic model usually used consists of a nonlinear network with two coupled nonlinear LC

transmission lines. Each line contains a finite number of cells which consist of two elements: a

linear inductor of inductance Lj in the series branch and a nonlinear capacitor of capacitance Cj

in the shunt branch, where the subscript j designates the line number and can take the values 1

and 2. From now on we shall use this notation. The two lines are connected by an intermediary

linear capacitor C, as shown in Figure 1. The capacitance Cj are voltage dependent and are

biased by constant voltages Vbj [7]:

Cj (Vbj + Vjn) ≡dqj,n

dVj,n= C0j

(

1 − 2αjVj,n + 3βjV2j,n

)

, (j = 1, 2), (1)

3

with C0j = Cj(Vbj) and where αj and βj are the nonlinear positive coefficients of the electrical

charge qj,n stored in the nth capacitor of line j. Denoting by Vj,n the voltage across the nth

capacitor of line j and using the Kirchhoff’s laws, the circuit equations are then given by:

d2qj,n

dt2=

1

Lj(Vj,n−1 − 2Vj,n + Vj,n+1) + C

(

d2V3−j,n

dt2−

d2Vj,n

dt2

)

, n = 1, 2, ...N, (2)

By inserting the above expansion of the capacitance (1) into Eq.(2), we obtain the following

coupled equations governing wave propagation in this nonlinear network:

d2Vj,n

dt2+ Ω2

0j (2Vj,n − Vj,n−1 − Vj,n+1) −

− γj

(

d2Vj,n

dt2−

d2V3−j,n

dt2

)

= αj

d2V 2j,n

dt2− βj

d2V 3j,n

dt2, n = 1, 2, ...N, (3)

where the characteristic frequencies of each line, Ω0j , and the coupling coefficients γj verify the

following relations:

Ω0j = 1/LjC0j , γj = C/C0j . (4)

The linear properties of the network can be studied by assuming a sinusoidal wave of the

form:

Vj,n = Vj exp[i(kn − ωt)] + c.c., (5)

where Vj is the constant amplitude, k and ω are, respectively, the wave number and the angular

frequency, and c.c. stands for complex conjugate. Substituting Eq.(5) into Eq.(3) yields the

following linear dispersion relation

ω2 = ω(ℓ)2c sin2(k/2), ω(ℓ)

c = 2ω0ℓ, (ℓ = 1, 2) (6)

where the characteristic frequencies of the network ω0ℓ are given by:

ω0ℓ =(1 + γ2)Ω

201 + (1 + γ1)Ω

202 + (−1)ℓ

√

[

(1 + γ2)Ω201 − (1 + γ1)Ω

202

]2+ 4γ1γ2Ω

201Ω

202

2(1 + γ1 + γ2). (7)

Relations (6) explain that there are two elementary waves (modes) which coexist on each line

at the same frequency ω but with different wave vectors. The mode corresponding to ℓ =

2 has a higher group velocity compared to the group velocity of the mode ℓ = 1: it’s the

fast-mode. Accordingly, the mode ℓ = 1 is called slow-mode. Figure 2 shows, for different

situations, the linear dispersion of these two modes of propagation. When the two coupled

lines have identical linear characteristic parameters, i.e. Ω01 = Ω02 ≡ Ω0 and γ1 = γ2 ≡ γ,

the fast mode reduces to the standard mode of propagation of an isolated single line with the

characteristic frequency ω02 = Ω0, while the characteristic frequency of the slow mode reduces

to ω01 = Ω0/(1+2γ)1/2 . The amplitudes of the signal voltage propagating along the two coupled

lines are linearly dependent and verify the following relation (obtained by inserting Eq.(5) into

Eq.(3)):

V(ℓ)2 = λ(ℓ)V

(ℓ)1 , with λ(ℓ) = 1 +

1

γ1

(

1 +Ω2

01

ω20ℓ

)

, (ℓ = 1, 2), (8)

4

where the superscript ℓ stands for the mode of propagation. For the sake of convenience, hereafter

we shall use this notation where the case ℓ = 1 stands for the slow-mode while the case ℓ = 2

corresponds to the fast-mode. Due to the fact that the coefficient λ(2) > 0 and λ(1) < 0, the

signal voltages in the two lines are always in phase for the fast-mode whereas they are always

180 out of phase for the slow-mode. Furthermore, it appears from (7) and (8) that when the

two lines have identical linear characteristic parameters, the amplitudes of the signal voltages

on both lines are equal.

3 Coupled NLS equations: fast-and slow-nonlinear modulated

waves

3.1 Coupled NLS equations

To describe modulated waves in the network, we consider waves with a slowly variation of

envelope in time and space with respect to a given carrier with angular frequency ω = ωp and

wave vector k = kp. Then, in order to use the reductive perturbation method in the semi-discrete

limit [24, 25, 26, 27, 28], we introduce the slow envelope variables x = ǫ(n − vgt) and τ = ǫ2t

where ǫ is a small parameter and vg a constant. Hence, the solution of Eq.(3) is assumed to

have the following general form:

Vj,n(t) = ǫAj(x, τ)eiθ + ǫ2[

φj(x, τ) + Bj(x, τ)e2iθ]

+ c.c., (9)

with θ = kpn − ωpt. Substituting Eq.(9) into the coupled equations (3) and keeping terms of

order ǫ3 proportional to eiθ, one obtains the following coupled NLS equations:

iAj,τ + iΓjA(3−j),τ + Pj,jAj,xx + Pj,(3−j)A(3−j),xx +

+ Qj,jAj | Aj |2 +Qj,(3−j)Aj | A3−j |

2 +QjA∗

jA23−j = 0, (10)

where the coefficients verify the following relations:

Γj = −γj/ (1 + γj) , Pj,(3−j) = −Γjv2g/2ωp, (11a)

Qj,(3−j) = −ωpαjξ(j)3−j/(1 + γj), Qj = −ωpαjη

(j)3−j/(1 + γj), (11b)

Pj,j =(1 + γj)v

2g − Ω2

0j cos kp

2ωp(1 + γj), Qj,j = −

αjωp

(1 + γj)

[

ξ(j)j + η

(j)j −

3βj

2αj

]

(11c)

with

η(j)j = −

αj

δ

[

Ω203−jv

2g

ω40ℓ

− (1 + γ3−j)

]

, η(j)3−j = −

α3−j

δγj, (12a)

ξ(j)j = 2

αj

∆

[

(1 + γ3−j) − Ω203−j/v

2g

]

, ξ(j)3−j = 2

γj

∆α3−j , (12b)

∆ =[

(1 + γ1) − Ω201/v

2g

] [

(1 + γ2) − Ω202/v

2g

]

− γ1γ2, (12c)

δ =[

(1 + γ1) − Ω201v

2g/ω

40ℓ

] [

(1 + γ2) − Ω202v

2g/ω

40ℓ

]

− γ1γ2. (12d)

5

Furthermore, the dc and the second harmonic terms φj(x, τ) and Bj(x, τ) are, respectively,

related to the fundamental terms Aj(x, τ) by:

φj = ξ(j)j | Aj |

2 +ξ(j)3−j | A3−j |

2, Bj = η(j)j A2

j + η(j)3−jA

23−j , (13)

obtained by keeping terms proportional to ǫ4(eiθ)0 and ǫ2(eiθ)2, respectively. The group velocity

appearing in Eq.(11) has two distinct analytical expressions since there are two different modes

of propagation:

vg =dw

dkp= ω0ℓ

(

1 − ω2p/ω

(ℓ)2c

)

. (14)

Accordingly, Eqs.(10) constitutes two sets of two coupled NLS equations corresponding to the

two different modes of propagation of the network. However, the two modes are entirely inde-

pendent each other. This allows us to consider, separately, each mode of propagation.

Let us mention that coupled NLS equations of different forms have been obtained in var-

ious domains of physics. It can be used to describe: interactions of solitons in a baroclinic

atmosphere [29], the superposition of forward and backward propagating modulated waves in

a single electrical transmission line [30], the evolution of two linear polarization components in

nonlinear birefringent optical fibers [6], to cite just a few. On the contrary to the above cited

studies, in the context of coupled electrical transmission lines, the coupled NLS equations are

easily unlashed by means of the linear dependence relation between A(ℓ)1 and A

(ℓ)2 :

A(ℓ)2 = λ(ℓ)A

(ℓ)1 (15)

obtained from the terms proportional to ǫ2eiθ in the series expansion of Eq.(2), where λ(ℓ) verifies

Eq.(7). Because of this linear dependence of envelopes A(ℓ)1 and A

(ℓ)2 , the two sub-equations of

each coupled NLS equations have to be equivalent and lead to the same NLS equation. This

condition is satisfied for different frequencies, if and only if the two coupled transmission lines

have the same linear and nonlinear characteristic parameters, i.e.

Ω01 = Ω02 ≡ Ω0, α1 = α2 ≡ α, and β1 = β2 ≡ β (16)

and then γ1 = γ2 = γ. In this limiting case, the signal voltages propagate along the two coupled

lines with identical amplitudes and phase for the fast-mode (i.e. A(2)1 = A

(2)2 ), and with identical

amplitudes but with opposite phases for the slow-mode (i.e. A(1)1 = −A

(1)2 ). Setting u(ℓ) = A

(ℓ)1 ,

we obtain the NLS equation describing the propagation of each mode in the network:

iu(ℓ) + Pu(ℓ)xx + Qu(ℓ) | u(ℓ) |2= 0, (ℓ = 1, 2) (17)

where the dispersive and the nonlinear coefficients are, for the slow-mode

Q(1) = −αΓωp

(

b(1)0 + b

(1)2 − a

)

, P (1) = −ωp/8, (18)

with

b(1)0 =

2[

1 − 1/Γ(1 − ω2p/ω

(1)2c )

] , b(1)2 =

Γ

Γ − 1 + ω2p/ω

(1)2c

, (19)

6

and for the fast-mode

Q(2) = −αωp

(

b(2)0 + b

(2)2 − a

)

, P (2) = −ωp/8, (20)

with

b(2)0 = 2(1 − ω(2)2

c /ω2p), b

(2)2 = ω(2)2

c /ω2p. (21)

The parameters a and Γ denote the nonlinear coefficients of the coupled NLTL and the normal-

ized coupling coefficients between the two lines, respectively, and whose expressions are given

by:

a = 3β/2α2, Γ = 1/(1 + 2γ). (22)

Note that, in the absence of the coupling between the two lines, i.e. γ = 0 and then Γ = 1,

the two modes of propagation reduce to the standard single mode of propagation of modulated

waves in an isolated NLTL [30].

3.2 Fast- and Slow-mode envelope solitons

It is well known that the Benjamin-Feir instabilities exhibited by a dispersive nonlinear

medium constitute the proof of its capacity to support envelope solitons in certain domains of

propagation. From Eq.(17), it is easy to show that, a continuous slowly modulated plane wave

should be unstable if P (ℓ)Q(ℓ) > 0. This instability leads to the formation of small wave packets

or envelope pulse solitons train, solution of the NLS equation (17) and whose explicit expression

is given by [2]:

u(ℓ) = A0sech[

(x − P (ℓ)veτ)/L(ℓ)s

]

exp[

ive(x − P (ℓ)vcτ)/2]

(23)

where ve and vc are respectively the envelope and phase velocities while L(ℓ)s =

√

2P (ℓ)/Q(ℓ)/(ǫA0)

designates the spatial soliton extension. The superscript ℓ can take the values 1 and 2, corre-

sponding to the two modes of propagation of the network, i.e., the fast and slow-mode envelope

solitons.

According to the above mentioned, our electrical network exhibits the propagation of the

fast-mode envelope soliton if P (2)Q(2) > 0 , i.e., when the carrier angular frequency ωp belongs

to the domain [ω(2)1 , ω

(2)c ] with

ω(2)c = 2Ω0, ω

(2)1 = ω(2)

c (2 − a)1/2. (24)

With the help of relations (9), (20) and (23), the general expressions of the signal voltage, in

the two lines, in terms of the fundamental dc and second harmonics can be readily obtained.

For the slow-mode envelope soliton, several frequency domains for which P (1)Q(1) > 0 may

exist and depend on the numerical values of the following characteristic angular frequencies, ωa,

ωb, ω+, and ω− defined as follows:

ωa = ω(1)c

√

1/Γ − 1, ωb = ω(1)c

√

1 − Γ, ω± = ω(1)c

√

1 −3Γ − a(1 + Γ2)/Γ ±

√

∆

2(2 − a), (25)

7

with ∆ = [3Γ − a(1 + Γ2)/Γ]2 − 4(1 − a)(2 − a) and the coefficient

γc1 = (1/Γ+ − 1)/2, Γ± =

(

a2 − 3a + 4)

± 2√

2(2 − a)(1 − a)

(3 − a)2. (26)

Using the numerical values described in Sec.5, the following classification is obtained:

• for γ ∈]0, γc1], there are two domains allowing the propagation of envelope soliton: Domain

I corresponds to ωp ∈ [ωb, ω+] and Domain II to ωp ∈ [ω−, ω(1)c ].

• For ]γc1,∞], we have only one domain allowing envelope soliton propagation: ωp ∈

[ωb, ω(1)c ].

The above expressions achieve the study of modulated nonlinear waves in the basic model,

hereafter referred to Model I. The simultaneous presence of the fast- and slow-modes in the

network is at the origin of the crucial problem of the wave mixing, since each input wave at the

frequency ωp generates in each line two waves with different wave vectors. This situation renders

more complex the study of modulated waves in the network and eventually its applications. In

the next section we give the answer to this problem.

4 Suppression of the wave mixing: Influence of parallel self

4.1 Model description and NLS equations

In this section we modify the coupled electrical network of Figure 1 by introducing, in each

unit cell, the self L2 in parallel with the nonlinear capacitor Cj , j = 1, 2. Thus, each unit cell of

the new network hereafter referred to Model II contains three elements: two self and a nonlinear

capacitor as illustrated in Figure 3. The basic linear properties of this unit-cell are analogous to

those of the band-pass filter. In the following, we restrict our analysis on the case where the two

lines have identical linear and nonlinear characteristics. This restriction is based on the results

of the preceding section for which the coupled electrical network exhibits NLS solitons only in

this limiting case. Equations governing wave propagation in this nonlinear network are then

given by:

d2Vj,n

dt2+ Ω2

0 (2Vj,n − Vj,n−1 − Vj,n+1) + u20Vj,n −

− γ

(

d2Vj,n

dt2−

d2V3−j,n

dt2

)

= αd2V 2

j,n

dt2− β

d2V 3j,n

dt2, n = 1, 2, ...N, (27)

where Ω20 = 1/L1C0, u2

0 = 1/L2C0 and γ = C/C0 are characteristic parameters of the network.

We should have also the occasion to use the constant ωc = (u20 + 4Ω2

0)1/2.

For the linear waves of the network whose dynamics is governed by Eq.(27), we have identified

two modes of propagation whose angular frequency and wave number k are described by the

dispersion relation of a typical band-pass filter. For the fast-mode, we have:

ω2 = u20 + 4Ω2

0 sin2(k/2) (28)

8

with the upper cut-off frequency ω(2)c = ωc, and for the slow-mode

ω2 = Γ[

u20 + 4Ω2

0 sin2(k/2)]

, Γ = 1/(1 + 2γ), (29)

with the upper cut-off angular frequency ω(1)c = Γ1/2ωc . In the following, as in the preceding

section, we use the superscript ℓ = 1 to designate the slow-mode and the superscript ℓ = 2 refers

to the fast-mode. Similarly, the subscript j = 1 designates line 1 while j = 2 stands for the

second line.

For the fast-mode propagation, the waves on the two coupled lines are in phase and have

equal amplitudes while for the slow-mode propagation they are in opposite phase and propagate

with equal amplitudes. Figure 4 displays the dispersion relation of the two modes of propagation

for the coupling coefficient γ = 8. It follows that, when the value of the coupling coefficient

exceeds the critical value

γcr = 2Ω20/u

20, (30)

the band-pass of the two modes of propagation are entirely separated. As a consequence, each

input signal at frequency ω generates one and only one mode of propagation in each line. In

addition, it is also possible to create the gap between the linear spectrums of the two modes of

propagation. As we shall see below this result is a first step toward the suppression of the slow-

and fast- nonlinear modulated wave mixing.

Let us come to nonlinear excitations in the network. In order to describe the slowly modu-

lated nonlinear waves, we use the same reductive perturbation method in the semi-discrete limit

as presented in Sec.3. Thus the insertion of Eq.(9) into the coupled differential equation (27)

leads to the following standard NLS equation for the envelope evolution:

iu(ℓ) + Pu(ℓ)xx + Qu(ℓ) | u(ℓ) |2= 0, (31)

where the variable u(ℓ) is related to A(ℓ)1 and A

(ℓ)2 as follows: u(ℓ) = A

(ℓ)1 = (−1)(ℓ)A

(ℓ)2 . The

dispersion coefficient of the NLS equation is given by:

P (ℓ) =

(

ω(ℓ)2cr − ω2

p

) (

ω(ℓ)2cr + ω2

p

)

8ω3p

(32)

with

ω(1)cr = (Γu0ωc)

1/2, ω(2)cr = (u0ωc)

1/2, (33)

while the nonlinear coefficients are given by

Q(1) = Γα2ωp(a − b(1)), Q(2) = α2ωp(a − b(2)) (34)

with a defined by Eq.(22) and

b(1) =4Ω2

0ω2p

Ω20[3u

20 − 4(1 − Γ)ω2

p/Γ] + (u20 − ω2

p/Γ)2, b(2) =

4Ω20ω

2p

3u20Ω

20 + (u2

0 − ω2p)

2, (35)

9

The functions B1(x, τ) and φ1(x, τ) are related to u(ℓ) as B(ℓ)1 (x, τ) = B

(ℓ)2 (x, τ) = αb(ℓ)u(ℓ)2(x, τ)

while the dc terms vanish due to the existence of low-frequency forbidden band, i.e. φ(ℓ)1 (x, τ) =

φ(ℓ)2 (x, τ) = 0 .

4.2 Fast-mode envelope soliton

As pointed out above, the existence of the envelope soliton in the network is conditioned

by the positive value of the product P (2)Q(2). Thus, using Eqs.(32) and (34) we are able to

predict the frequency range of the existence of the fast-mode envelope soliton. Accordingly,

the fast-mode should propagate on the network if the angular frequency of the carrier wave lies

within the interval [ω(2)cr , ω

(2)c ]. The width of the band-pass of the fast-mode envelope soliton is

then given by:

∆ω(2) = ωc

(

1 −√

u0/ωc

)

(36)

By comparing the band-pass of the fast-mode envelope soliton for the two models of coupled

electrical transmission lines (Model I and Model II), we note that the band-pass of soliton in

Model II is larger than that of the Model I. Consequently, the introduction of the self in parallel

with the nonlinear capacitor enhances the frequency range of the envelope soliton propagation.

Note also that the coupling coefficient has no effect on the characteristic parameters of the

fast-mode soliton. The explicit expression of the signal voltage corresponding to the fast mode

envelope soliton can be easily obtained by means of the relations (9), (28) and (35).

4.3 Slow-mode envelope soliton

In this regime of propagation, the product P (1)Q(1) takes positive values in two domains

of frequencies depending on the numerical values of ω(1)c and ω

(1)cr , and also on the following

characteristic frequencies ωa, ωb, ω+, and ω−, defined by:

ω2a = Γu2

0

[

1 + 2(Ω20/u

20)(1 − Γ) − δ1/2

]

, ω2a = Γu2

0

[

1 + 2(Ω20/u

20)(1 − Γ) − δ1/2

]

, (37)

and

ω2± = Γu2

0

[

1 + 2(Ω20/u

20)(1 − Γ + Γ/a)±∆1/2

]

, (38)

where

δ =

[

1 + 2Ω2

0

u20

(1 − Γ)

]2

−

(

1 + 3Ω2

0

u20

)

, ∆ =

[

1 + 2Ω2

0

u20

(1 − Γ +Γ

a)

]2

−

(

1 + 3Ω2

0

u20

)

. (39)

It depends also on the normalized coupling coefficient

Γ± = 1 +1

2

[

Ω20/u

20 ±

√

(Ω20/u

20)(3 + Ω2

0/u20)

]

, (40)

from which we define the parameter γc1 = (1/Γ− − 1)/2. Using the numerical values of the

preceding section in addition with L2 = 0.470mH, we obtain the following hierarchy:

10

• For γ ∈]0, γc1], we have one domain of propagation: ωp ∈ [ω(1)cr , ω

(1)c ].

• For γ ∈ [γc1, γcr], we have two domains of propagation: ωp ∈ [ωa, ωb] and [ω(1)cr , ω

(1)c ].

• For γ >> γcr, there are also one domain of propagation: [ω0, ω(1)cr ].

On the contrary to the fast-mode envelope soliton, the existence of the slow-mode is strongly

dependent on the combined effects of the adding self L2 and the coupling capacitor. This

dependence allows to manage in our convenience the frequency range where the slow-mode

soliton can propagate. For example for γ > γcr, the network exhibits slow-mode envelope

soliton out of the frequency-band of the fast-mode. Consequently, the two modes are completely

separated and the wave mixing is then avoided.

Apart the envelope soliton, the network described in this section may exhibit the hole soliton

(fast-mode and slow-mode), in the frequency domain where the product P (ℓ)Q(ℓ) has negative

values. Its explicit expression could be found by mean of the hole soliton solution of the NLS

equation (31) for different modes of propagation.

To conclude this section, we point out that, the merits of the introduction of the additional

self in parallel with the nonlinear capacitor in each unit cell of the network results in (i) the

creation of the possibility to dissociate the two modes of propagation (fast and slow-mode) by

a convenient choice of the coupling capacitance of the two electrical transmission lines and then

to avoid the wave mixing in the network; (ii) the enlargement of the band-pass width of the

fast-mode envelope soliton.

5 Numerical experiments

According to the analytical calculations presented in the preceding sections, it is possible

to determine in the spectrum of the coupled NLTL the frequency range for which the network

supports the propagation of envelope solitons. In order to check the validity of these predictions,

we first present the numerical experiments on the propagation of slowly modulated waves in the

network since their unstable propagation for certain parameter regime allows us to conclude

about the possibility of the network to support envelope soliton in this frequency domain and

then experiments on the fast- and slow-envelope solitons propagation are presented in Model II

in order to verify the achievement of the suppression of the wave mixing.

5.1 Slowly modulated plane waves: Modulational Instability

5.1.1 Basic Model: Model I

The numerical experiments are carried out on the exact equation (3) describing the propa-

gation of waves in the two coupled electrical transmission lines of Figure 1. The parameters of

the network are chosen to be L = L1 = L2 = 0.220mH, C0 = C01 = C02 = 320pF and γ = 1/2

which implies the cut-off frequency f(1)c = 848kHz for the slow-mode and f

(2)c = 1200kHz

11

for the fast-mode of propagation. The characteristic parameters of the reversed biased diode

are: α1 = α2 = α = 0.21V −1 and β1 = β2 = β = 0.197V −2 for a constant biased voltage

Vb1 = Vb2 = Vb = 2V [26, 27, 28] . The fourth-order Runge-Kutta scheme is used with normal-

ized integration time steep ∆t = 5×10−3 corresponding to the sampling period Ts = 1.33×10−9s.

Similarly, the number of cells is variable in order to avoid wave reflection at the end of the line

and also to run the experiments with sufficiently large time (here tmax = 4ms).

The input of line 1 (cell n = 0) is supplied by a slowly modulated signal of the form

V1,0(t) = Vm [1 + m cos(2πfmt)] cos(2πfpt) (41)

where Vm is the amplitude of the unperturbed plane wave (carrier wave) with frequency fp, and

where m and fm are respectively the rate and the frequency of the modulation. The input of

line 2 is matched by a resistor with variable resistance. The experiments have been made over

the whole carrier wave frequency range, i.e fp < f(2)c and for different values of the modulation

frequency less than 20kHz. The result is sketched in this table:

Mode Modulational InstabilityAnalytical Predictions (kHz) Numerical Results(kHz)

Slow-mode 600-848 Mixing of waves

Fast-mode 1040-1200 1050-1200

Figure 5 shows an example of the MI exhibited by the network where the corresponding mode

of propagation is the fast-mode. The input signal parameters are Vm = 0.5V , fp = 1180kHz,

fm = 4.8kHz and m = 0.01. This sine wave with a small modulation applied at n = 0 on line

1, exhibits some nonlinear distortions of the envelope as one can easily observe in the Figure 5.1

at cell number 1500 of both lines. As time goes on and the wave propagates along the network,

the modulation increases and the sine wave breaks into a periodic envelope pulse train. It is a

typical example of modulational instability phenomenon. It is also found that the amplitude of

the resulting sine wave is equal to 0.5V (see Figure 5.2: at cell 20 of lines 1 and 2), the algebraic

sum of the amplitudes of each sine wave in the two lines which would be 0.25V . This result is

in accordance with the analytical treatment of Sec.2 for linear waves. However when instability

occurs in the system, the amplitude of the resulting wave packet is larger than the amplitude of

the input wave 0.5V , the envelope pulse wave in each line behaves with the amplitude slightly

equal to 0.6V (see Figures 5.2 c1 and c2). Let us mention that, it is not possible to observe,

separately, the propagation of the slow-mode of modulated waves because of the simultaneous

presence of the fast-mode.

5.1.2 Suppression of the fast and slow modulated waves mixing: Model II

The numerical experiments are carried out on the exact equation (41) describing the prop-

agation of waves in the two identical coupled electrical transmission lines of Figure 3. The

12

parameters of the network are chosen to be L1 = 0.220mH and L2 = 0.470mH which implies

the critical coupling coefficient γcr = 4.56. The characteristic parameters of the reversed biased

diode are the same as in Model I. The corresponding cut-off frequencies are f(2)c = 1268kHz and

f(2)0 = ω0/2π = 411kHz for the fast-mode of propagation. As in the preceding paragraph, the

fourth-order Runge-Kutta scheme is used with the same normalized integration time steep and

with the same number of cells. Similarly, at the input of the line 1, we apply a slowly modulated

signal (41). Two particular cases are investigated according to the magnitude of the coupling

normalized coefficient. For the first case γ = 8 corresponding to these two cut-off frequencies

of the slow-mode f(1)c = 307kHz and f

(1)0 = Γ1/2u0/2π = 100kHz, while for the second case,

γ = 40 and the cut-off frequencies are f(1)c = 141kHz and f

(1)0 = Γ1/2u0/2π = 46kHz. The

theoretical predictions and the corresponding numerical results are shown in the following tables:

• For γ = 8

Mode Modulational InstabilityAnalytical Predictions (kHz) Numerical Results (kHz)

Slow-mode 99.47-175.01 and 302-307 100-175 and 300-308

Fast-mode 721-1268 750-1267

• For γ = 40

Mode Modulational InstabilityAnalytical Predictions (kHz) Numerical Results (kHz)

Slow-mode 45.52-80.05 50-80

Fast-mode 721-1267 750-1268

Figure 6 shows an example of MI developed by the network with coupling coefficient γ = 8 by

a slowly modulated wave with carrier frequency fp = 150kHz and initial amplitude Vm = 0.8V .

Frequency modulation and modulation rate are: fm = 3.47kHz and m = 0.01, respectively. In

this domain of frequency only the slow-mode can propagate. It appears that, the mechanism

of development of this instability is different to the well-known mechanism of MI described by

Benjamin-Feir [31] and presented in Figure 5 for the fast-mode.

5.2 Envelope solitons

In the preceding paragraph, we have shown that, in certain range of frequency, if a sine wave

applied at one end of the coupled lines is slowly modulated, we may expect modulation growth

and formation of wave packets which propagates along the network. This transformation of

sine wave into envelope soliton is interpreted by the fact that, in this domain of frequencies, the

modulated plane wave is not solution of the wave equations (3). When this wave is applied as the

input voltage, as the wave propagates along the network, the nonlinear system improves their

profiles and adapts them to the exact profile solution of the wave equations. In this paragraph

we show that this exact profile is given by the solution of the NLS equation (31). For this

13

purpose, we take as the input voltage, the profile of a modulated soliton given by

V(ℓ)1,0 (t) = Vmsech

[

v(ℓ)g t/L(ℓ)

s

]

cos (2πfpt) (42)

where fp and v(ℓ)g are the carrier frequency and the group velocity of the wave packet, respec-

tively, and where L(ℓ)s is the soliton width defined in (23). The upper-script (ℓ) stands for the two

modes of propagation. The parameters of the signal voltage are fp = 1180kHz belonging to the

domain of MI and to the frequency range of the fast-mode, and Vm = 2ǫA0 = 0.4V . With these

values, the group velocity is v(2)g = 1367cells/ms and the soliton width L

(2)s = 15cells. However,

since the amplitude of the signal generated in each line is the initial one divided by 2, we take

L(2)s = 30cells which is the best estimation of the soliton generated in the network. Figure 7

shows the propagation of this fast-modulated soliton along the line 1. The same behavior (not

presented) is observed on line 2. The phase plane plot diagnostic sketched in Figure 9(a) is 45o

line confirming the fact that the waves on the two lines are in phase. In addition, this result con-

firms the fact that the modulated soliton generated in the network corresponds to the fast-mode.

When we consider the following numerical values for the parameter of the signal voltage,

fp = 150kHz, Vm = 2ǫA0 = 0.4V and γ = 8, implying that v(1)g = 631cells/ms and L

(1)s =

58cells, we obtain the picture of Figure 8 and the diagnostic phase plane plot of Figure 9(b)

which produces the −45o line, which evidences the propagation of the slow-modulated soliton.

These numerical experiments confirm the fact that the use of the coupled NLTL described by

model II allows to avoid the wave mixing due to the existence of two modes of propagation in

the network.

6 Conclusion

In this paper, we have investigated the dynamics of modulated waves in two coupled discrete

nonlinear electrical transmission lines (NLTL). More precisely, we have first shown that, in the

limiting case where the two coupled lines are identical, the network may support the propagation

of envelope solitons. The modes of propagation of these modulated solitons have been detected;

the fast- and the slow-mode. The fast mode corresponds to the mode of propagation of a sin-

gle isolated line while the slow mode results in the coupling between the two lines. However,

the simultaneous presence of these two modes in each line of the coupled discrete NLTL is at

the origin of the undesirable wave mixing effects. We have shown that, the use of the coupled

discrete NLTL in which the unit-cells have the characteristics of the band-pass filter allows to

avoid any wave mixing.

Next, the analytical studies are completed by the numerical experiments preformed in the

network. The obtained results confirm the validity of the analytical approach. In fact, to each

14

input signal in either line of the network, one and only one mode of propagation is created

in each line; the nature of the generated mode (fast or slow) depends on the magnitude of the

carrier frequency. Finally, we mention that, while this study is crucial for the best understanding

and interpretation of experimental results of modulated waves in the coupled discrete NLTL,

it can also be viewed as the first step toward the study of modulated waves interactions on

two-dimensional discrete NLTL where the low-frequency regime case has been issued by Dinkel

et al. [32]. This work is now under consideration.

Acknowledgments. David Yemele is grateful to the Abdus Salam International Centre for

Theoretical Physics (ICTP), Trieste, Italy, for hospitality, where part of this work was done

during his visit under the Associate Federation Scheme and to the Swedish International Devel-

oping Cooperation Agency (SIDA) for financial support. He also thanks Pr. Jean Marie Bilbault

and Pr. Patrick Marquie for helpful discussions and for the critical reading of the manuscript.

References

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15

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16

Figure Captions

Figure 1: Schematic representation of the coupled nonlinear LC transmission lines (Model

I).

Figure 2: Linear dispersion curves of the two coupled LC transmission lines for the fast-

mode (1) and slow-mode (2) of propagation, for two particular cases:

a) The two coupled lines are different: C = 160pF , C01 = C02 = 320pF , L1 = 0.220mH and

L2 = 0.470mH. curves (3) and (4) are the dispersion curves of the two lines taking separately.

It appears that the coupling capacitance lower the cut-off frequency.

b) The two coupled lines are identical: C = 160pF , C01 = C02 = 320pF , L1 = L2 =

0.220mH. In this case, the fast-mode reduces to the mode of propagation of a single line. Each

curve is divided in two regions concerning the modulational instability (MI) described by the

NLS equation.

Figure 3: Schematic representation of the unit cell of two coupled LC transmission lines

with an additional linear inductor in parallel with the nonlinear capacitor (Model II).

Figure 4: Linear dispersion curves of the network (Model II):

a) For the two modes of propagation; fast-mode (2) and slow-mode (1).

b) Zoom on the dispersion curve of the slow-mode.

The coupling coefficient is taken to be γ = 8 > γcr . One note that the intersection between the

curves of the two modes observed in Model I has disappeared, indicating the entire separation

of the two modes of propagation on the network.

Figure 5: Example of MI in the coupled network. The initial condition corresponding to

the input wave is given by Eq.(41) with carrier frequency fp = 1180kHz and initial amplitude

Vm = 0.5V . The frequency modulation and modulation rate are, fm = 4.81kHz and m = 0.01,

respectively.

5.1 Signal voltage (in Volts) in the network as a function of cell number n, (a) for line 1 and

(b) for line 2, at given arbitrary times t1, t2 and t3 (with t1 < t2 < t3), respectively. At this

carrier frequency, the fast-mode only is generated by the network. As time goes on the wave

exhibits a modulation of its amplitude and phase, on the two lines, which lead to the formation

of wave packets.

5.2 Signal voltage (in Volts) as a function of time (the time unit is t0 = 0.6ms) at different

cells (20, 1500 and 2250).

Figure 6: MI exhibits by the propagation of the slow-mode in the Network. The input wave

17

is given by Eq.(41) with carrier frequency fp = 150kHz belonging to the MI domain and initial

amplitude Vm = 0.8V , frequency modulation fm = 4.81kHz and modulation rate m = 0.01.

6.1 Signal voltage (in Volts) in the network as a function of cell number n, (a) for line 1 and

(b) for line 2, at given arbitrary times t1, t2 and t3 (with t1 < t2 < t3), respectively. The plane

wave on the two lines exhibits a MI.

6.2 Signal voltage (in Volts) as a function of time and at different cells (100, 500), showing

also the MI.

Figure 7: Propagation of fast-mode envelope soliton on the network in line 1. Signal voltage

(in Volts), (a) at a given cell as a function of time (in ms), (b): at a given time (t1 = 1.50ms,

t2 = 2.29ms and t3 = 3.06ms) as a function of cell number n. The input signal is given by

Eq.(42) with carrier frequency fp = 1180kHz belonging to the MI domain and initial amplitude

Vm = 0.4V .

Figure 8: Propagation of slow-mode envelope soliton on the network in line 1. Signal voltage

(in Volts), (a) at a given cell as a function of time (in ms), (b): at a given time (t1 = 1.50ms,

t2 = 2.29ms and t3 = 3.06ms) as a function of cell number n. The input signal is given by

Eq.(42) with carrier frequency fp = 150kHz belonging to the MI domain and initial amplitude

Vm = 0.4V and γ = 8.

Figure 9: Signal voltage at a given cell in line 1 as a function of the signal voltage of the

same cell in line 2,(a) fast-mode and (b) slow-mode, with the initial conditions described in

Figure 7 and 8 for the fast and slow mode, respectively.

18

Figure 1

19

Figure 2

20

Figure 3

21

Figure 4

22

Figure 5.1

23

Figure 5.2

24

Figure 6.1

25

Figure 6.2

26

Figure 7

27

Figure 8

28

Figure 9

29