PhysicsLettersA166 (1992) 116—122North-Holland PHYSICSLETTERS A

Time dependentsolutionsof wavecollapse

Luc Bergeand DenisPesme’Cornmissariat al’EnergieAtomique,Centred’EtudesdeLirneil- Valenton,94195Villeneuve-Saint-GeorgesCedex,France

Received6 January1992; acceptedfor publication26February1992Communicatedby A.P. Fordy

We investigatethecollapsedynamicsof multidimensionalradially symmetricsolutions obeyingthenonlinearSchrodingerequation.Thecollapsingsolutionshavinga finite massareshownto beconstitutedof a self-similarpartwhosespatialextentislimitedby acutoff radiusandof a non-self-similartail ensuringthemassfiniteness.

ThenonlinearSchrodingerequation(NSE) yi(r, t)= [g(r)]’~Ø(c~, ~)exp[i(1r—~a~2)]

i~+r~rd_1~+I~I2cW=O, (1) (3)0t or Or inwhich the newspaceandtimecoordinatesarede-

with a> 0, arisesin variousareasof nonlinearphys- fined by ~ r/f( t) = r/g( r) and r~J~f2(u) du;ics concernedwith wave envelopemotions [1—5]. herethe scalingfactorg(’r)~f(t) refers,on the oneOneof its mostsalientpropertiesconsistsin theself- hand,toa radialsizethatdecreasesto zeroasS tendsfocusingandtheultimatecollapseof a radially sym- toa finite timedenotedast*; on theotherhand,thismetric structurewheneverits initial massexceedsa scalingfactorgoesquickly enoughto zerofor thenewcriticalvalue.Thiscollapsephenomenonoccurspro- time~to tendto infinity in thelimit t— t”. Thisself-vided that the spacedimensiond satisfiesthe in- contractingbehaviorthencausesa divergingampli-equalityd>~2/a [41,andwewill restrictourselvesto tudein thesolution (3) ast—~.t”,which characterizesthe physicalsituationd~3. The system(I) partic- a collapsingstate.In expression(3) therealconstantularly admits two invariants, namely the mass ,~ andthe quantitya(S)denoterespectivelya posi-integral tive eigenvalueof eq. (1) andthe positivefunction

a(S) —JJ= —8. in g. Inserting solution (3) intoeq.(1) leadsto thefollowing evolutionequationfor

N{~}~J IWI2rd_Idr,0

andthe Hamiltonian i~ +~d ~d-1 + øI2aø+e(~~2_~4)ø=0,(4)

H{~}~$ [IVwI2—(a+l)’ IWI2~+l)1rd_ldr. where e(r) is the time dependentfunction r)~

o ~(a2+a~)and where ~T(�) denotesthe complex(2)

quantitydefinedby the relationIn the following, theseinvariantswill be assumedto

[~T(e)]2~ [2+ia(~d—l/a)]/ebebothfinite.Thecollapsingsolutionsof eq.(1) areinvestigatedin termsof substitutionsof the form Underthissubstitutionthe massinvariantN trans-

forms into

Permanentaddress:Centrede PhysiqueThéorique,Ecole N{Ø} = [g( ~r)] 2”~”N{w}Polytechnique,91128PalaiseauCedex,France.

116 0030-4018/92/S05.00© 1992 ElsevierSciencePublishersB.V. All rights reserved.

Volume166,number2 PHYSICSLETI’ERS A 15 June1992

on the other hand by multiplying eq. (4) by with ~ = ~a~ thesesolutions00 havebeenanalyti-~d ~ andby integrating the imaginarypart of the cally shownto representzeroenergystatessatisfying

result from zero to ~, one obtainsthe continuity H’{00}=0 [4,12,15]. Within this self-similar as-

equation[6,7] sumption,the valueof theconstanta0 canbe easilyfound by inserting into eq. (5) the BKW approxi-$ ~ I OCo, r) I 2pdt dp matedfunction

0~ ~2)

+2~’ Ø(~r) 12 arg [O(~, r) I computedin the spatial range~>> ~T with

IC(~O)I2c~exp(_7A/2\/~).=a(d—2/a) J I0(p,t)I

2p~’ dp. (5)Let us summarizethe main resultsconcerningthe

0exactly self-similar solutions that have been ob-

Theenergyintegral (2) mustbealsochangedthrough tamedin the past,in this way:the transformation(3) into (i) At thesupercriticaldimensiond> 2/a, theself-

H{~}= [g(r)1d—2/a—2 H’{Ø} similarsolution is characterizedby a nonzerovalue

of a0, which leadsto the contractionscalef(s)=

with J~j*_()The latterratehasbeenconfirmedby

H’{Ø} S ( 101 2 (~+1) severalnumericalcomputations[9—13],from which= 1V012— +a~Im(ØVq) the constanta

0hasbeenfoundas beingof theordera+ 1of unity; moreprecisely,thesenumericalresultsleadto the ratio k a0/A 1 whosevalue is independent

+ ~a2~2 012) ~d—1 d~. (6) of the initial data (for instancein the cased= 3 and

a0=a=l, the value k=0.9l7 hasbeenfound, and

The problem of a wave collapse reducesto the theeigenvalue)= 1.091 appearsto be a “universal”nonlineareigenvalueproblem consistingin detei’- constant[141). Letuspointoutthateq.(7) is adif-mining the function a( r) in sucha waythat the ~- ficult complexnonlineareigenvalueproblemwhichlution 0~ x) satisfiesthe boundaryconditions0(~ hasstill receivedno rigorousproofconcerningther)—~0for ~ and 8~Ø(0,r)=0: solving the dif- existenceof 00(~,e~)for d>’2/a, in spiteof severalferential equationfJ= — a(t) thenyieldsthescaling attempts[6,141. In whatfollows, we shallthereforelaw g(r) associatedto the spaceandtime evolution simply assumeon the basisof numericalevidenceof the self-focusingsolution (3). For two decades, that suchself-similarsolutionsexist andsatisfy thethe contractionrate g( r) hasbeensearchedfor by eigenvalueproblem (7) in the supercriticalcase.assumingtheexistenceof self-similarsolutionsvalid (ii) At the critical dimensiond= 2/a, the proce-within the whole spatial domain.Hereby self-sim- dureoutlinedaboveleadsto a0= 0, so thateq. (7)ilar, it is meantthat the function 0(~, r) variessuf~ takestheformof a NSEwitha realpotentialandad-ficiently slowly in time as r—* ~ for the approxi- mits for A = 1 a soliton-like eigenfunctionthat ex-mationO~Ø= 0 to be valid in this limit. Theexactly ponentially decreasesto zero as ~ ~. Unfortu-self-similarsolutionsare definedas thosefor which nately, the value a0= =0 is not sufficient tothe functiona( r) reachesa steady-statevaluea0 to- determinecorrectly the contractionrate f(t) be-getherwith a~-+0;in this case0(~,r) reducesto the causethetermin ~2 in eq. (7) vanishesat this orderstationarysolution,denotedas 0o(~, to), satisfying of perturbation.Forthis reasonit is necessaryto ex-the following equation, tend the previousanalysisto the class of quasiself-

similar solutionsdefinedas follows: the quasiself-1 —d ..~ ~d— + I Oo 2a

0~+ e0~

2Ø0 similar solutions correspondto the adiabatic ap-

proximationconsistingin keepinge(r) asa slowly

— [A+ ~ia0(d—2/a)100 = 0, (7) varyingfunction of t while maintainingtheequality

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= 0. We will denoteaccordingly the quasi self- vation N= constsincethe latterhasto remainfinitesimilar solution as Oo( ~, e ( r)). In addition to this at any time. We show in this Letterthat this failureadiabaticapproximation,one also assumesthe or- follows from the quasi self-similar approximationdering I a~I ~za2 which canbe checkedto be satis- O~Ø= 0 performedin the previousinvestigations.Wefled by the solutionsdiscussedlater. The analytical provethat retainingthe time derivativein eq. (4)procedureusedby severalauthors [7—121in order leadsto a function 0( ~, r) which is constitutedofto determinef(s) then consistsin expandingthis two connectedcomponents,namely the quasi self-quasiself-similarsolution0 Oo(‘~, e(r)) into a cen- similar part of the solution, whosespaceextensiontral core, correspondingto the domain ~< ~--= is limited by acutoff radius,anda non-self-similar1 /\/~, and an asymptotictail whosespatial exten- tail, whoseexpressionsatisfyingO~O� 0 will be ex-sion is the complementaryrange~> ~. Assuminga plicitly computed.priori a finite massN{Oo( ~, e ( r) ) } <X, these au- Letusfirst commentuponthedivergenceproblemthorsfind that the massexchangedbetweenthecore that is associatedto the quasiself-similarsolutions:and the tail of the solution is given by the equation within the approximation8~Ø=0,eq. (4) exhibitsa

/ quadraticpotential in e~2introducedby the substi-~j ~ ~ ~ l~O) tution (3), so thatthe coordinate~T maybeviewedwhich follows from the estimate 0o(~,e(r)) as a complexturningpointwhich movesto infinity[1 +aC(r)]0

0(~),where0o(~)denotesthe asymp- for a critical collapseandremainsconstantfor a su-totic self-similarform Oo( ~) O~(~,~= 0),a beinga percritical one. The spectralproblem which is as-positiveconstant(see,e.g.,refs. [7,8]). Insertingthis sociatedto eq. (4) thenconsistsin connectingtheresult (8) into eq. (5) leadsto the relation quasiself-similarsolutionOo(~,C) definedin the vi-

2 cinity of the origin to its asymptoticbehaviorde-IC(e)I2=—Ne~a=iraN/2rln3r fined for large ~. The latter correspondsto a tail

and finally yields the scalinglaw function 0T(~,e) whoseexpressionmay be com-“ t’ — ~2 ~ t’ ‘1 1 Fl It* ~ ~ 1/2 putedwithin the spatial range~‘>Max(~T, 1). As

.‘ / — ~ — ‘I ~ / ‘~ — ~f ‘ the nonlinearpotential 01 2a canbe ignoredin thiswith e(t) (it/2 ln r)2 which characterizesa criti- latterrange,OT(~,e) is simplygivenby alinearcorn-cal collapse, as numerically confirmed [11—13]. binationof two BKW solutionssatisfyingthelinearThus, sincethe functione(r) goes to zero as ~ versionof eq. (4). Among theselattersolutions,thethe tail domain is removedto infinity asr—~,and time invarianceof N togetherwith the finitenessofthe solutionreducesto the self-similarform Oo(~5)in H imposeto selectthe following [6,121,the core domain,as expected. ~t• r ~2

Summarizingthe above-mentionedresults, one 0T(~,C)=C(C)

seesthat theexactlyself-similarsolutioncorrespond- � C

ing to a supercriticalcollapsecan be regardedas a ~particularcaseamong the whole classof the quasi X (9a)self-similarsolutions0o(~,e(r)): e(r) hassimply anon-vanishinglimit �~ for t-3 ~ in the supercritical wherethe complexamplitudefactor C( e) is explic-case,by contrastwith a critical collapsefor which itly givenbyonehas�(r)—*0 in the samelimit. For the sakeof IC(e I = IOo(0, e) I exp[—1r2/4\/~simplicity, wehenceforthusetheterminology quasiself-similar” for both supercritical and critical +a(d—2/a)(1 + 2 ln2)/8~/~collapses. —Aar ~ /2 c]

The solutionsresulting from thesepreviousana- g T

lytical investigations[5—15]howeversufferfrom the argC( �)= {it~./~ [1 — a(d—2/a)/2~J~Iimportantfailure lying in the spatialdivergence?“ +A (2 in I ~~T I — 1) — a(d—2/a) argtheir L normN which appearsfor anyfinite r. Thisis in obvious contradictionwith the massconser- + arg[Ø

0(0, e)] , (9b)

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withrepresentstheturningpoint ~T in thenewspacevan-

arg~T = ~arctg[a (d—2/a)/2A I , able. In the spatial rangex>> XT, the lastcontnibu-

in sucha waythat thetail OT ( ~, ~)definedby expres- tion in x4/x2of eq. (11) is an infinitesimal correc-sion (9) is normalizedin agreementwith thebound- tion whosetime variation with respectto r’any conditionO~(0,e). canbe ignored.Eq. (11) hasto be integratedunder

Inserting these expressionsinto N{O}, one can the boundary conditions O’(x0(t’), t’)=

checkthat the masscontributioncorrespondingto Ø~(x0(t’),t’) andO~0’(x—~oo,r’) =0, wherethe co-the tail exhibitsa spatial divergenceat bothcritical ordinatex0 (t’) ~ e’ ~

4~oand the functionandsupercriticaldimensions,sinceonerespectively exp(~ix2—~ix4 ln x) (12)obtains Ø~(x)~C’(�)

$ I0TI2~dln(~/~T) withC’(e)=0

0(0,e)

and < exp{~x4[—it+i(2ln(~xT)—l)+~i1t}

J I ~ ~ d~~ (~/I~TI ~ representthepoint~andthe:ail solution~throughthe substitution (10) respectively.Laplacetrans-

that tendto infinity for ~-+oo. Wethusconcludethat forming eq. (11) thenyields two maincomponentsthe timederivativeO~Oin eq. (4) hasto be retained for 0’~namely 0’=0~+Ø~,which areassociatedtowithin thelongspatialdomainof thetail in orderto a complexLaplacevariablerespectivelysmallerandremovethesedivergences, greaterthan the quadraticterm ~x2(1 — x4/x2) of

Wenow considerthe timedependentproblem(4) eq. (11).Returningto theoriginal coordinatesc~anddefinedin the spatial range~ where ~(e) de- t, we thusfind that0(~,t) is constitutedof two con-notesan arbitrarypoint in thetail domaindefined tributions,0=0~+0~~,whichcanbecomputedunderby c~c~(C) ~A I ‘~-r(C) I with A>> 1 being a numerical the basicapproximationa/2..J~ 1. The main partconstant.For technicalconvenience,we set 0~is evaluatedfor large~by thefollowing expression,

(1—d)/2

0(~,r)=(~~,~) 0i(~,r)~~ ~ ~ ë) (~)(d~l)/20~r(’~o,~)~ H(~max~),(13a)

x exp(_i 32f) O’(x, i’). (10) whereH(x) denotesthe usualHeavisidefunction,

forwhich onehasH(x) = 1 for x> 0, andwheretheIn the substitution(10), x and ~‘ denotenew space functions~o=~o(~)and~ �(i) respectivelydenoteandtimevariablesdefinedby x =~ ~ ~ and thequantities~o(�)ande( t) in which thetimevari-

able~hasbeenreplacedby the newvariable

t’=2J~du~ln[l/g(x)],

In the critical casefor which �(r) is explicitly timerespectively.Inserting(10) into eq. (4) leadsto thefollowing equationfor 0” dependent,expression(13a) only remainsvalid in

the spatial range ~ ~ where ~ denotestheit9~,O’+OX2O’+~x2(1—x~/x2)Ø’=0, (11) cutoffpoint definedby

in which theinequalitiesIa~I/a2i 1 <<x2 havebeen ~~max o(�)/g(t) . (l3b)takeninto account;the quantity

Thus, in the latter spacedomain,0~is nothingbutx4=2A/.J~+ia(d—2/a)/..J~ the quasiself-similarsolution 0T(~,~).

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The remainingstate0j~corresponds to the non-self- that primarily occurredin the quasi self-similarsimilar part of the solution andcanbe expressedin analysis.the limit ln[ 1 /g(r)]>> 1 as follows, On the otherhand,the massintegral

~ ~)

— ~d/2 (z1(r)(o+2~)exp[i(~v~2)I N{0}=N{Oo}+M{01}+$ I011I2~d~,I’(v+l)

+ Z2(t) exp(_i{~~2ln[l/g(r)]}) with~2v+3/2 ), (14)

with N{Ø0}~$ 100 I2~’d~0

and

M{o1}~ $ IO1I2~d_1d~,

Z2 ( r) mainly reduces to its quasiself-similarcontributionln[l/g(r)]

N{Ø} ~N{Ø0} +M{01}, because the mass contribu-Here I’(x) denotesthe usual Euler function the tion of the residualtail Ø~satisfiesthe inequalityquantity~i reads

L j0t2~<< $and the exponentv takesthe values v= — ~ at the It finally resultsthat thecontinuityequation(5) cancritical dimensionand v=0 at the supercniticalone. be written in the limit Ia~I.~(a

2as follows,Estimating now the mass densities corresponding ~

to the solutions (13) and (14), one finds81012~d d~= 2Ic(�)12

I0~~ lfl3[lfl(~max/~)]}2~~

in the spaceinterval ko, ‘~max] (a similarresult hasrecentlybeenderivedby Malkin [7] in the critical +a(d—2/a)$ I0I

2~~d~, (15)case with v= — ~); regarding I 0111 2 oneobtains o

101112 = ~ dp2 (~~) , where ~ is constrained by the inequality ~T << ~~4Z~~maxin the critical case,andwhere~belongs to the spatial

with range ~ in the supercritical case.p( 5, r) {Z

1 [1n2(~)+ i~2j — 1/2+ z

2 ~— 2p—3/2} , Let us nowapply theseresultsto a critical collapse

so that the inequality I Oii 12/I 01 I 2 ~ 1 holds within and a supercritical one separately:(i) At the critical dimension d= 2/a, N{Ø} is foundthe spatial range ~~<ç~maxin the asymptotic limitg( r) -~ 0, implying therefore that 01 dominates over to be given by its central quasi self-similar part N{00},

since one has ~max~00 andOii in the latter domain.Consequently0k, r) simplyreduces in the whole range ~‘~< ~ax to the quasi self- M{Ø1}~~t

2{ln ln[l/g(r) ]}_2 ~ N{00}.

similar solutionOo(~,�). In the complementary do-main ~>> ~ 0 equalsthe residualstate011 whose As the masscontributionof theresidualstateØ’~hasspatial dependencein ~d/2p(~, r) ensuresthe L

2 beenshown to be negligible,the estimate(8) madeconvergenceof thewhole solution0k~t) foranyfi- afterthe a priori assumptionN{0

0} <oc by previousnite r, which solvesthe spatialdivergenceproblem authors[8—121,is nowjustified a posteriori.On the

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otherhand,setting~= ~ in eq. (15), it is foundthat in time, in contrastwith thecritical case.On theotherthe 1.h.s. of the continuity equation(15) doesno hand,setting ~ ~max in eq. (15) allowsoneto corn-longerdependon any spacevariable,which corre- putethe cutoffradiusrmax in termsoftheinitial dataspondsto auniform masstransferin thetaildomain; N{w}; in the limit g( r) —~0, one finds namelyassaid before,the contractionrate 2 d—2/a

21C(�0)I /‘rmax\ (16)f(t) ={27~(t*_t)/lnln[ 1/(t*_t) ]} 1/2 N{~}= a0(d—2/c)

resultsfrom insertinginto eq. (15) expression(9b) Besides,the total massin which one can use the estimate

N{~} [g(r)]d_

2/~(N{0o} +M{01 })

Ø~(~=0,�) [1 +a�(r) ]00k=0)mainlycorrespondsto the tail contribution,because

Returningto the function ~(r, t), it canbe seen M{Øj}cc [g(r)I2/~d diverges as t—oc whenthecore

that ~T and ~max transforminto contributionN{00} remainsconstantin time. Thus,

rT ( t) = (2/~,/~){ (t~— t) ln in [1/ (t* —5)] } 1/2 the massof the collapsingcore decreasesas r—+oc,andthemassN is finally confinedin thetail domain.

and In conclusion,our resultsjustify theso-called“weak”rmax( t) = (2A/it) ln ln [1/ (t* —5)] collapsescenariodevelopedin the Soviet literature

[15], inwhich acutoffradiuswasintroduceda priori,which respectivelytendto zeroandto infinity in the so as to preventthe L

2 divergenceof the solutionasymptoticlimit t—’ t”. Thistimeevolutionof rmax ( t) Oo(~,e

0). We haveprovedthe existenceof suchamaythereforebeviewedascorrespondingtothefol- cutoffradiusandfurthermorewe haveexhibitedthelowing physical process:the local decreaseof the masstransfer from the core to the tail, a behaviorcentralnonlinearcoreofthesolution0(~,r) towards which hasbeennumericallyobservedin thephysicalthe steady-statefunctionOok) is associatedto a uni- cased= 3 and a= 1 [16,17].form mass input into the tail, from which the be-havior rmax(t)-+oo follows. This evolutionpredictsthat besidesthe collapsing core whose amplitude Referencesgrows up in time in the vicinity of the origin r=0,the remainingtail has a spaceextensiondefinedin [1] R.Y. Chiao, E.GarmireandC.H. Townes,Phys.Rev. Lett.

the range rT(t)~r~rmax(t), which increasesas 13(1964)479.[2] V.E.Zakharov,Zh. Eksp.Teor. Fiz. 62 (1972) 1745 [Soy.tendsto the collapsetime t~. Phys.JETP35(1972)9081.

(ii) In thesupercriticalcased> 2/a, ~ simply re- [3] L. Bergé,G. PelletierandD. Pesme,Phys.Rev.A 42 (1990)

mainsconstantand 4962.[4] J.J. RasmussenandK. Rypdal, Phys.Scr.33 (1986)481.

~max(t)(AI~TI/~)(t*t)I/2 [5]K.RypdalandJ.J.Rasmussen,phys.Scr.33(l986)498.

[6] V.M. Malkin and E.G. Shapiro,Zh. Eksp. Teor. Fiz. 97tends to infinity as t~~~*t*; conversely the radius (1990)183 [Soy.Phys.JETP70 (1990) 102].

rT(t)=l~T,/2(t*_t) tendsto zeroin thesamelimit, [7]V.M. Malkin, Phys.Lett. A 151 (1990)285.whereasrmax=AIc~TI>> 1 is a finite constant. [81GM. Fraiman,Zh. Eksp. Teor. Fiz. 88 (1985) 390 [Soy.

Although the self-similar solution in the super- Phys.JETP61(1985)228];

critical casehasbeenentirelydeterminedby solving A.I. SmirnovandG.M. Fraiman,PhysicaD 52 (1991)2.[9] N.E. Kosmatov,yE.ZakharovandV.F. Shvets,PhysicaD

the eigenvalueproblem (7), it is interestingto look 52(1991)16.

at the information provided by eq. (15). One can [lO]D.W. McLaughlin,G.C. Papanicolaou,C. SulemandP.L.first checkthat in the limit I aT I <<a

2~ 1, the l.h.s. of Sulem,Phys.Rev.A 34 (1986)1200.

eq. (15) is of orderaTN{OO} andthereforenegligible [11] M.J. Landman, G.C. Papanicolaou,C. Sulem and P.L.

as comparedwith the last term of the r.h.s. of eq. Sulem,Phys.Rev.A 38 (1988)3837.[12] B.J. LeMesurier,G.C. Papanicolaou,C. Sulem and P.L.

(15), namelya(d—2/a)N{00}. Thismeansthat the Sulem,PhysicaD3l(1988)78; 32(1988)210.

masstransferredinto the tail canbe ignored,from [13] M.J. Landman,G.C. Papanicolaou,C. Sulem, P.L. Sulem

which onerecoversthe factthat rmax doesnot move andX.P. Wang,PhysicaD 47 (1991) 393.

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[14] P. Lochak, The blow-up of the supercritical collapse [16] SN. Vlasov, L.V. PiskunovaandVI. Talanoy, Zh. Eksp.nonlinearSchrddingerequationand a class of spectral Teor. Fiz. 95 (1989) 1945 [Soy. Phys.JETP68 (1989)problems(1986),unpublished. 1125].

[15] V.E. ZakharovandE.A.Kuznetsov,Zh. Eksp. Teor. Fiz. 91 [17] DL. Newman,P.A. Robinson andMV. Goldman,Phys.(l986)13i0[Sov.Phys.JETP64(l986)773]; Rev.Lett.62(l989)2l32.V.E. Zakharov,E.A. Kuznetsoyand S.L. Musher,Pis’maZh.Eksp.Teor.Fiz.4l (1985) 125 [JETPLett.41 (1985)155].

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