On Recursion Operator of the q-kp Hierarchy

Transcription

1 Commun. Theor. Phys. 66 ( Vol. 66 No. 3 September On Recursion Operator of the -KP Hierarchy Ke-Lei Tian ( 田可雷 1 Xiao-Ming Zhu ( 朱晓鸣 1 and Jing-Song He ( 贺劲松 2 1 School of Mathematics Hefei University of Technology Hefei China 2 Department of Mathematics Ningbo University Ningbo China (Received May ; revised manuscript received June Abstract It is the aim of the present article to give a general expression of flow euations of the -KP hierarchy. The distinct difference between the -KP hierarchy and the KP hierarchy is due to -binomial and the action of -shift operator θ which originates from the Leibnitz rule of the uantum calculus. We further show that the n-reduction leads to a recursive scheme for these flow euations. The recursion operator for the flow euations of the -KP hierarchy under the n-reduction is also derived. PACS numbers: I Key words: -KP hierarchy flow euations recursion operator 1 Introduction The -deformed Kadomtsev Petviashvili (-KP hierar chy [112] is an interesting subject of intensive study in the integrable systems which can be defined formally [1314] by means of the -derivative instead of the usual derivative with respect to x in the Kadomtsev Petviashvili (KP hierarchy. [1516] As 1 goes to then -KP reduces to KP hierarchy. So main motivation of the research of the -KP hierarchy is to explore two aspects: integrable properties inherited from KP hierarchy and the difference (or called -effect between -KP and KP hierarchy. On the one side it has been shown that -KP hierarchy inherits some integrable properties such as bi-hamiltonian structure infinite conservation laws τ function additional symmetries and symmetry constraint sub-hierarchy [23579] from KP hierarchy. On the other side the -effect can be observed [9] intuitionally single -soliton of the -KP euation: - deformation does not destroy the profile of soliton it is just similar to an impulse to soliton which can be thought of a non-uniform translation [5] of dynamical variables t i between τ KP and τ KP i.e. τ KP (t i = τ KP (t i t i = t i + ((1 i /i(1 i x i. This translation t i t i is also re-delivered in the Virasoro constraint of the -KP hierarchy. [12] Because uantum calculus [14] is one ind of calculus in non-commutative plane these results of the -KP hierarchy just demonstrate the internal properties of the dynamical evolution of the noncommutative plane. H-calculus is another one of the calculus in non-commutative plane. What is more the - deformed differential-difference systems [1719] are corresponding to the -calculus and h-calculus respectively. Up to now there are still two uestions in the study of the -KP hierarchy: (i What is the distinct difference between the flow euations of the -KP and KP hierarchy? (ii Is there recursion operator for the -KP hierarchy under n-reduction? The central wor of this paper is to solve these uestions starting from the Lax euations of the - KP hierarchy. For the KP hierarchy noncommutative KP hierarchy and Dispersionless KP Hierarchy there are several methods to construct the recursion. [2024] Here we use Strampp and Oevel s method to get a general formula of the recursion operator. The organization of this paper is as follows. After reviewing some basic results of -KP hierarchy we give two general expressions of flow euations in Sec. 2. In particular a single component -KP family euation about the dynamical coordinate u 1 of -KP hierarchy is obtained and the n-reduction is also discussed in this section. The recursion operators for the n-reduced -KP hierarchy are given in Sec. 3. Section 4 is devoted to conclusions and discussions. At the end of this section let us recall some useful facts of the -calculus [14] in the following to mae this paper be self-contained. The -derivative is defined by (f(x = and the -shift operator is f(x f(x x( 1 (1 θ(f(x = f(x. (2 It is worth pointing out that the -shift operator θ does not commute with indeed the relation ( θ (f = θ ( f Z holds. (f(x recovers the ordinary differentiation x (f(x as goes to 1. Let 1 denote the formal inverse of. In general the following -deformed Leibnitz rule holds n f = ( n θ n ( f n n Z (3 0 Supported by the National Natural Science Foundation of China under Grant Nos and and Anhui Province Natural Science Foundation under Grant No MA04 Corresponding author ltian@ustc.edu.cn c 2016 Chinese Physical Society and IOP Publishing Ltd

2 264 Communications in Theoretical Physics Vol. 66 the -number and the -binomial are defined by (n = n 1 1 ( n = (n (n 1 (n + 1 (1 (2 ( For example ( n = 1. 0 f = ( f + θ(f (4 2 f = ( 2 f + ( + 1θ( f + θ 2 (f 2 (5 3 f = ( 3 f + ( θ( 2 f + ( θ 2 ( f 2 + θ 3 (f 3. (6 By comparing with Leibnitz rule for n f the difference in E. (3 is due to -binomial and the action of -shift operator θ. This simple observation is crucial to find the difference between the flow euations of the -KP and KP hierarchy. Moreover for a -pseudo-differential operator (-PDO of the form n P = p i i i= we separate P into the differential part and the integral part P + = i 0 p i i P = p i i. i 1 2 Flow Euations of the -KP Hierarchy Similar to the general way of describing the classical KP hierarchy [1516] we first give a brief introduction of -KP hierarchy based on Ref. [5]. Let L be one -PDO given by L = u 1l l = + u 1 + u u (7 l 1 with u 0 = 1 the operator is called Lax operator of -KP hierarchy. There exist infinite number of - partial differential euations related to dynamical variables {u i (x t 1 t 2 t 3... i = } and can be deduced from the generalized Lax euations L = [B n L] n = (8 t n which are called -KP hierarchy. Here B n = (L n + = n i=0 b i i and L n = L n L n +. We fix a natural number n and define the sets of functions p j j n and j j n as the coefficients in the n-th power on of L: L n = ( + u 1 + u u n = p j (n j = j j (n (9 j n j n p n (n = n (n = 1. There is a one to one correspondence between (u 1 u 2 u 3... and (p n1 (n p n2 (n p n3 (n... also (u 1 u 2 u 3... and ( n1 (n n2 (n n3 (n.... Moreover we note that the functions j (n (and p j (n are uniuely determined as functions of {u 1 u 2... u nj1 u nj } as j (n = nj1 =j θ (u nj + f jn (u 1 u 2... u nj1 (10 f jn are differential polynomials with of {u 1 u 2... u nj1 u nj }. For example 1 (2 = θ 2 (u 3 + θ(u 3 + θ(u 1 u 2 + u 1 θ(u 2 + θ( (u 2 (11 2 (2 = θ 3 (u 4 + θ 2 (u 4 + θ 2 (u 2 θ(u 2 1 θ 2 (u 2 ( (u 1 + u 1 θ 2 (u 3 + θ 2 (u 1 u 3 + θ 2 ( (u 3 + θ( 2 (u 2 + θ 2 ( (u 3 + θ( (u 1 u 2 + θ( (u 2 θ 1 (u 1 + θ( (u 3. (12 By comparing with h (n of the KP hierarchy [20] it is easy to now that the distinct difference of E. (10 is due to -binomial and the action of -shift operator θ. The differential part of L m + is m m L m + = p j (m j = j j (m (13 j=0 and the integral part is j=0 L m = p j (m j = j j (m. (14 j<0 j<0 By a straightforward calculation the relation between p j (n and j (n is given as nj ( j + r p j (n = θ j ( r j+r (n. (15 r Theorem 1 The flow euations of the -KP hierarchy can be expressed as u 0tm = 0 (16 u jtm = j A jh h (m j = (17 h=1 the differential operator jh ( ( 1 r h A jh = u r j h r ( h j h r θ 1j jhr θ rj jhr (u r jh ( ( 1 r h A jh h (m = u r j h r ( h j h r θ rj jhr (u r h (m θ 1j jhr ( h (m. (18 Proof The first flow euation u 0tm = 0 is obvious because of the fact u 0 = 1. Note that L = r 0 u r 1r L m = h h (m h>0

3 No. 3 Communications in Theoretical Physics 265 following the Lax euation of the -KP hierarchy L tm = [L m + L] = [L L m ] we have = [L L m ] = LL m L m L L tm = (u r 1r r 0 h>0 = r 0 h>0 0 ( h = 0 h>0 ( h r = j 1 h=1 h h (m h h (mu r 1r ( ( 1 r h u r θ 1rh ( h (m θ h ( h (mu r h+1r ( ( 1 r h u r r θ 1h r ( h (m θ h+r r ( h (mu r 1h j jh ( ( 1 r h u r j h r ( h j h r θ 1j jhr ( h (m θ rj jhr ( h (mu r 1j = r + and j = + h. Combining the above euation with L tm = j 0 u jtm 1j finishes the proof. Remar 1 We can see that the main difference between flow euations of the KP hierarchy and -KP hierarchy is due to -binomial and the action of -shift operator θ which is appeared in A jh and h (m. This distinct difference originates from the essential property of uantum calculus i.e. -deformed Leibnitz rule E. (3. We write out several explicit forms of operators A jh as A jj = θ 1j θ j U(n = u 1 u 2 u n1 u n Q(n m = 1 (m 2 (m n+1 (m n (m A jj1 =(2 jθ 1j (1 jθ j + u 1 θ 1j θ 1j u 1 A jj2 = 1 2 (3 j(2 jθ1j (2 j(1 jθj 2 + (2 ju 1 θ 1j (2 jθ 1j u 1 + u 2 θ 1j θ 2j u 2. The first two flow euations of the -KP hierarchy can be given by u 1tm = A 11 1 (m = 1 (m θ 1 ( 1 (m u 2tm = A 21 1 (m + A 22 2 (m = θ 2 ( 1 (m + u 1 θ 1 ( 1 (m θ 1 (u 1 1 (m + θ 1 ( 2 (m θ 2 ( 2 (m. Let m = 1 2 and using u i substitute for j (m we will get the flow euations of u 1 u 2 on t 1 t 2 respectively as u 1 t 1 = θ(u 2 u 2 u 2 = u 2 + θ(u 3 u 3 + u 1 u 2 u 2 θ 1 (u 1 t 1 u 1 = θ( u 2 + θ 2 (u 3 + θ(u 1 θ(u 2 + u 1 θ(u 2 t 2 ( u 2 + u 1 u 2 u 2 θ 1 (u 1 + u 3. Solving these above euations we arrive at the following euation of a single component -KP family ( u1 2 u 1 u 1 + 2u 1 t 2 ( 1x t 1 t 1 ( 2 2 u 1 = + ( 1x t 2. (19 1 Let us consider the n-reduced -KP hierarchy let L be a purely differential operator i.e. L n = L n +. Theorem 2 The Lax euations of n-reduced -KP hierarchy can be written in the finite matrix form as A(n = U(n tm = A(nQ(n m (20 A A 21 A A n11 A n12 A n13 A n1n1 0 A n1 A n2 A n3 A nn1 A nn. Proof From E. (10 and L n = L n + it is easy to now j (n = 0 for j < 0. n θ (u n+1 = f 1n (u 1 u 2... u n j = 1 =1 n+1 θ (u n+2 = f 2n (u 1 u 2... u n+1 j = 2 =2 n+p1 =p θ (u n+p = f pn (u 1 u 2... u n+p1 j = p. The above euations show that (u n+1 u n+2... u n+p depending only on (u 1 u 2... u n. This means that in the case of n-reduction only the first n dynamical coordinates (u 1 u 2... u n are independent. Euation (10 tells us the coordinates ( n1 (n n2 (n n3 (n... become to be finite ( 0 (n 1 (n 2 (n... n1 (n and (p n1 (n p n2 (n p n3 (n... are also finite as (p 0 (n p 1 (n p 2 (n... p n1 (n. The n-reduced -KP

4 266 Communications in Theoretical Physics Vol. 66 hierarchy defined by E. (19 is a closed time-evolution system of (u 1 u 2... u n. Now we can write the Lax euations E. (19 of the n-reduced -KP hierarchy as U(n tm = A(nQ(n m. The above two theorems are based on the coordinates { 1 (m 2 (m... n (m} we also can express the flow euations of the -KP hierarchy by another family of coordinates {p 1 (m p 2 (m... p n (m}. Theorem 3 The flow euations of the -KP hierarchy can be expressed as u 0tm = 0 (21 j u jtm = B jh p h (m j = (22 h=1 the differential operator jh ( ( 1 r B jh = u r j h r θ 1j+h jhr ( h j h r θ rj ( jhr u r. (23 Theorem 4 The Lax euations of n-reduced -KP hierarchy can be written in the finite matrix form as P (n m = p 1 (m p 2 (m p n+1 (m p n (m B(n = U(n tm = B(nP (n m (24 B B 21 B B n11 B n12 B n13 B n1n1 0 B n1 B n2 B n3 B nn1 B nn 3 Recursion Operator of n-reduced -KP Hierarchy In this section we shall give a recursion formula relating P (n n+m and P (n m and a recursion operator mapping the lower order flow euations to higher order flow euations of n-reduced -KP hierarchy. To this end we first state two useful Lemmas. Lemma 1 n p j (m + n = C js (np js (m j 1 (25 Proof C js (n = 1 s=j+1 n =max(0s ( ((L m (L n = p (m ( = = 1 0 =0 n ( j s θ j ( s p (n. s n p j (n j j=0 n ( p (m n j 1 s=j+1 =max(0s ( j s s ( = 1 =0 n p (m p (n θ ( p (n + θ j ( s p (np js (m j. Since (L m+n = ((L m (L n comparing with ((L m+n = j 1 p j(m + n j yields the Lemma 1. Lemma 2 n p j (m + n = C s (np js (m j 1 (26 Proof ( n ((L n (L m = p (n =0 =0 s=0 ns ( + s C s (n = p +s (n θ s (. 1 p (m ( n = p (n p (m =0 1.

5 No. 3 Communications in Theoretical Physics 267 = ( = 1 0 =0 j 1 s=0 =0 n ( p (n n ns ( ( + s p +s (n θ ( p (m + θ s ( p js (m j. Since (L m+n = ((L n (L m comparing with ((L m+n = j 1 p j(m + n j yields the Lemma 2. Theorem 5 The n-reduced -KP hierarchy possess a recursion operator Φ(n such that U(n tm+n = Φ (nu(n tm (27 for = and m n = Proof Note that (L m+n = ((L m (L n = ((L n (L m E. (25 in Lemma 1 and E. (26 in Lemma 2 for j = n together give the next euations. P (n n + m = P (n n + m = E(nP (n m + F (n P (2n m = G(nP (n m + H(n P (2n m (28 p 1 (n + m p 2 (n + m p n+1 (n + m p n (n + m P (2n m = p n1 (m p n2 (m p 2n+1 (m p 2n (m C 10 (n C 11 (n C 1n2 (n C 1n1 (n C 21 (n C 20 (n C 2n3 (n C 2n2 (n E(n = C n+1n+2 (n C n+1n+3 (n C n+10 (n C n+11 (n C nn+1 (n C nn+2 (n C n1 (n C n0 (n C 1n (n C 2n1 (n C 2n (n 0 0 F (n = C n+12 (n C n+13 (n C n+1n (n 0 C n1 (n C n2 (n C nn1 (n C nn (n C 0 (n C1 (n Cn2 (n Cn1 (n C n (n C0 (n Cn3 (n Cn2 (n C n1 (n Cn (n 0 0 G(n = H(n = 0 0 C0 (n C1 (n C0 (n C 2 (n C3 (n Cn (n 0 C 1 (n C2 (n Cn1 (n Cn (n. Then we obtain P (2n m = (H(nF (n 1 (E(nG(nP (n m. (29 Inserting the vector P (2n m = (p n1 (m p n2 (m... p 2n+1 (m p 2n (m T from E. (29 into E. (28 i.e. with P (n n + m = R(nP (n m (30 R(n : = E(n+F (n(h(nf (n 1 (E(nG(n.(31 Introducing Φ(n : = B(nR(nB(n 1 (32 B(n is the same in Theorem 4. Finally the recursive scheme for the Lax euations of the n-reduced -KP hierarchy could be written out as U(n tm+n = B(nP (n m + n = B(nR(nP (n m + ( 1n = B(nR(nB(n 1 B(nP (n m + ( 1n = Φ(nB(nP (n m + ( 1n = Φ(nU(n tm+(1n = Φ 2 (nu(n tm+(2n = Φ (nu(n tm for = and m n = Remar 2 From Theorem 5 one can find that the recursion operator Φ(n for the n-reduced -KP hierarchy cannot be reduced to the classical situation as 1. The

6 268 Communications in Theoretical Physics Vol. 66 reason is that in general we cannot let θ = 1 i.e. θ(f = f in the recursion operator because there is (θ n 1 1 in the (H(nF (n 1 of Φ(n. Therefore we can not thin that the recursion operator E. (32 is a -extension of the recursion operator of the KP hierarchy. [20] 4 Conclusions and Discussions To summarize we have derived two general expressions of the flow euations of the -KP hierarchy in Es. (17 and (22 in Theorem 1 and Theorem 3 respectively. We have pointed from these general expressions that the main difference between the flow euations of the KP hierarchy and the -KP hierarchy is due to -binomial and the action of -shift operator θ which reflects an essential character of uantum calculus in some senses. The another main result is a general formula of the recursion operator for the flow euations of the -KP hierarchy under the n- reduction which is given in Theorem 5. We would lie to stress that the recursion operator Φ(n for the n-reduced -KP hierarchy cannot be reduced to the classical situation as 1 which shows that the recursion operator E. (32 is not a -extension of the recursion operator of the KP hierarchy. [20] These results also show that -KP hierarchy is not a trivial formalism generalization of the KP hierarchy and thus it deserves further investigation in the near future. Acnowledgments One of the authors (K.T. is supported by Erasmus Mundus Action 2 EXPERTS and would lie to than Professor Jarmo Hietarinta for many helps. References [1] E. Frenel Int. Math. Res. Notices 2 ( [2] J. Mas and M. Seco J. Math. Phys. 37 ( [3] Z.Y. Wu D.H. Zhang and Q.R. Zheng J. Phys. A: Math. Theor. 27 ( [4] P. Iliev J. Phys. A: Math. Theor. 37 ( [5] P. Iliev Lett. Math. Phys. 44 ( [6] P. Iliev J. Geom. Phys. 35 ( [7] M.H. Tu Lett. Math. Phys. 49 ( [8] K. Taasai Lett. Math. Phys. 72 ( [9] J.S. He Y.H. Li and Y. Cheng SIGMA 2 ( [10] R.L. Lin X.J. Liu and Y.B. Zeng J. Nonlinear Math. Phys. 15 ( [11] K. Taasai J. Geom. Phys. 59 ( [12] K.L. Tian J.S. He and Y. Cheng AIP Conf. Proc ( [13] A. Klimy and K. Schmüdgen Quantum Groups and Their Represntaions Springer Berlin (1997. [14] V. Kac and P. Cheung Quantum Calculus Springer- Verlag New Yor (2002. [15] E. Date M. Kashiwara M. Jimbo and T. Miwa Transformation Groups for Soliton Euations in Nonlinear Integrable Systems-Classical and Quantumtheory World Scientifc Singapore ( [16] L.A. Dicey Soliton Euations and Hamiltonian Systems 2nd ed. NJ: World Scientifc Singapore (2003. [17] S.W. Liu and Y. Cheng J. Phys. A: Math. Theor. 43 ( [18] H.M. Li Y.Q. Li and Y. Chen Commun. Theor. Phys. 62 ( [19] R.G. Zhou and J. Chen Commun. Theor. Phys. 63 ( [20] W. Strampp and W. Oevel Lett. Math. Phys. 20 ( [21] M. Gurses A. Karasu and V.V. Soolov J. Math. Phys. 40 ( [22] D.J. Zhang and D.Y. Chen J. Phys. Soc. Jpn. 72 ( [23] J.S. He J.Y. Tu X.D. Li and L.H. Wang Nonlinearity 24 ( [24] Q.S. Cheng and J.S. He Commun. Theor. Phys. 58 (