Volume66A, number1 PHYSICSLETTERS 17 April 1978

PERTURBATIONS CONSERVING SOLITONS

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

Received1 March1978

A condition is derivedfor theexistenceof a soliton-like solutionof theperturbedKorteweg—deVries (KdV) equation.A similar condition is writtenfor theperturbedmodifiedKdV equation.

Let usconsider,first, theperturbedKdV equation = 4k2 —---f-- JR [u5(z)J(zsech

2z+ tanhz)dz. (6)

— 6uu~+ = eR[u], (1) dt 4k2

wheree is small andR is an operator,suchthatR[u] Substitutingeq.(4) into eq.(1) we have—~Oif u -÷0.It wasshownin ref. [1] that the pertur-bation eR[u] actingon the soliton produces,generally, d2w/dz2+ (12 sech2z— 4)w = F(z), (7)a growingtail which takesawaythesolitonmass IC J~[U

5(z)] dz(f~0udx) andthis leadsto the destructionof the soli- F(z)ton. 2k

5 1. zThe growingtail doesnotappear,however,if [1]

+ 2kk~(1— tanhz — z sech2z)(8)

JR [U~(z)]tanh2zdz = 0, (2) + 2k3(~t— 4k2) sech2z

u~(z) —2k2 sech2z, z k(x (3) The right-handside of eq. (7) mustsatisfythe condi-tion

In this note we showthat, underthe condition(2),the eq.(1) hasa solution describinga soliton-likepulsewith slowly varyingparameters(up to first order f F(z) f(z) dz= 0, (9)me).

To prove that,we look for the solution of eq.(1) wheref(z) is a regularsolutionof the homogeneouseq.havingthe form (7):

u = u5(z)— 2k

2w(z), (4) f(z) = tanhzsech2z. (10)

where w(z) ‘-~ e,andk = k(t), ~ = ~(t), andthesefunc. Substitutingeqs.(5), (6), (8), and(10) in eq. (9), onelions are assumedto satisfy the adiabaticequations obtainseq. (2). It canbeshown,also, that thereexists[2,1] a solutionof eq.(7)which vanishesat zI —~~. As far

asw is smallenoughanddependsonly on z,we con-dt — — ~-~- JR [u

5(z)] sech2zdz, (5) dude that w(z) describesthe smallcorrectionto the

unperturbedsolitonprofile and,thus,eq.(4) canbeconsideredas a soliton-likesolutionof theperturbedKdV equation.

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Volume66A, number1 PHYSICSLETTERS 17 April 1978

If, besideseq.(2), theperturbationobeysthe addi- For theperturbednonlinearSchrodingerequation,

tional condition ~, ~ +U~= i�R[u], (13)

thesoliton-like solution,up to first orderin e,existsJR[u5(z)] dzO (11) jf[1]

R[u(x, t)e~(t)i= e~8(t)R[u(X, t)], (14)

(whichholds,e.g., whenR [u5(z)] is an odd function

of z) then,asis clear from eq.(5), the soliton anipli- where~(t) is an arbitraryfunction. This condition istudeis constantand the perturbationresultsonly in sufficient,at least. It is fulfilled, in particular,if R[u]stationaryvariationsof thesolitonvelocity 1~ is alinear operator,for example,if R [uI = yu, which

= — 4k2 andshape~5u= —2k2w(z).The simulta- is in agreementwith the resultsof ref. [4]

neousfulfillment of theconditions(2) and(11) takesplace,in particular,for thedressedsolitonsconsidered The authorappreciatesdiscussionswithin ref. [3]. E.M. Maslov.

Similar resultscanbe obtained,also,for otherevolutionequations.In particular,for the perturbed Referencesmodified KdV equation,

u~.+ 6u2u~+ ~ = eR[u], (12) [1] VI. KarpmanandEM. Maslov, Zh. Eksp. Teor. Fir. 73(1977)537.

with realu,R [u] , thecondition of existenceof the [21 VI. KarpmanandF.M. Maslov, Phys.Lett. 60A (1977)307; 61A (1977)493.

soliton-like solution hasaform coincidingwith eq. [31K. Konno,T. MitsuhashiandY.H. Ichikawa,J.Phys.Soc.(11). Japan43 (1977)669.

[41N.R. PereiraandL. Stenflo,Phys.Fluids 20 (1977)1733.

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