Computable Integrability. Chapter 2 Riccati equati

arXiv:math-ph/0509024v112Sep2005ComputableIntegrability.Chapter2:RiccatiequationE.Kartashova,A.ShabatContents1Introduction22GeneralsolutionofRE22.1a(x)=0..............................22.2a(x)6=0..............................32.3Transformationgroup.......................52.4Singularitiesofsolutions.....................73DifferentialequationsrelatedtoRE93.1Linearequationsofsecondorder.................93.2Schwarzianequation.......................143.3ModifiedSchwarzianequation..................154Asymptoticsolutions184.1REwithaparameterλ......................194.2Soliton-likepotentials.......................224.3Finite-gappotentials.......................255Summary296ExercisesforChapter23011IntroductionRiccatiequation(RE)φx=a(x)φ2+b(x)φ+c(x)(1)isoneofthemostsimplenonlineardifferentialequationsbecauseitisoffirstorderandwithquadraticnonlinearity.Obviously,thiswasthereasonthatassoonasNewtoninventeddifferentialequations,REwasthefirstonetobeinvestigatedextensivelysincetheendofthe17thcentury[1].In1726RiccaticonsideredthefirstorderODEwx=w2+u(x)withpolynomialinxfunctionu(x).Evidently,thecasesdegu=1,2corre-spondtotheAiryandHermitetranscendentfunctions,respectively.BelowweshowthatHermitetranscendentisintegrableinquadratures.AstoAirytranscendent,itisonlyF-integrable1thoughthecorrespondingequationit-selfisatthefirstglanceasimplerone.Thus,newtranscendentswereintroducedassolutionsofthefirstorderODEwiththequadraticnonlinearity,i.e.assolutionsofREs.SomeclassesofREsareknowntohavegeneralsolutions,forinstance:y′+ay2=bxαwherealla,b,αareconstantinrespecttox.D.Bernoullidiscovered(1724-25)thatthisREisintegrableinelementaryfunctionsifα=−2orα=−4k(2k−1),k=1,2,3,.....BelowsomegeneralresultsaboutREarepre-sentedwhichmakeitwidelyusablefornumerousapplicationsindifferentbranchesofphysicsandmathematics.2GeneralsolutionofREInordertoshowhowtosolve(1)ingeneralform,letusregardtwocases.2.1a(x)=0Incasea(x)=0,REtakesparticularformφx=b(x)φ+c(x),(2)1SeeEx.32i.e.itisafirst-orderLODEanditsgeneralsolutioncanbeexpressedinquadratures.Asafirststep,onehastofindasolutionz(x)ofitshomogeneouspart2,i.e.z(x):zx=b(x)z.InordertofindgeneralsolutionofEq.(2)letusintroducenewvariable˜φ(x)=φ(x)/z(x),i.e.z(x)˜φ(x)=φ(x).Then(z(x)˜φ(x))x=b(x)z(x)˜φ(x)+c(x),i.e.z(x)˜φ(x)x=c(x),anditgivesusgeneralsolutionofEq.(2)inquadraturesφ(x)=z(x)˜φ(x)=z(x)(Zc(x)z(x)dx+const).(3)Thismethodiscalledmethodofvariationofconstantsandcanbeeasilygeneralizedforasystemoffirst-orderLODEs~y′=A(x)~y+~f(x).Naturally,forthesystemofnequationsweneedtoknownparticularso-lutionsofthecorrespondinghomogeneoussysteminordertousemethodofvariationofconstants.Andthisisexactlythebottle-neckoftheprocedure-indistinctionwithfirst-orderLODEswhichareallintegrableinquadratures,alreadysecond-orderLODEsarenot.2.2a(x)6=0InthiscaseoneknownparticularsolutionofaREallowstocon-structitsgeneralsolution.Indeed,supposethatϕ1isaparticularsolutionofEq.(1),thenc=ϕ1,x−aϕ21−bϕ1andsubstitutionφ=y+ϕ1annihilatesfreetermcyieldingtoanequationyx=ay2+˜by(4)with˜b=b+2aϕ1.Afterre-writingEq.(4)asyxy2=a+˜by2seeEx.13andmakinganobviouschangeofvariablesφ1=1/y,wegetaparticularcaseofREφ1,x+˜bφ1−a=0anditsgeneralsolutioniswrittenoutexplicitlyintheprevioussubsection.Example2.1Asanimportantillustrativeexampleleadingtomanyap-plicationsinmathematicalphysics,letusregardaparticularREinaformyx+y2=x2+α.(5)Forα=1,particularsolutioncanbetakenasy=xandgeneralsolutionobtainedasaboveyieldstoy=x+e−x2Re−x2dx+const,i.e.incase(5)isintegrableinquadratures.IndefiniteintegralRe−x2dxthoughnotexpressedinelementaryfunctions,playsimportantroleinmanyareasfromprobabilitytheorytillquantummechanics.Forarbitraryα,Eq.(5)possessremarkableproperty,namely,afteranelementaryfraction-rationaltransformationˆy=x+αy+x(6)ittakesformˆyx+ˆy2=x2+ˆα,ˆα=α+2,i.e.formoforiginalEq.(5)didnotchangewhileitsrhsincreasedby2.Inparticular,afterthistransformationEq.(5)withα=1takesformˆyx+ˆy2=x2+3andsincey=xisaparticularsolutionof(5),thenˆy=x+1/xisaparticularsolutionofthelastequation.Itmeansthatforanyα=2k+1,k=0,1,2,...generalsolutionofEq.(5)canbefoundinquadraturesasitwasdoneforthecaseα=1.Infact,itmeansthatEq.(5)isform-invariantunderthetransformations(6).FurtherwearegoingtoshowthatgeneralREpossesssimilarpropertyaswell.42.3TransformationgroupLetuscheckthatgeneralfraction-rationalchangeofvariablesˆφ=α(x)φ+β(x)γ(x)φ+δ(x)(7)transformsoneRiccatiequationintotheanotheronesimilartoExample2.1.Noticethat(7)constitutesgroupoftransformationsgeneratedby1φ,α(x)φ,φ+β(x),thusonlyactionsofgeneratorshavetobechecked:•ˆφ=1/φtransforms(1)intoˆφx+c(x)ˆφ2+b(x)ˆφ+a(x)=0,•ˆφ=α(x)φtransforms(1)intoˆφx−a(x)α(x)ˆφ2−[b(x)+(logα(x))x]ˆφ−α(x)c(x)=0,•ˆφ=φ+β(x)transforms(1)intoˆφx−a(x)ˆφ2+[2β(x)a(x)−b(x)]ˆφ−ˆc=0,whereˆc=a(x)β2(x)−b(x)β(x)+c(x)+β(x)x.Thus,havingonesolutionofasomeRiccatiequationwecangetimmedi-atelygeneralsolutionsofthewholefamilyofREsobtainedfromtheoriginaloneundertheactionoftransformationgroup(7).ItisinterestingtonoticethatforRiccatiequationknowinganythreesolutionsφ1,φ2,φ3wecanconstructallothersolutionsφusingaverysimpleformulacalledcross-ratio:φ