Volume88A, number4 PHYSICSLETTERS 8 March1982

BREATHER DECAY INTO FLUXON—ANTIFLUXON PAIRAND ITS TRANSFORMATION INTO BUNCHED FLUXONS IN A LONG JOSEPHSONJUNCTION

V.1. KARPMANIZMIRAN,AcademicCity, MoscowRegion,142092, USSR

Received4 January1982

A simpleanalyticaltheoryof thedecayof a perturbedsine-Gordonbreatheris developedwith applicationto long butfinite Josephsonjunctions.Thedecayappearsif thebiascurrentexceedsa critical valuefor which anequationis obtained.After a numberofreflections,theproducedfluxon andantifluxonaretransformedinto abunchedpail of fluxons.

A theory of fluxon interaction in a longJosephson supposethat solitonsare far from eachother.Thenjunction with biascurrentand losses[1,21leadsto we may write in the zerothorder approximationtheconclusionthat a numberof fluxonswith the =

samepolarity (e.g.,kink—kink pair)may form a con- v(X, T) v15 + u2~, (2)gealed(bunched)statein which distancesbetween wherefluxons increaseso slowly that this stateis observed

U = 2cr arcsintanhZ + iT, (3)asstablein a ratherwide rangeof parameters.This flS fl fltheoryappearsto be in a goodagreementwith ana- n = 1, 2, a,~is the polarity number,a1 = ~ ±1,logue and numericalsimulationsof theperturbedsine- andGordon(SG) equation[3—7], aswell aswith the ex- Z = (1 — V,~Y”

2[X — X~(T)I. (4)perimentsin realJosephsonJunctIons[8]

In this note we extendthe approachof refs. [1,2] Theequationsfor X~(T)and V~(T),obtainedfromto breathersandunboundedkink—antikinkpairs. In perturbationtheory, aregiven,e.g.,in refs. [1,2] . In-particular,a simpleanalyticaltheoryof the breather troducingthe distancebetweenthe solitonsr(T)decayinto unboundedfluxon—antifluxonsystem = X

1 (T) — X2(T),an averagevelocity of thesystem(FF) is developed.Thisdecayappearsif thebiascur- V = (V1 + V2)/2,and therelativevelocity P = V1rent exceedssomecritical value for which an asymp- — V2, and usingequationsfor dX~/dTanddV~/dTtotic equationis obtained.The fluxon and antifluxon from refs. [1,21, we cometo the following equationsthen bouncebetweentheendsof thejunction and, in the lowest approximationafter a certain time, form a systemof two bunched 3 2

solitonswith equalpolarities if, of course,the losses dP/dT— —(8/7 ) exp(—yIrI)— a(l — 3V )Pare sufficiently small(otherwisethe dampingcomes 3 (5)

beforethetransformation). + ira1f/2yWe start with the perturbedSG equation

dr/dT=P, (6)a

2v/aT2— a2v/aX2+ sin v = —a au/aT—f, (1)

in which all variablesare dimensionless,beingnormal- dV/dT= —aV(l — V2)—*iT(l — V2)”2 VPa1f, (7)

ized as in refs. [1,2]. It is assumedthat the biascur- where~= (1 — V2)—1/2,andit is assumedthat

rent!and dissipativecoefficienta are smallconstants.Considerthe kink—antikinksolution of eq. (1) and

0 031-9163/82/0000—0000/S02.75© 1982 North-Holland 207

Volume 88A, number 4 PHYSICS LETTERS 8 March 1982

71r(T)I ~ 1, !(V+P/4)P172 <1, tion (11) shouldbecontinuedasfollows: r(T’ + ~T)= —r(T’ — ~T)(0 < L~T<T’), and at T< 0: r(T)

(V+P/4)PIy3r~1. (8) r(—T). Therefore,the oscillationperiodof our “par-ticle” is r = 4T’.

Eq. (7) canbe easily integratedand the solutionshows Returningto thetwo-solitonsystem,we obtainthat the averagevelocity V(T) vanisheswith damping from eqs.(2)—(4), with V

1 = —V21, thefollowing cx-ratea, if V(0) � 0. To simplify further analysis,we as- pressionsumehere,at thebeginning,that V(0) = 0. ThenV(T)as ~, asit is easilyseenfrom the eq. (7). Numerical in- sinh[(Z1 — Z2)/2]

arctanvestigationswith V(0) * 0 [91give thesamephysical u = 4cr, cosh[(Z1 +Z2)/2]resultsasobtainedbelow.SubstitutingV 0 into eq. (12)(5), we have sinh[r(T)/2]

as —401 arctan coshXdP/dT —8exp(—IrI) +4ira1f— csP. (9)

The systemof eqs.(9) and (6) has a simple interpreta- According to conditions(8), one shouldassumein eq.tion: it describesthemotion of a particleof unit mass (12) that r(T) ~ 1, Le., E ~ 8 cos

2[(4IEI )l/2TI.Therefore, eq. (12) takes the formdriven by the force F = F~+ Ffwhere F~= —au/ar,

u as —~°~ arctan ~ ~ ~ —} . (13)with theattractivepotential 2 )1/2 cos[(4 IEI)”2T]Ufr)—8exp(—IrI)—ira,fr/2, (10)

andF1 = —aP is a friction force.A plot of u~r)is pre- On theotherhand,the generalform of the breathier

sentedin fig. 1. A characterof thestateof the system equationisdependson thevalueof “particle” energyE = P

2/2+ U(r). v = —4cr, (14)

Consider firstly, a f 0. Then, if E <0, we have (V cos[Øi/N)(T—VX)(l—V2)~12+01a boundstate.Solving eqs.(9) and (6) in that case, we X arctan ~ cosh[(v/N)(X_VT)(l_V2y1/21)obtainr(T) = ln~8/IEI) cos2[(4Ej)”2TJ } (0< T< T’). whereN2 = p2 + 172, and V= (I — 4N2)(l + 4N2)_1.

We see that eq. (13) describes approximately a breath-(11) erwithV0,O=0,172IEI/8,andvasl/2_-r~2.

Here, it is assumed that r(0) = r,, where r1 ln(8/IEI) This shows that a breather with 17/V ‘~ I may be repre-

is the turningpoint, andr(T’) = 0. At T> T’, the solu- sentedwith good accuracyby thesuperposition(2) oftwo solitons regarded as interacting quasiparticles (cf.ref. [101).

1 Consider nowf* 0 andassume, for definiteness,___________________________________________ thatf> 0 and a~= 1. From fig. 1, one sees that the

bound state now exists only if E <Em, where Em

= max U(r) is the marginal energy,

/7Em= U~r0)=(rrf/2)[ln(irf/l6)— 11, (15)r0 = —ln(irf/l6). (16)

The statewith E = Em (r = r0) correspondsto a sys-tem of two motionlesssolitons.It is describedby theequation

Fig. 1. Potential energy U(r) [eq. (10)1. Broken line and turn- v as —4 arctan[2(iTf)112 sechX] + ~v. (17)

ing pointsr~and —r1 correspond to the case f = 0. The full

line and the turning points r2 and r3 correspondtof+ 0. Here, the first term is obtained from eq. (12)with

208

Volume 88A, number 4 PHYSICS LETFERS 8 March 1982

r(T) = r0, andöv = 0(f) correspondsto deformationsof sohitonshapesdueto their interactionand thebiascurrent. It may be calculatedasin ref. [11] andwill 14 -~

notbe discussedhere.We only mentionthat, to thefirst order,.Sv-÷ —arcsinfasfif IXI —~~. As is easy 12

to see from fig. 1, thesolution (17) is unstable. It is2worth to note that the abovesituationis very similar ,o -~

to that of the doubleSGequation(cf. ref. [10]). Inthe last case,however,~v= 0. 8

Theexistenceof themarginalenergyEm indicates 7

that breatherdecaysif the bias currentfexceedssome 6

critical value. Assumethatfadiabaticallyincreasesandf= 0 at t = 0. Thenthe energyEchangesand the func-tion E(f) is definedby conservationof theadiabatic

2invariantr2 _________________________________________

ICE, f) = IT ~f P dr, (18) 0 i 2 5 4 5 6 7 8 g jo /E~1~1o~r3

Fig. 2. The dependenceoffcr on E0.Thecurve 1 correspondswherer2 andr3 are theturningpoints (fig. 1). It is to eq. (20), andthecurve2 to theapproximateeq. (21).

easyto obtain

imate,onecanfully useeq.(21) for estimations.IT ~ In the presenceof small dissipationtheenergyde-

— — f ~ = —4 iT F, (19) creasesfaster,becausein this caser3

wherer is the periodof oscillationand bardenotes dE_ 1 df ~averageover theperiod. We seethatE decreasesif the dT — 21TF ~ — a(~j~). (22)biascurrentincreases.On theotherhand,themarginal As a> °‘fcr appears to be larger than the solution ofenergyEm(f),beingequalto zero atf= 0, also de- eq. (20), andfcr increasestogetherwith a. However,creases.From eq. (15)we havedEm/df —irr0/2, and one shouldbearin mind that a ratherlargedissipationsinceF <r0 (seefig. 1), we concludethat I dE/dfI may lead to breatherannihilationbeforeit decaysinto

dE~/dfI.Thus, thereexistsa critical valueof the separatedivergingsohitons.The equationsforfcr, (20)biascurrent,fcr, suchthatE(fcr) = Em(fcr), and the and (21), are ingood agreementwith numericalsolu-breatherdecays~ff>fcr. An equationforfcr may be tions of eq. (1) [91~.

written as Forinstance,direct numericalsolutionsof eq.(1)

I(E0, 0) = I(Em(fcr)~~“cr~ (20) for a= 0 andE0 = —0.157givef~~as 0.0135 [9]which is in good agreementwith eq.(21).

This definesfcrasa function ofE0.IfE0 is small Considernow theeffect of a finite lengthofenoughthenfer is alsosmall, andE(f) asE0.There- Josephsonjunction,assuming that it is sufficientlyfore, in this case,insteadof eq.(20), one maywrite long andtheboundaryconditionsat the endsare au/Em(fcr) as E0,and an equation for sufficiently small ax = o. it hasbeenshown that the reflection of a

fcr takesa more simple form breatherundersuchconditionspreservesits identity,

fiTfcr(1 — ln ~cr~as —E0. (21) iff<f~5 anda is sufficiently small [9] . H’f>fcr,thebreatherdecaysinto kink and antikink.Eachsoli-

Theplot Offcr(E), definedby eq. (20), is showninfig. 2 by curve 1, while the line 2 representseq.(21). *1 The breather decayfor eq. (1) hasbeen demonstratednu-

Both curvesarecloseto eachotheratsufficiently mericallyby Inoue et al. [121 ,but theygave no criteria

smallE0. Sinceour approach,on thewhole,is approx- for that phenomenon.

209

Volume88A, number4 PHYSICSLETTERS 8 March 1982

tonbouncesbetweenthe endsof thejunction mdc- The authorthanksMrs. N.A. Ryabovafor numeri-pendentlyof the otherandchangesits polarity after a cal calculationsand V.V. Solov’ev for discussionsandreflection.Due to that,thereare time intervalswhen technicalhelp.both solitonshaveequalpolarities(cr1 = cr2). If suchan interval is sufficiently long, the solitons are Referencesbunchedtogether,formingacongealedstate,accord-ing to the refs. [1,21 . It is clear that the probability, [1] VI. Karpman,N.A. RyabovaandV.V. Solov’ev,Phys.

that solitonshaveequalpolarities,increaseswith the Lett. 85A (1981)251.numberof reflections.Therefore,the kink—antikink [21V.1. Karpman,NA. RyabovaandV.V. Solov’ev,Zh.

Exp. Teor. Fiz. 81(1981)1327.pair in a finite Josephsonjunction, sooneror later, [31K. Nakajima,T.YamashitaandY. Onodera,J. Appl.

transformsinto asystem of two bunchedsolitons Phys.45 (1974)3141.

with 01 = 02. Thereupon,this systemcannotbe de- [41K. Nakajima,Y. Onodera,T.NakamuraandR. Sato, J.

stroyedby the reflections[1,2]. From the above Appl. Phys.45 (1974)4095.[5] S.N. Erné andR.D. Parmenticr, J. Appi. Phys.52 (1981)

stated,we concludethat after the bias currentfcx- 1091.ceedsfcr,thebreathertransformsinto a pairof [6] M. Cirio, R.D. Parmentier and B. Savo,Physica3D

bunchedfluxons with equalpolarities.Thisconclusion (1981)549.

hasbeenconfirmedby numericalexperiments[9] , [7] P.L. Christiansen,P.S.Lomdahl, A.C. Scott,

whichwill be publishedin aseparatepaper. O.H. SoerensenandJ.C. Eilbeck, Appl. Phys. Lett. 39

According to ref. [8],the transformationof a (1981) 108.[8] B. Dueholm et al., Phys.Rev. Lett. 46 (1981) 1299.kink—antikink pair into two bunchedsolitons wasob- [9] V.1. Karpman,N.A. RyabovaandV.V. Solov’ev,Prcprint

servedin laboratoryexperiments(see,especially,corn- IZMIRAN No. 52a (1982).mentsto fig. 2 in ref. [8]). However, from that paper [10] V.1. KarpmanandV.V. Solov’ev,Physica3D (1981)487.

one cannot see whether the kink—antikink pair was a [11] V.1. Karpman andV.V. Solov’ev,Phys.Lett. 84A (1981)39.

resultof abreatherdecayor hadanotherreason.[12] M. Inoueand F.G. Chung,J. Phys.Soc.Japan46 (1979)

1594;M. Inouc J. Phys.Soc. Japan47 (1979)1723.

210