COMPRESSION OF ENVELOPE SOLITONS IN NONLINEAR ELECTRICAL LINES

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COMPRESSION OF ENVELOPE SOLITONS IN NONLINEAR ELECTRICAL LINES

J.-M. Bilbault, D. Barday

To cite this version:

J.-M. Bilbault, D. Barday. COMPRESSION OF ENVELOPE SOLITONS IN NONLIN- EAR ELECTRICAL LINES. Journal de Physique Colloques, 1989, 50 (C3), pp.C3-157-C3-162.

�10.1051/jphyscol:1989324�. �jpa-00229465�

COMPRESSION OF ENVELOPE SOLITONS IN NONLINEAR ELECTRICAL LINES

J.-M. BILBAULT and D. BARDAY

Laboratoire ORC, Université de Bourgogne, Faculté des Sciences, 6 Bel Gabriel F-21100 Dijon, France

Résumé : La propagation des solitons enveloppe dans les lignes non linéaires de transmission électrique et de transmission Josephson est étudiée sous l'aspect théorique et en simulation numérique. Dans l'approximation des milieux continus et la limite des faibles amplitudes, les équations caractéristiques de ces lignes se ramènent à l'équation NLS . La solution "à deux solitons enveloppe" se propage parfaitement dans les lignes considérées avec des phénomènes de récurrence et de compression d'enveloppe. Ceci est observé également pour des profils d'excitation non solution de NLS, ce qui est d'un grand intérêt pour les applications pratiques.

Abstract: We study theoretically and numerically the properties of envelope solitons in a nonlinear electrical transmission line and in a Josephson transmission line, in a small amplitude limit. In the continuum limit and weak amplitude limit, we reduce the characteristic equations of these systems to NLS equation and we study

"the two solitons envelope" solutions which propagate with recurrence phenomena along the two types of lines. Compression of envelope is observed for the academic solutions but we show that, even with a non- academic profile, the full width at half maximum is reduced, which is of a great interest for practical applications.

INTRODUCTION

Solitons propagate along the electrical transmission line with a constant velocity and a constant shape. Some authors, (Chu et al., 1978) proposed information transmission systems with the Pulse Code Modulation system (PCM) where solitons transmit information without distortion.

By another way, researchers in optical fibers have discovered the possibility of pulse narrowing (Mollenauer et al., 1980). For example it is possible to compress picosecond pulses into the femtosecond regime.

So, it is justifiable to ask the question : is it possible to compress solitons in an electrical transmission line or in a Josephson line ? This paper reports the theoretical calculations made with a reductive perturbation method and the numerical results .

I- Theory

1.1. Transmission electrical lines.

We consider the circuit illustrated in figure 1 . In the following analysis to a first approximation we shall neglect the losses due to the resistance of the inductor, as well as the conductance of the capacitance, which are small.

A set of the fundamental equations for the line is : d l

L — 1 = v - V ( 1 . 1 ) d t " - ' n

d Q i t ->\ 1 = i _ i ( i - 2 ) dt n n + 1

Here, Qn denotes the charge stored in the capacitor, Vn the AC voltage across it and ln the current in the inductance (Kofane et al. 1988). The differential capacitance C(V) = dQ/dV depends non-linearly on the voltage V as represented in figure 2. An expansion around the bias DC voltage V0, with vn = Vn/ V0 gives with scaled variables :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989324

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Fig.1. Cell model for the nonlinear electrical transmission line.

Fig.2 Nonlinear capacitance C(v) versus voltage V.

Here V represents the DC bias voltage ( f e r r o e f e c t r ~ c dielectric).

2

C ( V ) = 1 + 2 a v n + 3 b v n +

. . .

Eqs (1 . l ) and (1.2) give :

with equation (1.3a), we obtain the nonlinear equation which describes the voltage along the line :

Following the reductive perturbation method (Taniuti et al. 1969), we obtain the voltage v,(t) v n ( t ) =

f: 2

^rm

^{) v:} ( c , ~ I

e x p (140)

m = l

5

⁼ e (n-V,t), t = e 2 t, €I= k n - a t (1.7)

If we rewrite eq (1.5) by means of eqs (1.6) and (1.7), we obtain a series of equations for the different orders of

e .

Replacing these relations in the equation of order € 3 gives the Nonlinear Schrodinger Equation (NLS) :

( 1 ) ( 1 ) ( 1 ) 2 ( 1 )

i ( v ,

+

P(vl I t \ + Q l v l

I

v l = O (1.9)

2 2 3

(1.10) and Q = a o[ (-1 - 2 1

+ -

^bo

w i t h : P = -

-

( 1 . 1 1 )

8 o 2

In this case, it is well known that envelope solitons can propagate along the line .We used a nonlinear capacitance (figure 2) with a = 0 and b < 0 which leads to PQ > 0 in eq (1.9). Then (Satsuma et al. 1974), envelope soliton propagate in the line, and

( 2 ) ( 2 )

v o ( \ , % I

⁼

v2 ( % , % I

^{= 0} (1.12)

The well known one soliton solution of NLS eq (1.9) is :

( 1 ) e

v l

( 5 , ~ )

= A s e c h [ A

-

u e P t )

I

exp [i

-

(f,-u, P i l l ( 1 . 1 3 ) 2

with the relation

This solution belongs to a set of solutions which are characterized by the following shape for ⁱ

= 0 (Satsuma et al. 1974) : (\,s = 0)

I

= NA s e c h A

For N = 1, we have the "one soliton-envelope solution" for the nonlinear electrical line : ( n - V e t )

vn(n,t) = 2 e A s e c h

C

c

I

cos (Kn - a t ) L e

for N = 2, we get the "two solitons-envelope solution" :

where V,, K and Le are the sarnes than in eq. (1.17) and

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e"

2 2

a

= a + -

1 cos

+

e ( u 2 P - A Q )

2 2 2 e 2

The shape and the width of the solution (1.1 8) are not constant but change, for a given cell, between to = 0, where the envelope has an only maximum 4cA and t i = n / % where the envelope presents a narrow and high central peak with the amplitude ~ E A between two wide lateral bumps. This evolution in time is periodical with a recurrence periodicity of TR = 2 n / R ~ . 1.2. Jose~hson transmission lines in the weak am~litude limit

The Josephson transmission line can be modelled by the structure indicated in fig. 3 in the lowest level of approximation. (Lonngren et al. 1978).

...

^{I I}

^I..

TH

%

T H

N-1 CELL

i

N CELL

1

Z Z

Fig.3 Transmission line model of the lossless Josephson line.

According to Josephson, the voltage V is related to the phase difference 0 b y :

- - - -

d t Z e V 45

Applying Kirchoff's law to this model gives for the fundamental equation (Scott et al. 1976) :

These can be combined elementary operations a single equation

2 2

(O,)tt- c o ( O n - ,

+ -

20,)

+

^W, s i n 0, = 0 with co2 = 1 / L C

and o , Z = 2 e I, / 4C

Eq (2.4) then becomes :

2 2 3

(Onlit

-

C ~ ( O , - ~ + On+,

-

20,) + ~ ~ ( 0 , - 0,161 = 0 ( 2 . 6 )

Following the reductive perturbation method as developped in $3 1 .I. (Remoissenet, 1986) we obtain NLS

( 1 ) ( 1 ) 2 ( 1 )

i ( 0 : I 1 ) < + P(Ol + B I B 1

1

0, = 0 (2.7) with

2 2 2

C o c o s k

-

V a,

P = (2.8) and Q =

-

2 0 4 0

Therefore, we can use the solutions (1.16) and (1.18) for O n ( t ) . II. Numerical Results.

2.1. Numerical method

The numerical method used to simulate envelope soliton propagation basically consists in solving with a fourth-order Runge-Kutta method the equations of motion in a discrete transmission line. The total number of units cells in the system is 900. The time step At is choosen so that we have enough points during an oscillation (typically 80 ptslosc).

2.2. Pulses comoressions

We have excited the extremity of the line with the exact "two-solitons" solutions (1.18), and observed the evolution of the tension in time. This "two solitons-envelope" solution undergoes recurrent deformations, which are in complete agreement with theoretical predictions with respect to time as well as to voltage amplitude (figures 4 and 5).

We can therefore deduce that reductive expansion method and NLS eq. are very well adapted to the propagation of envelope solitons in the lines.

In second time, we have excited the extremity of the Josephson line with the "one-soliton"

solution (eq 1.16) in phase but with the amplitude multiplied by two, in object to observe whether this pulse compressed itself. This pulse is compressed later (in time and space) than the "two-solitons" solution with the same parameters, but there is no real reformation of the initial excitation (Figure 6).

REFERENCES

.

Chu P.L. and Whitebread T. Electron Lett. (1 978)

-

^14-531.

.

Mollenauer L.F., Stolen R.H. and Gcrdlon J.P. (1980) Phys. Rev. Lett.

4.

1095.

.

Kofane T., Michaux B. and Remoissenet M. (1988) J. Phys. C : Solid State

21 -

1395.

.

Taniuti T. and Yajima N. (1969) Journal of Math. Phys. JQ-1369.

.

Muroya K., Saitoh N. and Watanabe S. (1982) Journal of Phys. Soc. Japan a-1024.

.

Satsuma J. and Yajima N. (1974) Supplement of the Progress of Theor. Physics a - 2 8 4 . . Lonngren K. and Scott A. "Solitons in Aztion" Academic Press (1978) -173.

.

Scott A., Chu F.Y. and Reible S.A. (1 976) J. Appl. Phys. 47-3272.

.

Remoissenet M. (1 986) Phys. Rev. B

33-

^2386.

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Fig.4 Numerical simulation of a "two-soliton envelope"

solution propagation in a nonlinear electrical transmission line with C(v) given by fig.2;

e d . 1 u e =1.0 A

=

7 k z 2 . 8 8

0 0

Fig.5 Propagation of the "two-soliton" solution in the Josephson line;

w, ~ 2 1 0 0 cellsls c,

=

3200 cellsls

Fig.6 Propagation of the "one-soliton" solution with the amplitude mutiplied by 2.

(same parameters as fig.5)