Using the algebraic method of Gardner's deformations for completely
integrable systems, we construct the recurrence relations for densities of the
Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending
the Magri schemes for these systems, we obtain new integrable equations adjoint
with respect to the initial ones and describe their Hamiltonian structures and
symmetry properties.Comment: Proc. Workshop `Quantizaion, Dualities, and Integrable Systems,'
  23-27 January 2006, Pamukkale University, Turkey; 7 pages. MSC 35Q53, 37K05,
  37K10, 37K3

Karasu, Atalay

Kiselev, Arthemy V.

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Gardner's deformations of the Boussinesq equations

Kiselev, Arthemy V

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Kiselev, A.

Karasu, A.

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Using the algebraic method of Gardner's deformations for completely integrable systems, we construct recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magri schemes for these equations, we obtain new integrable systems adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry properties

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arXiv:nlin/0603029v2  [nlin.SI]  8 Apr 2006ISPUmath-1/2006GARDNER’S DEFORMATIONS OF THE BOUSSINESQ EQUATIONSATALAY KARASU AND ARTHEMYV.KISELEVAbstract. Using the algebraic method of Gardner’s deformations for completelyintegrable systems, we construct the recurrence relations for densities of the Hamilto-nians for the Boussinesq and the Kaup-Boussinesq equations. By extending the Magrischemes for these equations, we obtain new integrable systems adjoint with respect tothe initial ones and describe their Hamiltonian structures and symmetry properties.Introduction. In this paper we consider the most efficient way to prove the completeintegrability of evolutionary systems. Namely, we apply the method of Gardner’s de-formations [1]–[6] to the Boussinesq and Kaup-Boussinesq equations. By construction,the deformations consist of the (multi-)Hamiltonian scaling non-invariant parametricextensions Eε of the original systems E0 and the parameter-dependent Miura transfor-mations Eε → E0. Inverting the Miura transformations from the new systems, oneobtains the recurrence relations for the infinite sequences of densities of the Hamil-tonian functionals. We improve the result of Kupershmidt in [3] by showing that thesequences of conserved densities for the Boussinesq equation satisfy two different re-currence relations simultaneously. We demonstrate that the extended equations mustnot necessarily interpolate (as it is assumed in [7]) between the equations E0 and themodified systems that provide the canonical factorizations for the higher Poisson struc-tures [8]. The recurrence relations we obtain are inherited by the modified Boussinesq [9]and Kaup–Boussinesq [10] equations.Next, we separate the flows at the deformation parameters and determine the adjointBoussinesq equations whose Magri schemes are coupled with the non-extended ones. Weshow that the new systems are also integrable.The examples discussed in this paper are essentially used in the general algebraicapproach to the problem of Gardner’s deformations and integrable extensions of theMagri schemes. This approach is based on the notion of coverings over PDE [11] andtheir parametric families constructed using the Frölicher–Nijenhuis bracket [12, 13, 14].This will be the object of a subsequent publication.The paper is organized as follows. In section 1 we obtain two Gardner’s deformationsfor the Boussinesq equation using the dispersionless case [15] as the starting point.We specify the adjoint Boussinesq systems and discuss their integrability. In section 2we construct the deformation of the Kaup-Boussinesq equation and describe the bi-Hamiltonian adjoint system.Date: March 21, 2006.2000 Mathematics Subject Classification. 35Q53, 37K05, 37K10, 37K35. PACS 02.30.Ik.Key words and phrases. Gardner’s deformations, integrable extensions, Boussinesq equation, Kaup-Boussinesq equation.12 A.KARASU AND A.V.KISELEV1. The Boussinesq equationFirst we consider the Boussinesq equation with dispersion and dissipation, see [3],Ut = Vx + αUxx, Vt = Uxxx + UUx − αVxx, α ∈ R. (1)We note that for any α system (1) is transformed to the equation Utt =(Uxxx · (1 +α) +UUx)x, which is scaling-equivalent to the Boussinesq equation utt =(uxxx+ uux)xwhenever α 6= −1. Therefore for α 6= −1 equation (1) is reduced to the Boussinesqequationut = vx, vt = uxxx + uux. (2)In the sequel, we consider the problem of Gardner’s deformation for system (2).Next, we observe that a Gardner’s deformation for (2) (and for the Kaup–Boussinesqequation (7) as well) is obtained from the deformation of the dispersionless reduction byadding higher order terms to the equations Eε and to the Miura transformations Eε → E0.Indeed, the zero order terms in the conserved densities, which are obtained recursivelyby inverting the Miura substitutions, originate from the relations for the dispersionlesssystems. The deformation for the dispersionless Boussinesq equation has been describedin [15]; it corresponds to the case α = −1 in (1). Using this result, we construct twodeformations for (2) and get two distinct recurrence relations for both sequences of theconserved densities. Thus we improve the result of [3], where a recurrence relation foronly one sequence was obtained. The adjoint Boussinesq equations that appear in theextended systems are the by-products of our reasonings.Theorem 1. There are two deformations E±ε for the Boussinesq equation (2). Both ofthem are Hamiltonian w.r.t. the structure(0 DxDx 0)(3)and the functionals with the densitiesH±(ε) =16ũ3 +12ṽ2 + ε3 ·(16ṽ3 +12ũ2xṽ ±12ũxṽ2 ±16ũ3x),respectively.The deformed equation E+ε isũt = ṽx + ε3 ·(ũxũxx + ũxxṽ + ũxṽx + ṽṽx),ṽt = ũxxx + ũũx − ε3 ·(ũ2xx + ũxũxxx + ũxxxṽ + 2ũxxṽx + ũxṽxx + ṽ2x + ṽṽxx). (4)The Miura transformation from E+ε to (2) isu = ũ− 2εũx + 2ε2 · (ũxx + ṽx) + ε3 · (ũṽ + ũũx),v = ṽ − 2εṽx + 2ε2 · (ṽxx + ũxxx + ũũx) + ε3 ·(13ũ3 + ṽ2 + ũũxx + ṽṽx + ũxṽ)− 2ε4 · (ũxũxx + ũxṽx + ũxxṽ + ṽṽx) + ε6 ·(13ṽ3 +13ũ3x + ũ2xṽ + ũxṽ2).The deformed equation E−ε isũt = ṽx + ε3 ·(ũxũxx − ũxxṽ − ũxṽx + ṽṽx),ṽt = ũxxx + ũũx + ε3 ·(ũ2xx + ũxũxxx − ũxxxṽ − 2ũxxṽx − ũxṽxx + ṽ2x + ṽṽxx). (5)GARDNER’S DEFORMATIONS OF THE BOUSSINESQ EQUATIONS 3The Miura transformation from E−ε to (2) is given throughu = ũ+ 2εũx + 2ε2 · (ũxx − ṽx) + ε3 · (ũṽ − ũũx),v = ṽ + 2εṽx + 2ε2 · (ṽxx − ũxxx − ũũx) + ε3 ·(13ũ3 + ṽ2 + ũũxx − ṽṽx − ũxṽ)+ 2ε4 · (ũxũxx − ũxṽx − ũxxṽ + ṽṽx) + ε6 ·(13ṽ3 −13ũ3x + ũ2xṽ − ũxṽ2).Extended equations (4) and (5) consist of the original Boussinesq flows and the adjointflows at ε3, which will be further discussed in more detail. The Poisson structure (3)for the extended equations E±ε together with the Miura transformations E±ε → E0 in-duce [8] the Poisson structures Â1∓ε3Â2 for Boussinesq’s equation (2), here Â1 is giventhrough (3) andÂ2 =8D3x + uDx +Dx ◦ u 3vDx + vx3vDx + 2vxD5x + 5 · (uD3x +D3x ◦ u)−− 3 · (uxxDx +Dx ◦ uxx) + uDx ◦ uare its first and second structures, respectively (see [9] and references therein). TheHamiltonians for the extensions E±ε are inherited from the original functionals, whichare described in Proposition 1 below, by using the Miura substitutions.Proposition 1. The densities of Hamiltonian functionals for the Boussinesq equationcan be obtained using two different recurrence relations, which areũ0 = u, ṽ0 = v, ũ1 = ±2ux,ṽ1 = ±2vx, ũ2 = 2uxx ∓ 2vx, ṽ2 = 2vxx ∓ 2uxxx ∓ 2uux,ũn = ±2Dx(ũn−1)− 2D2x(ũn−2)∓ 2Dx(ṽn−2) +∑k+ℓ=n−3[−ũkṽℓ ∓ ũkDx(ũℓ)],n ≥ 3,ṽ3 = ±2Dx(ṽ2)∓ 2D3x(ũ1)− 2D2x(ṽ1)∓ [uDx(ũ1) + ũ1ux]−13u3 − v2 − uuxx ∓ uvx ∓ uxv,ṽn = ±2Dx(ṽn−1)∓ 2D3x(ũn−2)− 2D2x(ṽn−2)∓∑k+ℓ=n−22ũkDx(ũℓ)−∑k+ℓ+m=n−313ũkũℓũm+∑k+ℓ=n−3[−ṽkṽℓ − ũkD2x(ũℓ)∓ ũkDx(ṽℓ)∓Dx(ũk)ṽℓ]+∑k+ℓ=n−42 ·[±Dx(ũk)D2x(ũℓ) +Dx(ũk)Dx(ṽℓ) +D2x(ũk)ṽℓ ± ṽkDx(ṽℓ)],n = 4, 5,4 A.KARASU AND A.V.KISELEVṽn = ±2Dx(ṽn−1)∓ 2D3x(ũn−2)− 2D2x(ṽn−2)∓∑k+ℓ=n−22ũkDx(ũℓ)−∑k+ℓ+m=n−313ũkũℓũm+∑k+ℓ=n−3[−ṽkṽℓ − ũkD2x(ũℓ)∓ ũkDx(ṽℓ)∓Dx(ũk)ṽℓ]+∑k+ℓ=n−42 ·[±Dx(ũk)D2x(ũℓ) +Dx(ũk)Dx(ṽℓ) +D2x(ũk)ṽℓ ± ṽkDx(ṽℓ)]+∑k+ℓ+m=n−6[−13ṽkṽlṽm ∓13Dx(ũk)Dx(ũℓ)Dx(ũm)−Dx(ũk)Dx(ũℓ)ṽm +Dx(ũk)ṽℓṽm], n ≥ 6.The ambiguity of signs of the differential terms does not affect the nontrivial conserveddensities ũ3k and ṽ3k. The densities with subscripts 3k+1, 3k+2 are trivial for all k ≥ 0.The recurrence relations listed in Proposition 1 are also valid for the Hamiltonians ofthe modified Boussinesq equation [9] which are obtained using the Miura substitutionsto (2). The modified fields themselves do not start these sequences of conserved densitiesand thus cannot be regarded as their negative terms ũ−3, ṽ−3.Further, consider the flow at ε3 in the r.h.s. of equation (5); the second case relatedto (4) is analogous. We thus obtain the systemuτ = vvx + uxuxx − uxxv − uxvx,vτ = vvxx + v2x − uxvxx − 2uxxvx − uxxxv + uxuxxx + u2xx. (6)It is Hamiltonian w.r.t. the structure (3) and the functional with densityH̄ =16v3 +12u2xv −12uxv2 −16u3x.Note that system (6) is homogeneous w.r.t. the set of weights |u| = 2, |v| = 3, |τ | = −5,|x| ≡ −1. We claim that equation (6) admits two infinite sequences of Hamiltoniansymmetries (us, vs) = ϕ[s] for all weights |s| = −(6k + 3 ± 2), k ∈ N≥0. These flowsreproduce (with proper modifications) the scheme for the symmetries of the originalBoussinesq hierarchy. Namely, they are arranged according to the diagramϕ[1] → ϕ[7] → ϕ[13] → · · · , ϕ[5] → ϕ[11] → · · · .The flow ϕ[5] is the right-hand side of the adjoint Boussinesq equation (6) itself. Thetwo sequences of Hamiltonians for the higher symmetries of (6) are obtained from thefunctionals for system (5) by separating the coefficients of the highest powers of ε.Example 1. The components ϕ1,2[7] of the symmetry ϕ[7] areϕ1[7] = vvxxx − uxvxxx + 2vxvxx − 2uxxvxx − 2uxxxvx − u4xv + uxu4x + 2uxxuxxx,ϕ2[7] = vv4x − uxv4x + 3vxvxxx − 3uxxvxxx + 2v2xx − 4uxxxvxx − 3u4xvx − u5xv+ uxu5x + 3uxxu4x + 2u2xxx.GARDNER’S DEFORMATIONS OF THE BOUSSINESQ EQUATIONS 5The symmetry ϕ[11] that succeeds the equation in the hierarchy has the componentsϕ1[11] = vv7x − uxv7x + 4vxv6x − 4uxxv6x + 10vxxv5x − 10uxxxv5x + 16vxxxv4x− 16u4xv4x − 16u5xvxxx − 10u6xv2x − 4u7xvx − u8xv + uxu8x+ 4uxxu7x + 10uxxxu6x + 16u4xu5x,ϕ2[11] = vv8x − uxv8x + 5vxv7x − 5uxxv7x + 14vxxv6x − 14uxxxv6x + 26vxxxv5x− 26u4xv5x + 16v24x − 32u5xv4x − 26u6xvxxx − 14u7xvxx − 5u8xvx− u9xv + uxu9x + 5uxxu8x + 14u3xu7x + 26u4xu6x + 16u25x.2. The Kaup–Boussinesq equationNow we construct the Gardner deformation for the Kaup–Boussinesq equationut = uux + vx, vt = (uv)x + uxxx. (7)Theorem 2. The integrable extension Eε of the Kaup–Boussinesq equation (7) is thesystemũt = ũũx + ṽx + ε ·(ũũxx + ũ2x + (ũṽ)x),ṽt = (ũṽ)x + ũxxx − ε ·(2ũxũxx + ũũxxx + ũxṽx + ũṽxx − ṽṽx). (8)System (8) is Hamiltonian w.r.t. the structure (3) and the functionalH(ε) =∫(12ũ2ṽ +12ṽ2 −12ũ2x +12ε ·[ũũ2x + 2ũũxṽ + ũṽ2])dx.The Miura transformation Eε → E0 is given throughu = ũ+ ε ·(ũx + ṽ), v = ṽ + ε ·(ũũx + ũxx + ũṽ + ṽx). (9)The recurrence relations upon densities of the Hamiltonian functionals for (7) areũ0 = u, ṽ0 = v, ũk = −Dx(ũk−1)− ṽk−1,ṽk = −D2x(ũk−1)−Dx(ṽk−1)−∑ℓ+m=k−1[ũℓDx(ũm) + ũℓṽm], k > 0. (10)Relations (10) do not produce any auxiliary trivial conserved densities.Relations (10) provide the formulas for the Hamiltonians of the (twice- and thrice-)modified Kaup–Boussinesq equations [10]. The modified fields are conserved; neverthe-less, we note that they are not obtained from the Hamiltonians for (7) by the Miurasubstitutions and that the modified fields cannot be used as the negative terms in thegeneral scheme of (10).Substitution (9) provides the canonical factorization of the trivially extended Poissonstructure Â1 + 2εÂ2 for the Kaup–Boussinesq equation, where Â1 is defined in (3) andÂ2 =(Dx12Dx ◦ u12uDx D3x +12Dx ◦ v +12vDx).The two sequences of Hamiltonians, see (10), for the deformed Kaup-Boussinesq equa-tion (8) are inherited from the original functionals for the Kaup–Boussinesq equation (7)by using the Miura substitution (9). The correlation between the higher flows for (8)and the symmetries of (7) is standard, see [3] and references therein.6 A.KARASU AND A.V.KISELEVTaking the flow at ε in extension (8), we obtain the adjoint Kaup–Boussinesq equationuτ = uuxx + u2x + uxv + uvx,vτ = −(2uxuxx + uuxxx + uxvx + uvxx − vvx). (11)System (11) is bi-Hamiltonian w.r.t. two compatible local Poisson structures Γ1 and Γ2,where Γ1 is (3) andΓ2 =0 Dx ◦ ux +Dx ◦ vuxDx + vDx−uxxDx −Dx ◦ uxx −− vxDx −Dx ◦ vx (12)We conclude that the adjoint Kaup–Boussinesq equation (11) is completely integrable.Remark 1. Both Poisson structures (3) and (12) for (11) are of differential order 1.This indicates that the system can be further extended with a higher order symbolsuch that its complete integrability is preserved. The situation is analogous to theGardner extension of the Korteweg–de Vries equation [1]; we recall that the extensionof KdV resulted in the dispersionless modified KdV equation whose Poisson structuresare of order 1.Remark 2. The ‘minus’ Kaup-Boussinesq equation ut = uux + vx, vt = (uv)x − uxxxadmits a unique real quadratic extensionũt = ũũx + ṽx + ε ·(ũũxx + ũ2x),ṽt = (ũṽ)x − ũxxx − ε ·(ũxṽx + ũṽxx)− ε2 ·(ũ2ũxxx + ũũxũxx + ũ3x). (13)System (13) is assinged by the first structure (3) to the HamiltonianH′(ε) =∫(12ũ2ṽ +12ṽ2 +12ũ2x + ε · ũũxṽ +12ε2 · ũ2ũ2x)dx.The invertible Miura transformation from the deformed equation to (7) is{u = ũ, v = ṽ + εũũx} ⇐⇒ {ũ = u, ṽ = v − εuux}. (14)The recurrence relation obtained from (14) provides only the Casimirs∫u dx and∫v dx;all the conserved densities ṽk are trivial if k > 0. Therefore deformation (13–14) of the‘minus’ Kaup–Boussinesq equation is trivial.Acknowledgements. The authors thank E.V. Ferapontov, B.A.Kupershmidt, andA.V.Mikhailov for helpful discussions and are also grateful to M.Brodovsky for com-puter programming. The content of this paper was reported at the 5th Workshop onQuantization, Dualities, and Integrable Systems at Pamukkale University. The researchwas partially supported by the Scientific and Technological Research Council of Turkey(TUBITAK). A part of this research was carried out while A.V.K. was visiting atMiddle East Technical University (Ankara).References[1] Miura R M, Gardner C S, and Kruskal M D 1968 Korteweg–de Vries equation andgeneralizations. II. Existence of conservation laws and constants of motion J. Math.Phys. 9 1204–1209.GARDNER’S DEFORMATIONS OF THE BOUSSINESQ EQUATIONS 7[2] Kupershmidt B A 1982 On algebraic models of dynamical systems Lett. Math. Phys.6 no.2, 85–89.[3] Kupershmidt B A 1983 Deformations of integrable systems Proc. Roy. Irish Acad.A83 no.1, 45–74.[4] Fordy A P 1983 Projective representations and deformations of integrable systemsProc. Roy. Irish Acad. A83 no.1, 75–93.[5] Mathieu P 1988 Supersymmetric extension of the Korteweg–de Vries equationJ. Math. Phys. 29 no.11, 2499–2506.[6] Andrea S, Restuccia A, and Sotomayor A 2005 The Gardner category and non-localconservation laws for N = 1 super KdV J. Math. Phys. 46 no.10, 103517, 11 p.[7] Labelle P and Mathieu P 1991 A new N = 2 supersymmetric Korteweg–de Vriesequation J. Math. Phys. 32 no.4, 923–927.[8] Manin Yu I 1978 Algebraic aspects of non-linear differential equations Itogi Naukii Tekhniki. Ser. Sovremennye Prob. Mat. 11 5–151; transl. in J. Sov. Math. 111–122.[9] Pavlov M V 2004 The Boussinesq equation and Miura-type transformations, Fun-dam. Prikl. Mat. 10 no.1, 175–182.[10] Pavlov M V 2002 Integrable systems and metrics of constant curvature, J. Nonlin.Math. Phys. 9 suppl.1, 173–191.[11] Bocharov A V, Chetverikov V N, Duzhin S V et al 1999 Symmetries and Conser-vation Laws for Differential Equations of Mathematical Physics AMS ProvidenceRI. Krasil’shchik I and Vinogradov A (eds).[12] Igonin S and Krasil’shchik I S 2003 On one-parametric families of Bäcklund trans-formations Adv. Stud. Pure Math. 37 99–114.[13] Kiselev A V 2002 On Bäcklund autotransformation for the Liouville equation Vest-nik Moskovskogo Univ. 6 22–26.[14] Kiselev A V and Golovko V A 2004 On non–abelian coverings over the Liouvilleequation Acta Appl. Math. 83 no.1-2, 25–37.[15] Kiselev A V and Wolf T 2006 Supersymmetric representations and integrablefermionic extensions of the Burgers and Boussinesq equations Symmetry, Integra-bility and Geometry : Methods and Applications (SIGMA) 2 no.030, 19 p.Department of Physics, Middle East Technical University, 06531 Ankara, Turkey.E-mail address : karasu@metu.edu.trPermanent address (A.V.K.): Department of Higher Mathematics, Ivanovo State PowerUniversity, Rabfakovskaya str. 34, Ivanovo, 153003 Russia.Current address : Department of Physics, Middle East Technical University, 06531 Ankara, Turkey.E-mail address : arthemy@newton.physics.metu.edu.tr

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Gardner's deformations of the Boussinesq equations

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