Some behaviors of a lattice soliton in nonlinear L-C ladder lines with abrupt parameter change are investigated. The results of computer and circuit experiments show that in the case of a soliton incident upon the line of larger characteristic impedance and of higher phase velocity in linearized-line-limit, the transmitted wave evolves into larger number of solitons. The experimental results can be well explained by use of both linearization approximation for the line near the junction and the theory by Gardner, Greene, Kruskal and Miura

Fukui, Kiyoshi

Nogi, Shigeji

Okayama University Digital Information Repository

Behaviors of a Soliton in Nonlinear L-C Lines with Abrupt Parameter Change

Okayama University Scientific Achievement Repository

Memoirs of the School of Engineering,Okayama University Vol. 14-1, November 1979Behaviors of a Soliton in Nonlinear L-C Lineswith Abrupt Pammeter Change* *Shigeji NOGI and Kiyoshi FUKUI(Received November 2, 1979)SynopsisSome behaviors of a lattice soliton in nonlinearL-C ladder lines with abrupt parameter change areinvestigated. The results of computer and circuitexperiments show that in the case of a soliton incidentupon the line of larger characteristic impedance andof higher phase velocity in linearized-line-limit, thetransmitted wave evolves into larger number of solitons.The experimental results can be well explained by useof both linearization approximation for the line nearthe junction and the theory by Gardner, Greene, Kruskaland Miura.1. IntroductionRecently much attention has been paid to "soliton"in various fieldof science and technology because of its interesting and peculiarproperties [lJ. Soliton is a pulse-like solitary wave which can existin nonlinear dispersive media and, due to the contribution by R. Hirotaand K. Suzuki can easily observed in nonlinear L-C ladder line [2J-[5J.The soliton is found originally as a stationary solitary wavesolution of the constant-coeficient Korteweg-de Vries equation [6Jwhich describes wave propagation in homogeneous media with nonlinearityand dispersion. But actual media usually have some inhomogeneities,and regarding their effects on the soliton propagation, considerably* Department of Electronics3940 Shigeji NOGI and Kiyoshi FUKUImany studies have been made to investigate the effects of gradualparameter change in medium through computer experiments arid analysesby use of perturbat.ionmethodfor several sorts of waves: shallowwater waves [7J-[9], nonlinear lattice waves [lOJ, and magneto-acousticwaves in plasma [llJ. On the other hand, there seems to be only afew reports for the effects of abrupt parameter change [12J [13J.In this paper, some behaviors of a lattice soliton in nonlinearL-C ladder lines with abrupt parameter change are stUdied. Computerexperiments are performed on the time evolution of a soliton incidenton a junction of two uniform lines with different parameter values,and the results are interpreted by use of both linearization approxi-mation for the line near junction and Gardner-Green-Kruskal-Miuratheory [14J. Experiments using electrical network are also carriedout and compared with the computer results.2. A lattice sDliton on a nonlinear ladder lineConsider a nonlinear L-C ladder line as shown in Fig.l. The lineeN)L L L Lvoo nrv.Fig.l Nonlinear L-C ladder line. Fig.2 Solitonis composed of linear inductors of inductance L and nonlinear capacitorswhose differential capacitance C(V) is given as a function of appliedvoltage V =Va + VnbyVC(V) =Col [ 1 + Vn J. awhere Va is the bias voltage.The line equation are written, from Fig.l, as(1)Behaviors of a Soliton in Nonlinear L~C Lines 41v -Vn-l nandI - In-l n(2)(3)Eliminating I s from the above equations givesnV +1 - 2V + V 1n n n- (4 )From Eqs.(l) and (4), we obtain ~ stationary solitary wave solutionsuch thatwhereV -Vn 002V* sech (B*t - em), (5 )andB* __1_ j1+ ~oo sinh uILea 0V00 = lim v .n-+±oo n(6 )(7)(8)Eq.(5) expresses a waveform as shown in Fig.2, and is called a latticesoliton [15J. As Eq.(5) shows that a soliton travels at a velocity ofB*/u sections per unit time, it can be seen from Eqs. (6) and (7) thatthe larger amplitude has a soliton, the faster it travels. It is alsowell-known that solitons pass through with each other without losingtheir identities in spite of the wave propagation on a nonlinear line[l6J.Now, if the line is not uniform, steady propagation in the form ofa soliton will be impossible. In such case, an incident soliton maysplit into two or more solitons, or may disintegrate into a train ofripples.42 Shigeji NOGI and Kiyoshi FUKUI3. Behaviors of a soliton on a line with abrupt parameter changeAs a simple case of a line with spatially varying parameters, weconsider a line which is composed of two parts with different parametervalue from each other (see Fig.3). Let us denote the first part ofCC (V) = __10_1 Vn1+-VoJUNCTIONCC(V)=~2 V1 + ~VoFig.3 Junction of two uniform lines with differentparameter values.the line as Line-l, and the second part as Line-2, and be the parametersof each line denoted as L. and C. O with i=1,2.l lSuppose that a soliton propagating on the Line-l is expressed by2 2S sech (ST-cm),S sinha. T = t/lLl COl (10 )Our concern is what will be the time evolution of the soliton afterpassing the boundary of Line~l and Line-2. Some results of computerexperiments on this problem are shown in Fig. 4, for the case a. = 0.5.In the first two of the figure, (a) and (b), linearized propagationvelocity is equal on the both line. In (a), the linearized character-istic impedance of Line~2, Z02' is small than that of Line-l, ZOl' andreflection at reversed polarity results as in the usual transmissionline. In this case, the transmitted wave of considerably small amplitudedisintegrates into one soliton and ripple. In (b),Z02 is twice aslarge as ZOl and it is seen that a wave of slightly larger amplitudeis generated just after passirig the boundary, which in turn is evolvedBehaviors of a Soliton in Nonlinear L-C Lines 43200 210 220 230T ~O T~1040 50 590 600 610 620 630Fig. 4 Time evolution of a soliton incident on thejunction (J) of two uniform lines.into one large soliton and another small soliton accompanied by anoscillating trail. In the last two of Fig.4, (c) and (d), the inputwave to the Line-2 is broadened. In (c), the input wave has slightlylarger amplitude and disintegrates into two solitons and ripple. In(d), the transmitted wave has sufficiently width so that more than44 Shigeji NOGI and KiyoshiFUKUIthree solitons are generated through its time evolution. We are muchinterested especially in the case (d), in which both lines are almostperfectly matched as in the linear problem, and incident disturbanceis broadened only because of the increase in propagating velocity.4. Consideration for the transmitted waveLet us now introduce continuum approximation for the L-C lineshown in Fig.l under the assumption of small nonlinearity and small,dispersion. If Vns have gradual spatial variation, we can expandVn±l in a Tailor's series :(ll)where, denoting the spatial length of the line per section as D,andx = nD (12)(13 )In the case of Vn/Va~n«l, where n is a constant, Eq.(1) is writtenin the form,C(V )nv VCa{ 1 - .-!!. + Qf( .-!!. )}Va Va (14 )where 0 is a small parameter. Substitution of Eqs.(ll) and (14) intoEq.(4) giveswhere v=VIVa'Let us introduce the co-ordinate transformation such as,Behaviors of a Soliton in Nonlinear L-C Lines 45x-T_D_ tILe OEt} (16 )where E is a small parameter relating to the wave number. Substituting4Eq.(16) into Eq.(IS) and neglecting O(E, DIE, 8IE,n), we obtain0,whereDa =and (18 )bEq.(17) is the Korteweg-de Vries (K dV) equation.Puttingu = -v!;2aDb!;3T2 (9: r1.- Tb Ii)Eq.(17) can be rewritten as(20)Gardner, Greene, Kruskal and Miura (GGKM) showed that if u(!;2' T2 )46 Shigeji NOGI and Kiyoshi FUKUIvaries with time according to Eq.(20), the eigenvalues Em (m=0,1,2,",N-l) for the time-independent Schrgdinger equation are time invariant[l4]2 (m) ( ).L:L _ [u (I; , T ) - E ] 1jJ m = 0 (21)31; 2 22m2Thus, the bound-state eigenvalues E «0) are preserved and are asso-mciated with the amplitudes A of the solitons that are produced fromman arbitrary initial wave u(1;2' 0) asA = 2Em mIf the initial wave is given by2v(I;,O)=A*sech (I;/R.)(22)in the co-ordinate (I;,T), the eigenvalues for the corresponding"Schrodinger equation are easily found [17], and the amplitudes of theproduced solitons are determined. The result is that, puttinga 2DbA*R. = P (p +1) ,Ithe amplitudes of the produced solitons, Ak s, are2A* (k)2p(p+l)' p- ,(k=0,1,2,· .. ·, N -1)P(24)(25)where N is the largest integer satisfying the inequality N < p+l.P p-The initial wave is disintegrated into N solitons or N solitonsp paccompanied by ripple according as p is integer or not. Fig.6 showsthe relation between P2 and normalized amplitudes Ak/A* of producedsolitons.Next, we must obtain the values of p for incoming wave to Line~2,that is, for the initial perturbation. Denote the distributed-line-limit of the characteristic impedance of Line-l and Line-2 whenlinearized as ZOl and Z02' respectively. Then the transmissionBehaviors oj a Soliton in Nonlinear L-C Linescoefficient from Line-l to Line-2 is given as47T (26)*and the amplitude of transmitted wave into Line-2, A20*where AIO is the amplitude of the incident soliton andis(28)As the propagation velocity can be approximated by ll/L.C O' (i=1,2)l lon each line, the width of transmitted wave into Line-2, ~2' can berelated to that of the incident soliton on Line-I, ~l' by1 ~/aL ac 1(29)Let us denote the p values f6r the ~ncident soliton and transmittedwave as PI and P2' respectively. Since Pl=l in the continuumapproximation, Eq.(24) gives2. (30 )Assuming that the shape of the wave which has just entered into Line-2is almost the same as a soliton, we have from Eq.(24)(31 )From Eqs. (27), (29), (30), and (31), we obtain(32)48 Shigeji NOGI and Kiyoshi FUKUIFor given Gcand GL , P2 is determin~d by Eq.(32). The number of solitonsproduced from the transmitted wave is equal to the largest integerincluded in P2+l,*and the individual soliton has the amplitude givenby inserting A=A20 and P=P2 into Eq.(25). A contour map for P2 inGc-GL plane is given in Fig.5. The results of the computer experimentsare shown in Fig.6, where the amplitudes of the produced solitons*normalized by A20 of Eq.(27) are plotted for eleven values of (Gc,GL )which are shown in Fig.5, for the case a=O.5. Fig.6 shows that thevalues of the computer experiments agree well with the calculatedresults.For an incident soliton of large amplitude, the nonlinearity anddispersion is so strong that the line equation (4) can not be led tothe Korteweg-de Vries equation such as Eq.(17). The results ofcomputer experiments is plotted in Fig.6 for the case that theincident soliton has a phase constant a=1.5, therefore a normalizedamplitude sinh2a=4.53. It can be seen from Fig.6 that the values ofthe computer experiments agree fairly well with the calculated resultsobtained from Eqs. (27), (29), and (32) even in such strong nonlinear1O..--~---~----~----------,50.50.5 5 10Fig.5 Contour map for p.Behaviors of a Soliton in Nonlinear L-C Lines 49V1 2.0zar---JaV1ClUJ~ 1.5Cla0::c.....aV1UJ§ 1.0r---Jc..~ClUJN--J 0.5~0::azox()( =0.5 () }I EXPERIMENTAL' VALUEoc. =1.5 [ ]I IJ~] (G)(El (F) [G,][C [F](Cl (D~[A] (B) [B] I[D lSI Ix2 3(I) (J)I(K)[H]IPFig.6 The number and the normalized amplitudesof the produced solitons.and dispersive cases. Agreement is relatively good when Z02/Z0l~1.These results indicate the possibility that the approximate calculationjustifiable for weak nonlinearity and dispersion is also applicableto the case of relatively strong nonlinearity and dispersion.5. ExperimentsExperimental confirmation of some results obtained in the previoussections was carried out using a circuitry as shown in Fig.7. TheIJL' L,/2 Il2_......L_~_---'____ C JCO2tJUNCTIONFig.7 Network used for experiment.50 Shigeji NOGI and Kiyoshi FUKUIinductance Ll is 39 ~H with error of at most 10 percent. The voltage-VI -1dependent capacitance is Cl (V) = 108(l+v) pF with error of at most5 percent, where V=VO+Vl is the appliedOvoltage and Vo=4 volt is the biasvoltage. The Qof the inductor is almost 115, while the capacitorQ is almost130 at 2 MHz. The inductance L2 and the capacitance C2 were obtainedby making series or parallel combinations of LI and CI , respectively.The pulse generator provides the L-C line with a single pulse whichresembles a soliton.Some of the experimental results on behaviours of a solitontraveling on a line composed of two parts with different parametervalues are shown in Figs.8 and 9, for the case (L2/LI =2.0, CO2 /C Ol =0.5) and (L2/Ll =0.5, CO2 /C Ol=0.5), respectively. It is seen fromthese both figures that the width of the wave just after passingthe boundary is equal to that of the incident soliton, because thesefigures represent the time variation of voltage at each section, whileFig.4 represents the spatial variation of voltage. Figs.18(d) and19(d) show that the transmitted wave evolves to produce one solitonand two solitons, respectively. In the both figures, so small solitonscannot be found because they are masked by small reflective wavescaused by irregular variation of the line parameter.(a) n=47(Incident soliton)•II I•(b) n=51(c) n=lOO (d) n=225 1 ~secFig.8 Time evolution of a soliton incident on ajunction (at n=50) of two lines for the caseL2/Ll =2.0 and CO2 /C Ol=0.5 .Behaviors of a Soliton in Nonlinear L-C Lines 51(c) n=lOO (d) n=225Fig.9 Time evolution of a soliton incident on ajunction (at n=50) of two lines for the caseL2/L1=0.5 and CO2 /C 01=0.5.Losses of the line used in the experiments reduce the amplitudeof a soliton by about 4 percent as the soliton travels 100 L-C units.If this attenuation is taken into account, the experimental resultsalmost agree with the results of the computer experiment.6. ConclusionsBehaviors of a soliton in nonlinear L-C ladder lines with dis-continuity in parameter values have been investigated both by computerand circuit experiments, and given a theoretical interpretation.Typical results obtained include the followings. When a solitonpasses a junction with the second line of larger characteresticimpedance and of higher phase velocity in linearized-line-limit, itevolves to disintegrate generally into larger number of solitons.Whatever parameter values has the second line, the transmitted waveproduce one or more solitons ultimately. These experimental resultscan be well explained by using both linearization approximation forthe line near the junction and Gardner-Greene-Kruskal-Miura theoryfor the K-dV equation which describes the continuum limit of nonlinear52 Shigeji NOGI and Kiyoshi FUKUIladder line. Computer experiment has also shown that the same idea ofexplaining results is also available for a soliton of large amplitude,though the linearization and the continuum approximation can not bevalidated for such case.Further interesting problems to be studies will be i) similarinvestigation for the case of inhomogeneity in nonlinearity of thecapacitor, 2) time evolution of the reflected wave produced at thediscontinuities, and 3) propagating characteristics of a solitonincident on a junction of several lines.References[lJ A. Scott, F. T. F. Chu, and D. W. Mclaughlin, "The soliton, anew concept in applied science", Proc. IEEE, 61, p.1443(1973)[2J R. Hirota and K. Suzuki, "Studies on lattice solitons by usingelectrical networks", J. Phys .. Soc. Japan, ~, p.1366(l970)[3J R. Hirota and K. Suzuki, "Theoretical and experimental studiesof lattice solitons in nonlinear lumped networks", Froc. IEEE,61, p.1483(1973)[4J A. Noguchi, "Solitons in a nonlinear transmission line", Trans.IECE Japan, section A, 2l-A, p.85(1974)[5J S. Kawamoto, M. Kido, and S. Kuroda, "Solitons along K-dV trans-mission line with loss", Trans. IECE Japan, section A, J61-A,p.896(l978)[6J D. J. Korteweg and G. de Vries, "On the change of form of longwaves advancing in a rectangular canal, and on a new type of longstationary waves", Phil. Mag., 12., p.422(1895)[7J O. S. Madson and C. C. Mei, "The transformation of a solitarywave over an uneven bottom", J. Fluid Mech., 12., p.781(1969)[8J T. Kakutani, "Effect of an uneven bottom on gravity waves", J.Phys. Soc. Japan, 30, p.272(1971)[9J P. S. Johnson, "On the development of a solitary wave movingover an uneven bottom", Proc. Camb. Phil. Soc., 73, p.183(1973)[lOJ H. Ono, "Wave propagation in an inhomogeneous anharmonic lattice",J. Phys. Soc. Japan, ~, p.332(1972)[llJ T. Kakutani, "Weak nonlinear magneto-acoustic waves in an inhomo-geneous plasma", J. Phys. Soc. Japan, 31, p.1246(197l)[12J F. Yoshida and T. Sakuma, "Computer-simulated scattering of latticesolitons from a mass interface in a one-demensional nonlinearBehaviors of a Soliton in Nonlinear L-C Lines 53lattice", J. Phys. Soc. Japan, ~, p.1412(1977)[13J N. Yajima, "Reflection and transmission of lattice solitons",Prog. Theor. Phys., 58, p.1114(1977)[14J C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura,"Method for solving the Kerteweg-de Vries equation", Phys. Rev.Letters, 19, p.1095(1967)[15J M. Toda, "Wave propagation in anharmonic lattice", J. Phys. Soc.Japan, 23, p.501(1967)[16J M. Toda, "Mechanics and statistics of nonlinear chains", Suppl.to J. Phys. Soc. Japan, ~, p.235(1969)[l7J N. J. Zabusky, "Solitons and bound states of the time-independentSchr8dinger equation", Phys. Rev., 168, p.124(1968)

Behaviors of a Soliton in Nonlinear L-C Lines with Abrupt Parameter Change

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