Singularity Analysis and Integrability of a Burgers-Type System of Foursov
aa r X i v : . [ n li n . S I] J a n Symmetry, Integrability and Geometry: Methods and Applications SIGMA (2011), 002, 5 pages Singularity Analysis and Integrabilityof a Burgers-Type System of Foursov
Sergei SAKOVICH †‡†
Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus
E-mail: [email protected] ‡ Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
Received October 28, 2010, in final form December 24, 2010; Published online January 04, 2011doi:10.3842/SIGMA.2011.002
Abstract.
We apply the Painlev´e test for integrability of partial differential equationsto a system of two coupled Burgers-type equations found by Foursov, which was recentlyshown by Sergyeyev to possess infinitely many commuting local generalized symmetrieswithout any recursion operator. The Painlev´e analysis easily detects that this is a typical C -integrable system in the Calogero sense and rediscovers its linearizing transformation. Key words: coupled Burgers-type equations; Painlev´e test for integrability
The system of two coupled Burgers-type equations w t = w xx + 8 ww x + (2 − α ) zz x ,z t = (1 − α ) z xx − αzw x + (4 − α ) wz x − (4 + 8 α ) w z − (2 − α ) z , (1)where α is a parameter, was discovered by Foursov [1] as a nonlinear system which possessesgeneralized symmetries of orders three through at least eight but apparently has no recursionoperator for a generic value of α . Foursov [1] noted that two systems equivalent to the cases α = 0 and α = 1 of (1) had already appeared in [2] and [3], respectively, and found a recursionoperator for the system (1) with α = 1 /
2. Very recently, Sergyeyev [4] proved that the system (1)does possess an infinite commutative algebra of local generalized symmetries but the existenceof a recursion operator – of a reasonably “standard” form – for a generic value of α is disallowedby the structure of symmetries. Sergyeyev [4] found that the algebra of generalized symmetriesof (1) is generated by a nonlocal two-term recursion relation rather than a recursion operator.In the present paper, we explore what the Painlev´e test for integrability, in its formulation forpartial differential equations [5, 6, 7], can tell about the integrability of this unusual system (1)with α = 1 /
2, which possesses infinitely many higher symmetries without any recursion operator.The Painlev´e test easily detects that this is a typical C -integrable system, in the terminology ofCalogero [8]. In Section 2, we show that the singularity analysis of the Burgers-type system (1)naturally suggests to introduce the new dependent variable s ( x, t ), s = z , (2)to improve the dominant behavior of solutions. The system (1) in the variables w and s passesthe Painlev´e test for integrability successfully: positions of resonances are integer in all branches,and there are no nontrivial compatibility conditions at the resonances. In Section 3, we show S. Sakovichthat the truncation of singular expansions straightforwardly produces the transformation w = φ x φ , s = a (4 − α ) φ (3)to the new dependent variables φ ( x, t ) and a ( x, t ) satisfying the triangular linear system a t = (1 − α ) a xx , φ t = φ xx + a . (4)This linearizing transformation was found in an inverse form in [9] and used in a form closeto (2), (3) in [4]. Section 4 contains concluding remarks. First of all, let us note that the cases α = 1 / α = 1 / α = 1 /
2, thesystem (1) is the triangular system w t = w xx + 8 ww x , z t = − zw x − w z, (5)where the first equation is the linearizable Burgers equation possessing the Painlev´e proper-ty [5], whereas the second equation simply defines a function z ( x, t ) by the relation z = f ( x ) exp R (cid:0) − w x − w (cid:1) dt , with f ( x ) being arbitrary, for any solution w ( x, t ) of the Burgersequation. Thus, integrability of this case is obvious.In the generic case of the Burgers-type system (1) with α = 1 /
2, we substitute into (1) theexpansions w = w ( t ) φ σ + · · · + w r ( t ) φ σ + r + · · · , z = z ( t ) φ τ + · · · + z r ( t ) φ τ + r + · · · , (6)where φ x ( x, t ) = 1, in order to determine the dominant behavior of solutions near a movable non-characteristic manifold φ ( x, t ) = 0 and the corresponding positions of resonances. In this way,we obtain the following four branches, omitting the ones corresponding to the Taylor expansionsgoverned by the Cauchy–Kovalevskaya theorem: σ = τ = − , w = 12 , z = ± r α − , r = − , − , ,
2; (7) σ = τ = − , w = 1 , z = ± r α − , r = − , − , − ,
2; (8) σ = − , τ = − , w = 14 , ∀ z ( t ) , r = − , , ,
2; (9) σ = − , τ = 12 , w = 14 , ∀ z ( t ) , r = − , − , , . (10)We see that the system (1) does not possess the Painlev´e property because of the non-integervalues of τ in the branches (9) and (10). Nevertheless, the positions of resonances are integerin all branches, and we can improve the dominant behavior of solutions by a simple power-typetransformation of the dependent variable z , just as we did for the Golubchik–Sokolov systemin [10]. We introduce the new dependent variable s given by (2), and this brings the Burgers-typesystem (1) into the form w t = w xx + 8 ww x + (1 − α ) s x , ingularity Analysis and Integrability of a Burgers-Type System of Foursov 3 ss t = (1 − α ) ss xx −
12 (1 − α ) s x − αs w x + (4 − α ) wss x − (8 + 16 α ) w s − (4 − α ) s . (11)This form is hardly simpler than the original one, but the studied system (1) in this form (11)will pass the Painlev´e test.We substitute into (11) the expansions w = w ( t ) φ σ + · · · + w r ( t ) φ σ + r + · · · , s = s ( t ) φ ρ + · · · + s r ( t ) φ ρ + r + · · · , (12)with φ x ( x, t ) = 1, and find the following four branches: σ = − , ρ = − , w = 12 , s = 14 α − , r = − , − , ,
2; (13) σ = − , ρ = − , w = 1 , s = 32 α − , r = − , − , − ,
2; (14) σ = ρ = − , w = 14 , ∀ s ( t ) , r = − , , ,
2; (15) σ = − , ρ = 1 , w = 14 , ∀ s ( t ) , r = − , − , , . (16)Now the exponents of the dominant behavior of solutions, as well as the positions of resonances,are integer in all branches.The next step of the Painlev´e analysis is to derive from (11) and (12) the recursion relationsfor the coefficients w n and s n ( n = 0 , , , . . . ) and then to check the compatibility conditionsarising at the resonances. Omitting tedious computational details of this, we give here only theresult. The compatibility conditions turn out to be satisfied identically at the resonances ofall branches (13)–(16), hence there is no need to introduce logarithmic terms into the expan-sions (12) representing solutions of the system (11). The function ψ ( t ) in φ = x + ψ ( t ) remainsarbitrary in all branches. Also the following functions remain arbitrary: s ( t ), and either s ( t )if α = 1 or w ( t ) if α = 1, in the branch (13); either s ( t ) if α = 3 / w ( t ) if α = 3 /
2, in thebranch (14); s ( t ), s ( t ) and w ( t ) in the branch (15); and s ( t ) and w ( t ) in the branch (16).The generic branch is (15): the expansions (12) contain four arbitrary functions of one variablein this case, thus representing the general solution of the system (11).Consequently, the Burgers-type system (1) in its equivalent form (11) has passed the Painlev´etest for integrability. There is a strong empirical evidence that any nonlinear differential equation which passed thePainlev´e test must be integrable. The test itself, however, does not tell whether the equationis C -integrable (solvable by quadratures or exactly linearizable) or S -integrable (solvable by aninverse scattering transform technique). Often some additional information on integrability ofthe studied equation, such as its linearizing transformation, Lax pair, B¨acklund transformation,etc., can be obtained by truncation of the Laurent-type expansion representing the equation’sgeneral solution [5, 11, 12, 13, 14, 15, 16, 17].Let us apply the truncation technique to the system (11). We make the truncation in thegeneric branch (15) which corresponds to the general solution. In what follows, the simplifyingreduction φ = x + ψ ( t ), w n = w n ( t ) and s n = s n ( t ) ( n = 0 , , . . . ) is not used. We substitutethe truncated expansions w = w ( x, t ) φ ( x, t ) + w ( x, t ) , s = s ( x, t ) φ ( x, t ) + s ( x, t ) (17) S. Sakovichto the coupled equations (11), equate to zero the sums of terms with equal degrees of φ , and inthis way obtain the definitions w = φ x , s = φ t − φ xx − w φ x − α (18)for the coefficients w and s , as well as a system of four nonlinear partial differential equationsfor three functions, w ( x, t ), s ( x, t ) and φ ( x, t ). Two of the four equations of that system are thesame initial equations (11) with w and s replaced by w and s , respectively, which means thatthe obtained system and the relations (17) and (18) constitute a so-called Painlev´e–B¨acklundtransformation relating a solution ( w , s ) of (11) with a solution ( w, s ) of (11). The other twoequations of the obtained system are fourth-order polynomial partial differential equations – letus simply denote them as E and E because it is easy to obtain them by computer algebratools but not so easy to put them onto a printed page – they involve the functions w , s and φ and contain, respectively, 55 and 110 terms.Fortunately, there is no need to study the obtained complicated system of four nonlinear equa-tions for compatibility in its full form. Instead, let us see what will happen if we take w = 0and s = 0, which means that we apply the obtained Painlev´e–B¨acklund transformation to thetrivial zero solution of the system (11). The reason to do so consists in the following empiricallyobserved difference between C -integrable equations and S -integrable equations, which, as far aswe know, has never been formulated explicitly in the literature. A Painlev´e–B¨acklund trans-formation of a C -integrable equation, being applied to a single trivial solution of the equation,produces the whole general solution of the equation at once. Examples of this are the Burgersequation [5] and the Liouville equation in its polynomial form uu xy = u x u y + u [12]. On thecontrary, numerous examples in the literature show that a Painlev´e–B¨acklund transformation ofan S -integrable equation, being applied to a single trivial solution of the equation, produces onlya class of special solutions of the equation, usually a rational solution or a one-soliton solutionwith some arbitrary parameters (see, e.g., [14, 15, 17, 18]).Taking w = 0 and s = 0, we find that the equation we denoted as E is satisfied identically,whereas the equation we denoted as E is reduced to( φ t − φ xx )( φ t − φ xx ) t + 12 (1 − α )( φ t − φ xx ) x − (1 − α )( φ t − φ xx )( φ t − φ xx ) xx = 0 . (19)The general solution of this fourth-order equation contains four arbitrary functions of one vari-able, which is exactly the degree of arbitrariness of the general solution of the system (11). Forthis reason, we conclude that the system (11) must be C -integrable. Now it only remains tonotice that, if we introduce the new dependent variable a ( x, t ) such that φ t − φ xx = a , (20)the equation (19) becomes linear: a t = (1 − α ) a xx . (21)Finally, combining the relations (17), (18), (20), (21) and w = s = 0, we obtain the exactlinearization (3) and (4) for the system (11). In the present paper, we used the Painlev´e test for integrability of partial differential equations tostudy the integrability of a system of two coupled Burgers-type equations discovered by Foursov,which possesses an unusual algebra of generalized symmetries as was shown by Sergyeyev. Theingularity Analysis and Integrability of a Burgers-Type System of Foursov 5Painlev´e analysis easily detected that the studied Burgers-type system is a typical C -integrablesystem in the Calogero sense and rediscovered its linearizing transformation. As a byproduct, weobtained a new example confirming the empirically observed difference between the Painlev´e–B¨acklund transformations of C -integrable equations and S -integrable equations. In our opinion,the Painlev´e test deserves to be used more widely to search for new integrable nonlinear equa-tions, because with its help one can discover new equations possessing such new properties whichlook unusual from the point of view of other integrability tests. Acknowledgements
This work was partially supported by the BRFFR grant Φ10-117. The author also thanks theMax Planck Institute for Mathematics for hospitality and support.
References [1] Foursov M.V., On integrable coupled Burgers-type equations,
Phys. Lett. A (2000), 57–64.[2] Svinolupov S.I., On the analogues of the Burgers equation,
Phys. Lett. A (1989), 32–36.[3] Olver P.J., Sokolov V.V., Integrable evolution equations on associative algebras,
Comm. Math. Phys. (1998), 245–268.[4] Sergyeyev A., Infinitely many local higher symmetries without recursion operator or master symmetry: inte-grability of the Foursov–Burgers system revisited,
Acta Appl. Math. (2010), 273–281, arXiv:0804.2020.[5] Weiss J., Tabor M., Carnevale G., The Painlev´e property for partial differential equations,
J. Math. Phys. (1983), 522–526.[6] Tabor M., Chaos and integrability in nonlinear dynamics. An introduction, John Wiley & Sons, Inc., NewYork, 1989.[7] Ramani A., Grammaticos B., Bountis T., The Painlev´e property and singularity analysis of integrable andnon-integrable systems, Phys. Rep. (1989), 159–245.[8] Calogero F., Why are certain nonlinear PDEs both widely applicable and integrable?, in What is Integra-bility?, Editor V.E. Zakharov,
Springer Ser. Nonlinear Dynam. , Springer, Berlin, 1991, 1–62.[9] Tsuchida T., Wolf T., Classification of polynomial integrable systems of mixed scalar and vector evolutionequations. I,
J. Phys. A: Math. Gen. (2005), 7691–7733, nlin.SI/0412003.[10] Sakovich S.Yu., A system of four ODEs: the singularity analysis, J. Nonlinear Math. Phys. (2001), 217–219, nlin.SI/0003039.[11] Weiss J., The Painlev´e property for partial differential equations. II. B¨acklund transformations, Lax pairs,and the Schwarzian derivative, J. Math. Phys. (1983), 1405–1413.[12] Steeb W.-H., Kloke M., Spieker B.M., Liouville equation, Painlev´e property and B¨acklund transformation, Z. Naturforsch. A (1983), 1054–1055.[13] Musette M., Conte R., Algorithmic method for deriving Lax pairs from the invariant Painlev´e analysis ofnonlinear partial differential equations, J. Math. Phys. (1991), 1450–1457.[14] Karasu-Kalkanlı A., Sakovich S.Yu., B¨acklund transformation and special solutions for the Drinfeld–Sokolov–Satsuma–Hirota system of coupled equations, J. Phys. A: Math. Gen. (2001), 7355–7358,nlin.SI/0102001.[15] Karasu-Kalkanlı A., Karasu A., Sakovich A., Sakovich S., Turhan R., A new integrable generalization of theKorteweg–de Vries equation, J. Math. Phys. (2008), 073516, 10 pages, arXiv:0708.3247.[16] Conte R., Musette M., The Painlev´e handbook, Springer, Dordrecht, 2008.[17] Hone A.N.W., Painlev´e tests, singularity structure and integrability, in Integrability, Editor A.V. Mikhailov, Lecture Notes in Physics , Vol. 767, Springer, Berlin, 2009, 245–277, nlin.SI/0502017.[18] Est´evez P.G., Prada J., Lump solutions for PDE’s: algorithmic construction and classification,
J. Nonlinear Math. Phys.15