Volume84A,number2 PHYSICSLETTERS 13 July 1981

THE INFLUENCE OF PERTURBATIONS ON THE SHAPE OF A SINE-GORDON SOLITON

V.1. KARPMAN andV.V. SOLOV’EVIZMIRAN, AcademicCity, MoscowRegion, 142092, USSR

Received29 April 1981

The deformationof a perturbedsine-Gordonsolitonis calculatedin first-orderperturbationtheory. Someexamplesofphysicalinterestareconsidered.

A small perturbationactingon a soliton leadsto a slowtime evolutionof the solitonamplitudeand velocityand to somedeformationof its shape(seee.g.ref. [l}).

In the presentpaperwe considerthe shapeof a soliton describedby the perturbedsine-Gordon(SG)equations

vxt+SinVeR[V] , (1)

VTT xx+si=d1~[u1 , (2)

whereeq.(2) is obtainedfrom eq.(1) by the transformationX = x — t, T= t +x. The perturbedsoliton of eq.(1)maybe written as

v(x, t) = u~(z)+ 6v(z), u5(z) 2asin~tanhz+ ir(2n + 1) , (3)

z(x, t) = 2v(et) [x — ~(t)] , (4)

wheren = 0, ±1,..., a = ±1,6u describesthe distortionof the soliton shapeby theperturbationeR[u], and e is asmallparameter.The evolutionof the parametersv and~ is definedby the equations[2,3]:

~ (5)dt dt 4p

2 2v2

i=~afR[u5(z)]sechzdz, I~~a fR[v5(z)]zsechzdz. (6)

Thedistortion6 v in case(1) may be easily found if one takesinto accountthateq.(1) is associatedwith the samespectralproblemasthenon-linear Schrödinger(NLS) equation.Thus6v may beobtainedfrom theappropriateformulaefor the perturbedNLS equation(see e.g.ref. [1]), by using a simple “correspondencerule” betweentheNLS andSGequations,which wasutilized in ref. [2] in thederivationof eqs.(5) and(6).

As a result we have*l

6v(z)= 2a f w(y)dy, (7)

~ This is theadiabaticpart of the distortion.A term causedby a sharp“switch-on”of theperturbationis neglectedherebecause

it vanishesat larget (for moredetailsseeref. [1]).

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Volume84A, number2 PHYSICSLETTERS 13 July 1981

w(z) = ~ f ~)riQ2 + 2i~t~nhz - I + 2 sech2z)e~zPdp , (8)4iii~2 (p2 + 1)3

A(p) = a~2f (p2 2ip tanh y 1)R [v~(y)j e~PYdy. (9)

After integrationby partsin (8) and(9) and sometransformation,we obtain:

~v(z)=4

6h [F1(z) zF2(z)+F3(z)] +4eF4(z). (10)

Here

F1(z)= f (y2 -2zy+z2 I)R[v~(y)1 --~-. F~(z)fR1v~(y)} dy fREv~(v)1~

F3(z) = f (e~+ ~ ) R [u~(y)jdy +~(e~i’— R [v~~)] dy,

F4(z) = ez fR[v~Cv)] ±~+ ezfR[vs(y)] ~

In thederivationof(l0), it hasbeenassumedthat IR [v~(z)I1<00 for Iz —* oo~We seethat theperturbedSG soli-ton for this classof perturbationshasno tail (or shelf). Formula (10) givesthe generalsolution of the problemunderconsiderationfor eq.(1). Now let us turn to eq.(2). It hasbeenshown in refs. [4,5] that the perturbedsoliton of eq.(2) maybe presentedas

v(Z,eT)=2asintanh(Z+e&~)+ir(2n+l)+~v(Z), n0,±l,..., (11)

Z(X,T) = [X X(T)] [1 — V2(eT)] 1/2 , ~l(Z, eT)= [1 + V(eT)IIiZ ~ [1 V2(eT)]1Z2+ 0(e) , (12)

where~2

dV/dT= e [1 V2(eT)13121, dX(T)/dT= V(cT) + e[l V2(eT)]11 . (13,14)

Herev~t~Z),I(eT), I~(eT) and6u~Z)areobtainedfrom v5(z),I(et),I (et),and~v(z) by substitutingz -~

R [v~(z)] —~R[u~(Z)]in (3),(6),and (10),respectively.Therefore,eqs.(11)—( 14) give the solution of theproblemfor eq.(2).

Now let usapplythegeneralequation(10) to the two concretecaseswhich appear.for example,in thedescrip-tion of spin wavesin liquid

3Heandin the theoryof Josephsonjunctions.(a) Let us assumethat

eR[v]—~Xsin~v (e=—~X). (15)

This leadsto thedoublesine-Gordonequation.(Manyresultsfor this equationaredescribedin ref. [7].) From(6)and(10), it follows that

1 ay/4, 1~0, 6v(Z)uXhz(~~2+Z2)1, (16)

$2 Eq. (13)wasfirst obtainedby McLaughlin andScott 161.Yet, theequationfor dX(T)/dTinref. 161 differs from eq. (14).

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Volume84A,number2 PHYSJCSLETTERS 13 July 1981

where7 = (_l)~l.Substituting(16) into (11) onemay write theperturbedsoliton,to thesameorder asin (11), in the form:

u(Z,eT)= 2cr sin~tanh{Z +~X[.~pr2+(vz)2]1} +yir+ 0(X2). (17)

(b) Now let ustakein eq.(2)

CR[v]=—avr+IIUXXT—f, a,13>0, fconst., (18)

wheree = max (a,~, If I). Thefirst two termsin (18) describethe dissipativeeffectsin a Josephsonjunction, thethird one is a constantbiascurrent [8]. In this case,it follows from (6) and (10) that

e1aV(l—V2)I2+V(I—~)312—a~irf, IiO,

6v(Z)=ahZ{e(~7T2+Z2)I+b[l_2ln(2coshZ)]_a17rf(~7T2_l) (19)

+ a ~fZtan~ sinhZ — a~fJ(Z)}—~f(1+ eZ tan_leZ+ eZ tan1 e~),

where

b=V(1V2)312, J(Z) =f~~5~~~

Substituting(19) in (11) one may write theperturbedsoliton in first orderin � as

v(Z,eT) = 2a sin~tanh(Z+ et,t’) + ir —~f(I+ e_~tan_leZ+ eztan~e_Z)+ 0(e2) , (20)

where

es/i ~‘~eh~ir2+(VZ)2]I+b[l —21n(2coshZ)]_a~1r(~7T2— 1)f+~afZ tan’sinhZ —~ afJ(Z) . (21)

In this case,theperturbation(18) leads,in particular,to a changeof thesoliton asymptote

v-+—f(mod2ir), lzI~+oo, (22)

which is, of course,in agreementwith theasymptotefound directly from eqs.(2) and(18),

v-+—sin~f(mod27r), IZI—,’°°, (23)

if fI~1.Expressions(19)—(21)are useful,in particular,in someproblemsrelatedto soliton interactionswhichwill be

consideredin forthcomingpublications.

References

[1] V.1.Karpman, Phys.Scr. 20 (1979)462.[21 V.1.KarpmanandV.V. Solov’ev,A perturbationalapproachto two-solitonsystems,IZMIRAN preprints, Part I: No. 34 (262),

PartII: No. 35 (263) (Moscow,1979);PhysicaD, to bepublished.131 K.H. Spatschek,Z. Phys.32B (1979)425.[4J V.1.Karpman and V.V. Solov’ev,The influenceof externalperturbations on solitonsin Josephsonjunctions, IZMIRAN pre-

print, No.28(294)(Moscow, 1980).[51V.1.Kaxpman andV.V. Solov’ev,Phys.Lett. 82A (1981)205.[61 D.W. McLaughlin andA.C. Scott, Phys.Rev.A18 (1978) 1652.[7] R.K. Bullough and PJ. Caudry, in: Nonlinear evolutionequations,ed. F. Calogero (Pitman, 1978)p. 180.[8] A.C. Scott,Activeand nonlinear wave propagation in electronics(Wiley—Interscience,New York, 1970).

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