Singularity Analysis and Integrability of a Burgers-Type System of Foursov

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 7 (2011), 002, 5 pages Singularity Analysis and In te grabilit y of a Burgers-T yp e System of F ourso v Ser gei SAKO VICH †‡ † Institute of Physics, National A c ademy o f Scienc es, 220 072 Minsk, Belarus E-mail: saks@tut.by ‡ Max Planck Institute for Mathematics, Vivatsgasse 7, 53 111 Bonn, Germany Received Octob er 28, 2010, in f inal form December 24, 2 010; Published online J anuary 04, 20 1 1 doi:10.38 42/SIGMA.20 11.002 Abstract. W e apply the Painlev´ e test for integrability of partial dif ferential equations to a system of tw o coupled Burgers-type equations found by F oursov, whic h w as recently shown by Sergyey ev to p oss ess inf initely many co mmuting lo ca l g e ne r alized sy mmetr ies without an y recur sion op erator . The Painlev ´ e analysis easily detects that this is a typical C -in tegrable system in the Calog ero sense and rediscovers its linearizing transforma tio n. Key wor ds: co upled Bur g ers-type equations; Painlev ´ e test for integrability 2010 Mathematics Subje ct Classific ation: 35K55 ; 37K10 1 In tro duction The system of tw o coupled Burgers-t yp e equations w t = w xx + 8 ww x + (2 − 4 α ) z z x , z t = (1 − 2 α ) z xx − 4 αz w x + (4 − 8 α ) wz x − (4 + 8 α ) w 2 z − (2 − 4 α ) z 3 , (1) where α is a parameter, was discov ered by F ourso v [1] as a n on lin ear system w h ic h p o ssesses generalized symmetries of orders three through at least eight bu t apparently has no recursion op erator for a generic v alue of α . F o urso v [1] noted th at tw o sys tems equiv alent to the cases α = 0 and α = 1 of (1 ) had already app eared in [2] and [3], r esp ectiv ely , an d f ou n d a recur sion op erator for th e system (1) with α = 1 / 2. V ery recent ly , Sergye y ev [4] p ro v ed that the system (1) do es p o ssess an inf inite comm utativ e algebra of lo cal generalized symmetries b u t the existence of a recursion op erato r – of a reasonably “standard” form – f or a generic v alue of α is disallo w ed b y th e structur e of s ymmetries. Sergye y ev [4] foun d that th e algebra of generalized symm etries of (1) is generated b y a nonlo c al t w o-t erm r ecursion relation rather than a recur sion op erato r. In the present pap er, we explore what the Painlev ´ e test for in tegrabilit y , in its formulat ion for partial dif feren tial equations [5, 6, 7], can tell ab out the integrabilit y of this un usual system (1) with α 6 = 1 / 2, w hic h p o ssesses inf initely many higher symmetries without any recursion op erator. The P ainlev ´ e test easily detects that this is a t ypical C -inte grable sys tem, in the terminology of Calogero [8]. In Section 2, we sh o w that the singularit y analysis of the Burgers-type system (1) naturally suggests to in tro duce the new dep endent v ariable s ( x, t ), s = z 2 , (2) to im p ro v e the d ominan t b ehavio r of solutions. Th e system (1 ) in the v ariables w and s passes the P ainlev ´ e test for in tegrabilit y successfully: p ositions of r esonances are integ er in all branches, and ther e are no nontrivial compatibilit y conditions at the resonances. In Section 3, we show 2 S. Sak o v ich that the truncation of sin gular exp ansions straigh tforw ardly pro d uces the transformation w = φ x 4 φ , s = a 2 (4 − 8 α ) φ (3) to th e new dep end ent v ariables φ ( x, t ) and a ( x, t ) satisfying the tr iangular linear system a t = (1 − 2 α ) a xx , φ t = φ xx + a 2 . (4) This linearizing transformation was found in an in v erse form in [9] and used in a form close to (2 ), (3) in [4]. Section 4 contai ns concluding r emarks. 2 Singularit y analysis First of all, let us note that the cases α = 1 / 2 and α 6 = 1 / 2 of the system (1) are essenti ally dif ferent, at least b ecause the total ord er of the s y s tem’s equations is dif feren t in these cases, and the general s olution of th is t w o-dimensional system conta ins dif ferent num b ers of arbitrary functions of one v ariable in these cases, three and four , resp ec tiv ely . When α = 1 / 2, the system (1) is the triangular sys tem w t = w xx + 8 ww x , z t = − 2 z w x − 8 w 2 z , (5) where the f irst equ ation is the linearizable Burgers equ ation p o ssessing th e P ainlev ´ e prop er- t y [5], w hereas t he second equation simply def ines a fun ction z ( x, t ) b y the relation z = f ( x ) exp R  − 2 w x − 8 w 2  dt , w ith f ( x ) b e ing arbitrary , f or any solution w ( x, t ) of the Burgers equation. Thus, in tegrabilit y of this case is obvious. In th e generic case of the Burgers-t yp e system (1) with α 6 = 1 / 2, we subs titute into (1) the expansions w = w 0 ( t ) φ σ + · · · + w r ( t ) φ σ + r + · · · , z = z 0 ( t ) φ τ + · · · + z r ( t ) φ τ + r + · · · , (6) where φ x ( x, t ) = 1, in order to determine the dominant b eha vior of solutions near a mo v able non- c haracteristic manifold φ ( x, t ) = 0 and th e corresp o nding p ositions of resonances. In this wa y , w e obtain the follo wing four b r anc hes, omitting the ones corresp ond ing to the T aylo r exp ansions go v ern ed by the Cauch y–Ko v ale vsk ay a theorem: σ = τ = − 1 , w 0 = 1 2 , z 0 = ± r 1 4 α − 2 , r = − 2 , − 1 , 1 , 2; (7) σ = τ = − 1 , w 0 = 1 , z 0 = ± r 3 2 α − 1 , r = − 4 , − 3 , − 1 , 2; (8) σ = − 1 , τ = − 1 2 , w 0 = 1 4 , ∀ z 0 ( t ) , r = − 1 , 0 , 1 , 2; (9) σ = − 1 , τ = 1 2 , w 0 = 1 4 , ∀ z 0 ( t ) , r = − 1 , − 1 , 0 , 2 . (10) W e see that the system (1) do es n ot p ossess the Painlev ´ e prop ert y b ecause of the n on-in teger v alues of τ in the branches (9 ) and (10). Nev er th eless, the p osit ions of resonances are intege r in all b ranc hes, and we can improv e th e dominant b ehavi or of solutions b y a simple p o wer-t ype transformation of the dep e ndent v ariable z , j u st as w e d id for the Golub c hik–Sok olo v system in [10]. W e in tro duce the new dep e ndent v ariable s giv en b y (2), and this b rings the Burgers-t yp e system (1) in to the form w t = w xx + 8 ww x + (1 − 2 α ) s x , Singularit y Analysis and Integrabilit y of a Burgers-Type System of F ourso v 3 ss t = (1 − 2 α ) ss xx − 1 2 (1 − 2 α ) s 2 x − 8 αs 2 w x + (4 − 8 α ) wss x − (8 + 16 α ) w 2 s 2 − (4 − 8 α ) s 3 . (11) This form is hard ly simpler than the original one, b u t the studied sy s tem (1) in this form (11) will pass the Painlev ´ e test. W e sub stitute into (11) the expansions w = w 0 ( t ) φ σ + · · · + w r ( t ) φ σ + r + · · · , s = s 0 ( t ) φ ρ + · · · + s r ( t ) φ ρ + r + · · · , (12) with φ x ( x, t ) = 1, and f ind the follo wing four branches: σ = − 1 , ρ = − 2 , w 0 = 1 2 , s 0 = 1 4 α − 2 , r = − 2 , − 1 , 1 , 2; (13) σ = − 1 , ρ = − 2 , w 0 = 1 , s 0 = 3 2 α − 1 , r = − 4 , − 3 , − 1 , 2; (14) σ = ρ = − 1 , w 0 = 1 4 , ∀ s 0 ( t ) , r = − 1 , 0 , 1 , 2; (15) σ = − 1 , ρ = 1 , w 0 = 1 4 , ∀ s 0 ( t ) , r = − 1 , − 1 , 0 , 2 . (16) No w the exp onen ts of the d ominan t b e ha vior of solutions, as w ell as the p ositions of resonances, are in teger in all br anc hes. The next s tep of the P ainlev ´ e analysis is to derive from (11) and (12) the recurs ion r elations for the co ef f icien ts w n and s n ( n = 0 , 1 , 2 , . . . ) and then to c hec k the compatibilit y conditions arising at the resonances. Omitting tedious computational details of this, we giv e here only the result. The compatibilit y conditions tur n out to b e satisf ied identica lly at the resonances of all br anc hes (13)–(16), hence there is n o need to introdu ce logarithmic term s int o th e expan- sions (12) representing solutions of th e system (11). T he fun ction ψ ( t ) in φ = x + ψ ( t ) remains arbitrary in all b ranc hes. Also th e follo wing f unctions r emain arbitrary: s 1 ( t ), and either s 2 ( t ) if α = 1 or w 2 ( t ) if α 6 = 1, in the b ranc h (13); either s 2 ( t ) if α = 3 / 2 or w 2 ( t ) if α 6 = 3 / 2, in th e branc h (14); s 0 ( t ), s 1 ( t ) and w 2 ( t ) in the br anc h (15); and s 0 ( t ) and w 2 ( t ) in the br anc h (16). The generic branch is (15): the expansions (12) con tain four arbitrary fun ctions of one v ariable in this case, thus rep r esen ting the general solution of the system (11). Consequent ly , the Burgers-t yp e system (1 ) in its equiv al en t form (11) has passed the Pa inlev ´ e test f or in tegrabilit y . 3 T runcation tec hnique There is a strong emp ir ical evidence that an y nonlinear dif f eren tial equ ation wh ic h p assed the P ainlev ´ e test m ust b e in tegrable. Th e test itself, how ever, do e s not tell w h ether the equation is C -in tegrable (solv able b y quadratures or exactly linearizable) or S -in tegrable (solv able by an in v erse scattering transform tec hnique). Often some additional in formation on in tegrabilit y of the studied equation, s uc h as its lin earizing transformation, Lax p air, B¨ ac klund transformation, etc., can b e obtained by trun cation of the Laurent-t yp e expansion r epresent ing the equation’s general s olution [5 , 11, 12, 13, 14, 15, 16, 17 ]. Let u s apply the truncation tec hnique to the system (11) . W e make th e trun cation in the generic branch (15) whic h corresp onds to the general solution. In w h at follo ws, the simp lifying reduction φ = x + ψ ( t ), w n = w n ( t ) and s n = s n ( t ) ( n = 0 , 1 , . . . ) is not used. W e su bstitute the truncated expansions w = w 0 ( x, t ) φ ( x, t ) + w 1 ( x, t ) , s = s 0 ( x, t ) φ ( x, t ) + s 1 ( x, t ) (17) 4 S. Sak o v ich to the coupled equations (11), equate to zero the su m s of terms with equal degrees of φ , and in this wa y obtain the def initions w 0 = φ x 4 , s 0 = φ t − φ xx − 8 w 1 φ x 4 − 8 α (18) for the co ef f icients w 0 and s 0 , as we ll as a system of four nonlinear partial dif ferentia l equations for three functions, w 1 ( x, t ), s 1 ( x, t ) and φ ( x, t ). T w o of the four equations of that s ystem are the same initial equations (11) with w and s r eplaced by w 1 and s 1 , resp e ctiv ely , whic h means that the obtained sy s tem and the relations (17) and (18) constitute a so-called Pa inlev ´ e–B¨ ac klund transformation r elating a solution ( w 1 , s 1 ) of (11) with a solution ( w , s ) of (11). The other t w o equations of the obtained system are fourth-order p olynomia l partial dif f er ential equations – let us simply denote them as E 1 and E 2 b ecause it is easy to obtain them by computer algebra to ols but not so easy to put them on to a printe d page – they inv olve th e f unctions w 1 , s 1 and φ and conta in, resp ecti v ely , 55 and 110 terms. F ortunately , there is no need to study the obtained complicated system of four n onlinear equa- tions for compatibilit y in its full form. In stead, let u s see what will happ en if we tak e w 1 = 0 and s 1 = 0, whic h means that w e apply the obtained Pa inlev ´ e–B¨ ac klund transf orm ation to the trivial zero solution of the system (11). The reason to do so consists in the f ollo wing empirically observ ed dif ference b etw een C -in tegrable equations and S -in tegrable equations, whic h, as far as w e kno w, has never b ee n f orm ulated explicitly in the literature. A Pa inlev ´ e–B¨ ac klund trans- formation of a C -in tegrable equation, b eing app lied to a single tr ivial solution of the equation, pro du ces the whole general solution of the equation at once. Examp les of this are the Burgers equation [5] and the Liouville equation in its p ol ynomial form uu xy = u x u y + u 3 [12]. On the con trary , numerous examples in the literature sho w that a P ainlev ´ e–B¨ ac klund tr an s formation of an S -integ rable equation, b eing applied to a single trivial solution of the equation, pro d uces only a class of sp eci al solutions of th e equ ation, usu ally a r ational solution or a one-soliton solution with some arbitrary p arameters (see, e.g., [14, 15, 17, 18]). T aking w 1 = 0 and s 1 = 0, w e f ind that the equation we denoted as E 1 is satisf ied iden tically , whereas the equation we d enoted as E 2 is r educed to ( φ t − φ xx )( φ t − φ xx ) t + 1 2 (1 − 2 α )( φ t − φ xx ) 2 x − (1 − 2 α )( φ t − φ xx )( φ t − φ xx ) xx = 0 . (19) The general solution of this f ou r th-order equation cont ains four arbitrary functions of one v ari- able, whic h is exactly the degree of arbitrariness of the general solution of the system (11). F or this reason, we conclud e that the system (11) must b e C -in tegrable. No w it only remains to notice that, if we introdu ce the n ew dep enden t v ariable a ( x, t ) s u c h th at φ t − φ xx = a 2 , (20) the equation (19) b ecomes linear: a t = (1 − 2 α ) a xx . (21) Finally , com bining the relations (17), (18), (20), (21 ) and w 1 = s 1 = 0, we obtain the exact linearizatio n (3) and (4) f or the s y s tem (11 ). 4 Conclusion In the present pap er, w e used the Pa inlev ´ e test for int egrabilit y of partial dif ferential equations to study th e in tegrabilit y of a system of t w o coupled Burgers-t yp e equ ations disco v ered by F ourso v, whic h p ossesses an unusual algebra of generalized symm etries as w as s ho wn by Sergye y ev. The Singularit y Analysis and Integrabilit y of a Burgers-Type System of F ourso v 5 P ainlev ´ e analysis easily d etected that the stud ied Burgers-t yp e s ystem is a typica l C -inte grable system in the Calogero sense and redisco v ered its linearizing transformation. As a bypro duct, we obtained a n ew example conf irming th e emp irically observ ed dif fer en ce b etw een the Pa inlev ´ e– B¨ ac klund transformations of C -integ rable equations and S -in tegrable equations. In our opinion, the Pa inlev ´ e test deserves to b e used more widely to searc h for new int egrable n onlinear equa- tions, b ecause with its h elp one can disco v er new equations p ossessing su c h new prop ertie s whic h lo ok unusual from th e p oin t of view of other integrabilit y tests. Ac kno wledgemen ts This w ork w as p artially supp orted b y the BRFFR grant Φ 10-117 . The author also thanks the Max Planc k In stitute for Mathematics for hospitalit y and supp o rt. 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