On High-Frequency Soliton Solutions to a (2+1)-Dimensional Nonlinear Partial Differential E

OnHigh-FrequencySolitonSolutionstoa(2+1)-DimensionalNonlinearPartial

DifferentialEvolutionEquation

KuetcheKamgangVictor1∗,BouetouBouetouThomas2,3∗∗,TimoleonCrepinKofane1,3∗∗∗

DepartmentofPhysics,FacultyofScience,UniversityofYaoundeI,POBox.812,CameroonEcoleNationaleSup´erieurePolytechnique,UniversityofYaoundeI,POBox.8390,Cameroon

3

TheAbdusSalamInternationalCentreforTheoreticalPhysics,P.O.Box586,StradaCostiera,II-34014,Trieste,Italy

21

(Received25November2007)

A(2+1)-dimensionalnonlinearpartialdifferentialevolution(NLPDE)equationispresentedasamodelequationforrelaxinghigh-rateprocessesinactivebarothropicmedia.WiththeaidofsymboliccomputationandHirota’smethod,sometypicalsolitarywavesolutionstothis(2+1)-dimensionalNLPDEequationareunearthed.Asaresult,dependingonthedissipativeparameter,singleandmultivaluedsolutionsaredepicted.

PACS:05.45.Yv

Innonlinearscience,solitontheorymayplayanessentialroleandmaybeappliedinalmostallthenaturalsciences,[1]inwhichmanydynamicalphenom-enaaremodelledby(1+1)-dimensionalNLPDEequa-tions.Asanillustration,recently,Vakhnenko[2]hasderivedanevolutionequationgivenby

∂x˜∂xu+u˜=0,˜+u˜(∂t˜)˜

(1)

2

p=pewithvelocitiesvedefinedbyve=

dp

.Thehy-dρ

drodynamicequationsrelatedtosuchamediumaregivenby

∂tV−

1

∇·v=0,ρ0

∂tv+

1

∇p=0,ρ0

(2)

asamodelequationofrelaxinghigh-frequencybarothropicmedia.Thequantityu˜maystandfora

˜mayrepresentspace-likephysicalobservable,x˜andt

andtime-likeindependentvariables.

Ontheotherhand,invariouscases,therealnat-uralphenomenaaretoointricatetodescribeonlybyvirtueofthe(1+1)-dimensionalNLPDEequa-tions.Forinstance,innature,thereexistverycom-plicatedphenomenasuchasfoldedprotein,[3]foldedbrainandskinsurfacesandotherkindsoffoldedbio-logicsystems.[4]Thus,onemajorquerythatmayarise:whichkindofpatternformationsmaybefoundinhigherdimensionalsystems(ifintegrable)especiallyin(2+1)-dimensionalNLPDEequations,andcanthesolitonpropertiesbefoundin(1+1)-dimensionalNLPDEequationsstillsurvivinginhigherordersys-tems?

Inordertofindananswertothisquery,weexam-inethedynamicalbehaviourofarelaxingbarothropicmediaunderhigh-frequencyperturbations.Usingthedynamicstateequationofsuchmedia,wederiveanovel(N+1)-dimensionalNLPDEequation(N>1).ParticularinterestsarepaidtotheN=2case.

Weconsiderabarothropicmediumcharacterizedbyp=p(ρ,ξ)wherepisthepressure,ρisthedensity,andξisanadditionalparameterofthemedium.Forfastprocesses,ξ=1andp=pfwithvelocitiesvf

dp2

definedbyvf=.Forlowprocesses,ξ=0and

dρ

∗Email:∗∗

wherethequantitiesv,t,Vand∇maystandforvelocity-vector,time,specificvolumeandgradientop-erator,respectively.Equation(2)mayarisefromthelawsofconservationofmassandmomentum.Forsmallperturbationsp󰀂,i.e.p󰀂󰀅p0andp0,beingthepressureattheunperturbedstate,afteranex-pansionofthespecificvolumeVaspowerseriesofp󰀂withaccuracyo(p󰀂2),thedynamicstateequationmaybegivenby

󰀆󰀁12󰀂2󰀂2󰀂22󰀂τ∂t∂xmxmp−2∂tp+αf∂tp+∂xmpmxvf

12󰀂2󰀂2

p+αe∂tp=0,(m=1,...,N),−2∂t

ve

(3)whereτistherelaxationtime;constantsαeandαfmaybetheexpansionsecond-ordercoefficientsofVeandVf,respectively;quantityxm≡(x,y,z,...)

22

maystandforposition-vectorand∂xm=∂x1x1+mx22∂xItisnoted2+∂xx3+···,(m=1,...,N).2x3

thatanyrepeatedindexreferstosummationwithrespecttoEinstein’snotation.Inordertoinvesti-gateEq.(3),themultiscalemethod[5,6]maybeuseful.Thus,definingthequantity󰀃=τω(ωbeingthefre-quencyoftheprocesses)chosentobesmall(large)pa-rameter,afterintroducingtheindependentvariables

mmm−4

T0=tω,T−4=tω󰀃−4,X0=xmω,X−,4=xω󰀃(m=1,...,N)intoEq.(3),sevencoupledequationsmaybederived.Fromthiscoupledsystem,theremaybetwoleadingequationsexpressedintermsofT0and

vkuetche@yahoo.frEmail:tbouetou@yahoo.fr∗∗∗Email:tckofane@yahoo.com

c2008ChinesePhysicalSocietyandIOPPublishingLtd󰀁

426

KuetcheKamgangVictoretal.Vol.25

X0

m(m=1,...,N),describinglow-frequencypertur-bations,andthetwoothersexpressedintermsofT−4

andX−m

4(m=1,...,N)describinghigh-frequencyperturbations.Focusingourinterestonlyonthehighfrequencyperturbations,wemayderivethefollowingevolutionequation:

(∂xx)(p󰀂+αfv2fp󰀂2

n+∂m)(∂xn+∂xm)−v−22p󰀂f∂tt+βf∂xmJm+γfp󰀂=0,

(n(4)

whereJm=p󰀂(m=1,...,N),andthequantitiesβfandγfareexpressedasfollows:

244

β=vf

−v2efτv2γvf−ve

f=ev,f

2τ2v4ev2.

(5)

f

Equation(4)maybeobtainedinthefollowingway.Adispersionrelationforthelinearizedequation(4)

maybewrittenintheformv−2fω2

=(kn+km)(knk)+jβ󰀂+

mfN

m=1km−γfwith(n(∂x+∂y)2p󰀂−v−22󰀂αfv2

f∂ttp+f(∂x+∂y)2p󰀂2

+βf(∂x+∂y)p󰀂+γfp󰀂=0.

(6)

Inthehigh-frequencyperturbationsregime,itmaybeworthconsideringthefollowingaccuracy:

(∂x+∂y)2−v−22∂y+v−1

f∂tt≈2(∂x+∂y)(∂x+f∂t),(7)

tofurtherinvestigateEq.(6).Itmaybeclearthatthereexistmanytechniquestofindthesolitonsolu-tionstosomeNLPDEequations.Forexample,usingthe‘multilinearvariableseparationapproach’recentlyinvestigatedbyTangandLou[7]mayleadtomanyin-terestingsolutions.ApplyingtheextendedPainlev´e

truncation[8]

tothecurrentsystemmaybefullofin-terests.Thus,foragivenNLPDEequation,theremayexistmanysolutions.TheHirotabilinearmethod,[9]eventhoughlooksstandard,hasbeenshowntobepowerful.Thismethodmaybeusefultofindnontriv-ialsolutionstomanyNLPDEequations.[1]

Equation(6)isnowreducedto(∂x˜+∂y˜)[∂t+˜u(∂x˜+∂y˜)]˜u+α(∂x˜+∂y˜)˜

u+u˜=0,(8)providedthat

󰀅󰀅x˜=γf

(x−vf

2

ft),

y˜=

γ2

(y−vft),󰀅˜t

=vγf

ft,u˜=αfv2fp󰀂

2

,

α=󰀄βf

2γ,

(9)

f

hold.Theα-termmaystandforthe‘dissipative’

termasintroducedanddefinedbyVakhnenko[2]whoregardedthedissipationintermsofpartialderiva-tiveswithrespecttospacecoordinate.Beforelook-ingforsolutionstoEq.(8),wewouldliketomakeaconnectionbetweentheVakhnenkoequation(1)withotherequationswhichstillattractagreatdealofattentionfromthescientificcommunityduetothefactthattheymaydescribethedynamicsofagreatvarietyofbiological,chemicalandphysicalsys-tems.Ingeneral,theinterestinelucidatingdissi-pativeprocesses[10]onvariousspatialandtemporalscales,frommicroscopictomacroscopic,whichoccursinallmachinesandmechanismisjustifiedbymodernexperimentaltechnologiesthathavemadeitpossibletostudywearlessfrictionbetweencleanandatomi-callyflatsurface.[11]Inparticular,muchattentionhasbeenrecentlydevelopedwithinthefieldofnanotri-bologyinunderstandingthenatureoffrictionatthemicroscopicscale.[10,11]Shearedliquidsconfinedbe-tweentwoatomicallysmoothsolidsurfacesprovideagoodexampleofasystemwhereabroadrangeofphenomenaanddifferentbehaviourhavebeenexperi-mentallyobserved.Insuchsystems,thedissipationisintroducedintermsofpartialderivativewithrespecttotimecoordinate.Wewouldliketocallattentionofresearcherswhoarefamiliartodissipationintermsofpartialderivativewithrespecttotimecoordinatetothefollowingpoints.Therearesomephysicalsys-temswhereadissipativeeffectcoexiststogetherwithnonlinearanddispersiveeffects.Theionacousticsoli-ton,forexample,travelsinaplasmaundertheeffectsofLandaudampingand/orcollisionaldamping,andthesolitoninanonlinearlumpednetworkalsodampsgraduallyduetosomefinitevaluesofelements.[12]

DissipationterminnonlinearevolutionequationshasbeensubjecttomanyinterestsinperforminghighorderderivativeandnonlineartermstotheBurgersequation

ut+uux+···+up(ux)q+α1ux+α2uxx+···+αkukx+···+αnunx=0,

(10)

withp,q,nandαk(k=1,2,...,n)beingspace-andtime-independentparameters.Wenotethatukx≡∂ku

∂xk

(k=1,2,...,n).Thus,Burgers,Korteweg–deVries(KdV),KdV–Burgers,andBenney[13]equationsmaybegeneralizedtothefollowingform:

ut+uux+cmDm(u)+αu3x+βu4x=0,

(11)

wheremisanintegerwhilecmandDmmaystandforthedissipationcoefficientanddifferentialopera-tor,respectively.

No.2KuetcheKamgangVictoretal.427

Asadissipationterm,thefollowingcasesmaybeemployed:

1.D∂2

m≡

,cm=c1andα=β=0,theequation∂x2Burgersmaybederivedasfollows:

ut+uux+c1uxx=0,

(12)

2.Dm≡0,cm=0andβ=0,theKdVequationmaybederived[14]asfollows:

ut+uux+αu3x=0,

(13)

≡∂2

3.Dm∂x2

,cm=c1andβ=0,theKdV-Burgersequationmaybeexpressedas

ut+uux+αu3x+c1uxx=0,

(14)

4.Dm≡1,cm=c2andβ=0,akindofKdV

equationdescribingionacousticwaveswithweakion-neutralcollision[15]maybederivedas

ut+uux+αu3x+c2u=0,

(15)

5.D∂2

m≡equation[13]maybe∂x2andcm=c1,theBenney

derivedas

ut+uux+c1uxx(u)+αu3x+βu4x=0.

(16)

Ascanbeseen,theVakhnenkoequation(1)andEq.(8)mayserveasenrichingfeaturesinnonlinearevolutionsystemswheredissipationisintroducedintermsofpartialderivativeswithrespecttospaceco-ordinate.

Now,performingvariabletransformation,wein-troducenewindependentvariablesX,T1andT2asfollows:

󰀃X

x˜=T1+UdX󰀂+x˜0,

󰀃

−∞

X

y˜=T2+

UdX󰀂+y˜0,

−∞˜t

=X,(17)

wherex˜0andy˜0standforarbitraryconstantsand

u˜(˜x,y˜,˜t)=U(T1,T2,X).Thus,Eq.(8)maybetrans-formedto

WXXT1+WXWT1+WXXT2+WXWT2

+WX+α(WXT1+WXT2)=0,

(18)

whereithasbeensetU=WXsuchthatsubscriptsdenotepartialdifferentiation.Bytaking

W=6(lnF)X,(19)

Eq.(18)maybewrittenasthebilinearequation

F2(DT3T32

1DX+D2DX+DX)(F·F)

+α(DT)(D2

1+DT2X(F·F)·F2)=0,

(20)

whereDT1,DT2andDXdenoteHirota’soperators.[9]ThesolutiontoEq.(20)correspondingtoone-solitonmaybegivenby

F=1+exp(2η),

(21)

whereη=KX−ω1T1−ω2T2+η0.ThequantitiesK,ω1,ω2,η0maybeobservedasconstants.Thedispersionrelationmaybegivenby

2(ω1+ω2)(α+2K)=1.

(22)

SubstitutingEq.(21)intoEq.(19)gives

W(X,T1,T2)=6K[1+tanh(η)],

(23)sothat

U(X,T1,T2)=6K2sech2(η).

(24)Inaddition,Eq.(17)mayleadto

x˜=T1+6Ktanh(η),y˜=T2+6Ktanh(η),

˜t

=X,(25)wherex˜0=y˜0=−6Kandη0=0havebeentakentoreachthesymmetryinx˜-˜yspace.Inordertodiscussthedifferenttypesofsolutions,itisusefultoconsiderthesystem

∂T1=φ1∂x˜+ϕ1∂y˜,∂T2=φ2∂x˜+ϕ2∂y˜,

(26)

where

φ1=1+ϕ1=1+WT1,

ϕ2=1+φ2=1+WT2.(27)Fig.1.Loop-likepatternformationu˜vsx˜andy˜(α=

0.1).

428KuetcheKamgangVictoretal.Vol.25

Fig.2.Cusp-likepatternformationu˜vsx˜andy˜(α=5/6).

Fig.3.Hump-likepatternformationu˜vsx˜andy˜(α=1.1).

Assumingthatω1=Kv1andω2=Kv2suchthatv≡v1+v2>0,theparameterαmaysatisfythefollowingconditions:

1.for0≤α<√1

,

u˜u6vx˜and˜y˜maychangesignthreetimesandgoinfi-nitetwice.Thus,u˜vsx˜andy˜maydescribeatypicalfoldedsolitarywaveofloop-liketype(seeFig.1);

2.forα=√1

u˜6v,

x˜and˜uy˜maychangesignonceandgoinfiniteonce.Thus,u˜vsx˜andy˜maydescribeatypicalsingle-valuedsolitarywaveofcusp-liketype3.finally,forα>√1

(seeFig.2);

u˜6v,

x˜andu˜y˜

maychangesignonceandmaynevergoinfinity.Thus,u˜vsx˜andy˜maydescribeatypi-calsingle-valuedsolitarywaveofhump-liketype(seeFig.3);

Asisillustrated,wehavetakenv=0.24,andthethreetypicalsolutionsmentionedabovearedepicted

inFigs.1–3atinitialtime˜t

=0.Acharacteristicvalueofthedissipativeparameterisfoundtobeα=5/6correspondingtothecusp-likepatternformationsolu-tion.Recently,ithasbeenshownthatthedissipative

termwithadissipativeparameterlessthansomelimitvaluedonotdestroytheloop-likesolitonsolutionstotheVakhnenkoequation.[2]Thisobservationmaybemadehereinthecaseofloop-likepatternformationsolutiontothe(2+1)-dimensionalNLPDEEq.(8).Itisworthnotingthatifinteractionsamongfoldedsoli-tarywavesareelastic,thentheymaycalledfoldons.[8]In(1+1)-dimensionalcase,thesimplestfoldonsaretheso-calledloopsolitonswhichhavebeenfoundinmany(1+1)-dimensionalintegrablesystems.

Insummary,the(2+1)-dimensionalNLPDEEq.(8),whichmaybeobservedasa(2+1)-dimensionalversionoftheVakhnenkoequation,[2]mayhavesomescientificinterestsbothfromtheview-pointoftheinvestigationofthepropagationofhigh-frequencyperturbationsandfromtheviewpointoftheexistenceofstablewaveformations.Itmayactu-allybetruethatthereductiveperturbationmethodandtheHirotabilinearmethodmayseemtobestan-dard.Indeed,thisstudyaimstopointouttheprop-ertiesofthesystemunderinvestigationwhenhigherdimensionalspaceisconsidered.Itdoesnotspecializeonamethodortechniqueofinvestigation.Asmen-tionedabove,theremayexistmanyothertechniquestotackletheproblem.

Inaddition,relaxingmediaaresowideinthena-ture.Forinstance,onemayfindgas-media,steam-liquidmedia,electrons-gasinconductors,justtonameafew.Sinceconservativeflowequationshavebeenusedtodescribethebehaviourofsuchmedia,thesuitablefieldsforapplicationsofthepreviousresultsmaybefoundinsolitontheory,[1]geodynamics,[16]hydrodynamics,[17]justtonameafew.

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