Volume61A, number6 PHYSICSLETTERS 13 June1977

INVERSE PROBLEM METHOD FOR THE PERTURBEDNONLINEAR SCHRODINGER EQUATION

V.!. KARPMAN andE.M. MASLOVIZMIRAN, AcademicCity, MoscowRegion,142092,USSR

Received25 April 1977

A scatteringmatrix is definedfor eq. (1),anda completesetof equations,governingthe time evolutionof thescat-teringparameters,is derivedby a generalmethod.Theseequationsform a basisfor theperturbationtheory.

A perturbednonlinearSchrodingerequation(NSE) g(x,A) = a(X) T(x, A) + b(A)fix,A),(8)

~t ~ +iul2u=ieR[u] (1) f(x,X)=_a(X)g_(x,X)+b*(X)~x,X),

can bewritten in an operatorformwhere 1a12 + ibl2 = l.f(x, X),g(x, A) and a(A) can be

iaL~at + [L, A] = ie1~, (2) analytically continued into the upper half-plane of A,

wherethe operatorsL andA arethe sameas for � = 0 wheretheyhaveno singularities.The discreteeigen-valuesin theupperhalf-plane,A = ~r (r 1, 2, ...), are[1], in particular, rootsof a(X), andf(x, ~r) are eigenfunctionsof dis-

L(u) = iI~/ax) + ~(u), (3) cretespectrum.We supposethereare only nondegene-rate eigenvalues, i.e.

~/1 0\ 0 U” 0 R*[u]) ~X,~r)=prf(X,~r), (9)y, )~R=(

0 —1 —u 0 \—R[u] 0 wherePr areconstantcoefficients,and

(4) ,_aa =_iprIJ*(x,~)f(x,~r)dx�O.(10)Considernow theeigenvalueproblem ar

L {u(x, t)} ~‘(x,t) = X(t)i/i(x, t), (5)The quantitiesa(A, t), b(A,t), ~r(t) andpr(t)con-

andtwo associateequations stitute thecompleteset of parametersdefiningthe

scatteringmatrix for theeigenvalueproblem(5). TheV = L ~ = (6) knowledgeof them permitsto restorethe “potential’~

u(x, t) in a way similar to that for theKorteweg—(u(x, t) —~0, lxi -÷ oo). It is seenfrom (3)that~ = deVries (KdV) equation[2].

(~i~~2)~ then ~ = (id, ~4),~= (~, —ip~).Real?‘~ Now we deriveequationsgoverningthetimeconstitutethecontinuousspectrum,and,naturally, evolutionof thescatteringparametersin the presencethey areindependenton time. Following [1], we of perturbationR[u]. First,we observethat eq.(5)introducethe Jostfunctionsf(x,A) andg(x, ~ definesthem,in principle, asfunctionalsof u(x, t),whichare eigenfunctionsof (5) for realA, with and for anysuchfunctionalF{u}onehasasymptotics

dF{u} °° r 5F au ~F au*f(x,A)-+(0,l)e~, x-+oo, dt -f [~ ~ ~ (11)

(7) -~

g(x, A) —~(1, 0) ~ ~ ~ _c~. where~5F/~u(x)is thevariationalderivativeat thepoint x (t is consideredas fixed parameter).By substi-

Theyare connectedby relations tutingu~from (1)into (11)wehave

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Volume61A, number6 PHYSICSLETTERS 13 June1977

gate homogeneous equation, i.e. to f(x, ~r) (see(6)).dF~Iu}— (dF{u}) + � f ~‘ ~F From that, taking into account(10), we obtain

t’— R[u(x)]dt — dt (~u(x) ~r —

~ (17)ar6F *

~U*(X) [u(x)]) dx, (12) Then,after solvingeq.(14) for A = ~r’ and using (7),+ R

(9) and (17), one haswhere (dF[u}/dt)0 canbe expressedas functionalof ,~p ‘Pr ~u, thesameasfor � = 0. Therefore,accordingto [1],

~u(x) ~r-~ [f1(x, ~~)g1(x, A)(~)= =at ~ ~at10 —g1(x, ~r)fi(X, A)] ~. (18)

(13)*(d~r) (dPr\ To obtain the correspondingderivativesoveru (x),

-~ = 0 , = 2~~pr. it is sufficientto makethe following changesin thegivenaboveexpressions:1 —~2, 2 -÷ 1 and i -~

This heuristicprinciple,which is almostevident,can b —~ —b.beproveddirectly by usinga variationaltechnique After substitutionvariationalderivativesin expres-similar to thatdevelopedin [3]. However,theproof sionsof the type (12), andtaking into account(13),is ratherlong for this paper(cf. with a similar situation onehas the following equationsdefining thetime evolu-in [4], wherethe conservationlaws for the perturbed tion of scatteringparametersin the presenceof pertur-evolutionequationswere derived). bation

To calculateother termsin (12), we consider,first, d~r Prt1(~r,~r)

thevariationof(S)with ~1i f(x, A) (t is notwritten, = � . , , (19)dt Ia,.if it is unimportant)

(~,-~- + Q— A~6f(x, A) = dp,. = 2i~pr\ a~ i 6u(x) dt

1EP,. af(x, A) - 6(1)1~X~A). (14) +~ ~[øraO~, ~r) i3O~,~r)lx=~r’ (20)ar

For real A (continuousspectrum)onehas ~ A/~u(x’)= 0,aa(A, t) ie[aälA,A)+ba(A,A)] , (21)and thesolution of eq.(14), vanishingat x —~oo, is at

~f(x,A) — 0(x’—x)f1(x’,A)

~u(x’) - ia(A) [g1(x’,A)f(x,A) ab(A,t) = —2iA2b+ ie[aa*(A, A) — b~(A,A)] , (22)

at—f

1(x’,A)g(x,A)] . (15) where

Here0(x) = 1 (x >0), 0(x)= 0 (x <0), andfj, g~are

componentsof f, g (j = 1, 2). By puttingin (15) (A A’) = f f*(x, A)R[u(x)] f(x, A’) dx,x -~ —°°, andtaking into account(7), (8), onehas

~a(A)= _if1(x, A)g1(x, A), =

(16) ~(A,A’) = f 7*(x, A)1~[u(x)J fix, A’) dx,~b(A) *

&u(x)= if2(x, A)g1(x, A).

For discretespectrum,A = ~ theright-handside of 13(A A’) = f ~ *(x A) 1~[u(x)] f(x, A’) dx.eq.(14)mustbe orthogonalto the solutionof conju-

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Similar equationsmay be obtained,by thedescribed Referencesmethod,for otherevolutionequations,providedtheycanbesolved,in the absenceof perturbation,by the 11 V.E. ZakharovandA.B. Shabat,ZhETF 61(1971) 118

inverseproblemmethod.In particular,if we do this [translation in Soy.Phys.JETP 34 (1972)62].

for theKdV equation,we cometo resultspresented [2] C.S.Gardner,J.M. Greene,M.D. Kruskal and R.M. Miura,Phys.Rev.Lett. 19 (1967)1095.

in [5] (wheretheyhavebeenobtainedby different [3] V.E. ZakharovandL.D. Faddeev,Funkt. Analiz. 5

approachbasedon eq.(4) of [5]). As the correspond- (1971) 18 (in Russian)[translation in Funct.Anal. Appl.].

ing equationsin [5] , eqs.(19)—(22)may be assumed [4] V.1. Karprnan,ZhETF, Pis.Red.25 (1977)296

asa basisfor perturbationtheorywhich will bedes- [translation in JETPLett. (toappear)].

cribedin detail in [6]. [5] VI. Karpmanand E.M. Maslov, Phys.Lett. 60A (1977)307.[6] V.!. Karpmanand E.M. Maslov, ZhETF 73, No. 2(8)

(1977) [translation in Soy.Phys.JETP46, No. 2 (1978)].

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