Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations

Transcription

1 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 40, NUMBER 5 MAY 999 Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations Wen-Xiu Ma a) Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, Peoples Republic of China Benno Fuchssteiner b) Department of Mathematics, University of Paderborn, D Paderborn, Germany Received 7January 998; accepted for publication 0 January 999 An algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows ( t l,l 0) from the discrete spectral problem associated with agiven system of discrete evolution equations. Three examples are given. 999 American Institute of Physics. S I. INTRODUCTION The theory of integrable systems has various aspects, although the term integrable is somewhat ambiguous, especially for systems of partial differential equations. Symmetries are one of those important aspects and have adeep mathematical and physical background. When any special character, for example the Lax pair, has not been found for agiven system of continuous or discrete equations, among the most efficient ways is to consider its symmetries in order to obtain exact solutions. It is through symmetries that Russian scientists et al. developed some theories for testing the integrability of systems of evolution equations, both continuous and discrete, and classified many types of systems of nonlinear equations that possess higher differential or differential-difference degree symmetries for example, see Refs. and. Usually an integrable system of equations is referred to as asystem possessing infinitely many symmetries. 3,4 Moreover, these symmetries form nice and interesting algebraic structures. 3,4 For agiven system of evolution equations u t K(u), both continuous and discrete, avector field (u) is called its symmetry if (u) satisfies its linearized system, d u K, dt i.e., t K, ªK K, where the prime means the Gateaux derivative. Starting from alie-point symmetry, we can often construct the corresponding explicit group-invariant solutions. Asymmetry may, of course, depend explicitly on the evolution variable t. If asymmetry of the system u t K(u) not depending explicitly on tis apolynomial in t, i.e., then we have n t,u i 0 t i i! i u, n, a Electronic mail: mawx@cityu.edu.hk b Electronic mail: benno@uni-paderborn.de /99/40(5)/400/9/$ American Institute of Physics

2 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 40 i K, i, i n, 3 and ad K n 0 0, where ad K ) 0 K, 0. 4 Therefore the symmetry is totally determined by avector field 0 satisfying 4. This kind of vector field 0 has been discussed in considerable detail and is called amaster symmetry of degree nof u t K(u) by one of the authors BF in Ref. 5. The appearance of first degree master symmetries gives acommon character for integrable systems of continuous evolution equations, both in dimensions and in dimensions, for example, the KdV equation and the KP equation. The resulting symmetries are sometimes called -symmetries for more information, see Ref. 6, for example and usually constitute centerless Virasoro algebras together with time-independent symmetries. 7 9 Moreover these -symmetries may be generated by use of zero curvature equations or Lax equations, 0 and the corresponding master symmetry flows may also be solved by the inverse scattering method., Inthe case of systems of discrete evolution equations, there exist some similar results. For example, many systems of discrete evolution equations have -symmetries and centerless Virasoro symmetry algebras, 3 5 and the inverse scattering method may still be applied in solving themselves and their master symmetry flows. 6 9 Sofar, however, to the best of our knowledge, there has not been asystematic mathematical theory to explain why there exist -symmetries for systems of discrete evolution equations and how we can construct those -symmetries when they exist, from the point of discrete zero curvature equations. Throughout this paper, master symmetries is used to express the first degree master symmetries that generate -symmetries. Our purpose is to give an algebraic explanation of the first question above and to provide aprocedure to generate those master symmetries for agiven lattice hierarchy. The discrete zero curvature equation is our basic tool to give rise to our answer and procedure. The Volterra lattice hierarchy, the Toda lattice hierarchy, and asub-kp lattice hierarchy are chosen and analyzed as some illustrative examples, which have one dependent variable, two dependent variables, and three dependent variables, respectively. Let us now describe our notation. Assume that u (u,...,u q ) T, where u i u i (t,n), i q, are real functions defined over R Z in the case of the complex function, the discussion is similar, and let Bdenote all real functions P u P(t,n,u), which are C differentiable with respect to tand n, and C -Gateaux differentiable with respect to u. We always write Eas ashift operator and E m x n x m n x m n, where x:z R, m,n Z. 5 Note that x (m) here does not mean the mth derivative. Set B r (P,...,P r ) T P i B, i r, and denote by V r all matrix operators ( ij ) r r,where the entries ij ij (t,n,u) B, and by Ṽ r, all matrix operators depending on a parameter : U (U ij ) r r, where the entries U ij U ij (t,n,u, ) Bfor all, being C differentiable with respect to. We will need amultiplication operator, n :B B, P u n P u, n P u m m P u m, 6 which is often involved in the construction of master symmetries. This avoids an unclear expression np u,which may also mean (np u )(m) n(p u )(m).for example, it is absolutely clear that ( n P u )(m) mu(m ) mu(m), when P u E u u. We also need adifference operator E E,whose inverse operator may be defined by u n E E u n ª k u n k u n k, k 7

3 40 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner where uis required to be rapidly vanishing at the infinity. Moreover, we define Obviously, we can find that and thus / n, i.e., n / n, const. 8 E E, E E, 9 E n, E n, const., 0 which may also be viewed as adefinition of two inverse operators (E ) and ( E ). Note that here we have used the operator n so that two functions (E ) and ( E ) have the other clear expressions. The operators,(e ), and ( E ) often appears in the expressions of master symmetries, and thus master symmetries are usually nonlocal vector fields belonging to B q. In order to carefully analyze algebraic structures related to symmetries, we specify the definition of the Gateaux derivative X S of any vector-valued function X B r atadirection S B q asfollows: X S d d 0 X u S, which implies that X is an operator from B q tob r,and need the following two product operations: K,S K S S K, K,S B q, f,g f g f g, f,g C R, 3 where C (R) denotes the space of smooth functions defined over R. It is known that (B q,, ) and (C (R),, ) are all Lie algebras. We now assume that U Ṽ r and the Gateaux derivative operator U is injective throughout the paper. Let us consider the discrete spectral problem, where V Ṽ r.its adjoint system reads as E U U n,u,, t V V n,u,, 4 E U U n,u,, t EV EV n,u,. Their integrability conditions are given by the following discrete zero curvature equation: U t EV U UV. 5 If the operator equation 5 is equivalent to a system of discrete evolution equations u t K(n,u),K B q,then it is called adiscrete zero curvature representation of u t K(n,u). Evidently, U t U u t f U, if t f, where U U/. Therefore a system of discrete evolution equations u t K(n,u),K B q,is the integrability condition of 4 with the evolution law t f( ) if and only if U K fu EV U UV. 6

4 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 403 Note that the injective property of U is indispensable in deriving zero curvature representations of systems of evolution equations. The equation 6 exposes an essential relation between a system of discrete evolution equations and its discrete zero curvature representation. It will play an important role in the context of our construction of master symmetries. The paper is divided into five sections. The next section will be devoted to ageneral algebraic structure related to discrete zero curvature equations. Then in the third section we will establish an approach for constructing master symmetries by the use of discrete zero curvature representations, along with an explanation of why there exist master symmetries for systems of discrete evolution equations. In the fourth section, we will go on to illustrate our approach by three concrete examples of lattice hierarchies. Finally, the fifth section provides aconclusion and some remarks. II. BASIC ALGEBRAIC STRUCTURE We aim to discuss Lie algebraic structures of symmetries, including master symmetries, by using zero curvature equations. It is natural to ask what algebraic structure exists, related to zero curvature equations. To answer this question, we first plan to expose alie algebraic structure for the space B q,ṽ r,c (R). Let (K,V,f),(S,W,g) B q,ṽ r,c (R), in other words, K,S are vector fields, V,W are r r matrix operators, and f,g are smooth functions. We introduce their product: K,V,f, S,W,g K,S, V,W, f,g, 7 where K,S, f,g are defined by, 3, respectively, and V,W is defined by V,W V S W K V,W gv fw, 8 where V,W VW WV.The same product as 8 has been introduced for the continuous case in Ref. 0. Theorem : Lie algebra The space (B q,ṽ r,c (R)),, is alie algebra, the product, being defined by (7), i.e., where K,V,f, S,W,g K,S, V,W, f,g, K,S K S S K, V,W V S W K V,W gv fw, f,g f g f g. The proof of the theorem will be given in Appendix A. Upon looking at the product alittle bit more carefully, we can find that the Lie algebra (B q,ṽ r,c (R)),, has alie subalgebra (B q,ṽ r,0),, ),for which everything corresponds to the isospectral case. Moreover, the center of an element of this Lie subalgebra is often Abelian. The above theorem exposes that alie algebraic structure hidden in the back of vector fields, Lax operators,andspectralevolutionlaws. Usually we just touch Lie algebraic structures of vector fields while discussing symmetries. If we analyze symmetries from the point of zero curvature equations, it is natural that we need to find and handle the Lie algebraic structure for all triples (K,V,f) B q,ṽ r,c (R), where K, V, and fare related to each other by zero curvature equations. In other words, we need to observe how two triples (K,V,f),(S,W,g) that appear in zero curvature equations connect with each other. The following theorem tells us that such akind of connection can be reflected by the Lie algebraic operation of B q,ṽ r,c (R) in Theorem. Its proof can be found in Appendix B.

5 404 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner Theorem : Algebraic structure of representations Let V,W Ṽ r,k,s B q, and f,g C (R). If two equalities, EV U UV U K fu, EW U UW U S gu, 9 0 hold, then we have athird equality, E V,W U U V,W U T f,g U, T K,S, where V,W, K,S and f,g are defined by (8), (), and (3), respectively. According to this theorem, we can easily find that if asystem u t K(n,u)is isospectral, i.e., t f 0, then the product system u t K,S for any S B q can be viewed to be still isospectral because we have f,g 0,g 0, where gis the evolution law corresponding u t S(n,u). Actually, the above theorem gives adiscrete zero curvature representation for aproduct system u t K,S, which possesses the same order matrix operators as ones for the original systems u t K(n,u)and u t S(n,u) see Refs. 0 and for the continuous case. Combining two theorems above can show the following. Corollary : The space defined by K,V,f B q,ṽ r,c R U K fu EV U UV, is alie subalgebra of B q,ṽ r,c (R) under the Lie product (7). This corollary tells us alie algebraic structure about zero curvature equations, which will help us to establish Lie algebraic structures of symmetries, including master symmetries. However, for zero curvature representations, some interesting problems remain to be solved. For example, assuming that two initial systems u t K(n,u) and u t S(n,u) have zero curvature representations possessing different-order matrix operators, we want to know whether there exist any zero curvature representations for the product system u t K,S and what structures the resulting zero curvature representations possess if the answer is yes. It is likely to be helpful in solving this problem to use the Kronecker product, as in Ref.. III. LAX OPERATORS AND MASTER SYMMETRIES Assume that we already have ahierarchy of isospectral integrable systems of discrete evolution equations of the form or of the form u t K k k K 0, V q, K 0 B q, k 0, u t K k JG k MG k, J,M V q, G k B q, k 0, 3 associated with adiscrete spectral problem, E U,,..., r T. 4 The second form 3 occurs more often than the first form, although it is simpler to deal with the first form. Generally speaking, the operator above is ahereditary symmetry operator see Ref. 3 for adefinition determined by the spectral problem 4 and J,M constitute a bi-hamiltonian pair. 4,5 Ifwe choose MJ when Jis invertible, then the form 3 may be changed into the form. Usually involves nonlocal operators, for example,,but J,M often involves only local operators. Our examples are all local Hamiltonian systems.

6 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 405 A. Structures of Lax operators For agiven X B q org B q,let us introduce an operator equation of Ṽ r : E X U U X U X U X, 5 in the case of, or an operator equation of J Ṽ r : E J G U U J G U MG U JG, 6 in the case of 3. We call them the characteristic operator equations ofu. The introduction of the operator equation 5 or 6 is an important step in our manipulation. Obviously, we can choose J (G) (JG) when MJ.We demand that 5 or 6 has solutions, and (X) or J (G)] is aparticular solution at X or at G. Usually 5 or 6 has infinitely many solutions once one solution exists, because we can construct others (X) fvfor any f C (R) when V V r C, solves the stationary discrete zero curvature equation (EV)U UV 0. The existence of solutions of (EV)U UV 0may result from the existence of an isospectral hierarchy associated with E U. Theorem 3: structure of Lax operators Let two matrices V 0,W 0 Ṽ r and two vector fields K 0, 0 B q or 0 J 0, 0 B q )satisfy EV 0 U UV 0 U K 0, EW 0 U UW 0 U 0 U. 7 8 If we define l,l,v k,k, and W l,l, as follows: l l 0, l or l J l M l, l B q, l, 9 V k k V 0 i W l l W 0 j k l k i K i or J G i l, k, 30 l j j or J j, l, 3 then V k,w l, satisfy EV k U UV k U K k, EW l U UW l U l l U, k,l 0. 3 Therefore for any the systems of discrete evolution equations u t K k and u t l possess the isospectral ( t 0) and nonisospectral ( t l )discrete zero curvature representations, U t EV k U UV k, U t EW l U UW l, respectively. ThetheoremshowsthattheLaxoperators associated with two hierarchies of interesting vector fields can be constructed simply by aunified form. Its proof is left to Appendix C. We are successful, thanks to introducing acharacteristic operator equation. The difficulty is now transferred to seeking asolution to the characteristic operator equation. However, this can automatically be solved on the basis of the structure of Lax operators of isospectral hierarchies, which will be seen in the next Sec. IIIB. B. Amethod for constructing master symmetries Now we focus our attention on the construction problem of master symmetries. Theorem 3 already shows the structure of Lax operators associated with the isospectral and nonisospectral

7 406 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner hierarchies refer to Ref. 6 for the continuous case. When an isospectral hierarchy or 3 is known, the theorem also provides us with amethod to construct anonisospectral hierarchy associated with the discrete spectral problem 4 by solving an initial discrete zero curvature equation 8 and solving acharacteristic operator equation 5 or 6. However, asolution to 5 or 6 may easily be generated by observing the resulting Lax operators. In fact, we have K k or J G k V k V k. 33 This may be checked, say, for the case of, as follows: V k V k k V 0 i k k i K i k k V 0 i k i K i K k, by using 30. Now by the first equality of 3, we may compute the following: E K k U U K k EV k EV k U U V k V k EV k U UV k EV k U UV k U K k U K k U K k U K k, for example, for the case of. Therefore we see that apossible solution (X) to 5 or J (G)to 6 may be generated by replacing the element K k or G k )in the equality 33 with X or G. The Lax operator matrices V k and V k are known, when the isospectral hierarchy has already been found. Thus we do not have to directly solve the characteristic operator equations, and then the whole process of construction of the nonisospectral hierarchy becomes an easy task: finding 0,W 0 tosatisfy 8 and computing V k V k tofind asolution to 5 or 6. The nonisospectral hierarchy 9 is exactly the master symmetries that we need to find. The reasons are that the product systems between the isospectral hierarchy and the nonisospectral hierarchy are still isospectral by Theorem, or as we said before in Sec. II, and that usually all systems of the isospectral hierarchy commute with each other. Therefore it is because there exists anonisospectral hierarchy that there exist master symmetries for isospectral systems of discrete evolution equations derived from agiven discrete spectral problem. In the next section, we shall in detail illustrate our construction process by three concrete examples and establish the corresponding centerless Virasoro symmetry algebras. IV. APPLICATIONS We illustrate only by three examples how to apply the method in the last section to construct master symmetries for various lattice hierarchies. To make the process clearer, we introduce aconception for agiven discrete spectral problem E U, which has an injective Gateaux derivative U. That is a uniqueness property similar to the one in the continuous case: 7 if (EV)U UV U K,V V r C,,K B q, and V u 0 0, then V 0, and further K 0 by the injective property of U. It means that if an isospectral ( t 0) Lax operator V equals zero at u 0, then so does V itself. Actually, this property corresponds to the uniqueness of an integrable hierarchy associated with aspectral problem E U. That is to say, when initial conditions and constants of inverse difference operators are fixed for example, as in 7 and 8, the associated isospectral hierarchy is uniquely determined. Most of the discrete spectral problems share the uniqueness property. The following three spectral problems are exactly examples that share such aproperty.

8 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 407 A. The Volterra lattice hierarchy Let us first consider the following discrete spectral problem: 5 u E U, U 0,. The corresponding isospectral integrable lattice hierarchy reads as 34 Here the matrix u t K k k K 0 u a k a k, K 0 u u u, k uc i V i 0 a i i c i a i solves the stationary discrete zero curvature equation (EV)U UV 0, where we choose the initial conditions and the hereditary operator is given by a 0, c 0 0, a u, c, u E u E u E u, 36 where (E ) isdetermined by 9. It is worth pointing out that each system in 35 is local and polynomially dependent on u, although the hereditary operator has nonlocal and nonpolynomially dependent features. The first discrete evolution equation is the Volterra lattice equation, 8 u n t u n u n u n, which is significantly generalized by Bogoyavlensky. 9 The associated Lax operators are as follows: V k k V a k 0, k 0, 37 c k where (P) denotes the selection of the terms with degrees of no less than. In particular, the initial isospectral Lax operator reads as u u V u The result until here can be obtained from 34 by using apowerful method in Ref. 30. We easily obtain the corresponding quantities in the nonisospectral ( t ) initial discrete zero curvature equation 8 : 0 0 u, W 0, 39 0 a k and asolution to the characteristic operator equation 5 by 33 :

9 408 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner where ij (X),i,j,, are given by X X X X X, 40 X E u E u E u X, X ue E u X, X E u X, X E u X E E u E u E u X. Now by Theorem 3, we obtain ahierarchy of nonisospectral discrete evolution equations u t l l 0,l 0, associated with the spectral problem 34. Let us now consider how to compute the corresponding symmetry algebra. The idea below can be applied to other cases. We first make the following computation at u 0: K k u 0 0, l u 0 l 0 u 0 0, W l u 0 l 0 V k u 0 k 0 0, k 0, l l0 0 0 l 0, n n, k k 0 V k u 0 k W l u 0 l l 0 0 l0 0 0, k 0, 0 0 l l n l l n, l 0, where V k,w l, are given as in Theorem 3and l0 represents the Kronecker symbol. While computing W l u 0,we need to note that ( 0 ) u 0 0, but ( l ) u 0 0,l. The other two examples below have asimilar character, too. Now we can find by the definition 8 of the product of two Lax operators that V k,v l u 0 0, V k,w l u 0 k V k l u 0, 4 W k,w l u 0 k l W k l u 0, k,l 0. For example, we can compute that

10 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 409 V k,w l u 0 V k u 0,W l u 0 l V k u 0 k l 0, k k l k 0 0 l0 n n 0 0 k l 0 0 k l 0 k l 0 k k l 0 k k l u 0. k V 0 0 Because V k,v l, V k,w l (k )V k l, W k,w l (k l)w k l, are all isospectral ( t 0)Lax operators belonging to V C, by Theorem, based upon 4 we obtain alax operator algebra by the uniqueness property of the spectral problem 34, V k,v l 0, V k,w l k V k l, 4 W k,w l k W k l, k,l 0. Further, due to the injective property of U, we finally obtain avector field algebra of the isospectral hierarchy and the nonisospectral hierarchy, K k,k l 0, K k, l k K k l, 43 k, l k l k l, k,l 0. This implies that l,l 0, are all master symmetries of each lattice equation u t K k0 isospectral hierarchy, and the symmetries, inthe K k, k 0, and l k 0 t K k0, l l, l 0, constitute asymmetry algebra of Virasoro type possessing the same commutator relations as 43. B. The Toda lattice hierarchy Second, let us consider the discrete spectral problem: 30 E U, U 0 v p, u p v,. 44 The corresponding isospectral integrable Toda lattice hierarchy 3 reads as Here u t K k k K 0 a k a k b k v b k, K 0 v v v p p, k 0. 45

11 40 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner solves (EV)U UV 0, in which we choose V i 0 a i b i vb i a i i a 0, b 0 0, a 0, b, and the hereditary operator is determined by p v E v E v v E v pe p E v. 46 The first system of discrete evolution equations is the Toda lattice, 3 p n t v n v n, v n t v n p n p n, up to atransform of dependent variables. The lattice hierarchy above has alocal tri-hamiltonian structure, u t K k J H k u M H k N H k u u, k 0, where the Hamiltonian operators J,M,N and the conserved quantities H k,defined by J 0 E v v E 0, M J J Ev ve v E p p E v v E E v, N M M p ve Ev ve Ev p p E v ve Ev E v v E ve Ev v E p v E p pe v, H 0 p lnv, H k b k, k, k where denotes the conjugate operator of. Note that this tri-hamiltonian structure may be established through atrace identity. 30 The corresponding Lax operators read as V k k V b k 0 k 0, , where the subscript denotes selecting the non-negative part. Hence, in particular, V 0 p v. 48 It is easy to find the corresponding quantities in the nonisospectral ( t ) initial discrete zero curvature equation 8 :

12 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 4 0 p v, W 0 n 0 0 n, 49 where n is the multiplication operator defined by 6, and asolution to the characteristic operator equation 5 by 33 : X X X X X, X X X, 50 where ij (X),i,j,, are given by X E E X p E v X, X E v X, X ve E v X, X E X. In this way, we obtain ahierarchy of nonisospectral systems of discrete evolution equations l l 0,l 0, associated with the spectral problem 44. In order to construct avector field algebra, we make asimilar computation at u 0: K k u 0 0, l u 0 l 0 u 0 0, V k u 0 k 0 n 0 W l u 0 l 0 n l0, k 0, n n, l 0, 0 0 k k 0 V k u 0 k, k 0, 0 0 n 0 W l u 0 l l 0 n l0 l l n l l n, l Now we can find through the product definition of, in 8 that V k,v l u 0 0, V k,w l u 0 k V k u 0, 5 W k,w l u 0 k l W k l u 0, k,l 0. Asimilar argument yields alax operator algebra by the uniqueness property of the spectral problem 44,

13 4 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner V k,v l 0, V k,w l k V k, 5 W k,w l k l W k l, k,l 0. And then because of the injective property of U, we obtain asemiproduct Lie algebra of the isospectral hierarchy and the nonisospectral hierarchy, K k,k l 0, K k, l k K k, 53 k, l k l k l, which gives rise to asymmetry algebra of the Virasoro type for the isospectral Toda hierarchy 45. C. Asub-KP lattice hierarchy Let us finally consider the discrete spectral problem: E U, b a U u a b c 0 0, c, 3, 54 which is equivalent to ( E b ae E c),asub-kp discrete spectral problem. 34 The corresponding isospectral integrable lattice hierarchy reads as u t K k JG k MG k, k 0, 55 where ahamiltonian pair J, Mand G,G 0,G are defined by 0 0 J E E 0 0 E c, 0 c E 0 M Eb be a a EcE E c a c ce E ce E ac ace b c c a c b c c, G 0 0, G 0 c b a, G c Eb b b ac E ac a Eb b Ec E c, where, are the difference operators: E, E.The first nonlinear system of discrete evolution equations is a n t c n c n, b n t a n c n a n c n, c n t c n b n b n.

14 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 43 We easily find the corresponding quantities in 7 and 8 : E E c 0 0 K 0 E ac, V 0 c 0 0 c E b E ac E c b, 0 J 0 M J a 0 n 3 0 c b M a b n c 3 3 c, n 0 0 W 0 0 n n. We can also obtain asolution to the characteristic operator equation 6 by 33 : G G 3 G J G G G 3 G G G G 3 G 3 G 33 G, G 3, 56 where ij (G),i,j,,3, are determined by G E E cg 3 EaG, G E G, 3 G G, G ceg b G, G E cg 3 EaG ag, 3 G G, 57 3 G E ce G E acg, 3 G E cg, 33 G E E cg 3 EaG ag b G. By Theorem 3, we get ahierarchy of nonisospectral systems of discrete evolution equations u t l l 0,l 0, associated with the spectral problem 54. In order to generate avector field algebra of the isospectral hierarchy and the nonisospectral hierarchy, we need the following quantities, which may be directly worked out: K k u 0 0, 0 u 0 J 0 u 0 0, l u 0 J l u 0 M l u 0 0, k 0, l, V k u 0 k , l n W l u 0 0 n 0 l0 l n 0 0 n, 3

15 44 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner k 0 0 k V k u 0 k , l n l W l u 0 l 0 n 0 l0 0 l n n. 0 0 l 3 l n Now we easily find, according to the product definition of,, that V k,v l u 0 0, V k,w l u 0 k V k l u 0, W k,w l u 0 k l W k l u 0, k,l 0. The same deduction leads to alax operator algebra, V k,v l 0, V k,w l k V k l, 58 W k,w l k l W k l, and further avector field algebra, K k,k l 0, K k, l k K k l, 59 k, l k l k l, which may generate amaster symmetry algebra possessing the same algebraic structure as 59. V. CONCLUSION AND REMARKS We have established an algebraic structure related to discrete zero curvature equations and further introduced asimple but systematic approach for constructing master symmetries of the first degree for isospectral lattice hierarchies associated with discrete spectral problems. The resulting algebraic structures also leads to an explanation of why there exist master symmetries of the first degree. Some complicated calculation in our construction is saved by using acharacteristic operator equation 5 or 6 and auniqueness property of discrete spectral problems. The crucial step is the construction of the corresponding nonisospectral lattice hierarchies, which can be found by solving an initial nonisospectral discrete zero curvature equation. Three lattice hierarchies are shown as illustrative examples, and the corresponding master symmetry algebras of the centerless Virasoro type are exhibited. Some of the results in this paper have been reported at SIDE II, UK. 35 It is worth noting that three examples described in the last section possess the same commutator relations between their isospectral and nonisospectral vector fields. In general, we have K k, l (k )K k, const., but the other two equalities of the whole Virasoro algebra do not change. This is also acommon phenomenon for continuous integrable hierarchies. 36,37 Furthermore, we may add anonisospectral master symmetry with t to the whole Virasoro symmetry algebra, but this often requires additional checking. For example, anonisospectral master symmetry with t of the sub-kp lattice hierarchy 55 is J (0,,0) T.On the

16 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 45 other hand, similar to the theory in Ref. 37, we may also choose an operator solution (X) or J (G)]satisfying (X) X 0 0 or J (G) G 0 0] all three examples in the last section have this property, and then we only need to compute V 0,W 0 u 0 soas to give Lax operator algebras at u 0 and finally give Lax operator algebras generally. In our discussion, in fact, we have not used the hereditary property of the recursion operator or thebi-hamiltonianpropertyofjand M, while we construct Virasoro symmetry algebras for integrable lattice hierarchies, and thus it can also be applied to lattice hierarchies that possess nonhereditary recursion operators. The advantage of our scheme is to fully utilize discrete zero curvature equations so that the whole process to generate master symmetries of the first degree becomes an easy task. There were also an algorithm implemented in MuPAD 38 and other direct tricks 3 5,39 tocompute master symmetries of first degree for systems of discrete evolution equations. However, our theory focuses on seeking an answer to the existence and structure problem of master symmetries of the first degree. We should mention that there exists alarge variety of other theories or methods to discuss integrable properties of systems of nonlinear discrete equations, which include Hamiltonian theory, 40,4 Bäcklund Darboux transformation, 4,43 The R-matrix method, 34,44 symmetry reduction, 45 etc. Moreover, we can consider the time discretization problem 46 and periodic initial and boundary value problems of time discretizations 47 for symmetry flows of systems of discrete evolution equations. The resulting difference equations and mappings should be useful in discussing the integrability of the underlying systems of discrete evolution equations themselves. We are also curious about the following natural problem: Are there any higher degree master symmetries for systems of discrete evolution equations that do not depend explicitly on the evolution variable? If the answerisyes,canweestablishany relations between those higher degree master symmetries and discrete zero curvature equations as we did for the first degree master symmetries? ACKNOWLEDGMENTS The authors are indebted to the referee for invaluable comments. One of the authors W. X. Ma would like to thank the Alexander von Humboldt Foundation of Germany, the City University of Hong Kong, and the Research Grants Council of Hong Kong for financial support. He is also grateful to J. Leon, W. Oevel, and W. Strampp for their helpful and stimulating discussions, and to R. K. Bullough and P. J. Caudrey for their warm hospitality during his visit at UMIST, UK. APPENDIX A: PROOF OF THEOREM Let (K i,v i,f i ) B q,ṽ r,c (R), i 3. Because the bilinearity and the skew symmetry of the product 7 are self-evident and we already know that the products defined by and 3 are Lie products, we only need to prove the following Jacobi identity: V,V,V 3 cycle,,3 0. A Let us first compute by 8 that V,V,V 3 V,V K 3 V 3 K,K V,V,V 3 f 3 V,V f,f V 3 V K K 3 V K K 3 V,V K 3 f V K 3 f V K 3 V 3 K,K V K,V 3 V K,V 3 V,V,V 3 f V,V 3 f V,V 3 f 3 V K f 3 V K f 3 V,V f f 3 V f f 3 V f f 3 V f f 3 V f,f V 3. A We need to use the following fundamental equalities: V K V K, V Ṽ r, K B q,

17 46 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner V,W V,W V,W, V,W Ṽ r, V,W K V K,W V,W K, V,W Ṽ r, K B q, V T V K S V S K, T K,S, V Ṽ r, K,S B q, which may be shown by adirect computation and the last equality of which is asimilar result as in Ref.. Now we can go on to compute that a 3 ª V K K 3 V K K 3 V 3 K,K V K K 3 V K K 3 V 3 K K V 3 K K, b 3 ª V,V K 3 V K,V 3 V K,V 3 V K 3,V V K 3,V V K,V 3 V K,V 3, c 3 ªf V K 3 f V K 3 f 3 V K f 3 V K f V K 3 f V K 3 f 3 V K f 3 V K, 3 d ªf V,V 3 f V,V 3 f 3 V,V f V,V 3 f V,V 3 f 3 V,V f 3 V,V, 3 e ªf f 3 V f f 3 V f f 3 V f f 3 V f,f V 3, f f 3 V f f 3 V f f 3 V f f 3 V f f V 3 f f V 3. Adirect check can result in that 3 cycle,,3 0, where * * a,b,c,d or e. Noting A, it follows therefore that V,V,V 3 cycle,,3 3 a 3 b 3 c 3 d 3 e V,V,V 3 cycle,,3 0, which is exactly the Jacobi identity A and thus completes the proof. APPENDIX B: PROOF OF THEOREM The proof is an application of the equalities 9 and 0 and the third equality, U K S U S K U T, T K,S, B which has been mentioned in the proof of the first theorem. We observe that Eq. 9 S Eq. 0 K g Eq. 9 f Eq. 0. The resulting equality reads as U K S U S K f,g U EV S U EV U S U S V UV S EW K U EW U K U K W UW K g EV U g EV U gu V guv f EW U f EW U fu W fuw. B

18 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 47 On the other hand, we have immediately E V,W U U V,W EV S U EW K U EV EW U EW EV U g EV U f EW U UV S UW K UVW UWV guv fuw. B3 It follows, therefore from B, B, and B3 that E V,W U U V,W U T f,g U E V,W U U V,W U K S U S K f,g U EV EW U V S gu EW EV U U K fu UVW UWV gu V fu W U S V U K W EV UW EW UV UVW UWV gu V fu W U S V U K W EV U UV fu U K W EW U UW gu U S V 0, which is what we need to prove. APPENDIX C: PROOF OF THEOREM 3 We prove two equalities in 3. The rest is obvious. We compute that EV k U UV k k EV 0 U UV 0 k i { E K i U U K i } i k k k U K 0 k i U K i U K i i k k U K 0 k i U K i U K i U K k, i k ; EW l U UW l l EW 0 U UW 0 l j E j U U j j l l l U 0 U l j U j U j j l l U 0 U l j U j U j j U l l U, l. Note that we have used the characteristic operator equation 5, but the situation in the case of 6 is completely similar. The proof is therefore finished. A.V. Mikhailov, A. B. Shabat, and V. V. Sokolov, in What is Integrability?, edited by V. E. Zakhalov Springer-Verlag, Berlin, 99, pp D.Levi and R. I. Yamilov, J. Math. Phys. 38,

19 48 J. Math. Phys., Vol. 40, No. 5, May 999 W. X. Ma and B. Fuchssteiner 3 B.Fuchssteiner and A. S. Fokas, Physica D4, A.S. Fokas, Stud. Appl. Math. 77, B.Fuchssteiner, Prog. Theor. Phys. 70, W.X. Ma, J. Phys. A3, H.H. Chen and Y. C. Lee, in Advances in Nonlinear Waves, Research Notes in Mathematics, edited by L. Debnath Pitman, Boston, 985, pp A.Y. Orlov and E. I. Schulman, Lett. Math. Phys., Y.S. Li and G. C. Zhu, J. Phys. A9, W.X. Ma, Phys. Lett. A79, ; Proceedings of the st International Conference on the Differential Geometry Methods in Theoretical Physics, edited by M. L. Ge World Scientific, Singapore, 993, pp ; J. Phys. A5, L79 L F.Calogero and A. Degasperis, Lett. Nuovo Cimento, W.L. Chan and Y. K. Zheng, Lett. Math. Phys. 4, W.Oevel, H. W. Zhang, B. Fuchssteiner, and O. Ragnisco, J. Math. Phys. 30, W.Oevel, B. Fuchssteiner, and H. W. Zhang, Prog. Theor. Phys. 8, H.W. Zhang, G. Z. Tu, W. Oevel, and B. Fuchssteiner, J. Math. Phys. 3, M.J. Ablowitz and J. F. Ladik, J. Math. Phys. 6, ; 7, D.Levi, M. Bruschi, and O. Ragnisco, Nuovo Cimento A58, M.Bruschi, D. Levi, and O. Ragnisco, J. Math. Phys., M.Bruschi and O. Ragnisco, Lett. Nuovo Cimento 3, W.X. Ma, J. Phys. A6, W.X. Ma, J. Phys. A5, W.X. Ma and F. K. Guo, Int. J. Theor. Phys. 36, B.Fuchssteiner, Nonlinear Anal. TMA 3, F.Magri, J. Math. Phys. 9, I.M. Gel fand and I. Y. Dorfman, Funct. Anal. Appl. 3, W.X. Ma, Chin. Sci. Bull. 38, W.X. Ma and W. Strampp, Phys. Lett. A85, V.Volterra, Lecons sur la Theorie Mathematique de la Lutte Pour la Vie Gauthier-Villars, Paris, O.I. Bogoyavlensky, Phys. Lett. A34, ; Math. USSR Izv. 3, G.Z. Tu, J. Phys. A3, B.G. Kupershmidt, Discrete Lax equations and differential-difference calculus, Asterisque, 3 Soc. Math. France, Paris, M.Toda, Theory of Nonlinear Lattices, nd enl. ed. Springer-Verlag, Berlin, Y.T. Wu and X. G. Geng, J. Math. Phys. 37, M.Blaszak and K. Marciniak, J. Math. Phys. 35, B.Fuchssteiner and W. X. Ma, An approach to master symmetries of lattice equations, in Symmetries and Integrability of Difference Equations, edited by P. A. Clarkson and F. W. Nijhoff Cambridge University Press, London, Y.Cheng and Y. S. Li, Chin. Sci. Bull. 36, W.X. Ma, J. Math. Phys. 33, B.Fuchssteiner, S. Ivanov, and W. Wiwianka, Math. Comput. Modell. 5, I.Y. Cherdantsev and R. I. Yamilov, Physica D87, M.Bruschi and O. Ragnisco, J. Math. Phys. 4, O.Ragnisco and P. M. Santini, Inverse Probl. 6, ; MBruschi, O. Ragnisco, P. M. Santini, and G. Z. Tu, Physica D49, M.Bruschi and O. Ragnisco, Phys. Lett. A9, W.Oevel, in Nonlinear Physics: Theory and Experiment, edited by E. Alfinito, M. Boiti, L. Martina, and F. Pempinelli World Scientific, Singapore, 996, pp C.Morosi and L. Pizzocchero, J. Math. Phys. 37, D.Levi and P. Winternitz, Phys. Lett. A5, ; J. Math. Phys. 34, Yu. B. Suris, J. Phys. A9, ; J. Math. Phys. 37, V.G. Papageorgiou, F. W. Nijhoff and H. W. Capel, Phys. Lett. A47, ; H. W. Capel, F. W. Nijhoff, and V. G. Papageorgiou, ibid. 55,