Pacific Journal of Applied Mathematics Volume 1 umber 4 pp. 61 65 ISS 1941-3963 c 008 ova Science Publishers Inc. A OTE O r-matrix OF THE PEAKO DYAMICS Zhijun Qiao Department of Mathematics The University of Texas Pan-American 101 West University Drive Edinburg TX 78539 USA Abstract This paper deals with the r-matrix of the peakon dynamical systems. Our result shows that there does not exist constant r-matrix for the peakon dynamical system. Keywords: r-matrix Structure Peakon Systems. AMS Subject: 35Q53; 58F07; 35Q35 PACS: 03.40.Gc; 03.40Kf; 47.10.+g In 1993 Camassa and Holm proposed a shallow water equation and discussed the peaked-soliton (peakon solution of the equation [1]. Later in 1996 Ragnisco and Bruschi [] showed the integrability of the finite-dimensional peakon system through constructing a constant r-matrix. Their starting point is the following Lax matrix (1. The r-matrix is usually dynamical in the framework of the r-matrix approach located in the fundamental Poisson bracket [3]. Ragnisco and Bruschi claimed that for a particular choice of the relevant parameters in the Hamiltonian (the one corresponding to the pure peakons case the r-matrix becomes essentially constant []. In ref. [4] Qiao extended the Camassa-Holm (CH equation to the whole integrable CH hierarchy including positive and negative members in the hierarchy and studied r-matrix structures of the constrained CH systems and algebraic-geometric solutions on a symplectic submanifold through using the constraint approach [5]. In this note what we want to show is no constant r-matrix for the CH peakon system. Let us discuss below. For the peakon system let us consider the Lax matrix which is given in ref. []: where L L i j E i j (1 i j1 E-mail address: qiao@utpa.edu L i j p i p j A i j ( A i j A(q i q j e 1 q i q j. (3
6 Zhijun Qiao In Eq. (3 A(x e 1 x (4 and A(x has the following properties: A (x 1 sgn(xa(x A i j A ji A ii 1 A i j A (q i q j A (q j q i A ji A ii 0 ( x + y A(xA(y A (xa(y + A(xA (y ( x + y A(xA(y y x 0. We work with the matrix basis E i j : 1 A(xA(y[sgn(x + sgn(y] (E i j kl δ ik δ jl i jkl 1... To have the r-matrix structure we consider the so-called fundamental Poisson bracket [3]: where {L 1 L } [r 1 L 1 ] [r 1 L ] (5 L 1 L 1 L 1 L r 1 r 1 {L 1 L } i j1 kl1 L i j E i j 1 L kl 1 E kl r lk E lk (E lk + E kl lk1 r lk (E lk + E kl E lk lk1 {L i j L kl }E i j E kl. i jkl1 Here {L i j L kl } is of sense under the standard Poisson bracket of two functions 1 is the unit matrix and r lk are to be determined. In Eq. (5 [ ] means the usual commutator of matrix.
A ote on r-matrix of the Peakon Dynamics 63 ow let us calculate the left hand side of Eq. (5. L i j q m p i p j A i j(δ im δ jm L kl p m A kl p k p l (p l δ km + p k δ lm ( Li j L {L i j L kl } kl L kl L i j m1 q m p m q m p m 1 [ pi p j A A kl i j (δ im δ jm (p l δ km + p k δ lm m1 pk p l ] p k p l A A i j kl (δ km δ lm (p j δ im + p i δ jm pi p j 1 ( pi p j p l δ ik A i j p A pk kl k 1 ( p j pi p l δ jk A i j p A kl + k i pk j where the supscript means A (x with the argument. Thus we obtain {L 1 L } 1 {L i j L kl }E i j E kl i jkl1 jkl1 + 1 ( pi p j p k δ il A i j p A pl kl + l i pl j 1 ( p j pi p k δ jl A i j p A kl l [ pk p j A jl (E jl E kl E kl E jl + p k p j A lka l j (E lk E l j E l j E lk + ] p k p j (A lk A jl (E lk E jl E jl E lk. ext we compute the right hand side of Eq. (5. Before doing that let us give some simple tensor product of the matrix basis E i j : So we have (E i j E st (E kl 1 δ jk E il E st (E kl 1(E i j E st δ il E k j E st (E i j E st (1 E kl δ tk E i j E sl (1 E kl (E i j E st δ ls E i j E kt E kl E st δ ls E kt. [r 1 L 1 ] [r 1 L ] r lk L i j ([E lk (E lk + E kl E i j 1] [(E lk + E kl E lk 1 E i j ] i jkl1
64 Zhijun Qiao r lk L i j (δ ik E l j (E lk + E kl δ ik (E lk + E kl E l j i jkl1 +δ jl (E lk + E kl E ik δ jl E ik (E lk + E kl ( r lk L k j E l j (E lk + E kl (E lk + E kl E l j jkl1 + jkl1 r kl L jk ( E jl (E lk + E kl + (E lk + E kl E jl r lk L jk ((E l j + E jl (E lk + E kl (E lk + E kl (E l j + E jl jkl1 where we set r lk r kl and used L jk L k j. After comparing both sides of the fundamental Poisson bracket (5 we should have the following equalities: r lk 1 A jl A jk (6 r l j r lk 1 (A lk A jl. A jk (7 In fact the nd one is a natural result derived from the 1st one. Thus for the CH peakons case we have r lk 1 A jl A jk 1 4 sgn(q k q l A kla jl A jk 1 4 sgn(q l q k e 1 ( q l q k + q j q l q j q k j Z +. (8 This equality holds for arbitrary j Z +. Obviously only in the cases of j > l > k or j < l < k Eq. (8 becomes constant namely ± 1 4. But for other j apparently Eq. (8 is OT constant. So we think that the constant matrix given in ref. [] r 1 a lk1 sgn(q l q k E lk (E lk + E kl a constant is not an r-matrix for the CH peakon dynamical system. Discussions for more general case of Lax matrix are seen in ref. [6]. Acknowledgments This work has been supported by the UTPA-FRC.
References A ote on r-matrix of the Peakon Dynamics 65 [1] R. Camassa and D. D. Holm An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71(1993 1661-1664. [] O. Ragnisco and M. Bruschi Peakons r-matrix and Toda Lattice Physica A 8(1996 16-149. [3] L. D. Faddeev and L. A. Takhtajan Hamiltonian Methods in the Theory of Solitons (Springer-Verlag Berlin 1987. [4] Zhijun Qiao The Camassa-Holm hierarchy -dimensional integrable systems and algebro-geometric solution on a symplectic submanifold Comm. Math. Phys. 39 309-341. [5] Zhijun Qiao Generalized r-matrix structure and algebro-geometric solutions for integrable systems Rev. Math. Phys. 13(001 545-586. [6] Zhijun Qiao r-matrix structure for general even functions in finite-dimensional Hamiltonian systems preprint 008.