Symmetries and Lie algebra of the differential-difference Kadomstev-Petviashvili hierarchy

Symmetries and Lie algebra o f the differen tial-difference Kadomstev-P etvia s h vili hierarc h y Xian-long Sun ∗ , Da-jun Zhang † , Xiao-ying Zh u, Deng-yuan Chen Dep artment of Mathematics, Shanghai U niversity, Shanghai 200444, P.R.China No v em b er 18 , 2018 Abstract By int ro ducing suitable non-isosp ectra l flows we construct tw o sets of symmetries for the isosp ectral differential-difference Kadomstev-Petviashvili hierarch y . The symmetries form an infinite dimensional Lie algebr a. Keyw ords: non-isosp ectra l flows, the differential-difference Kadomstev-Petviashvili equa- tion, symmetries, Lie alg ebra. 1 In tro duc tion Searc hing for symm etries and Lie algebraic str u cture is an imp ortan t and i nteresting topic in in tegrable systems[1]. V ariet y of metho d s ha v e b een dev elop ed to obtain infinitely man y symmetries and their Lie algebraic s tructures for Lax integ rable systems[2]-[15], f or b oth (1+1)- dimensional and high-dimensional cases. One of efficient w a ys is to use Lax repr esen tation of isosp ectral and non-isosp ectral flows (cf. [4]-[7],[11, 12]), r ather than recursion op erators, and this approac h has b een extended to high dimensional contin uous integrable systems[15]. In this p ap er, w e consider the symmetries and their Lie algebra f or the differential- differen ce Kadomstev-P etviash vili (D∆KP) h ierarc hy b y means of Lax representati on approac h. The (isosp ectral) D∆KP hierarch y is deriv ed follo wing the basic fr ame of Sato’s theory starting from a quasi-difference op erator [16, 18]. Ho w ever, d ue to the lac k of a neat form of discrete deriv ativ es, the non-isosp ectral flows d o not ha v e regular asymptotic p rop erties as the contin u- ous n on-isosp ectral flo ws d o. W e ha v e to c ho ose suitable time ev olution for sp ectral parameter so that w e can get suitable non-isosp ectral fl o ws whic h can b e u sed to constru ct symm etries. This pap er is organized as follo ws. In Sec.2, we construct isosp ectral and -non-isosp ectral D∆KP hierarc hies from a quasi-difference op erator. In Sec.3, w e construct t w o sets of symmetries and their Lie algebra for the isosp ectral D∆KP h ierarc hy . 2 The isosp ectral and non-isosp ectral D ∆ KP hierarc hies Let us consider the difference anal ogy of a qu asi-differential op erator[16] L = ∆ + u 0 + u 1 ∆ − 1 + · · · + u j ∆ − j + · · · , (2.1) ∗ E-mail add ress: xlongs@yahoo.cn † Corresponding auth or. E-mail address: djzhang@staff.sh u.edu.cn 1 where u s =: u s ( n, t ) = u s ( n, t 1 , t 2 , · · · ) ( s = 0 , 1 , 2 , . . . ), t = ( t 1 , t 2 , · · · ), eac h u s v arnishes r apidly when | n | → ∞ , ∆ denotes the forwa rd difference op erator defin ed b y ∆ f ( n ) = ( E − 1) f ( n ) = f ( n + 1) − f ( n ) and the shift op erator E d efined b y E f ( n ) = f ( n + 1). The operators ∆ and E are connected by ∆ = E − 1 and ∆∆ − 1 = ∆ − 1 ∆ = 1. Ob viously , the r th -p o wer of L can b e expressed as L r = X j ≤ r p r,j ( u )∆ j . (2.2a) where the co efficien ts p r,j ( u ) are uniquely determined by the coord inates u j ( j = 0 , 1 , 2 , . . . ) and their differences. Here by u we d enote ( u 0 , u 1 , · · · ) T . L r can b e sep erated into ( L r ) + = n X j =0 p r,j ( u )∆ j , ( L r ) − = L r − ( L r ) + , (2.2b) where ( ) + denotes the nonnegativ e p art of ∆ an d ( ) − the residual p art. In general, th e isosp ectral flo ws can b e obtained from the compatibilit y of Lφ = η φ, (2.3a) φ t s = A s φ ; (2.3b) i.e L t s = [ A s , L ] , (2.4a) where η t s = 0, A s = ( L s ) + and obvio u sly A s satisfies the b ound ary cond ition A s | u =0 = ∆ s . (2.4b) The fir st few explicit forms of A s and equations giv en by the Lax equation (2 .4a ) are[18] A 1 = ∆ + u 0 , (2.5a) A 2 = ∆ 2 + (∆ u 0 + 2 u 0 )∆ + (∆ u 0 + u 2 0 + ∆ u 1 + 2 u 1 ) , (2.5b) A 3 = ∆ 3 + a 1 ∆ 2 + a 2 ∆ + a 3 , (2.5c) · · · , with a 1 = ∆ 2 u 0 + 3 ∆ u 0 + 3 u 0 , (2.6a) a 2 = 2∆ 2 u 0 + 3 ∆ u 0 + 3 u 2 0 + 3 u 0 ∆ u 0 + (∆ u 0 ) 2 + 3 u 1 + 3 ∆ u 1 + ∆ 2 u 1 , (2.6b) a 3 = ∆ 2 u 0 + 5 u 0 u 1 + 3 u 0 ∆ u 0 + u 3 0 + (∆ u 0 ) 2 + ∆ u 0 ∆ u 1 + 3 u 0 ∆ u 1 + u 1 ∆ u 0 + u 1 E − 1 u 0 + 2 ∆ 2 u 1 + 3 ∆ u 1 + 3 u 2 + 3 ∆ u 2 + ∆ 2 u 2 ; (2.6c) u 0 ,t 1 = q 10 = ∆ u 1 , (2.7a) u 1 ,t 1 = q 11 = ∆ u 1 + ∆ u 2 + u 0 u 1 − u 1 E − 1 u 0 , (2.7b) u 2 ,t 1 = q 12 = ∆ u 3 + ∆ u 2 + u 0 u 2 + u 1 E − 1 u 0 − u 2 E − 2 u 0 − u 1 E − 2 u 0 , (2.7c) · · · ; 2 u 0 ,t 2 = q 20 = ∆ 2 u 1 + 2 ∆ u 2 + ∆ 2 u 2 + u 1 ∆ u 0 + 2 u 0 ∆ u 1 +(∆ u 0 )∆ u 1 + u 0 u 1 − u 1 E − 1 u 0 , (2.8 a) u 1 ,t 2 = q 21 = ∆ 2 u 1 + 2 ∆ u 2 + 2∆ 2 u 2 + 2 ∆ u 3 + ∆ 2 u 3 + 2 u 0 ∆ u 1 + ∆ u 0 ∆ u 1 +2 u 0 u 2 + u 2 ∆ u 0 + 2 u 0 ∆ u 2 + ∆ u 0 ∆ u 2 + u 1 ∆ u 0 + u 1 u 2 0 + u 2 1 + u 1 ∆ u 1 − u 1 E − 2 u 0 + u 1 E − 1 u 0 − u 1 E − 1 u 1 − u 2 E − 1 u 0 − u 2 E − 2 u 0 − u 1 E − 1 u 2 0 , (2.8b) · · · . F rom (2.7 ), w e obtain u 1 = ∆ − 1 ∂ u 0 ∂ t 1 , (2.9a) u 2 = ∆ − 2 ∂ 2 u 0 ∂ t 2 1 − ∆ − 1 ∂ u 0 ∂ t 1 − E − 1 u 0 ∆ − 1 ∂ u 0 ∂ t 1 + ∆ − 1 ( u 0 ∂ u 0 ∂ t 1 ) , (2.9b) · · · . Eliminating u 1 , u 2 , · · · fr om (2.7a), (2.8a ) , · · · , one can obtain ( u 0 = u, t 1 = y )[16, 18] u t 1 = K 1 = u y , (2.10a ) u t 2 = K 2 = (1 + 2∆ − 1 ) u y y − 2 u y + 2 uu y , (2.10b) · · · , whic h are isosp ectral D∆KP hierarc h y where Eq.(2.10b) is the w ell-kno wn D∆KP equati on. T o get the τ -symmetries w e need to in tro d uce the non-isosp ectral D∆KP hierarch y . In this case, we set ∗ η t r = η r + η r − 1 . (2.11) Then the Lax equation turns out to b e L t r = [ B r , L ] + L r + L r − 1 , (2.12a ) where B r = b 0 ∆ r + b 1 ∆ r − 1 + · · · + b r , ( r > 0) (2.12b) ∗ One may wonder that (2.11) is a linear combination and so is the non- isosp ectral flow σ r . Actually , in the Lax representation approach w e need σ r | u =0 = 0. Supp ose th at w e start from a general form η t r = aη α + bη β , with constants a, b and integers α, β . Then the Lax equation is L t r = [ B r , L ] + aL α + bL β . Noting th at L | u =0 = ∆ and (2.12c ) the r.h.s. of th e ab o ve equ ation b ecomes − ∆ r − ∆ r − 1 + a ∆ α + b ∆ β when u = 0, whic h means we have to take a = b = 1 , α = r, β = r − 1 so that it v anishes. H ence we need th e time evol ut ion (2.11). 3 in whic h b i ( i = 0 , 1 , 2 , · · · , r ) are un determined fun ctions of co ordinates u j ( j = 0 , 1 , 2 , . . . ) and their differences. B r is imp osed the b oundary condition B r | u =0 = t 1 ∆ r + n ∆ r − 1 , (2.12c ) and then the b oth sides of the Lax equ ation (2.12a) go to zero w hen u → 0. The fir st few of B r and equations give n by (2.12a) are B 1 = t 1 A 1 + n, (2.13a ) B 2 = t 1 A 2 + n ∆ + nu 0 + ∆ − 1 u 0 , (2.13b) B 3 = t 1 A 3 + n ∆ 2 + (2 nu 0 + n ∆ u 0 + ∆ − 1 u 0 )∆ + u 0 ∆ − 1 u 0 + 2 nu 1 + n ∆ u 0 + n ∆ u 1 − 2 u 1 − ∆ u 1 + nu 2 0 − u 2 0 + ∆ − 1 ( u 1 − u 0 + u 2 0 ) , (2.13c ) · · · ; u 0 ,t 1 = t 1 q 10 + u 0 , (2.14a ) u 1 ,t 1 = t 1 q 11 + 2 u 1 , (2.14b) u 2 ,t 1 = t 1 q 12 + u 1 + 3 u 2 , (2.1 4c) · · · ; u 0 ,t 2 = t 1 q 20 + n ∆ u 1 + u 2 0 − u 0 + 3 u 1 + ∆ u 1 , (2.15a ) u 1 ,t 2 = t 1 q 21 + n ∆ u 1 + n ∆ u 2 + ( n + 1) u 0 u 1 + u 1 ∆ − 1 u 0 + 2 u 1 + ∆ u 2 +3 u 2 + (2 − n ) u 1 E − 1 u 0 − u 1 E − 1 ∆ − 1 u 0 + ∆ u 1 , (2.15b) · · · ; u 0 ,t 3 = t 1 q 30 − ∆ 2 u 0 + nu 0 ∆ u 1 + n ∆( u 0 u 1 ) + ∆ − 1 u 0 ∆ u 1 + nu 0 u 1 + n ∆ 2 u 1 + u 1 ∆ − 1 u 0 − u 0 ∆ u 0 − 4 n ∆ u 1 − 2 u 1 − ∆ u 0 − ∆ u 1 + u 0 − 3 u 2 0 − nu 1 E − 1 u 0 + u 1 E − 1 u 0 + u 0 u 1 − u 1 E − 1 ∆ − 1 u 0 + n ∆ 2 u 2 +2∆ u 2 + 2 u 2 , (2.16) · · · . Here A l and q ij are describ ed b y (2.5 ), (2.7) and (2.8) resp ectiv ely . Then su bstituting (2.9) with t 1 = y into (2.14a) , (2.15a) and (2.1 6 ) yiel d s( u 0 = u ) u t 1 = σ 1 = y K 1 + u, (2.17a ) u t 2 = σ 2 = y K 2 + (1 + n ) u y + 3 ∆ − 1 u y + u 2 − u, (2.17b) · · · , in whic h, K s are giv en by (2.1 0 ). These equations constitute the non-isosp ectral h ierarc hy of the D∆KP system. The obtained isosp ectral and non-isosp ectral D∆KP hierarc hies can b e expr essed through Lax equations in the follo wing form L ′ [ K s ] = [ A s , L ] , (2.18a ) A s | u =0 = ∆ s ; (2.18b) 4 L ′ [ σ r ] = [ B r , L ] + L r + L r − 1 , (2.19a ) B r | u =0 = t 1 ∆ r + n ∆ r − 1 , (2.19b) whic h w e call Lax r epresen tations of flo ws. 3 Lie algebra structure of the D ∆ KP system In this section, we b egin with a discussion of Gateaux der iv ativ e concerning the quasi- difference op erator. Let ∂ t j = ∂ ∂ t j and F denote a linear space constructed b y all real functions f = f ( u ) dep ending on n, t and d eriv ati ves and differences of u . f ( u ) is C ∞ differen tiable w.r.t. t and n , and v anishes r apidly when | n | → ∞ . The Gateaux deriv ativ e of f ( u ) ∈ F in direction h ∈ F w.r.t. u is defined as f ′ [ h ] = d dε f ( u + εh ) | ε =0 , (3.1) from wh ich, F form s a Lie alg ebra according to the follo wing Gate aux comm utator J f , g K = f ′ [ g ] − g ′ [ f ] , (3.2) where f , g ∈ F . F or a quasi-difference op erator P ( u ) = X j ≤ s p j ( u )∆ j , (3.3) its Gateaux deriv at ive in d irection h with resp ect to u is defined by P ′ [ h ] = X j ≤ s p ′ j [ h ]∆ j . (3.4) Besides, usin g ( p ′ j ) ′ [ f ] g = ( p ′ j ) ′ [ g ] f , (3.5) one can get[19 ] ( P ′ [ f ]) ′ [ g ] − ( P ′ [ g ]) ′ [ f ] = P ′ [ J f , g K ] . (3.6) In addition, it is easy to p ro ve the follo wing le mm a. Lemma 1. F or the quasi-differ enc e op er a tor L define d in (2.1) , B in the form (2.12b) and X ∈ F , the e quation L ′ [ X ] = [ B , L ] , B | u =0 = 0 (3.7) only admits zer o solution X = 0 , B = 0 . Then, fr om the Lax represen tations (2.18)-(2.19 ), we hav e the follo wing prop ert y . Theorem 1. Supp ose that h A s , A r i = A ′ s [ K r ] − A ′ r [ K s ] + [ A s , A r ] , (3.8a) h A s , B r i = A ′ s [ σ r ] − B ′ r [ K s ] + [ A s , B r ] , (3.8b) h B s , B r i = B ′ s [ σ r ] − B ′ r [ σ s ] + [ B s , B r ] , (3.8c) 5 then we have L ′ [ J K s , K r K ] = [ h A s , A r i , L ] , (3.9a) L ′ [ J K s , σ r K ] = [ h A s , B r i , L ] , (3.9b) L ′ [ J σ s , σ r K ] = [ h B s , B r i , L ] + ( s − r ) L s + r − 1 + 2( s − r ) L s + r − 2 + ( s − r ) L s + r − 3 , (3.9c) and h A s , A r i| u =0 = 0 , (3.10a ) h A s , B r i| u =0 = s ∆ s + r − 1 + s ∆ s + r − 2 , (3.10b) h B s , B r i| u =0 = ( s − r )  t 1 ∆ s + r − 1 + ( t 1 + n )∆ s + r − 2 + n ∆ s + r − 3  . (3.10c ) Pr o of. W e only pro v e the equalities (3.9c) an d (3.10c) , the others can b e obtained in a similar w a y . T aking the Gatea u x deriv ativ e of (2.19a) in th e direction σ s with resp ect to u , and noting L r ′ [ σ s ] = [ B s , L r ] + r L s + r − 1 + r L s + r − 2 (3.11) and [[ B s , B r ] , L ] = [ B s , [ B r , L ]] − [ B r , [ B s , L ]] , (3.12) w e ha v e ( L ′ [ σ r ]) ′ [ σ s ] =[ B ′ r [ σ s ] , L ] + [ B r , [ B s , L ]] + [ B r , L s ] + [ B r , L s − 1 ] + [ B s , L r ] + r L s + r − 1 + r L s + r − 2 + [ B s , L r − 1 ] + ( r − 1) L s + r − 2 + ( r − 1) L s + r − 3 . (3.13) Similarly , ( L ′ [ σ s ]) ′ [ σ r ] =[ B ′ s [ σ r ] , L ] + [ B s , [ B r , L ]] + [ B s , L r ] + [ B s , L r − 1 ] + [ B r , L s ] + sL s + r − 1 + sL s + r − 2 + [ B r , L s − 1 ] + ( s − 1) L s + r − 2 + ( s − 1) L s + r − 3 . (3.14) Then (3.13) co u p led with (3.14 ) yield ( L ′ [ σ s ]) ′ [ σ r ] − ( L ′ [ σ r ]) ′ [ σ s ] = [ h B s , B r i , L ] + ( s − r )( L s + r − 1 + 2 L s + r − 2 + L s + r − 3 ) , (3.15) whic h giv es (3 .9c ) by us ing (3.6 ). Next, noting that K s , σ r ∈ F , i.e., K s | u =0 = σ r | u =0 = 0, from (3.8c) we obtain (3.10 c ) im m ediately . W e complete the pro of. With the ab o v e theorem in hand , the algebraic relati on of flows K s and σ r can b e d eriv ed. Theorem 2. Th e flows K s and σ r form a Lie algebr a with structur e J K s , K r K = 0 , (3.16a ) J K s , σ r K = sK s + r − 1 + sK s + r − 2 , (3.16b) J σ s , σ r K = ( s − r )( σ s + r − 1 + σ s + r − 2 ) , (3.16c ) wher e s, r ≥ 1 and we set K 0 = σ 0 = 0 . 6 Pr o of. In the ligh t of (3.7) only admitting zero solution, (3.9a) coupled with (3.10a) p ossesses the same p rop erty as w ell, which means (3.16a) holds. Next, taking θ = J K s , σ r K − sK s + r − 1 − sK s + r − 2 , (3.17a ) ˜ A = h A s , B r i − sA s + r − 1 − sA s + r − 2 , (3.17 b) it then follo ws from (3.9b), (3.10b ) and the isosp ectral Lax represen tation (2.18) that L ′ [ θ ] = [ ˜ A, L ] , ˜ A | u =0 = 0 , (3.18) whic h has only zero solution θ = 0 and ˜ A = 0, and then means (3.16b ) is true. Similarly , taking ω = J σ s , σ r K − ( s − r )( σ s + r − 1 + sσ s + r − 2 ) , (3.19a ) ˜ B = h B s , B r i − ( s − r )( B s + r − 1 + B s + r − 2 ) , (3.19b) and noting that ˜ B | u =0 = 0 together with (2.19), (3.9c) and (3.10c), we then hav e L ′ [ ω ] = [ ˜ B , L ] , ˜ B | u =0 = 0 . (3.20) Hence w e get ω = 0 and ˜ B = 0, wh ic h shows that (3.16c) is also correct. Thus w e complete th e pro of. Based on Th eorem 2, the symmetries and their algebraic structure for the isosp ectral D∆KP hierarc hy u t s = K s can b e derive d immediately . Theorem 3. The isosp e ctr al D ∆ KP hier ar chy u t s = K s c an have tow sets of symmetries, K - symmetries { K l } and τ -symmetries, τ s r = st s K s + r − 1 + st s K s + r − 2 + σ r ( l = 1 , 2 , . . . , r = 1 , 2 , . . . ) , which form a Lie algebr a with structur e J K l , K r K = 0 , (3.21a ) J K l , τ s r K = l ( K l + r − 1 + K l + r − 2 ) , (3.21b) J τ s l , τ s r K = ( l − r )( τ s l + r − 1 + τ s l + r − 2 ) , (3.21c) wher e l , r , s ≥ 1 and we set K 0 = τ s 0 = 0 . Esp e ci al ly for the D ∆ K P e quation (2.1 0b ) its symmetries ar e K l and τ r = 2 tK r +1 + 2 tK r + σ r . W e end up this section b y the follo wing t wo remarks. First, the new time-dep endence (2.11) of the sp ectral parameter η leads to the new algebra s tructure (3.21), w h ic h is different fr om the cen treless Kac-Mo o dy-Virasoro alge br a (cf.[20]) of the D∆KP equation giv en in [18], and also d ifferen t from the cen treless Kac-M o o d y-Virasoro alge b ra of the KP hierarch y obtained in [15]. Besides, { K 1 , K 2 , τ s 1 } comp ose a subalgebra. Th is agrees with the s y m metry algebra of the D∆K P equation obtained in [21] where τ 2 1 pro vides an in v ariabilit y for the D∆KP equation under a combined Galilean-scalar transf ormation, and no w τ 2 1 has got its clear con text in the Lax r epresen tation approac h. The second remark is on the r elation b et w een th e D∆KP equation and the KP equation. In fact, the D∆KP equation w as originally prop osed b y Date , et.a l.[22 ]. It wa s derive d from a bilinear identit y (discretized b y partially imp osing Miw a’s transformation on con tinuous exp onent ial functions) which is related to the KP hierarc hy . How ever, since in the discrete exponential the discrete v ariables (eg. n, m, l ) app ear sym metrically and do not repre- sen t disp ersion relatio n as in the cont inuous one, th erefore there are (sometimes complicated) 7 v ariable com bination and transformation inv olved in the con tinuous limit pr o cedure. There ha ve b een many results on th e D∆KP equation, suc h as bilinear form[22], Sato’s approac h [16]-[18], Casoratian solutions[21, 24], gauge transformation and d ouble Casoratian solutions[23], and also sym metries in the pr esen t pap er. T he r elations b et wee n these results and those of the KP equation will b e in vestig ated in d etail elsewhere in terms of conti nuous limit. 4 Conclusion In this p ap er, by in tro d ucing suitable time-dep enden ce η t r = η r + η r − 1 for the sp ectral parameter η , we obtained non-isosp ectral D∆KP flo ws { σ r } whic h satisfy σ r | u =0 = 0. Th is enables us to construct K -symmetries and τ -symmetries for the isosp ectral D∆KP hierarc hy through the Lax represent ation approac h. The obta ined symmetries are pro v ed to form a Lie algebra. Ac kno wledgmen ts This pro j ect is su pp orted by th e National Natur al Science F oundation of Chin a (10671 121) and Shanghai Leading Academic Discipline Pro ject (No.J50101). 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