A BI-HAMILTONIAN INTEGRABLE TWO COMPONENT GENERALIZATION OF THE THIRD-ORDER BURGERS EQUATION ˙ TURHAN DARYOUSH TALATI AND REFIK Dedicated to Professor Yavuz Nutku who passed away on December 8, 2010.
arXiv:1210.6083v1 [nlin.SI] 22 Oct 2012
Abstract. We announce a new bi-Hamiltonian integrable two-component system admitting the scalar 3rd-order Burgers equation as a reduction.
1. Introduction Bi-Hamiltonian evolutionary equations are those which can be embedded in a Magri scheme uti = Fi [u] = KGi [u] = JGi+1 [u], Gi = E(hi [u]), i = −1, 0, 1, 2, 3, · · · constructed by two compatible Hamiltonian (skew-adjoint, Jacobi identity satisfying) operators J and K. Hamiltonian operators are defined to be compatible if their arbitrary linear combinations are Hamiltonian operators also. Here the differential functions Fi [u] are the (characteristics of) symmetries and the dual objects Gi are the conserved gradients which are Euler derivatives E(hi ) of the conserved densities hi . Very recently [1], it is rigorously proved that for a given two compatible Hamiltonian operators, construction of partial Magri scheme (Fi , Gi , hi ), i = −1, 0, 1, 2, · · · , N for some finite N implies existence of remaining infinitely many of them, and therefore integrability of the equations Fi in the sense of existence of infinitely many conservation laws. For a given pair of compatible Hamiltonian operators, construction of the Magri scheme is quite a straight forward task to do. The inverse problem of finding an appropriate compatible pair of Hamiltonian operators in whose Magri scheme a given system exists, is a nontrivial one. The Ablowitz-Kaup-Newell-Segur (AKNS)[2] system
ut vt
=
−uxx + 2u2v vxx − 2v 2 u
,
can be considered as a 2-component generalization of the scalar Heat equation (HE) in (1+1) dimensions, since u = 0 reduction of the system leaves the scalar HE vt = vxx behind. The second order HE cannot be written as a bi-Hamiltonian equation but the odd-order n members ut = u(2n+1)x , n = 0, 1, 2, 3, · · · , where unx = ∂∂xnu , in the (autonomous) symmetry hierarchy ut = unx , of the HE are bi-Hamiltonian equations as noted recently in [3] and given in a related variables in which the first member (x-translation symmetry of the HE) appears in the Riemann equation form in [4] (see Proposition 3.3 in [5]). Motivated by the observation that the second-order scalar HE, despite not being a biHamiltonian equation, has a bi-Hamiltonian 2-component generalization, i.e. the AKNS system, and that the odd-order HE’s are bi-Hamiltonian, we considered a class of third-order Burgers type two-component systems for integrability. We shall give the results of our classification elsewhere [6]. But here, starting from an example system which is related to a known integrable system, we introduce a new completely integrable 3rd-oder Burgers type system whose bi-Hamiltonian structure we constructed too. 1
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D. TALATI AND R. TURHAN
2. An example related to a known system Let us consider the following system ut uxxx + 3uxx u + 3u2x + 3u2 ux + vxx + 2ux v = , vt vxxx − 3vxx u + 6uxx v + 3ux vx + 3u2vx + 2vvx
(1)
which reduces to the scalar 3rd-order Burgers equation in v = 0. In terms of the new variables (w, z) which are defined as 3 z = v + u2 2 the above system becomes wt wxxx + zxxx + 2wx z + 2wzx = . zt zxxx − 9wwx + 6zwx + 3wzx + 2zzx w = ux ,
(2)
The transformed system (up to a linear change of variables) is the Karasu (Kalkanlı) equation which is obtained in Painlav´e classification [7]. A recursion operator is given for the system in [8] and its bi-Hamiltonian structure is constructed by Sergyeyev in [9]. 3. The new system The new Burgers type 2-component system we introduce here is ut uxxx + 3uxx u + 3u2x + 3ux u2 + vxx + ux v + uvx = . vt vxxx − 3vxx u + 3uxx v + 6uux v + 3u2vx + 4vvx
(3)
This system reduces to the 3rd-order Burgers equation by v = 0 also. It can be written as a bi-Hamiltonian system as given in the following proposition. Proposition 1. The system (3) is bi-Hamiltonian
ut vt
=K
δu δv
Z
v dx = J 2
δu δv
Z 1 2 1 2 vux + u v + v dx 2 3
with the compatible pair of Hamiltonian operators K1 K2 D 2 + 2Du − 31 D , K= , J= K3 K4 −D 2 + 2uD 4vD + 2vx where K1 = −D 3 + u2 D + uux − ux D −1 ux , K2 = D 4 + 4uD 3 + (9ux + 3v + 5u2 )D 2 + (7uxx + 5vx + 16uux + 3uv + 2u3 )D +(2uxxx + 6uxx u + 6u2x + 6u2 ux + 2vxx + 3ux v + 2uvx ) − ux D −1 vx , K3 = −K∗2 , K4 = 6vD 3 + 9vx D 2 + (7vxx + 12ux v − 12vx u + 12u2v + 9v 2 )D +(2vxxx − 6vxx u + 6uxx v + 12uux v + 6u2vx + 9vvx ) − vx D −1 vx , and D n =
dn , dxn
(∗) denotes the formal adjoint.
A proof follows from direct (and tedious) calculation that for arbitrary functions ξ(x), η(x) and linear combinations Hλ = K + λJ with constant λ, the functional trivector Z ξ (ξ η) ∧ Hλ dx Ψ = prV ξ η Hλ η
vanishes independently from the value of λ, where prVQ is the prolongation of the vector field with characteristic Q[u] [10].
A BI-HAMILTONIAN INTEGRABLE TWO COMPONENT GENERALIZATION OF ...
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The first few conserved densities of the hierarchy which are obtained first by the software in [11], are listed below. h−1 = u h0 = v2 h1 = vux + 21 u2 v + 13 v 2 h2 = −2uxx vx + vu2x − 6uuxvx + 3v 2 ux + 8u2 vux − 34 vx2 + u4 v + 2u2 v 2 + 79 v 3 .. . These densities suffice to write a partial Magri scheme proving integrability of the system (3). 4. Discussion Even though both of the systems (1) and (3) reduce to the same 3rd-order Burgers equation by v = 0, the Hopf-Cole transform u → ux /u of the system (1) is no more a local evolutionary equation. With the same transformation, system (3) transforms to a non-polynomial system having v = 0 reduction to the scalar 3rd-order HE. Most of the classified third order systems are generalizations of the KdV equation or equations related to it [12, 13, 14, 15] and the references therein. System (3) cannot be transformed to a form having a reduction to KdV or to a related equation without some additional constraint by differential substitutions. In the scalar case, the Hopf-Cole transformation relating the Heat and the Burgers equations transforms the conserved gradients of the odd-oder HE’s into some nonlocal objects. But in the 2-component case, Burgers forms of the systems seems more suitable to be written in Hamiltonian form. System generalizations of the Burgers equation are classified mainly by generalizing the 2ndorder scalar equation [16, 17, 18, 19, 20, 21] the references therein. They are expected and turn out to possess symmetries at all orders like their scalar version. However in the system we gave here, similar to the other known Magri schemes, the integrable equations are only in odd orders and there are no (local) symmetries with even order. The odd-order scalar Heat hierarchy seems an exception to this general rule: The even-order Heat equations persist to be the symmetries of the odd-order ones, even if the only place we can put them is the position of conserved gradients in the Magri scheme. If a multi-component system is to be named according to the scalar equation it reduces by some field reductions, the Karasu (Kalkanlı) system is not of KdV type since in its original form (2), it does not reduce to KdV. However the related system (1) we gave here is genuinely Burgers type because it has a reduction to the 3rd-order scalar Burgers equation without any extra constraint left. Acknowledgements D.T. is supported by TUBITAK PhD Fellowship for Foreign Citizens. References [1] A. de Sole and V.G. Kac, Non-local Hamiltonian Structures and applications to the theory of integrable systems I, arXiv:1210.1688v1. [2] A.C.Newell, Solitons in mathematics and physics, Philadelphia: Society for Industrial and Applied Mathematics, (1985). [3] A. de Sole, V.G. Kac and M. Wakimoto, On classification of Poisson vertex algebras, Transform. Groups 14 (2010), 883-907., arXiv:1004.5387v2. [4] P.J. Olver and Y. Nutku, Hamiltonian structures for systems of hyperbolic conservation laws, J. Math. Phys. 29 (1988), 1610-1619. [5] D. Talati and R. Turhan, On a Recently Introduced Fifth Order bi-Hamiltonian Equation and Trivially Related Hamiltonian Operators, SIGMA 7 (2011), 081, 8 pages. [6] D. Talati and R. Turhan, A class of 2-component 3rd-order Burgers equations, in preperation.
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[7] A. Karasu (Kalkanlı), Painlev´e classisfication of coupled Kortaweg-de Vries systems, J. Math. Phys. 38 (1997), 3616-3622. [8] A. Karasu (Kalkanlı), A. Karasu and S. Yu Sakovich, A strange recursion operator for a new integrable system of coupled Korteweg-de Vries equations, Acta. Appl. Math. 83 (2004) 85-94. [9] A. Sergyeyev, A strange recursion operator demystified, J. Phys. A:Math. Gen. 38 (2005), L257-L262. [10] P.J. Olver, Applications of Lie Groups to Differantial Equations, Second Ed., Graduate Texts in Mathematics, Vol. 107, Springer-Verlag, New York, (1993). [11] D. Poole and W. Hereman, Symbolic computation of conservation laws for nonlinear partial differential equations in multiple space dimensions, J. Symbolic Comput. 46 (2011), 1355-1377. [12] M. Antonowicz and A.P. Fordy, Coupled KdV equations with multi-Hamiltonian structures, Physica D 28 (1987), 345-357. [13] S.I. Svinolupov, Jordan algebras generalized Korteweg-de Vries equations, Teor. Mat. Fiz. 87 (1991), 391403. [14] W. X. Ma and B. Fuchssteiner, The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy, Phys. Lett. A 213 (1996), 49-55. [15] M. G¨ urses and A. Karasu, Coupled KdV systems, J. Math. Phys. 39 (1998), 2103-2111. [16] A.V. Mikhailov, A.V. Shabat and V.V. Sokolov, The symmetry approach to classification of integrable equations, in: V.E. Zakharov, ed., What is Integrability?, Springer Series in Non-Linear Dynamics, SpringerVerlag, New York, (1991), 115-184. [17] S.I. Svinolupov, On the analogues of the Burgers equatin, Phys. Lett. A 135 (1989), 32-36. [18] P.J. Olver and V. Sokolov, Integrable evolution equations on associative algebras, Commun. Math. Phys. (1998), 245-268. [19] M. Foursov, On integrable coupled Burgers-type equations, Phys. Lett. A 272 (2000), 57-64. [20] J.A. Sanders and J. P. Wang, On the integrability of systems of second order evolution equations with two components, J. Diff. Eqns. 203 (2004) 1-27. [21] T. Tsuchida and T. Wolf, Classification of polynomial integrable systems of mixed scalar and vector evolution equations:I, J. phys. A:Math. Gen. 38 (2005), 7691-7733. ˘ an, Ankara Department of Engineering Physics, Ankara University 06100 Tandog E-mail address:
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