Backlund transformation and special solutions for Drinfeld-Sokolov-Satsuma-Hirota system of
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arXiv:nlin/0102001v1 [nlin.SI] 1 Feb 2001B¨acklundtransformation
andspecialsolutionsfor
Drinfeld–Sokolov–Satsuma–Hirota
systemofcoupledequations
Ays¸eKarasu(Kalkanli)∗andSYuSakovich†
DepartmentofPhysics,MiddleEastTechnicalUniversity,
06531Ankara,Turkey
Abstract
UsingtheWeissmethodoftruncatedsingularexpansions,wecon-structanexplicitB¨acklundtransformationoftheDrinfeld–Sokolov–Satsuma–Hirotasystemintoitself.Thenwefindallthespecialsolu-tionsgeneratedbythistransformationfromthetrivialzerosolutionofthissystem.
Shorttitle:B¨acklundtransformationandspecialsolutionsPACSnumbers:02.30.Ik,02.30.Jr
Thesystemoftwocouplednonlinearevolutionequations
ut+uxxx−6uux−6vx=0vt−2vxxx+6uvx=0(1)
wasintroduced,independently,byDrinfeldandSokolov[1],andbySatsuma
andHirota[2].In[1],thesystem(1)wasgivenasoneofnumerousexamples
ofnonlinearequationspossessingLaxpairsofaspecialform.In[2],the
system(1)wasfoundasaspecialcaseofthefour-reductionoftheKPhier-
archy,anditsexplicitone-solitonsolutionwasconstructed.Recently,G¨urses
andKarasu[3]foundarecursionoperatorandabi-Hamiltonianstructurefor(1),whichprovidethesystem(1)withaninfinitealgebraofgeneralized
symmetriesandaninfinitesetofconservationlaws.
Inthispaper,weconstruct,usingtheWeissmethodoftruncatedsingular
expansions[4],anexplicitB¨acklundtransformationoftheDrinfeld–Sokolov–
Satsuma–Hirota(DSSH)system(1)intoitself,andthenfindallthespecial
solutionsgeneratedbythistransformationfromthetrivialzerosolutionof
thissystem.Inrelationwithourresults,weshouldmentiontherecentpaper
ofTianandGao[5],whodidnotfindaB¨acklundtransformationof(1)into
itself,contrarytotheirclaim,andproducedarationalspecialsolutionofthe
DSSHsystemwitherroneouslymanyarbitraryparameters.
Firstofall,wefinditusefultorewritetheDSSHsystem(1)intheform
ofthesinglesixth-orderequation
wtt−wxxxt−2wxxxxxx+18wxwxxxx+36wxxwxxx−36w2xwxx=0(2)
whichisrelatedto(1)bytheMiura-typetransformation
u=wxv=1
2φ−1xφxx−1,proposedbyConte[7],simplifiesthe
computationsveryconsiderably.Following[7],wesubstitutethetruncated
expansionw=−2χ−1+a(x,t)intotheequation(2),usetheidentitiesχx=
1+12(Cxx+CS)χ2andSt+Cxxx+2SCx+CSx=0,
whereS=φ−1xφxxx−3degreesofχ,andthusobtainacomplicatedsystemofequations,whichturns
outtobecompatibleandequivalenttothefollowingnormalsystemoftwo
equationsforφandaoftotalordersix:
ax+1
3S=0at+axxx−3
2axC+3
φ+φxx
φx+a(6)
isalsoasolutionof(2).Therefore(6)representsonemoreMiura-typetransformationbetween(4)and(2).ThesetwoMiura-typetransformations,(5)and(6),togetherwiththeauxiliarysystem(4),determineanimplicit
B¨acklundtransformationbetweenthe‘new’solutionwandthe‘old’solution
zoftheDSSHsystemwrittenintheform(2)(certainly,theseterms‘new’
and‘old’canbeusedintheoppositeorder).
Now,eliminatingφandafromtheequations(4),(5)and(6),weobtain
thefollowingexplicitB¨acklundtransformationoftheequation(2)intoitself:
pt+
4pxx−3p2x4p3
x=0(7)
qt−3
p+3
p2−1
2qxpxx4pxqxx8ppxx
+3
p2−9
2q2x−3
64p4=2k(8)
wherep=w−zandq=w+z.Theequations(7)and(8)constitutea
B¨acklundtransformationinthesenseofthedefinitiongivenin[8]:ifwe
eliminatezfrom(7)and(8),weobtainexactlytheequation(2)forw;and
ifweeliminatewfrom(7)and(8),weobtain(2)forz.Wethink,however,
thatitisimpossibletoprovethisbyhand-madecomputations:wediditby
meansoftheMathematicasystem[9].Asfarasweknow,theequations(7)
3and(8)arethemostcomplicatedexplicitB¨acklundtransformationinthe
literature.
Letusfindallthe‘new’solutionswoftheequation(2)generatedby
theobtainedB¨acklundtransformationfromthetrivial‘old’solutionz=0.
Settingz=0in(7)and(8),wegetanover-determinedsystemoftwoequa-
tionsforw,whichcanbesolvedexactly,usingthenewdependentvariable
ψ:w=−2φ−1φx,φx=ψ2.Thisgivesusthefollowingfivetypesofexplicit
specialsolutionsoftheDSSHsystemwrittenintheform(2):
w=−2
x3+12t(10)
w=−30(x2+σ)2Weshouldnotice,however,thattherearenosolitarywavesolutions
amongthesolutions(9)–(13).Indeed,theonlysolutionoftheformw=
f(x−ct)(c=constant),generatedby(7)–(8)fromz=0,isthesolution
(9),forwhichc=0.Forthisreason,wesuspect(7)–(8)ofbeingnotthesim-
plest(elementary)B¨acklundtransformationoftheequation(2)intoitself:
itmightbeaproductoftwoelementaryB¨acklundtransformations;sucha
phenomenonwasobservedin[10],wheretwodifferentB¨acklundtransforma-
tionsoftheCalogeroequationwerefoundandstudied.Thispointrequires
furtherinvestigation.
TheauthorsaregratefultotheScientificandTechnicalResearchCouncil
ofTurkey(T¨UB˙ITAK)forsupport.
References
[1]DrinfeldVGandSokolovVV1981Proc.SLSobolevSeminar,vol.2
(Novosibirsk)pp5-9(inRussian)
[2]SatsumaJandHirotaR1982J.Phys.Soc.Japan513390-7
[3]G¨ursesMandKarasuA1999Phys.Lett.A251247-9
[4]WeissJ1983J.Math.Phys.241405-13
[5]TianBandGaoY-T1995Phys.Lett.A208193-6
[6]WeissJ,TaborMandCarnevaleG1983J.Math.Phys.24522-6
[7]ConteR1989Phys.Lett.A140383-90
[8]AblowitzMJandSegurH1981SolitonsandtheInverseScattering
