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SymmetriesandintegrabilityofthemodifiedCamassa–Holm
equationwithanarbitraryparameter
ADURGADEVI1,KKRISHNAKUMAR2,∗,RSINUVASAN3andPGLLEACH4
1DepartmentofPhysics,SrinivasaRamanujanCentre,SASTRADeemedtobeUniversity,
Kumbakonam612001,India
2DepartmentofMathematics,SrinivasaRamanujanCentre,SASTRADeemedtobeUniversity,
Kumbakonam612001,India
3DepartmentofMathematics,VIT-APUniversity,Amaravathi522237,India
4DepartmentofMathematics,DurbanUniversityofTechnology,,Durban4000,
RepublicofSouthAfrica
∗-mail:******@
MSreceived5May2020;revised12August2020;accepted30December2021
–Holmequation(MCH),withan
arbitraryparameterk,oftheform
222
ut+k(u−uxx)ux−uxxt+(u−ux)(ux−uxxx)=0.
Thecommutatortableandadjointrepresentationofthesymmetriesarepresentedtoconstructone-dimensional
-dimensionaloptimalsystem,wereducetheorderornumberofindependent
variablesoftheaboveequationandalsoweobtaininterestingnovelsolutionsforthereducedordinarydifferential
,weapplythePainlevétesttotheresultantnonlinearordinarydifferentialequationanditis
observedthattheequationisintegrable.
–Holmequation;symmetry;Painlevétest;integrability.
;;;
-
tiesispresentedin[12,13].Especially,Liesymmetry
Linearandnonlinearpartialdifferentialequationshaveanalysis[14–23],Painlevéanalysis[24–29],Adomian
encompassedthelandscapeofmanydisciplinessuchasdecompositionmethod[30,31],variationaliteration
chemical,physical,biologicalsciences,engineeringandmethod,homotopyanalysismethod,homotopypertur-
mathematicsduetotheirpotentialabilitytoprovideade-bationmethodandvariationalhomotopyperturbation
aremanywaveequationsinengineeringandphysicsstudythenatureofintegrabilityofpartialdifferential
suchastheKorteweg–de-Vries(KdV)equation,nonlin-equations[32].Amongthesemethods,symmetryanaly-
earcoupledKdVequations,modifiedCamassa–HolmsisandPainlevéanalysishavebeenusedbyresearchers
(MCH)equationandsoon[1–11]thatarequitehelpfuleffectivelyduetotheirsystematicprocedureforfind-
,theclosed-formsolutionsofingthesolutionofthegivendifferentialequations
differentialequationsareveryhardtoachieveinmost[22,33,34].
,researchershavebeeninspiredtostudythe
Numeroussystematictechniqueshavebeendevel-integrabilityandsolutionsofvariousCamassa–Holm
opedtofindsolutionsofdifferentialequationsin(CH)typesofequationswithcubicorhigher-ordernon-
[35]andFuchssteiner
madeinidentifying/generatingsystemsofLiénard-type[8]derivedthewell-knownmodifiedCamassa-Holm
0123456789().:V,-vol
85Page2of8Pramana–.(2021)95:85
(MCH)equationwhereXiscalledthe‘infinitesimalgenerator’whichis
denotedby
yt+uyx+2uxy=0,y=u−uxx(1)
121
X=ξ(t,x)∂t+ξ(t,x)∂x+η(t,x)∂u.
byemployingthetri-Hamiltoniandualityapproachto
thebi-HamiltonianrepresentationoftheMKDVequa-Basedonthetheory,theinvariantconditionforeq.(3)
fluidanditspotentialdensity[36].TheMCHequationχ(t,x,u)=χ(t˜,x˜,u˜).
isawater-waveequationandisasuitableapproximation
-Itiswellknownthatonecanreducetheorderofdiffer-
quently,Qiao[36]discussedthenatureofintegrabilityentialequationsaswellasthenumberofindependent
someliterature,theMCHequationisotherwisecalledknownasasymmetryofeq.(3).
asFORQequation[37,38].Therefore,ifXisaLiepointsymmetryforequation
TheMCHequationiscompletelyintegrable[35].Itχ≡0,thenthefollowingconditionistrue:
hasabi-HamiltonianstructureandalsoadmitsaLaxpair[n]χ=λχ|,
[39]andhencemaybesolvedbytheinversescatteringXχ=0
[n]isthenthexten-
acubicextensionofthewell-knownCHequationwhichsionofXinthejetspace.
wasproposedasamodeltodescribetheunidirectional
propagationofshallowwaterwaves[40–42]andaxially
symmetricwavesinhyperelasticrods[43].
–Holm
Thepurposeofthisworkistostudythesymmetryand
equation
integrabilityofMCHequationwithanarbitraryparam-
Qiaodiscussedeq.(2)whichisintegrableonlywhen
2=
ut+k(u−uxx)ux−uxxtk2[36].Healsosuggestedtoexaminetheintegra-
bilityoftheaboveequationforallpossiblevaluesof
+(u2−u2)(u−u)=0,(2)
,weproposeamoregeneralformofeq.
wheretheparameterk∈Rcharacterisesthemagnitude(2)bytakingkask(t),
,k(t),asdis-
[36,44],canberepresentedas
In§2thegeneralprocedureofLie’+k(t)(u−u)2u−u
In§3wediscussLiepointsymmetriesandreductionstxxxxxt
+(2−2)(−)=.
oftheorderofeq.(2).Indeed,wereducetheorderofuuxuxuxxx0(4)
ordinaryorpartialdifferentialequationsbyusingtheWhenk(t)isanarbitraryfunctionoft,eq.(4)admitsa
,Liesymmetriesplaymajorrolesin
symmetry∂
§4,westudythePainlevéanalysisfor
for∂xisgivenby
eq.(2)andweareabletoprovethateq.(2)isintegrable.
=dx=du.
010
Thesolutionoftheauxiliaryequationgivessimilarity
’stheoryvariablesasα=tandu=f(t).Thisleadstothe
trivialsolutionu(x,t)=Ctoeq.(4).Whenk(t)=k,
Supposethatthegivenequationisoftheformanarbitraryparameter,thesymmetriesofeq.(2)are
=∂
χ(t,x,u,ut,ux,utt,utx,uxx,uxxx,...)=0,(3)X1t(5)
X=∂(6)
,2x
wheretxareindependentvariablesanduisadepen-=∂−∂.
,theinfinitesimalX32ttuu(7)
pointtransformationsaregivenasfollows:ThroughoutthisworktheMathematicaadd-onSym
[45–47]-
t˜(t,x,)=t+ξ1(t,x)+◦(2)=t+Xt+◦(2),
metriescanbeusedfortheclassificationofdifferential
x˜(t,x,)=x+ξ2(t,x)+◦(2)=x+Xx+◦(2),
equations,eq.(2)admitstheLiealgebra{2A1⊕sA1}
u˜(t,x,)=u+η1(t,x)+◦(2)=u+Xu+◦(2),whichisidentifiedfromtheMorozov–Mubarakzyanov
Pramana–.(2021)95:85Page3of885
-
acteristicequationforX1isgivenby
[XI,XJ]X1X2X3
dtdxdu
X1002X1==.
X2000100
−
X32X100Thesolutionofthecharacteristicequationgivesthesim-
ilarityvariablesas
.
ξ=xandu=U(x).
X
Ad[eI]XJX1X2X3
Thus,theinvariantsolutionofeq.(2)isu(x,t)=U(x).
X1X1X2X3−2X1Thisleadstoanordinarydifferentialequation
X2X1X2X3
2222
X3eX1X2X3(U−U)(U−U)−kU(U−U)=0.(8)
Theaboveequationcanberewrittenas
classificationscheme[48–51]basedontheLiebrackets(−)(−)
UU+k2UUU=,
oftheLiesymmetriesofeq.(2)asgivenintable1.0(9)
(U−U)2(U2−U2)
Olver[15]discussedtheadjointrepresentationand
optimalsystemwhichgivesuniquepossiblecombi-where-
nationofsymmetriestoperformthereductionsofgratingeq.(9)once,weget
,ZhaoandHan[52,53]
(−)(2−2)k/2+=,
,UUUUI10(10)
theadjointrepresentationsofthesymmetries,based
whereI1isaconstantofintegration.
XI=−[,]+
ontheformulaAdeXJXJXIXJAseq.(10)isanautonomousequation,ithasthetrivial
2[,[,]]−···∂.
2XIXIXJ,,xThecorrespondingcanonicalvariableis
Accordingtotheiridea,thecommutatortable1and
=().
adjointrepresentationtable2providetheoptimalsystemUVU(11)
ofeq.(2)asfollows:LetthegenericsymmetryvectorThisgivesU=VV.Thereforeeq.(10)becomes
fieldbetakenasX=l1X1+l2X2+l3X3.
22k/2
(VV−U)(U−V)+I1=0,(12)
=0
=0,thentheadjointrepresentationwheredenotesthederivativewithrespecttoUandalso
1eq.(12)canbewrittenas
givesthatX1andX2arelinearlyindependentsymme-
triesandthegenericsymmetryvectorfieldbecomesthe22k/222
(U−V)d(U−V)−2I1=0.(13)
,X1+cX2.
Thesolutionofeq.(13)isgivenby
=0,thenthegenericsymmetryvector
.22(k/2)+1
fieldisX2(U−V)−2I1U−I2=0.(14)
=
,thenthegenericsymmetryvectorThisimplies
fieldisX1.
22/(k+2)
V=±U−(2I1U+I2).(15)
=0
Fromeq.(11)theaboveequationcanbewrittenas
=0,thentheadjointrepresentation
givesthatX1=X3/2.Therefore,thegenericsymme-dU
.=±U2−(2IU+I)2/(k+2).(16)
tryvectorfieldbecomesX3dx12
Consequentlytheone-dimensionaloptimalsystems
areThisyields
U
X1,X2,X3andX1+
=±x+I3.(17)
22/(k+2)
U−(2I1U+I2)
-u(x,t)=cbecausethecanonicaltransformationbased
tionofeq.(2)throughtheinvariantfunctionwhichisonX2isu(x,t)=c.
85Page4of8Pramana–.(2021)95:85
2/(k+2)
(,)=−1/2()×expdx+I2
throughtheinvariantsolutionuxttUx2(U2−(dU/dx)2)
.(,)=
whichisgivenbyX3Withtheuseofuxt(27)
t−1/2U(x),eq.(2)canbereducedtotheequation
2Nowwediscussthetravelling-wavesolutionofeq.
2(U−U)(U2−U)
(2)
−(−)(−(−))=.12
UU12kUUU0(18)takethelinearcombinationofthesetwosymmetriesas
ThisequationcanberewrittenasX4=∂t+C∂
arer=x−Ctandu(x,t)=Q(r).Basedonthese
(U−U)k2U(U−U)1
+−=0,canonicalvariables,eq.(2)canbereducedto
(U−U)2(U2−U2)2(U2−U2)
(−2+2)(−)
(19)CQQQQ
+kQ(Q−Q)2=0,(28)
whererepresentsthedifferentiationwithrespecttox.
Integratingeq.(19)once,wegetwhereQisthefunctionofthenewindependentvariable
.(28)is∂rforwhichthecanonical
22k/21dx
(U−U)(U−U)+I1exp=0,variablesare
2U2−U2
(20)Q(r)=QandQ=W(Q).(29)
,QandQbecome
Aseq.(20)isanautonomousequation,ithasthetrivial
2
symmetry∂=WWandQ=W(WW+W).(30)
givenby
Basedonthesecanonicalvariables(29)andeq.(30),eq.
=().
UVU(21)(28)canbereducedto
=.
UVV(22)
(C−Q2+W2)(WW+W−1)
Therefore,(20)becomes
+k(WW−Q)2=0,(31)
(−)(2−2)k/2
UVVUV
-
−Iexp=0,(23)
1(2−2)triesofeq.(31),wearriveatthefollowingtwopossible
2VUVcases:
wheredenotesthederivativewithrespecttoUandeq.
Case1
(23)canbewrittenasIfk=−2,thenthesymmetriesofeq.(31)are
/
(U2−V2)k2d(U2−V2)
Q
1dU1=∂Q+∂W
−I=.W
21exp220(24)
2V(U−V)(−2+2)−k/2
=CQW∂
Integratingeq.(24)wehave2WW
(2−2)(k/2)+1Q(C−Q2+W2)−k/2
UV=∂
1dU3WW
−2IexpdU−I=0.(25)
12V(U2−V2)2(C−Q2+W2)
4=∂W
Thisprovides(k+2)W
22
22(C+(3+2k)Q+W)
V=U=2Q∂+∂
/(+)5Q(+)W
1dU2k2k2W
−2I1expdU+I2.(+(+)2+2)
V(U2−V2)2QCk1QW
26=Q∂Q+∂W
(26)(k+2)W
(C−Q2+W2)(k+2)/2
Byusingeq.(21)onecanfindUbysolvingthefollowing7=∂Q
equation:k+2
22(k+2)/2
2Q(C−Q+W)
dUdU+∂W
=U2−2I(k+2)W
dx1dx
Pramana–.(2021)95:85Page5of885
Q(C−Q2+W2)(k+2)/2Bytheuseofbackwardsubstitution,wecanobtain
=∂Q
8k+2thesolutionof(2)asfollows:
(C−Q2+W2)(k+2)/2(C+(k+1)Q2+W2)2=2+2,
+∂.WQ I2(37)
(+)2W
k2W22
W=±Q+I2.(38)
Case2
Ifk=−2,thenthesymmetriesofeq.(31)areNowbytheuseof(29)theaboveequationcanbewritten
as
Q
=∂Q+∂W
122
WQ=±Q+I2,(39)
(−2+2)
CQW
2=∂
Wsolutionofeq.(39)isgivenby
Q(C−Q2+W2)
=∂=±[+].
3WWQI2sinhrI3(40)
22
4=log[C−Q+W]∂QHence,thetravelling-wavesolutionofeq.(2)isgiven
Qlog[C−Q2+W2]by
+∂W
Wu(x,t)=±I2sinh[x−Ct+I3],(41)
(−2+2)[−2+2]
CQWlogCQWwhereIandIareconstantsofintegration.
5=∂W23
WInwhatfollows,wereducetheorderofanotherform
1222ofeq.(2)byusinganequivalenceformofthesymmetry,
6=4Q∂Q+(4Q+(C−Q+W)
WX3,,firstlyconsideranotherform
22
×log[C−Q+W])∂Wofeq.(2),
222
21322vt+k(t)vux+(u−u)vx=0,
7=
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