Volume143,number1,2 PHYSICSLETTERSA 1 January1990

SINGLE PHASE AVERAGING AND TRAVELLING WAVE SOLUTIONSOF THE MODIFIED BURGERS—KORTEWEG--DE VRIES EQUATION

Ian McINTOSHDepartmentofAppliedMathematicsandTheoreticalPhysics,UniversityofCambridge,SilverStreet,CambridgeCB3 9EW, UK

Received22June1989;acceptedfor publication31 October1989Communicatedby A.P. Fordy

Wedemonstratehowto describetheaveragebehaviouroftravellingwavesolutionsofthemodifiedBurgers—Korteweg—deVries(mBKdV) equationin thecaseof smalldissipation.Someexactkink-like solutionsarealsoobtained.

1. Infroduction Pitaevskii [1] doesnotmentionthe timescaleoverwhich their averagedequationsprovideagood ap-

We intend to~examinethe behaviourof someso- proximationto the true variationin the averagebe-lutions of the o~d.e. haviourof the travelling wave. In fact their equa-

2 tions (analogousto (2.7)below) arevalid for timesUu~=u~+cu~+6uu~ (1.1)£ of order 1/cU, with the solutionscorrectto order

which describe~the constantwavespeedsolutions, c.u= u (x+ Ut),o~’themodifiedBurgers—Korteweg—de However, it shouldbe stressedthat thisstatementVries (mBKdV) equation only appliestotheaveragebehaviourof thewaveand

2 not to the phaseof thewave.An explicit calculation~ (1.2)of the averagebehaviourpredictedby the method

We will be particularly interestedin the casewhere has beencarriedout numerically; this straightfor-0<c << 1, whenwe may consider(1.1) to be a per- wardcalculationis explainedin the appendix.turbationof the integrablesystemobtainedby set- We alsoprovidesomeexactsolutionsof (1.1) andtingc = 0 (correspondingto themodifiedKorteweg— of theBKdV travellingwaveequation,treatingbothde Vries (mKdiV) equation). asequationsin complexvariables.Theseresultsfol-

A recentpaperby GurevichandPitaevskii[1] has low easilyfrom the Painlevéanalysiscompletedbytreated the very similar Burgers—KdV equation Ince in his book [4]. In particular, we obtain(wherethe nor~linearterm in (1.2) is replacedby boundedsolutionsofbothequationswhich arekink-uu~)as a pertu~-bationof the KdV equation.They like. Thosefor (1.1) correspondto the casewhere,proposeda modification of Whitham’s method of in real variables,the sign of the nonlinearterm isaveraging[2] i~iorder to predict,in particular,the negative(the mBKdV solutionswerealsoknown tobehaviourof co~nstantwavespeedsolutions.Wewill M. Pytka,privatecommunication,whereastheBKdVsee,in the cour$eof ouranalysisof (1.1),that their solutionsarebelievedto be new).methodreducesto the classicalmethodof averagingas appliedto a perturbationof an integrableo.d.e.with a singlepl~asevariable (see,in particular,ref.[1]). Consequentlyit is possibleto makerigorous 2. Singlephaseaveragingestimatesof the averagebehaviourof solutions to(1.1) by virtue,of the theoremon singlephaseav- With c = 0 the o.d.e.(1.1) canbewritten asa firsteraging in ref. [3]. The paperby Gurevich and ordersystem

0375-9601/90/S03.50 © ElsevierSciencePublishersB.V. (North-Holland) 57

Volume143, number1,2 PHYSICSLETTERSA 1 January1990

u. = v, v~= w, w~= Uv—6u2v, (2.1) tativebehaviourof the integrablesystem(2.1) withthat of (2.2). We are going to describeapproxi-

which hasintegrals mately the averagebehaviourof a boundedspiral-

A= ~(v2+u4—Uu2—2Bu), ling solution of (2.2) by writing down equationswhich approximatelygovernthe variation in B and

B=w+2u3— Uu. (a coordinaterelatedto) A.

It is notdifficult to seethaton surfacesof constant First,we definethe averageof any function F( u,B the system (2.1) is Hamiltonian. Thus we can v, w) overa closedcontourof constantA, B to bechoosea phasevariable 0 complementaryto A; the _________ 1flow is thentotally determinedby thefrequencyo(A, F( u, v, w) = — F( u, v, w) dØ, (2.3)coA~B) of 0 on curvesof constantA andB.

Theperturbationcorrespondingto (1.1) whenc where

is small may be written as = dØ. (2.4)

cb~—w(A,B)+cf(A,B,O,c), CO

A~= ewu, B~= — ew. (2.2) Thislastquantityis clearly thewavelengthof theap-propriateperiodicsolution of the integrablesystem.

To apply singlephaseaveragingwe do not needto Thetheoremon singlephaseaveraging(in e.g.ref.know the explicit form of eitherco(A, B) orf(A, B, [31) statesthat if A(x), ~(x) are solutionsof the0~c). system

The phaseportraits in fig. 1 comparethe quali-A~=ci~ü,~ (2.5)with initial conditionsA

0=A(x0),B0=~(x0)(cor-V

respondingto smoothclosedcontoursA=A0, B= B0),thenthereexist constantT, C(T) for which

II (A(x), B(x))— (A(x),B(x))II <C(T)c

a an exact solution of the system(2.2) with initialfor 0< Ix—xoI <Tic, whereA(x), B(x) belongtoconditionsA(x0)=A0, B(x0)=B0.We can solve (2.5) exactlyafter a changeof co-

ordinateforA. Let us define

W ~vdu=~(2A+2Bu+Uu2_u4)hf2du.

(2.6)V Notice that this is the canonicalactionvariablefor

stant.Now,

_____________ ____________ ~=

1t~w~udØ=—~

b (/________~*••~_..~ the Hamiltonian systemobtainedby fixing B con-U (~th~.Jw=

wherewe haveused the fact that

Fig. 1. Phaseportraitsfor B+w=0. (a) is theintegrablesystem,~=0,(b)isfor~>0. W~VX

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Volume 143,number1,2 PHYSICSLETTERSA 1 January1990

on contoursof constantA andB. However, without detailed examination of theConsequentlythe averagedsystem(2.5) is equationgoverningthe phase0, it is not possibleto

c i~w\ —j say anythingmeaningful about the oscillatory be-= — W= -~ c ~ , = 0, haviourof u (x). GurevichandPitaevskiiproducea

wave profile by taking a periodic solution u(x, w0,andtherefore B0) of the integrable system and substituting

W — — c W .~ —0 (2 7 ~ W= W0exp[e(x0 — x)] for W0. There is no guar-— X — . / anteethat the phaseof this function bearsorder c

This has solutions W= W0exp[c (x~— x) j, B= B0 proximity to the realbehaviourfor x—x0 of the or-throughinitial points W0,~ which correspondto a der 1/c.smoothclosedcontour(indeedW0 istheareainside Remark.It is not difficult to seethat a similarthiscontour). analysiscould beperformedfor any perturbationof

It is not difficult to seethat this simpleanalysis themKdV (or KdV) travellingwaveo.d.e.,with cu~gives preciselythe samedescriptionas the method replacedby cK(u, u~,u~)for any polynomial K.suggestedby GurevichandPitaevskii [1] in thecase Further,we could attemptto tackle the travellingof constantwa~vespeedwaves.Of course,weare ul- wave solutionsof perturbationof other“completelytimately intereatedin the conclusionswhich canbe integrable” p.d.e.’s. It is well known (see, for ex-drawnconcern*ngsolutionsof the o.d.e.(1.1). We ample,ref. [5]) that thetravellingwaveo.d.e.’scor-mustbe carefulnotto claim too much from the av- respondingto, for example,any of the higherordereragingmethod.Certainlyeqs.(2.7) allow ustopre- KdV or mKdV equationscan be treated as corn-dict, with knos~’nerror,the behaviourof any quan- pletely integrableHamiltonian systemsin finitelytity which is independentofthephase0. In particular many dimensions.In generalit would benecessarywe maycalculatehowtheaverageü andtheenvelope to apply the methodof averagingadaptedto mul-of the decayingoscillationsof u both vary on the tiphasesystems(see, for example, ref. [6]). It istimescaleof order 1/c.Thequalitativebehaviourof worth mentioning,however,thatthe conditionsun-aandthewave]envelopeis representedin fig. 2. The derwhich averagingappliesto multiphasesystemsformulaefor a] andthe wave envelopeare given in are considerablyrestrictive. A good approximationthe appendix. (to orderc for a timescaleof order 1/c) to the true

2

‘.75

7; --

5.5

0.25

0 ~ØØ 150 200 250 300 350

Fig. 2. Theupper(a)andlower (b) waveenvelopesandtheaverageII, fore= 10_2.

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Volume143, number1,2 PHYSICSLETTERSA 1 January1990

behaviouris only obtainedfor certainsystems,and before,the substitution(3.3) reduces(3.7) toeventhennot for all choicesof initial conditions. 2 2 4 4

~ur=r—a . (3.8)

This equationhas precisely two rational solutions3. Exactsolutionsof mBKdV and BKdV obtainedby settinga= 0. Theseare

It is naturalto ask if thereare any known exact r(z)=+~(z+c) 1

solutions to (1.1). FortunatelyInce [4] has ana- forarbitraryconstantc, which wemaynormalizetolysed equationsof the form c= 1 sinceanyothervaluemerelyreparametrizesthe

solution.Thus (3.7) hastwo boundedsolutionsu~—F(x,u,u~), (3.1)

u(x)=e’~-”(l+e”-”)1, (3.9)

whereF is rational in u~,algebraicin u andanalyt-ical in x, consideredas complex variables.He de- which thereforeare solutionsofterminedunderwhat conditions suchan equation 2u~~+cu~~—6uu~—Uu~=0.couldbe reducedto a first orderequationwhoseso-lutionswere free of movablecritical points.We can In the phaseplanethesesolutionscorrespondtoapply Ince’s resultsto the first integral two saddle-nodeconnections,wherethe nodeliesat

3 the origin with the saddleson the u-axis at ±~c. Asu~~+cu~+2u—Uu—B=0 (3.2)a wave profile eachsolution representsa smooth

of (1.1). Incestates(ref. [4], pp. 333,334) that the monotonictransition from u=0 to u= ±~c as xonlycasewhere(3.2) canbereducedto an equation increases.whosesolutionshaveno movablecritical points is Similarly, it is possibletoconstructexactsolutionswhen B=0 and U= —2c2/g. In this case the of the first integralsubstitution 2

u~~+cu~—3u—Uu—B=0 (3.10)u=zr, z=e’~, ~t=—~c (3.3) .

ofthe BKdV travellingwaveequation.Herethesigntransforms(3.2) into of the nonlinearterm is irrelevant. We employ the

3 2 3 transformationz (ti r~7+2r )=0. (3.4)

u=z2r+b, z=e’~ (3.11)

After further reducingthis to~z2r2— — (r4—a4) (3 5) where ~= —ic, b=~U—u2.After setting B=

— b(3b—U) we discoverthat (3.10) can be trans-

with a a constant,we find the solutions formedinto

r(z)=acn(~/~ajz’z, l/~J~), r~=2~r2r3—a3, (3.12)

wherecn(~,k) is oneof theJacobielliptic functions, wherea is constant.With a= 0 we obtain rationalwith modulusk. For a real we obtain unbounded solutions for r(z) which lead to the boundedreal-valuedsolutionsof (3.2), solutions

u(x)=ae’~cn(~.f~air’e’~,l/,~h). (3.6) u—b=2~2e2~(l+e’~)2. (3.13)

On the (u, v) phaseplanethesesolutionscorrespond Clearly thesenew solutionsof the BKdV equationto a family of curvesspirallinginto a nodal pointat exhibit the samebehaviourasthesolutions(3.9) ofthe origin asx—~ce(for c>.0). the mBKdV equation.

An interestingpoint isthat, asanequationincom-plex variables,(3.2) is equivalentto

3 . Acknowledgementu~~+cu~—2u—Uu+iB=0 (3.7)underthe transformationu—l’iu. With U, B fixed as This work was carriedout while the authorwas

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Volume143,number1,2 PHYSICSLETTERSA 1 January1990

employedby ProfessorD.G.Crightonon a SERCre- Thedependenceof a, a, b uponx is given implicitlysearchgrant (GD/D/72525) ProfessorCrighton’s in the equationinterestin this work is gratefullyacknowledged. a

W=W0exp[c(x0—x)]= $vdu. (A.5)

Appendix This canbe rearrangedto give x as a function of k,

0<k<1. We usethe relationshipU=a2+b2 to de-

Theaverageaandthe upperandlowerboundsof terminea(k), b(k) from (A.4). Thuswehaveu(k),u (a, b respectively)are given by a(k), b(k) which canbe plottedas functionsof x.

a The graphs in fig. 2 use the values x0=0, U=2,

(A.l) c=102.

I would like tothankDr. L. Smithfor numencally

where evaluatingthe necessaryintegralandproviding the

v2=—u4+Uu2+2Bu+2A graphs.

=(a—u)(u—b)(u—c)(u—d), (A.2) References

with a~ b~ c~~. Forsimplicity weconsiderthecase [1] A.V. ciurevich and L.P. Pitaevskii, Soy. Phys. JETP 66

wherea+d=0,:b+c=0, inwhich caseuhastheform (1987)490.— [2JG.B.Whitham,Psoc.R. Soc.A 283 (1965)238.U = ~itK(k) —‘ . (A.3) [3] V.!. Arnol’d, Geometricalmethodsfor ordinarydifferential

equations(SpringerBerlin, 1983).HereK( k) dez)otesthe completeelliptic integralof [4] E.L. Ince,Ordinarydifferentialequations(Dover,NewYork,the first kind with modulus 1956).

[5] S.Novikov,S.P.Manakov,LP. PitaevskiiandV.E. Zakharov,a—b IA A~ Theoryofsolitons (ConsultantBureau,NewYork, 1984).

= “~‘~‘ [6] V.!. Arnol’d, ed.,Encyclopediaofmathematicalsciences,Vol.3. DynamicalsystemsIII (Springer,Berlin, 1988).

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