J. Phys. J Franc.e 4 (1994) 801-814 MAY 1994, PAGE 801

Classification

Physics Abst;acts

91.30F 03.40K 47.35

The semi-empirical construction of solitons using ordinarytime-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 ordinarydifferential 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 solitajywaveform 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

802 JOURNAL DE PHYSIQUE I N° 5

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 easilyfound 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 )=

gr2 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 oppositesituation, 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, specificallydrii>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.

N° 5 THE SEMI-EMPIRICAL CONSTRUCTION OF SOLITONS 803

[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 ordinaryoscillations 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 capacitorhas 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)

804 JOURNAL DE PHYSIQUE I N° 5

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) =

0dl 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 suppliedby the source to that stored in the maximally charged capacitor, obtaining

qm»

~0(Qmax ~<n)"

Vc(iÎ) ~iÎ (2)q,~

N° 5 THE SEMI-EMPIRICAL CONSTRUCTION OF SOLITONS 805

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 duringthe 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 significantlylarger 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 voltageoscillations 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~msD ~

Vadac

Fig. 4. The practical circuit, including the LC subcircuit, a diode D and a autotransformer, which canchange 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.

806 JOURNAL DE PHYSIQUE I N° 5

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, ofaielatively 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 appearwhen 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 strongenough. If the voltage applied to the LC circuit were to be gi~adually increased (not reachingthe 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 frequencyspectrum 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 steepnessof 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

N° 5 THE SEMI-EMPIRICAL CONSTRUCTION OF SOLITONS 807

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 :

LIi

+ R Ii+ vc (~(t))

- Iii+

Iii~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

808 jOURNAL DE PHYSIQUE I N° 5

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 c1related 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 inw

(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

N° 5 THE SEMI-EMPIRICAL CONSTRUCTION OF SOLITONS 809

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 assumingthat 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 amplitudenonlinearity.

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) orch-~ (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 )

w2( 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~ tjm

(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.

810 JOURNAL DE PHYSIQUE I N° 5

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

~q

(dq/di)-=

(2/L) [vo v~(z)j dz 17)

qj

where qj is the capacitor charge at the moment the integration begins. Equation (2) in

jqFig. 6. The purely geometncal constructions of : (1) v~(q) ; (2) vo v~(q) (3) [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 specificmterval 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.

N° 5 THE SEMI-EMPIRICAL CONSTRUCTION OF SOLITONS 811

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 easilyfound 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 onlyrelevant 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.

812 jOURNAL DE PHYSIQUE I N° 5

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-degreepolynomial :

(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.

N° 5 THE SEMI-EMPIRICAL CONSTRUCTION OF SOLITONS 813

((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

814 JOURNAL DE PHYSIQUE I N° 5

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).

[3] Scott A., Chu F. Y. F., McLaughin D. W., The soiiton a new concept in appiied science, Piot.-

IEEE 61 (1973) 1443-1483.

[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).

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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.

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[12] Barfoot J. C., Taylor G, W., Polar Dieiectncs and their Applications (Macmiian Press, London,

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

[14] Zabusky N. J.. Kruskai M. D., Interaction of 'sonnons" in a coilisionless plasma and the

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).

[17] Rizik I. M., Gradstem I S., Tables of 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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