Does the Supersymmetric Integrability Imply the Integrability of Bosonic Sector?

Transcription

1 Does the Supersymmetric Integrability Imply the Integrability of Bosonic Sector? Z I E M O W I T P O P O W I C Z Instytut Fizyki Teoretycznej Uniwersytet Wrocławski POLAND Shihezi China

2 Plan : 1.) Integrability. 2.) Classification of 2 component KdV like system. a.) Generalized symmetries b.) Recursion opertor c.) Lax operator 3.) New system of interacted two KdV equations and its Lax operator. 4.) Manin-Radul N=1 Supersymmetric Kadomtsev Pietvisahvilli hierarchy. a.) b.) MRSKP 3,0 MRSKP 5,0 as supersymmetric Sawada Kotera equation the bosonic sector gives us this new system

3 Korteweg de Vries equation Many different generalization to 2-component system. I.) Svinolupov 1991 where the constants a satisfy n a j, k j a n,k =u xxx 6uu x. u i i i t =u xxx a j, k r i a m, s a m,r II.) Gűrses and Karasu 1998 u j u x k r a n, s cyclic j,k, m=0 and hence they are the structural constants of Jordan algebra. It has the infinite number of generalized symmetries i =b j where b, s are constants i i i u xxx s j,k u j u x k

4 Investigated conditions on b,s in order to find the recursion operator of second and fourth order Decomposable hereditary operator Wen Xiu Ma, Fordy, Antonowicz a 0 v t = 0 a 0 a 1 H u H v = 0 M 0 1 M 0 M H u where H v M i=c 1 3 d i u i u i Definiton of Integrability of a system of equations: A.) If possesse the recursion operator. B.) Has infinite number of conserved quantities and generalized symmetries. C.) Has the Lax representation.

5 Foursov in 2003 investigated 2-component kdv system =F [u,v] v t =G [u,v] where F and G are polynomial functions of u,v, u x,v x Definition : A system of t-independednt evolution equation =Q 1 [u, v] v t =Q 2 [u, v] Is said to be a generalized symmetry if their flows commute D K Q D Q K =0 where Q=Q 1, Q 2 K [u,v]= F [u,v],g[u, v] and D K is a Frechet derivative

6 Foursov using CA found 5 systems with finite number of generalized symmetries and conserved quantities, 3 of them are known to be integrable A.) Hirota Satsuma B.) Ito system =u xxx 6 uu x 12vv x v t = 2 v xxx 6uu x =u xxx 3 uu x 3vv x v t =u x vuv x C.) Drinfeld Sokolov =u xxx 2 vu x uv x v t =uu x

7 Two of them are new D.) =u xxx v xxx 2 vu x 2uv x v t =v xxx 9 uu x 6 vu x 3uv x 2 vv x This system has been also found by Meshkov. E. =4u xxx 3v xxx 4 uu x vu x 2uv x v t =3u xxx v xxx 4uu x 2 uv x 2vv x Foursov found conserved densities of weight 2,4,8,10,12,14 and generalized symmetries of weight 9,11,13,15 and 19. We show that the last system is integrable because it has the Lax representation and has the Bi Hamiltonian structure

8 There is a possibility to construct 2 component system of interacted KdV equtions with the nonlocal generalized symmetries m t =m xxx 3n xxx 3m x 4m 9n3 n x 8m 15n n t = 3m xxx 4 n xxx 12m x n6 n x m 4 n Meshkov A.G Teor.Math.Phys. 156 (2008)

9 The Lax operator classification of 3 system :.) Hirota Satsuma L= 2 uv 2 u v = L KdV L KdV L t =[ L + 3 4, L ] Recursion operator of 4 order.) Drinfeld Sokolov Recursion operator of 6 order L= 3 u u x 2 3 v v x 2 = L L KK KK L t =[ L 2 +, L ] 1.) New equation Recursion operator of 10 order L= u 1 3 u x v =L L L =[ L 5 KK KdV t +, L] 3

10 L KdV L KK L KdV L KK Hirota Satsuma new equation new equation Drinfeld Sokolov Bi - Hamiltonian structure of new equation: =P H t u v H v = 3 3 uu v 2v H u H v H = dxv 2 4 u 2 6 uv.

11 The recursion operator has been found using the method of Gurses, Karasu and Sokolov. 3 3 L t 3 =[ L 5 +, L], L t13=[ L 2 L 5 +, L], L 2 L 5 =L 2 L 5 + L 2 L 5 3 L 2 L 5 + =Q= A A 0 L t 13=L 2 L t 3 [Q, L] R = terms, terms, terms and we can obtain the next Hamiltonian structure factorizing Z v t = terms, terms, terms R=P Z where v t = H u H v and H = dx21u 10x u26v 10x u8v 10x v158 terms

12 The supersymmetric Integrability for N=1. The Manin Radul supersymmetric N=1 KP hierarchy SKP r, m r 2 L :=D r i =0 where D= and v and For even r it is well known hierarchy. A. For r=3 and m=0 we have m v r i D i j D j j=1 are the superbosonic or superfermionic superfields =D 3 L t k =9 [, + k 3 ] where = u ξ is a fermionic, u is bosonic function. t,2 = x,7 = D xx 1 2 D2 2 x x t,10 = 5x 5 xxx D5 xx x 5 x D 2

13 There are several interesting observations : A.) t is usual time while Ƭ is an odd time B.) for k=10 we have the susy N=1 Sawada Kotera equation and in components it is =u 5x 5 u xxx u5 u x u xx 5 u 2 u x 5 xxx x t = 5x 5 u xxx 5u x xx 5u 2 x and has been recently considered by Tian and Liu. C.) The trace formula for Lax operator gives us conserved quantities which however are not reduced to the classical conserved charges of Sawada Kotera equation. H 1 = d dx x =2 dxu x H 2 = d dx3 D xx D x 2 D 3 =6 dx xxx u x u 2 D.) The odd time hirerachy is generated by even hamiltonian structure,7 = P H 1 = D5 2 2 D D H 1 P is the usual hamiltonian operator of the supersymmetric KdV equation which is connected with the N-1 supersymmetrical Virasoro algebra.

14 E. Second Hamiltonian Structure of susy Sawada Kotera equation is odd Proof: = D D D 1 D D D, A.) We have to check the Jacobi identity ' <,P P t = H 1. ' ><, P P <, >= dx d. ' ><, P P >=0, where P ' denote the Gatoux derivative along the vector α. B.) R- matrix approach to Bi-Hamiltonian systems L t = 1 F =L F + F L +, L t = F =L F L L F L, L t = 3 F =L L F L + L F L + L L F L + LL L F + L

15 where gradient is parametrized as H = k 1 D H H, k >0 a k b k L= k > 0 a k b k D k. We have to use the Dirac reduction, because we need to embed the Lax operator into a larger space. Ad 1. For Ad 2. For 1 it is impossible to carry out such reduction. 2 we have 3 dimensional matrix and Dirac reduction gives us P operator which generate odd time flows. Ad 3. For 3 we have 6 dimensional matrix and Dirac reductio gives us Ω which generates the even time flows. Second Hamiltonian structure. Is obtained from the factorization of the recursion operator which was derived by Tian and Liu J = 2 D 1 D x 1 xd x 1 D J t = H

16 The SKP hierarchy for r=5 and m=0. Results L=D Z Z 1 3 D P D where Z = 1 z 1, P= p 1 2 are superfermionic and superbosonic supermultiplet. Computing we obtain L t =5[ L, L ] Z t =4 Z xxx 3P xxx 2 Z x DZDPZ 6DZ x 2DP x P 3 DZ x DP x P x DP P t =3 Z xxx P xxx 8Z x DZ Z 8DZ x 6DP x P x 4DZDP P x 4DZ x 3DP x The bosonic sector in which Z =u, P= v. gives us new 2-component interacted KdV equations.

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