Integrals of open 2D lattices

In tegrals of op en 2D lattices Dmitry K. Demsk o i Academia Sinica , T aip ei, T aiwan e-mail: demskoi@math.sinica.e d u.t w Abstract W e present an explicit form u la for in tegrals of the op en 2D T o da lattice of t yp e A n . This form ula is applicable for v ario u s reductions o f this lattice. T o illustrate the concept w e fin d in tegrals o f the T oda G 2 lattice . W e also reveal a connection b et w een the op en T oda A n and Shabat-Y amilo v lattice s . 1 In tro d uction The most w ell kno wn re presen ta tiv es of the class of exactly solv able h yp er- b olic systems are open 2D T o da lattices u i,tx = exp  A i j u j  , j = 1 , . . . , n, (1) where ( A i j ) is the Cartan matrix o f a simple Lie algebra. A general metho d of in tegration of suc h systems was prop osed b y Leznov and Sa velie v [1]. The v ersion of lattice (1) c o r r esp onding to classical series A n w as kno wn to Darb oux who also fo und its general solution. Exactly solv able systems ha v e a few c haracteristic prop erties that set them apart from the m ultitude of all other systems. These include in part icular: finiteness of ch a ins of generalized Laplace inv arian ts [2, 3], presence of non-trivial in tegrals, and generalized symmetries. It is also know n that the latter tw o structures are related to eac h other b y means of a differential o perato r mapping integrals to symmetries [4]. Systems (1) ha v e long b een kno wn to p ossess the complete sets of integrals [5], ho w ev er, the explicit form ulas for them ha ve nev er been presen ted apart from a few particular cases. In this pap er w e suggest a solution of this problem for the A n T o da lattice and its reductions. The simplest h yp erb olic in tegrable equation is the d’Alem b ert equation w tx = 0 . (2) 1 It is not obvious, ho w ever, ho w this equation can b e generalized to the case of higher order equations o r s ystems of equations when it is written in this form. If w e in tro duce the new dep enden t v ariable w = log( u ), then (2) b ecomes det  u u t u x u tx  = 0 . (3) The ob vious generalisation of (3) is the 2 n -th order equation det      u u t . . . u t...t u x u tx . . . u t...tx . . . . . . . . . . . . u x...x u tx...x . . . u t...tx...x      = 0 (4) where u t...tx...x = ∂ n x ∂ n t u . F or brevit y w e write equation (4) as W n +1 ( u ) = 0 . This equation is cen tral for our further considerations. In the sequel it will b e referred a s t he higher d’Alem b ert equation. Equation (4) can b e view ed as the zero condition for the W ronskians: W ( u, u x , . . . , u x...x )( t ) = 0 , W ( u, u t , . . . , u t...t )( x ) = 0 . F rom this w e can deduce the gene r a l solution u = X 1 ( x ) T 1 ( t ) + X 2 ( x ) T 2 ( t ) + · · · + X n ( x ) T n ( t ) , (5) where X i , T i are arbitra ry functions. Apparen tly there a r e different w ay s of writing (4) as a system of equations. It w as shown by Darb oux that quan tities W j ( u ) satisfy the recurren t r elation (ln W j ) xt = W j − 1 W j +1 W − 2 j , W 0 = 1 , W 1 = u. (6) This relation initially app eared in connection with s t udying the Laplace in- v ariants of h yp erb olic equations. Up on introducing the new quan tities W j ( u ) = ex p ( w j ) , (7) equation (4) is transformed into the syste m w j,xt = exp( w j − 1 − 2 w j + w j +1 ) , w n,xt = 0 , j = 1 . . . n − 1 (8) 2 with the b oundary conditio n w 0 = 0. Note that w e can eliminate w n from this sys tem by means of the transformation w n − 1 → w n − 1 + 2 3 w n , w n − 2 → w n − 2 + 1 3 w n . The resu lt ing system reads w j,xt = exp( w j − 1 − 2 w j + w j +1 ) , w 0 = w n = 0 , j = 1 . . . n − 1 . (9) System (9 ) is often referred as the op en (finite, non-p erio dic) A n − 1 T o da lattice. Therefore equation (4) and the op en A n T o da lattice are arguably the simplest generalisations o f the d’Alembert equation for higher order equations and systems of equations. W e ma y w onder if there are other sy stems related to equation (4) whic h hav e reasonably compact form? Apparen tly any other system reducible to equation (4) m ust also b e related to (8). The other question w e are in terested in is: How other structures related the solv abilit y of the higher d’Alem b ert equation are related to those of corresp onding systems of equations. Ob viously the inte g rals of equation (4) are in tegrals of (8) as w ell. Surprisingly the f o rm ula fo r the scalar equation is muc h simpler than for the corresp onding sys tem of equations. 2 In tegrals of op e n T o da Lattices Let us review some pro p erties of higher d’Alem b ert equation. W e hav e al- ready indicated its general solution due to Darb oux, no w we wan t to sho w that it also p ossesses n + n indep enden t integrals. Note that due to the symmetry x ↔ t it suffices to presen t t − in tegrals only . Recall that a func- tion o f unkno wns and their deriv ativ es is called t − in tegr a l if it satisfies the c haracteristic equation D t ω = 0, see [2] for detailed exposition. Prop osition. Equation (4) a dmits n indep enden t t − in tegrals of the form ω i = W n,i W n , (10) where W n,i is the determinan t deriv ed from W n b y replacing its i -th r ow by ( ∂ n x u, ∂ n x ∂ t u, ∂ n x ∂ 2 t u, . . . , ∂ n x ∂ n − 1 t u ) . P erhaps the easie st w ay to prov e this statemen t is to sho w that in tegrals (10) b ecome functions of one v ariable after substituting expres sion of general solution (5) into (10). Indeed, after substituting w e get W n,i = W ( T 1 , . . . , T n )( t ) P j X ( n ) j C ij ( x ) , W n = W ( X 1 , . . . , X n )( x ) W ( T 1 , . . . , T n )( t ) , 3 where C ij ( x ) is the cof a ctor of the en try ( W ( X 1 , . . . , X n )( x )) ij . Therefore the integrals of (4) a re parametrized b y functions X i ( x ) the follo wing w a y ω i = P j X ( n ) j C ij ( x ) W ( X 1 , . . . , X n )( x ) . Indep enden ce of t hese in tegrals follo ws from form ula (10 ) itself. Remark. F orm ula (10) provid es us with the explicit expres sion for in te- grals not only of (4), but of (8) as w ell. Note that u = W 1 = exp( w 1 ), and hence ω i can be ex pressed in terms of the single quan tity w 1 = log( u ). Instead of (4) and (8) w e could hav e started with system (9 ). One can sho w that the latter is equiv alen t to the scalar equation [1] W n ( u ) = ( − 1) n ( n − 1) / 2 . In t his case the formulas for integrals should b e mo dified: ω i = W ∗ n,i W n , (11) where W ∗ n,i is the determinan t deriv ed from W n b y replacing its i -th r ow by ( ∂ n +1 x u, ∂ n +1 x ∂ t u, ∂ n +1 x ∂ 2 t u, . . . , ∂ n +1 x ∂ n − 1 t u ) . F ormulas (10) and (11) can b e used for finding integrals of v ar ious lattices deriv ed fro m equation (4). This includes the 2D T o da lattice corresp o nding to Cartan matrix of the Lie Algebra A n and its reductions, other v ersion o f the 2D T o da lattice giv en by (18), and also t he Shabat-Y amilov lat t ice (see b elo w). These formulas express inte g r als in terms of one unkno wn v ariable and its deriv ative s. The formulas a re therefore v a lid as long as this v ariable is not affected by reduction or b y a c hang e of v ariables. Otherwise the fo rm ulas m ust b e mo dified accordingly . Let us no w demonstrate this with ex a mple of the G 2 T o da lattice p tx = exp( − 2 p + q ) , q tx = exp(3 p − 2 q ) , (12) where w 1 = p, w 2 = q . Lattice (12) is kno wn to b e a reduction of A 6 T o da lattice, therefore its in tegrals are giv en by form ula (11) in whic h all mixed deriv ative s should b e replaced according to system (12). F rom form ulas (7) and (11) w e ha ve ω i = W ∗ 6 ,i (exp( p )) W 6 (exp( p )) 4 and th us ω 6 = q 2 x + 3 p xx − 3 q x p x + q xx + 3 p 2 x , ω 5 = 5 ω 6 ,x , ω 4 = 6 ω 6 ,xx − ω 2 6 , ω 3 = 4 ω 6 ,xxx − 3 ω 6 ω 6 ,x , ω 2 = ω ∗ 2 + ω 6 ,xxxx − ( ω 2 6 ) xx / 2 , ω 1 = ω ∗ 2 ,x / 2 , where ω ∗ 2 = 2 p 6 + 2 p 2 p 4 − 60 p 2 1 p 2 2 + 12 p 4 1 p 2 − 28 p 3 1 p 3 − 6 q 4 p 2 − 6 p 3 q 3 − 13 q 2 2 p 2 1 +2 p 2 q 4 1 − 2 p 1 ( q 5 − 2 p 5 ) + 14 q 3 p 3 1 − 10 p 2 q 2 2 + 4 q 2 1 q 2 p 2 + 26 p 1 p 2 q 3 + p 2 3 +2 q 4 p 2 1 + 30 p 3 q 1 p 2 1 − 24 p 3 2 − 4 q 2 p 4 + 30 p 1 p 3 q 2 − 6 p 1 q 2 1 p 3 − 14 q 3 p 2 1 q 1 +4 q 1 q 2 2 p 1 − 38 p 1 q 1 q 2 p 2 + 18 p 1 p 4 q 1 − 12 q 3 p 2 q 1 + 2 q 2 1 p 1 q 3 + 36 p 2 p 3 q 1 − 18 p 1 p 2 q 3 1 − 12 q 1 p 3 q 2 + 36 p 1 q 1 p 2 2 + p 2 1 ( q 1 − p 1 ) 2 ( q 1 − 2 p 1 ) 2 − 2 q 2 p 4 1 − 2 p 2 2 q 2 1 − 10 q 2 q 3 p 1 − 14 p 4 p 2 1 + 34 q 2 p 2 2 − 16 q 2 1 q 2 p 2 1 − 4 p 4 q 2 1 + 4 q 3 1 q 2 p 1 − 48 p 3 1 p 2 q 1 − 72 p 1 p 2 p 3 − 4 q 1 q 4 p 1 + 16 q 1 q 2 p 3 1 + 50 p 2 1 p 2 q 2 1 + 68 p 2 q 2 p 2 1 . T o sa v e space w e hav e denoted p i = ∂ i x p, q i = ∂ i x q . The other prop ert y wh ich is common among explicitly solv able eq uat io ns is presence of generalized symmetries. The generalized symme tr ies of the higher d’Alem b ert equation can b e obtained from its integrals b y means of the f orm ula u τ =  n − 1 2 uD x − u x  ω +  n − 1 2 uD t − u t  ¯ ω , (13) where ω and ¯ ω are t − and x − in tegra ls of (4). 3 Analogs o f op en T o da lattice s Previously w e ha v e raised the qu estion of whether there are other lattices as- so ciated with equation (4) whic h would ha v e reasonably simp le form. Belo w w e presen t tw o suc h examples of this sort: the Shabat-Y amilo v la ttice and the other form of the 2D T o da lattice. W e are una w a r e whether the connection b et w een these lattices and equation (4 ) has been men tioned elsewh ere. The following lattice w j,tx = w j,t w j,x  1 w j − w j − 1 − 1 w j +1 − w j  (14) w as introduced by Shaba t and Y amilov [7] as one the 2D a nolgs of the de- generations of the Landau-Lifshitz mo del. One can che ck that on solutions of (4 ), the quan tities w j = ∂ u log W j ( u ) , j = 1 , . . . , n − 1 , (15) 5 satisfy equations of lattice (14) along with the boundary conditions w 0 = 0 , w n = ∞ . Lattice (14) can therefore b e view ed as an ana lo gue of the op en A n T o da lattice. The Shabat-Y amilo v la ttice has some w ell kno wn particular cases, for example, for n = 3 w e ha ve the degenerate Lund-Regge ( complex sine- Gordon I) system v tx = w v t v x v w − 1 , w tx = v w t w x v w − 1 , (16) where w 1 = 1 /v , w 2 = w . System ( 16) was used in [4, 8] a s a w orking example for demonstrating prop erties of Liouville-t yp e systems . Note tha t equation (4) a dmits reductions t ha t mak e it p ossible to con- struct analogues of lattice (14) corresp onding to Lie algebras C n , B n , and p ossibly D n . This problem will b e considered elsewhere. Instead w e giv e one example of suc h la ttice that is akin to the T o da C 2 lattice. One can v erify that equation W 5 ( u ) = 0 admits the r eduction W 3 ( u ) = u . The latter equation can then b e written as the system v tx = w v x v t w v − 1 , w tx = v w x w t w v − 1 − ( w v − 1) 3 v v 2 t v 2 x , (17) where 1 /v = ∂ u log( W 1 ( u )) = 1 /u , w = ∂ u log( W 2 ( u )) = u tx u tx u − u x u t . Y et another example of a lattice related to equation (4) is given by v j,tx = exp( v j +1 − v j ) − exp( v j − v j − 1 ) , j = 1 , . . . , n − 1 (18) with the b oundary conditions v 0 = ∞ , v n = −∞ . This is ano t her w ell know n av atar of t he 2D T o da lattice. The transformation relating (9) and (18) is giv en b y v j = w j − w j − 1 . On the other hand system of equations (18) is r elat ed to (4) via the trans- formation v j +1 = − log  ∂ u j j log( W j +1 ( u ))  , j = 0 , . . . , n − 2 , 6 where u j j = ∂ 2 i u ∂ t j ∂ x j . There are examples of explicitly solv able systems whic h seem to b e related to equation (4) of a particular order. Consider, for ex a mple, the equation W 4 ( u ) = 0 . (19) In tro ducing the v ariables m = − 2 log( u ) , v = − 4 a W 2 ( u ) , w = ac 4 u ∂ log( W 3 ( u )) ∂ u tx w e ma y rewrite equation (19) as the sys tem m tx = a 2 v exp ( m ) , v tx = w v x w t v w + c , w tx = v w t w x v w + c + a 4 ( v w + c ) exp( m ) . (20 ) This system w as o bt a ined in [9] a s a degenerate v ersion of an S-integrable system. The g eneral solutions and in tegra ls of the ab o ve systems can b e easily deriv ed from form ulas (5) and (10). Of course, the tra nsformations giv en ab o v e do not exhaust all p ossible connections betw een equations o f t yp e (4) and open 2D la ttices. 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