The semi-empirical construction of solitons using ordinary time-dependent nonlinear oscillations

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The semi-empirical construction of solitons using ordinary time-dependent nonlinear oscillations

E. Gluskin

To cite this version:

E. Gluskin. The semi-empirical construction of solitons using ordinary time-dependent nonlinear oscillations. Journal de Physique I, EDP Sciences, 1994, 4 (5), pp.801-814. �10.1051/jp1:1994176�.

�jpa-00246947�

Classification Physics Abst;acts

91.30F 03.40K 47.35

The semi-empirical construction of solitons using ordinary time-dependent nonlinear oscillations

E. Gluskin

Electricai

Engineering

Department, Ben-Gunon University of the Negev, Beer-Sheva,

84105 Israel and

The

Applied

Eiectrical

Engineering

Program, The

Coliege

of Judea and Samaria, Ariel, B-P- 3, Israel

(Received 29 Oc.lober J992, ievised 20 Dec.ember J993, accepted ii

Januaiy

J994)

Abstract. The soliton-type solutions of _a nonlinear _wave equation ^and some nonlinear

oscillations described by an

ordinary

equation are considered and compared. A "construction" of the solitons tram the oscillatory

puises

of _a

lumped, penodically

driven eiectrical circ>Jit, using additionally _{a given}

dispersion

law Dia

physical

medium _is

suggested,

which leads _{te some}

predictions regarding

solitary waves and _some interestmg points ^for

analysis.

l. Introduction.

This paper suggests ^{and considers} a

semi-empincal modeling

of

sohtary

_waves

II -5] by

means

of _some

ordinary oscillations,

_i e. oscillations

generated by

a system ^descnbed

by

an

ordinary

differential

equation.

Trie

general similarity

of the form of sohtons and the

puises (which

_{are some} functions of

time) generated by

_an electncal circuit

[6-8]

and, _in

particular,

trie

specific

nonlinear

superposition,

discussed in

[7],

associated with the electrical

pulses,

_was the motivation for the

investigation

of trie connection between the

ordinary

oscillations and the

solitary pulses.

This

connection allows _{us, as} is shown here, to make _some

predictions

about trie

solitary

_waves,

using our expenence with _some

relatively simple

_systems where

ordinary

oscillations may be obtained. To find the connection between the

ordinary

oscillations and the _{waves we} have to

accept, at the first stage ^{of the}

analysis,

^that a moving

puise

_is not a

signal,

_as it

usually

_is, but

a self-oscillation of _{a continuous} medium, i e. that _a nonsinusoidal localized function of the argument >. ct(x _is the

spatial coordinate,

_{t is}

time,

and _{c is} the

velocity)

_may be _a solution of

a nonlinear _wave equation. ^{In fact the}

phase-type

variable _{x- ci}

(here

the variable

f)

is

widely

used in the

theory

of solitons, and in fact the sohtons _are,

usually,

autonomic waves.

Writing

the

profile

of the _{wave as}

f(x- cil

and

fixing time,

_we obtain the

solitajy

waveform _{as a} function of _a

spatial variable,

which is the usual way for _{us to}

imagine

a

waveform. The alternative

approach

is _to fix the

spatial

variable and to consider the function _as

a function of lime

only.

The latter

approach

allows _{us to} associate the solitons with time

dependent puises

of _some nonlinear

oscillations,

which _may be

easily

obtamed afid

investigated,

_e.g. in _afi electiofiics

labo;atoiy.

If, conversely,

_we obtain

physically

trie

time-dependent pulse f(t),

then for trie transfer from trie

time-dependent pulse

to the

running

wave we

change

the argument ^{of the}

pulse

to .; c-t, witt, _a relevant

velocity

c. In order _to find _{c we use} the "wave vector''

k,

in writing ^the

wave function

f(k(,<

ci )) and,

knowing

the

dispersion

Iaw of the medium for which the

modeling

is clone, consider

(which

is

possible,

_see Sect.

3)

this _{vector as a} direct function of _c then

fixing,;

_we obtain

f(ck(c)t)= f(fit),

with the

frequency-dimensional

parameter

fl(c)

=

c-k

(c).

Since 1l may be estimated

by

_a

simple

measurement

(Sect. 3),

the

knowledge

of

k(c)

allows _us to determine _c. This

approach

is considered in section 3

starting

from the

context of the known

II -5]

KdV

(Korteweg-de Vries)

equation.

As _a

simple introductory example

which shows that the

wave-oscillatory analogy

is _at ail

heuristically useful,

consider the influence of the po~l,ei lasses _on the form of the

puise.

A

puise

of

ordinary time-dependent oscillations,

_e-g-

electrical,

becomes

nonsymmetric[9]

because of the fosses. This asymmetry ^has a certain orientation the power Iosses make trie

puise

to the

right

of the maximum

(see Fig.

la) Iess steep ^than to the left of the maximum. This fact _is observed in many linear and nonlineai

(e.g. [7-9])

expenments, ^and may be

easily

found

analytically

in the linear _case for the

simplest

mortel of transient oscillations

f(t)~exp(- ai)

.sin _mi, with 0~ _c1«

w. The ratio of the absolute value of

df/dt

_at

wt =

3 gr/4 _to that _{at wt}

= gr/4 _is (w +

c1)

(w

c1)~ exp(- (c1/w

_gr ) ₌

gr 2 c1/w

~ I. With trie

replacement

of trie argument, t ~.< c-t, we obtain instead of

f (t

a

puise f (x

c-t

), moving

to the

nght

in space, whose

left fi-ont (the

back _one, relative _to the direction of the

movement)

is less steep ^{than the}

nght

_one

(sec Fig. lb)

the opposite

situation,

predicted

thus

by analogy,

to that of, _say, a wave

puise propagating

_on the surface of

a

Iiquid.

Since trie asymmetry ^{is also relevant for nonlinear}

ordinary oscillations,

trie

prediction

is also relevant to trie _case when trie moving

puise

is _a solution of _a nonlinear wave-equation, when the

velocity depends

on trie form of the

puise,

and when _a direct

analytical investigation

may ^not be easy.

Furthermore, since a

sufficiently large

resistance may make electrical oscillations _over-

damped,

the

analogy

with solitons suggests ^{that for} a too viscous

Iiquid solitary

_{waves are} not

possible,

which is _a less obvious

prediction.

Deeper analogies

^should be based _{on the}

comparative

mathematical

investigation

of the wave and the electrical processes

given

here in section 4.

For generation ^{of the}

ordinary

oscillations _{we are}

using

a

simple, Iocahzed, specifically

drii>en nonhnear _circuit which here will include _one choke and _one

capacitor.

There _{is no} direct

modehng

of the KdV _or any other wave-equation

by

_means of electrical LC ladder _or Iattice

circuits here. The latter topics are discussed in

[10,

II and

[4].

The

analogy

between the

oscillatory

^and _wave

problems

is

provided by

the

relatively

recent articles

[6-9],

and is associated with the

singulai

_input nonlinear

problem, investigated

in

t x-ct

(a) (b)

Fig.

i. (a) Schematically, ^the

asymmetncal

time-dependent electrical puise in a circuit with lasses.

(b) The _inversion of the lime-dependent

puise

with the transfer t _-.t c-t,

obtammg

the ^'

runnmg"

wave.

[6-9].

The

singular input necessanly

Ieads to the appearance in the system response of''trains''

or

''pockets''

^of

puises

which _are

compared

with the solitons. The concept ^of

singular puise

_is

basic in these studies _as it is in the

theory

of solitons. This is the link which leads _{to some new}

applications, suggested by

the

analogy,

and

provides

_an introduction into the

theory

of solitons for the reader who is familiar with the

theory

of

ordinary

oscillations and the relevant

modeling.

Some results, which _are obtained

forrnally

in the

theory

of nonlinear _wave equations, ^obtain

a

simple

_meaning in the

analogy oscillatory

_process which wiII be _{our concern} in _sections 2 and 3. The main conditions for the

similanty

of the

shapes

of the _waves and the

ordinary

oscillations _are denved in section 4.

2. The

ordinary

electrical oscillations.

To observe the

puise

to be used for trie "construction" of the solitons, consider _an LC _circuit with _a nonlinear ferroelectric

capacitor (see [12, 13]

and also

catalogs

of AVX for so-called

''skylab"

capacitors or those of _some other

producers

which include the relevant

information, though

not over the wide range of

nonlineanty

which would be desirable

here).

This

capacitor

has _a

monotonically increasing voltage-charge

characteristic

v~(q).

It _is

advantageous

to use

ferroelectnc capacitors for the

experiments

and _not diodes with

voltage-dependent

capacitances

(as

_in

[10, 1Ii),

because _we have then _a much greater capacitance, ^{and the}

frequency

_range of the oscillations

(the

basic

penod)

^is

usually

200-1500

Hz,

without the

problems

associated with noise.

2. I THE INTRODUCTORY CIRCUIT. In the

introductory

circuit shown _m

figure

2 the capacitor

C may be

initially uncharged,

_v~~~

=0,

_or

chargea,

which _can

greatly

^{influence the} capacitance ^{and the} oscillations which _are caused in the _circuit

by

the

suddenly applied

_step of the

input voltage v(t).

As with the average

depth

of _water in the channel m the KdV-

problem[1-5]

and the forrn of

solitary

waves, the

height

_v~ of the step

voltage,

or

vo v~,~,

strongly

influences the electncal oscillations. The usefuIness of the

analogy

between the

voltage

_stress and the

depth

here _is

supported by

several details.

s ~

vo

~

~

Fig. ^{2. The basic electncal circuit.} With the closing of the switch S the _constant voltage

v~ of the battery is

applied

_te the LC connection with the nonlinear capacitor, causmg transient oscillations _m the _circuit. Because of the nonhnearity ^of the capacitor ^{the forrn of these} oscillations may be different,

depending

on their

amplitude.

If the initial

voltage

on the capacitor _v~~~ is close to v~, ^{the stress caused}

by

v(t)

is weak and the

resulting

oscillations of the

capacitor's charge

and _even

voltage

_are weak and almost sinusoidal

(Fig. 3a).

This case is our

analogy

to the

small-amplitude,

almost sinusoidal

(nonsohtary)

solutions of trie KdV

equation [5].

In this _{case we} set m the circuit

equation

L

~j[~~

⁺

^v~(q(t))

⁼

^v(t)

^{= vo}

⁽¹⁾

vc (q(t))

la)

t

l~

v~

CnWX

=vc(q~)

-

Î

Î~

ig.

rge-amplitude, and solated scillations

of

v~(q(t)).

wntten for tinta, ^where t~ ^{is the} moment of the

closing

of the

switch; q(t)

= q~+

F(t),

with the _constant q~, found from the

equality v~(q~)

= v~, ^and some small function

e(t), obtaining

for

F(t)

the linear

homogeneous

equation ^with constant coefficients _:

L~~~)~~+ ~~ lF(t)

^{= 0}

dl Q

q=w

which results _in

small-amplitude

sinusoidal oscillations

e(t)

with the

cyclic frequency

dV

lt2

~- ^lt2 fi

~~

go

Since for trie sinusoidal oscillations trie lime average of trie

resulting voltage

on trie capacitor

is v~ (see

Fig. 3a),

the

amplitude

of these

voltage

oscillations _is v~ v~~~. For _{a more}

general

case the

amplitude

_may ^{be found from} the energy conservation Iaw _as follows. Since for the

electncal current

i(t)=dq/dt

the initial value and the value at the _moment when

q(t

is maximal _{are zero, i-e-} trie inductor's energy is zero, we can

equalize

the energy

supplied

by

the _source to that stored in the

maximally charged

capacitor,

obtaining

qm»

~0(Qmax ~<n) _"

q,~ Vc(iÎ) ~iÎ (2)

where q,~ is the initial

charge

_on the

capacitor,

^and q~~~ is the maximal

charge. Finding

from here q~~~

(which

_can

easily

be done

graphically, using

the

empirical v~(q), [13]),

we find v~~~~ =

v~(q~~~).

The relevant

nonlinear, strongly separated,

oscillations _are

schematically

shown in

figure

3b. See also

figure

5 below for _some

expenmentally

obtained oscillations.

Equation (2)

_means that the

aveiage

_{of v~}

_equals

_v~ for any v~

(q ).

^If we note

(see

also

Fig, 3)

that since the initial current is _zero, the

inequality

v~ ~ v~;~ results in v~ m v~~~ ail trie lime

during

the

oscillations,

then it follows from the condition for the average that with _an increase in v~ and the

resulting

_strong increase in the

amplitude

_{of v~} which becomes to be

significantly

larger

^than v~, ^the

puises

become

sharper

and _more

strongly

isolated.

For _more details associated with

(2)

and the

amplitude

of the

puises

see

[13],

where different

polynomial

models for

v~(q)

are considered. It is shown in

[13]

that for the _case of _a

really

strong stress, which is _most

interesting

here, trie

amplitude

of the nonsinusoidal

voltage

oscillations may be much

higher

than v~. ^For

example,

for _a cubic v~

(q

), _v~

~~~ may be close to 4

v~(v~~~~

_~ ⁴

v~),

_contrary to the _case of _a linear

v~(q),

^when v~~~~ = 2 v~ ^{for the Iossless}

system. For _a

fifth-degree polynomial v~(q),

_v~~~~ will be smaller than

6v~

^if all the

polynomial

coefficients _are

positive,

and may be

bigger

than 6 vo, ^for some range of vo, if the coefficient before the cubic _term is negative.

The

large amplitude, strongly separated ordinary

oscillations will be the

analogy

here of the

"packet"

of

sohtons,

obtained in

[14].

The connection with trie oscillations shown _in

[14]

wiII be closer if _we

imagine

both the mput ^{and trie} output waves of trie electrical circuit moving in

space, as a wave.

2.2 THE PRACTICAL CIRCUIT.- In the expenments ^with ferroelectnc

capacitors,

whose

nonlineanty

is

shown, usually,

as

starting

from

hyndreds

of volts, it is convenient to use a

system ^with

penodically provided

_zero "initial conditions"

by

_means of _a

penodic

input

voltage

function which has _a steep

jump,

but _is not constant afterward. If the rate of

change

of the input

voltage

after the

jump

_is much slower than that of trie oscillations

caused,

the

preceding

discussion

regarding

the oscillations _in the circuit shown in

figure

2, with the

voltage

constant, remams

basically

correct. This _{is so} for the

practical

circuit shown in

figure

4. This circuit, which includes a

voltage

autotransformer

(which

con

change

the

"scahng amplitude

factor" of the input

wave)

and has _on the input ^{the 220 V} r-m-s., 50

Hz,

sinusoïdal fine

voltage,

works _as follows. When the _current

through

the choke L is _m the

positive

direction for the diode D, the diode _is

conducting

and its

voltage drop,

which

equals

that of the

capacitor,

is

neghgible.

Because of the choke, the _{current in} the LD subcircuit is

Iagging compared

to the

voltage

on the input ^{of the LDC} circuit.

Thus,

when the choke's

current passes its +/-

zerocrossing,

and the diode

qmckly

switches off, there _is the situation when the input

voltage

_is close _{to its extreme}

(negative)

value. This value _is

suddenly applied

L

2201~ms

^{D ~}

Vadac

Fig.

4. The practical _circuit,

including

the LC subcircuit,

a diode D and _a autotransformer, which _can

change

the amplitude, but net the forrn of the mput (here the 220 V

r-m-s- sinusoidal fine) voltage.

Increasing the amplitude (vo), starting ^from zero value, _{we can} reveal _a very strong

nonlineanty

of the capacitor.

a) b)

Fig. 5. The

experimental

oscillations v~(t) obtained in the circuit, shown _in

figure

^4. If the diode _is inverted, the oscillations _are of positive polarity.

Compare

with

figure

3b, and with the

figure

of [14].

at this _{moment to} the LC subcircuit where both the initial _current and the capacitor

voltage

are zero which is the situation discussed in section 2.1. The

"packet"

of the

resulting

nonlinear

oscillations, caused

by

the stress, which is

repeated,

as a whole, _every 20

millisecond,

is shown in

figure 5,

_to be

compared

with

figure

3b. The

high-amplitude

nonlinear oscillations

are obtained

simply by setting

_v~,

by

means of the input autotransformer, to the range ^where

v~(q)

is

strongly

^nonlinear.

Inverting

the direction of the

diode,

_we would obtain oscillations of the _same forrn but of _a

positive polarity.

These oscillations _are very similar _to those of

II 4],

which _were obtained by a

computer simulation, which _are also in

general periodic,

also include

''packets''

of

spikes,

^and which show how _on the

fi.ont ofa

_moving

(becoming

_steeper and

steeper)

autonomic _wave,

ofa ielatively

small

amplitude,

'

solitaiy" sptkes of

_a

;igid form

_appear. The

initial, non-solitary

wave in

[14]

_is the

analogy

of the input

voltage

wave of the electrical LC subcircuit.

Despite

the

physically

_very different conditions for

producing

the electncal oscillations and the sohtons, there _{is an}

important

mathematical

similanty

here. The

sohtary puises

_appear

when the steepness ^of the _wave reaches _its critical value when

(see Eq. (3)

^{below and also}

[1- 5]1

the _term with the

highest

denvative

by

trie coordinate

representing dispersion

of the

medium in trie _wave

equation

becomes

significant

and compensates for the effect of the

increase of trie steepness ^caused

by

trie

nonlineanty.

For the

ordinary oscillations,

shown _in

figure 5,

the local transient

oscillations,

which here imitate trie

solitons,

also appear when the steepness of the

"input

function"

(the

rectified and _cut

sinej

of the LC subcircuit is strong

enough.

If the

voltage applied

to the LC circuit _{were to} be

gi~adually

increased (not

reaching

the

frequencies

of the nonlinear

self-oscillations),

the local transient oscillations would _not be

stimulated,

since the relevant

frequencies

would _not be

supplied.

The rote of the

frequency

spectrum ^{of the}

input

of the LC circuit

corresponds

thus to trie rote of _a

frequency dispersion

law _{in a}

solitary

_wave

equation,

and

considenng

this _we could

predict

the th;eshold

(by

the steepness of the front of the initial wave) character of the appearance of solitons.

On the other hand, the experiment ^{with the electrical} oscillations _says that after the steepness of trie exciting wave reaches the cntical value, further increase in the steepness of the stress

cannot

significantly change

the oscillations

caused,

_since ^the relevant

exciting frequencies

are

anyway present ^{in the} mput. ^This means, in the

analogy,

that if it _were

possible

to

change

in some way a parameter ^{of the medium where the} wave process is

observed,

_so that _a steepness,

higher

than the

previously

cntical _one, of the _wave could be obtained without appearance of the

clearly

exhibited

sohtary spikes,

and then return the parameter to the previous

value,

then

the

spikes appearing (for

the

extra-high steepness)

would be similar _to those obtained

directly (as usual,

_e-g-

[14]), by graduai

increase in the parameter.

As for the

important problem

of

stability

of

solitary spikes,

it is reduced in the electrical

modeling

to an introduction of a feedback which would

change

the

slope

of the input stress of the LC circuit.

Regarding

the form of the

puises,

there is _a very

interesting possibility

of

singular

nonlinear

modeling

_on the electronic

side,

which should be relevant to, for instance, the

original problems

^which led _to the fifth-order

soliton-equational

mortels in

[15]

and

[16]. Introducing

_a

(realizable,

_see

[13]

and

especially [18])

b;oken-fine nonhnear characteristic

v~(q)

of the

capacitor provides

_a

significant simplification

^{of the}

analysis,

and well imitates

[13]

the

properties

of _some of the

fifth,

or

higher

order

voltage-charge

charactenstics. It would be much

more difficult _to introduce and

analyze

such _a

singular nonlinearity

in a differential equation in

partial derivatives,

_even when the

physical

situation

(e.g.

a

layered

character of _a medium) would encourage introduction of the

singular nonlinearity.

As is shown in

[13],

the _ratio v~

~~/v~

^for the

amplitude

of the transient nonlinear oscillations _{in a} LC _circuit with _a

capacitive

unit

(connected

instead of the nonlinear

capacitor)

which is

automatically

switched at a certain

voltage level, strongly changing

the

capacitance,

_may

depend

_{on v~}

similarly

_as for the

polynomial

characteristic.

In

general,

^there is _a

significant advantage

in the

simplicity

of the expenments ^{with the}

electrical circuit. Thus, for instance, it is much easier to

change

the

damping

^and the

nonlinearity. Enlarging

the

experimental possibilities,

we can find _new

oscillatory

^effects

by

means of the

comparative analysis.

This may be _seen

if,

for

instance,

_we extend the

comparative

discussion of the

ordinary

oscillation and the _wave

problem

to patterns more

complicated

than

just

the forrn of _a

single

soliton. As _an

example,

the consideration in

il 9]

of

the _resonance reflection of shallow-water _waves due _{to a} certain form of the bottom of the

channel, suggests

(using

the

analogy

between v~ ^{and the}

depth

of the

channel)

consideration of trie electrical circuit with feedback which

synchronously

adds _some small

voltage

to the input,

obtaining

_a circuit

governed by

the

equation

:

L

Ii

+ R

Ii

_{+ vc}

_(~(t))

-

Iii

+

Ii i~i~

_jv~

_~~~i~~ v~j,

_~ o

,

with e~ ^« vo.

Use of the

synchronized

feedback with

"signum",

is a

simple

realization

by

_means of _a comparator.

It _is

înteresting,

in

particular,

to

clanfy,

_using ^trie electrical

model,

whether _or not the

nonzero

viscosity (here

R # 0) is necessary for limitation of the

amplitude (as

^it is in the usual

resonance)

of trie _wave process, and the conditions for

stability

of trie process.

3. On the "construction" of the santons.

Consider trie

ordinary-oscillatory/wave analogy

_on trie basis of trie well known KdV

equation

II -5],

which possesses

solitary

solutions.

3.1 THE KdV EQUATION. The relevant

dispersion

law of trie

physical

medium _in which the

wave is

propagating,

is (we ^{follow the} notation of

[5]) w(k)

=

uok~ pk3

where uo

(the velocity

for very

long waves)

and

fl

are some positive constants. The fact that this

dispersion

law is

independent

of

amplitude nonlinearity

is

important

^here.

Using

_a parameter c1 related to the

amplitude nonlinearity,

we can write the known KdV _wave

equation [5]

11+l~oll+fl1)+abll=0 (3)

for _a

physical quantity

b

(not necessarily

the

displacement

of the surface of a

liquid),

which has _a

nght-movmg

wave as a solution.

Introducing

a ₌ ab

(havmg

the dimension of

velocity)

and

f

= x uo t, we tum

(3)

into

ÎÎ~~Î~~$"°

which has

[1-5]

the autonomic

solitary

wave-solution

a(f, t)

_{= ai}

ch~~((x _ci) aj/(12

fl

))

with the

amplitude

_ai ^{and the}

velocity

c = uo +

aj/3

which

depends

on the

amplitude,

but is

independent

of time.

According

to the relation _c

= u~ ⁺

aj/3,

^the

expression (aj/(12 fl ))"~,which

_also

_depends

on the

amplitude,

is

directly

and

uniquely

defined

by

_c

(aj/(12

li

))"~

=

(l/2) (c l~o)/fl )"~

Returning

to the

original, physically meamngful

variable

b,

_we have

b(x,

i ₌

bj

^ch-

2((1/2) _{((c uo)/p )'/2 (x}

_ci

_)), (4)

with the

amplitude bi

₌

aj/c1,

and _c

= uo +

bj/(3 c1).

Since the

amplitude

_is an

easily

observed parameter,

usually

in

experimental investigation

of solitons the

amplitude

_is chosen _as the initial,

independent

parameter. ^Then c is defined

by bj

and _c1.

Altematively

_{we can} choose the

velocity,

which is also

easily

measured, _as the

independent parameter

; then

bj

_may be found _as

bj (c)

=

3 _c1-

'(c _u~),

and _{we can rewrite}

(4)

_as

b(>, t)

₌ 3 _a

~ic Ho) ch~~iil/2) _{(ic -l~o)/fl} )"~ix _{ct)). (4a)}

3.2 THE TRANSFER TO THE ORDINARY OSCILLATIONS.

Fixmg

x in

(4) (without

Ioss of

generahty

_we can set.< ₌

0),

_1e.

observing

the process ^at a certain

spatial

point,

using

the oddness of the function ch (z), and

introducing

trie notation

12

=

12

(c)

=

(c/2 ((c

_Ho

)lli

)~'~

,

which includes

solely

the parameters u~ and

fl

which _are included in

w

(k),

_we obtain from

(4a)

the

time-function

f(i)

m

b(o, i)

=

hi ch-2(nt)

whose

foi-m

_is that of trie sohton. If _{we can} observe

f(t)

_{as a}

time-dependent puise

on an

oscilloscope

_screen,

then, using

_a usual criterion

(see below),

we can estimate n

by

means of

a measurement. Then _we can find _{c as} the real-valued solution of the

equation

c~-u~c~-4pn~=0, (5)

which follows from the definition of

n,

and _{we can} find then

hi

_as

hi (c).

Let _{us note} that if

n _«

u('~ p-

^"~

(6)

then _c

m u~ + 4

pff ~/u(

m u~, and if the

inequality

_in

(6)

_is

inverted,

then

c w

(4 p

_)~'~ n ^~'~ w u~.

In any case the

dependence

of _{c on} n is not very sensitive. This _is

important,

since

assuming

that trie above mathematical

dependencies

_are

typical,

_we can consider

app;o>.imate

cntena for

finding

n.

3.3 ON THE DETERMINATION OF fl. In trie context of the "construction" of salirons from

empincally

obtained electnc

puises (as

in Sect. 2, _or

by

a different

circuit),

we assume that the

dependence

n

(c)

is known from the

physics

of trie _wave process, concentrating attention on

trie measu;ement of n. Not

thinking necessanly

about trie KdV

equation,

_we

keep

it _to be

important ^{that the}

dispersion

law may be

mdependent,

as in the KdV _case, of the

amplitude nonlinearity.

For the measurement we can use, for

example,

the most _common cnterion the width of the

puise

is defined _as z~ zj, where zj and _{z~ are} roots of the

equation f~ _(z

=

(1/2 ~f~~~

)~

Where

f(z)

is the form of the

generated puise.

Without assummg that

fis precisely

known,

we can suggest an approximate ^{critenon for} an estimate

of1l,

based _on the assumption ^{that the} relevant waveforrns do _not differ

strongly

from ch- ^~

(z)

_or

ch-~ (z). (Trie

latter function appears in trie soliton solution of _a

plasma problem [5]).

For this _we introduce _a standard

nondimensiona/ width A of the

puise. Rewriting

the

equahty ch~~(z)= Il,fi,

or

ch

(z

) ₌

2~'~,

_{as +}

z~/2

-

2"~,

we find z~

- 2

(2"~

l

)

w

2(

Il ) In 2, and z~ zj =

(2

In

2)"~.

For

ch~~ _(z)

we

similarly

obtain

(In 2)"~

The average of these _two values is

(1/2)

(1

+,fi),~.

_This

_value,

_{which will be chosen}

as A, _is very close _to 1.

The value A/ôt

-

(ôt)~

_~, where ôt _is the

puise

width obseri,ed _on the lime-axis, _may be

suggested

for _an estimate of n.

Summansing,

the

procedure

for the construction of the solitons may be _as follows _:

1) By

_means of v~

(Sect. 2)

_we set the relevant

amplitude

of the electrical

puise.

2) Observing

the

puise,

we measure on the horizontal

straight

hne _on the _screen of the

oscilloscope (or recorder)

which _crosses the

puise

at il

,fi

_{of its}

_height,

_{the distance between}

the crossing points.

Taking

into account the _lime scale of the device, _we find

(in seconds)

the width, ôt

= t~ ii, ^{of the}

puise.

3) Turning

to the nondimensional argument ^{of the}

function,

and

using

the standard width A, we find n from the equation

nt~ ntj

₌ A _; n

= Al

(t~

tj m

(t~ tj)~

^'

4)

After

deterrnining

the value of n _{we use} the

physical

data of the medium considered and find _c from the

equation

which appears as

equation (5).

Then for

mutually

interconnected

bj,

1l and _{c we construct} the

"soliton",

for the given

bj,

as

bj f((12/c)(x Lt)),

where

fis

the forrn of the observed electrical

puise.

A

justification

for the

simplified description

^of the

analog puise by

means of A and n follows e.g. from the fact that it is _not

always

_important to know the form of the soliton

precisely. Thus,

for the

coding-decoding applications

II of the

solitons,

the

amplitude

and the

velocity

of the solitons _are

clearly

_more

important

than the details of the

form,

and thus the

knowledge

of A and

n,

_may be

quite

sufficient for such _an

application.

However how similar is the

"synthetic"

sohton to the real _one which is observed in _a

physical

^{medium ? Is it}

possible

to assume that if the

physical velocity

of the real soliton

propagation

in trie medium is close _to that found

by

means of the introduction of

A and the measurement of

n,

then the form of the

puise

used is simflar to that of the real

sohton ?

To _come doser _to the _answers to these

questions,

we have to look _more

deeply

into the

equational analogy

of the _two processes,

considering

the realistic nonhnear characteristic v~

(q

^{of the} ferroelectnc capacitor. ^A step m this direction _is taken _in the next section where the

relevant requirements ^{for the} parameters are revealed.

4. The

equation

for the

amplitude

and the _comparison of the

shapes.

In order to _see the

equational analogy

_we

multiply (1) by dq/dt

^and

integrate by

q,

obtaining

~

(dq/di)-

q

=

(2/L) [vo v~(z)j

dz

17)

qj

where qj ^is the capacitor

charge

at the _moment the

integration begins. Equation (2)

in

Fig.

6. The purely geometncal constructions of _: (1)

v~(q)

_; (2) _vo

v~(q)

(3)

jq

[v~ u~(q)] dq

q

Only

the mterval (qj, q2) ^{with the} positive ^'

hump"

of the polynomiai (the curve 3") is relevant. The form of this hump _is most important. Though it is _a part ^of a fourth-degree polynomial, _in this specific mterval it can be (see Fig. 8 below) approximated, with _a proper choice of the parameters,

by

_a third

degree polynomial,

which is relevant _to the KdV equation.

section 2.1 is obtained from

(7)

for the condition

dq/dt

=

0,

which defines the

amplitude.

For

not-too-high voltages,

^{it is realistic} to assume that

v~(q)

₌

(1/Co)

_q +

yq~, _Co,

y ~ o

Figure

6

schematically

shows this

function, together

with the function v~ v~

(q )

and with the function of q obtained

by

the

integration

_in

(7).

Since

v~(q)

is

monotonic,

the

equation

vo v~

(q)

=

0 has

only

_one real-valued _root

(q *),

and, _{as a}

result,

the

right-hand

side of

(7)

is

a

"hump"

with _{one maximum at}

q*

and two real-valued _roots. The smallest _root of this function which is _a

fourth-degree polynomial

of _{q, is,}

obviously,

_qj ^from

(7). Denoting

the second real-valued _{root as} q~, we can write

(L/2)(dq/dt)~

=

(y/4)(q~

₊ à q +

K)(q qj)(q q2) (8)

with _some constants à and

K,

which _{are so} that q~ +

à.q

+K is

everywhere positive.

ô, K and q~ may be

easily

found

by comparison

between the _terms

having

^the same powers of q in

(8)

and the

following

form of the _same

polynomial,

found

directly

from

(7)

(L/2 )(dq/dt

_)~

=

y/4 )(q~ q() (1/2 Ci

^'

(q~ q()

_{+ vo(Q Qi}

Thus,

_we find à

= qj + q~, K

= 4 vo y

'(q

_{+ q)~}^' _{+ qj q~}

= 2

Ci

y~ ^' +

q(

₊

q(

_{+ qj q~,}

and the

following equation

for q~

QÎ+£Î1£Î1+(£Î1+~CO~Y ~)£Î2+~~0~Y ~£Îl~~~0Y ~"Ù,

which possesses ^a positive root because of the

negative

last term. This root may be

easily

found

graphically

for qj, vo,

Co

^and _{y given.}

The relevant interval for _q is qj ~ q ~ q~ where the

polynomial (8),

_1-e-

(dq/dt

)~, ^is

positive.

q~ has the meamng of the

amplitude

of the

ordinary

oscillations

(q~

in

(2)).

As the basic point we compare

(8)

with the

corresponding equation [5]

for the KdV _case p

(a')~

=

a~/3

_{+ uj}

a~

₊

c-j ^{a +} c~

(9)

1'"

_means denvative

by

the argument >.-

ci)

with _some constants:

fl,

_{uj, c-j} and c~ which _are

easily

^{found in}

[5], (uj

is denoted _as vo in

[5]).

The

nght-hand

side of

(9),

has

[5]

only

real _roots

(Fig. 7),

_{ai, a~, a~.} If ai = a~, this _is

[5]

the

solitary

wave, discussed in

section 3.1. If ail of the roots are

different,

there is

[5]

_a

penodic (nonsinusoidal)

solution of

the KdV

equation.

^{Both of the} cases may be relevant here. The

graph

of the

polynomial

of _a

~ a a

&2 ~3 ~l(2) a3

(a) (b)

Fig. 7. The _two main

cases for the polynomial of _{a m} the theory ^of KdV equation. (a) The _case of

three different real _roots.

Thji

_is the _case of

periodic,

generally nonlinear _waves. (b) The _case of _two similar _roots the case of santons. See [5] for details. The positive humps, which here aise _are only relevant for the solution, _are compared with the

hump

_m

figure

6. For this the requirements

specific

to the

polynomial

of q are made, as is illustrated by

figure

8.

in the

right-hand

side of

(9)

has _a '

hump''

in the relevant interval _a~

~ a ~ a ~, to be

compared

with the

''hump''

^of the

polynomial

of _q. The

possibility

of

making

the _two

''humps'' similar,

or at least

approximately

_so, is crucial for the

modeling

of the nonlmear _waves, using circuits with ferroelectric capacitors.

Considenng this,

_we choose the

(or

_some of

the)

parameters L,

Co,

vo ^and qj

properly (Co

_may be

changea by parallel

connection of similar

capacitors),

and _{we can}

make,

first of ail,

qjla~

=

q~la~,

I.e. we make

(ignoring

the difference _m the

physical

nature of the scale

variables)

the roots qj and a~

similar,

and also q~ and a~.

Further,

_we write the

nght-hand

side of

(9)

_as

(1/3)(a aj)(a a~)(a a~),

and, _companng ^with

(8),

note that _smce the extremum

(which

is at q =

à/2)

of the _curve of the expression

(y/4)(q~

_{+ à q +}

_K)

which appears in

(8)

_{as a} factor, is to the left of the interval qj ~ q ~ q~, then _{we can} lineanze

(see Fig. 8)

this expression in this interval,

and, introducing

the point

(q~

of trie

crossing

of the

approximating

^hne with the q-axis, ^can write in the interval

(qj, q~)

the

fourth-degree polynomial

of q

approximately

as the

third-degree

polynomial

_:

(y/4)(q qo)(q qi)(q q~),

which is similar to trie above

expression

for trie function of _a. The smaller the difference q~ qj, the _more precise is this representation,

obviously.

A

complete

connection between the variables _a and q, may be found

by integrating

the differential relation, which follows from

(8)

and

(9),

with

separated

variables

~ ~ ~ ^l/2 ~ ;- àa

(Q3 (~ ))

^~~~ ~~~~

'

~(~~Î)(~~~2)

'

,

o

Fig.

8. The

figure

shows

separately

the symmetnc

hump

of the polynomial (q _qj )(q

q~)

and the

everywhere

_positive

polynomial (y/4)(q~+

_.q+K), which _are factors _in (8). Lineanzation of (y/4

)(q~

_{+ q +} K) in the interval (qj, q2) is

suggested,

and the point q~ is thus mtroduced. This makes the polynomial _in (8) similar to that _m (9), _m this mterval.

((a')~

= c~

~(da/dt)~

for _x

fixed),

with the

polynomials

which express the

right-hand

sides of

(8)

and

(9) (now

with the factors _y

(2 L)~

^' and

(3 p )~

^'

respectively).

The differential relation

(10)

represents a precise ^connection between the forms of the electrical

q(t)

and the _wave

a(x

ct

puises.

If the condition for the

proportionality

of the _{roots is} satisfied

by

the choice of the parameters, we con,

according

to the above cubic approximation ^for

F4(q), simplify (10)

and obtain in the mterval of the

"hump"

_:

,

~~ j ~~/~ (q)j-

^"~ ₌ c~ ~~

~~~

~~_~~ ^~~~

with A

=

3

yp (2 L)~

_{~, or}

dq Q3 (q)~

^"~

=

c~ ^'

A"~

da

Q~(a

)~ ^"~

(l1)

Using

the formula

(see [17])

lq 1(X-Xj)(X-À2)(X-X3)) ^~~~dX~

i~

= 2

(xi

_x~)~ ^"~ F

(arcsin [(xj x3)"~ (xi x2)~

^"~

(q x~)"~ (q

_x3

)~

^"~

,

~~l X3)~~~ (Xj X

)~

l/2~

with Jacobi's

elliptic

function F

(~b, x),

and

using

the

proportionality

of the mtervals _in the q-

axes and _a-axes, which

provides

the second argument ^{of F}

(which

here _{is a}

parameter)

to be

the _same for bath sides of (1 1)

(and

thus _to be

ignored below),

_{we tutu}

(1 1), by

the

integration,

into

~ ~~~ ~~~~~~ ~~

~~~

_~~~~ ~ ~~~~

4'((a a~)"2 (~ ~~~_

^jj~

with the function #i which takes _{mto account}

only

the

dependence

of F _on the first of its arguments.

If _c~ ^' A ^"~

= l, _i e.

3

yp

₌ 2

Lc~, (13)

then from

(12)

(a

a2)(a a31~

^' _~

(q

_qi

)(q q2)~

^'

which admits

(since a~/qj

=

a~/q~) a/q

₌

a~/qj

=

a~/q~,

i-e- similar

shapes

of the functions

a

(x

c-t and

q(r

),

If

(13)

_is net

satisfied,

the

nonlinearity

of the function #i

by

the variable q

(or a)

_can make the

shapes strongly

different.

Since _c

~ uo,

(13)

_requires, in

particular,

that

y/L

_~ 2

u(/p

,

which is _a requirement ^{for the initial choice of the electronic} components.

Finally,

Iet _{us note} that

wcreasing

the

voltage

_{range m} the electronic experiment, we con came

(for

_many ferroelectric

capacitors)

to a

fifth-degree (monotonic) polynomial approxi-

mation of

v~(q)

^which would lead to a

sixth-degree polynomial

in the

right-hand

side of

(7).

This _can be

similarly

considered for

approximation

^{of the} wave

puise

of _a

solitary

_equation

which has _a

fifth-aider

derivative

by

the

spatial

coordinate. Such

equations,

which

correspond

to the

dispersion

law of the type

w(k)

=

pi

k~ ₊

p~k~,

_{appear, as was}

_already

noted in section 2.2, _m the

problems

of

gravitational-capillary

_waves

[15],

and in the

problems

of

long

waves m a

heavy liquid

under ice

[16].

References

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[2] A. C. Neweii Ed., Noniinear Wave Motion (Amer, Math, Soc., Provodence, 1974).

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[4] Toda M., Noniinear Waves and Soiitons (KTK Scient, Pubi., Tokio, and Kiuwer Acad, Pub].,

London, 1989).

[5] Lifshitz E. M., Pitaevskii I. I.,

Physical

Kinetics (Vol, 10 of the Course of Theoretical Physics by L, D, Landau and E. M. Lifshitz), §§ 38-39 (Pergamon Press, Oxford, 1981).

[6] Giuskin E., On the asymptotic pause state current response of _a nonhnear osciiiatory circuit to

rectanguiar voltage _waves, Int. J. Ele<W.on. 65 (1988) 251-254.

[7] Gluskin E,, The asymptotic

superposition

of steady-state eiectncal current responses of _a nonlinear

osciliatory

circuit _to certain

input voltage

waves,

Phys.

Lett. A 159 (1991) 38-46.

[8] Gluskin E., The internai _resonant relations in the pause states, Phys. Lett A 175 (1993) 121-132.

[9] Gluskin E., The symmetry argument m the analysis of oscillatory _processes, Phy.i. Lett. A 144 (1990) 206-210.

loi Hirota R., Suzuki K., Theoreticai and experimental studies of iattice solitons

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ii il Suzuki K., Hirota R., Yoshikawa K.,

Amplitude-modulated

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[12] Barfoot J. C.,

Taylor

G, W., Polar Dieiectncs and their

Applications

(Macmiian Press, London, 1979).

[13] Gluskin E., The _use of noniinear capacitors. I?it. J Elecn.on. 58 (1985) 63-81,

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recurrence of initial _states, Phys. Rei,. Lent. 15 (1965) 240-243.

[15] Zufira J. A., Symmetry breaking in penodic and soiitary gravity capiiary waves on water of finite

depth,

J. Fl~/id Me<h. 184 (1987) 183-206.

[16] Iiichev A. T., On the

properties

of _a nonlinear evoiutional equation ^of fifth order, which descnbes

wave processes m media with weak

dispersion.

Works of Stekiov Institute, vol. CLXXXVI (186), (1989) 222-261 (Nauka, Moscva, 1989).

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Integrals.

Serres, Sums and Products (Thechn,-Theoretic Literature, Moscva, 195 Ii

[18] Giuskin E.,

Dependence

of

puise-height

_{on circuit parameters} for _a generator ^with a switched capacitive circuit, I?it. J. Elecn.on. 60 (1986) 487-493.

[19] Yoon S, B., Liv P. L,-F., Resonant reflection of shaiiow-water _waves due _to corrugated boundanes.

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