Nonlocal symmetries of integrable two-field divergent evolutionary systems

Nonlo cal symmetries of i n tegrable t w o-field div er gen t e v o l utionary systems 1 A. G. Meshk o v Orel State T ec hnical Univ ersit y , Orel, Russia Abstract Nonlo cal symmetries for exactly in tegrable t w o-field ev olutionary systems of the third order h a v e b een computed. Differen tiatio n of the nonlo cal symmetries with re- sp ect to spatial v ariable give s a few n onev olutionary systems for eac h ev olutionary system. Zero cur v atur e represen tatio ns for some new nonev olution systems are p r e- sen ted. Keyw ords: conserv ed densit y , nonlo cal v ariable, nonlo cal symmetry , negativ e flow, exact in tegrabilit y . MSC: 37K2 0, 37K10, 35Q58 1 In tro du ction This pap er is dev oted to nonlo cal symmetries for the system s obtained in [1] via a symmetry classification. All calculations are simple a nd do not require a ny kno wledge of any zero curv ature repres en tation or Lax represen tation. The in ves tigation ga v e sev eral ne w integrable nonev olution systems b esides the know n T o da la t t ices. Kumei’s article [2] w as probably a pioneering w ork o n generalized symmetries fo r the sine-Gordon equation v tx = sin v . (1) It was disco v ered there that one of the symmetries coincides with the modified Kortew eg-de V ries equation (mKdV) v t = v xxx + 1 2 v 3 x (2) Rewriting t he sine-Gordon equation in the ev olution form u t = ∂ − 1 x sin u , one can sa y that this equation is a nonlo cal symmetry of (2). Let us consider t his problem in detail. T o simplify all fo rm ulas w e in tro duce the f unction u = iv x , that satisfies the following equation: u t = u xxx − 3 2 u 2 u x . (3) It is obv ious that u is the conserv ed densit y for (3) and one can intro duce the nonlo cal v ariable w = D − 1 x u . It can b e easily v erified that e w and e − w are the conserv ed densities for 1 This work is supp orted b y the RFBR gran t no 05- 01-00 775. Submitted to Theoretical and Mathematical Physics. (3) to o. This allows us to introduce tw o more nonlo cal v ariables: w 1 = D − 1 x e w , w 2 = D − 1 x e − w . Equation (3) p ossesse s the following nonlo cal symmetry u τ = c 1 e w + c 2 e − w + c 3 ( w 1 e − w + w 2 e w ) , (4) where c i are arbitrary constants (see [3], f or example). If c 3 = 0, then adopt ing w in (4) as a new unknown function and setting u = w x , w e obtain the follo wing w ell kno wn integrable equation w τ x = c 1 e w + c 2 e − w . (5) Differen tiation of equation (4) where c 3 = 0, giv es u τ x = u ( c 1 e w − c 2 e − w ). Excluding he re w with the help of the initia l equation, we obtain another in tegrable equation u τ x = u p u 2 τ − 4 c 1 c 2 . (6) Ob viously , the relation u = w x connects equations (5) and (6). Next, let c 3 6 = 0. Differen tiating equation (4) and com bining the result with t he initial equation, one can o btain ( u − 1 u τ x ) x = uu τ − 2 c 3 u − 2 u x . Using a dilatation of τ , one can adopt c 3 = − 1 / 2 and o btain u τ x x = u − 1 u x ( u τ x + 1) + u 2 u τ . (7) If one sets u τ x = uz τ , then the h yperb o lic system follo ws: z τ x = uu τ + u − 2 u x , u τ x = u z τ . (8) It can b e sho wn tha t nonlo cal equation (4) p ossesses a La x represen tation. Hence, all differen tial consequences of (4) hav e Lax represen ta tions to o. So, one in tegrable evolution equation (3) generates a set of in tegrable nonev olution equations (5) – (8). The canonical approac h to obtaining in tegrable hierarc hies is to fix a Lax op erat or L and consider v a rious op erators A . Usually , the op erator A is presen ted as a p olynomial with resp ect to p ositiv e or negativ e degrees of the sp ectral para meter. This gives p ositiv e and negativ e flow s of the Lax equations. There is a remark able pap er on this theme [4], where a gene ral construction of the Lax represen tations for the KdV- lik e equations has b een presen ted in terms of the affine algebras. Later in [5], this construction w as describ ed in detail with pro ofs and examples pro vided. The metho d that is used here is direct, b ecause it deals with the evolutionary system only . But to prov e in tegrabilit y of a nonlo cal symmetry o ne mus t construct the zero curv atur e represen tation. 2 2 Basic not i on and not ation Consider an ev olution system with tw o indep enden t v ariables t, x and m dep enden t v a r ia bles u α u t = K ( t, x, u , u x , . . . , u n ) , (9) where K = { K α } and u = { u α } , α = 1 , . . . , m a r e infinitely differentiable f unctions, u α = u α 0 , u α x = u α 1 , u α k = ∂ k u α /∂ x k . The set of whole dep endent v aria bles u α i is denoted a s u for brevit y . Definition 1. (see [6],[7]). If the v ector f unction σ ( t, x, u ) satisfies the equation ( D t − K ∗ ) σ = 0 , (10) where ( K ∗ ) α β = X k > 0 ∂ K α ∂ u β k D k x , D x = ∂ ∂ x + X α,k > 0 u α k +1 ∂ ∂ u α k , D t = ∂ ∂ t + X α,k > 0  D k x K α  ∂ ∂ u α k , (11) then it is said to b e the generalized symmetry of system (9). Here D x is called the total differen tiation op erato r with resp ect to x , D t is called the op erator of ev o lutionary differen tiation. The order of the differen tial op erator f ∗ is called the order of the (v ector-)function f . Generalized symmetries are often written as the ev olution systems u τ = σ ( t, x, u ) , (12) where τ is a new ev olution parameter. It is clear that to obtain lo cal in tegrable equations from (12) one m ust find the symm etries σ that do not dep end on t explicitly . Definition 2. (see [6 ],[7 ]). If fo r some differen tiable functions ρ and θ the following equation D t ρ ( t, x, u ) = D x θ ( t, x, u ) (13) is satisfied iden tically fo r an y solution u of system (9), then relation (13 ) is called the lo cal conserv ation law of system (9 ). The function ρ is said to b e the conserv ed densit y and θ is said to b e the densit y o f curren t. The pair ( ρ, θ ) is said to b e the conserv ed curren t. As the op erators D t D x are comm utativ e, then the vec tor ( ρ 0 = D x f , θ 0 = D t f ) with an y function f is t he conserv ed current for an y system. Suc h curren ts are called trivial. Conserv ed curren ts a re alw a ys defined b y mo dulo o f trivial curren ts. 3 Let ( ρ, θ ) b e the conserv ed current, then the follo wing system w x = ρ ( t, x, u ) , w t = θ ( t, x, u ) (14) is compatible for any u satisfying equation (9). The solution of (14) is formally written in the fo rm w = D − 1 x ρ . One can consider w a s a new dynamical v ariable. It is called w eakly nonlo cal or quasi-lo cal (see [8]). W e will call suc h v aria bles the first o rder nonlo cal v a riables. Let ( ρ i , θ i ) b e lo cal conserv ed curren ts and w (1) i = D − 1 x ρ i b e the corresp onding first order nonlo cal v ariables. If there exist conserv ed curren ts dep ending on w (1) i and, p ossibly , on lo cal v ariables, then one can construct new nonlo cal v ariables w (2) i and so on. The order of nonlo cal v aria bles is define d inductiv ely . Let the v aria bles w (1) , . . . , w ( n ) b e defined till the n -th order. If there exists a nontrivial conserv ed densit y ρ ( t, x, u, w (1) , . . . , w ( n ) ) and the n -th order v ar ia bles w ( n ) i can not b e remov ed by some gauge tra nsfor ma t io n ρ → ρ + D x f , θ → θ + D t f , then the v aria ble w = D − 1 x ρ ( t, x, u, w (1) , . . . , w ( n ) ) is called the ( n + 1) - th order nonlo cal v aria ble. Op erators (11) are to b e prolonged on the no nlo cal v ariables w i in accordance with the follo wing formulas ˆ D x = D x + ρ i ∂ ∂ w i , ˆ D t = D t + θ i ∂ ∂ w i , (15) where ( ρ i , θ i ) are the nonlo cal conserv ed curren ts corresp onding to the nonlo cal v a r ia bles w i . It is pro v ed that the op erators ˆ D x and ˆ D t are comm utativ e (see [9], for example). Hence, the equation fo r nonlo cal symm etries is (10) with the prolonged op erators ˆ D x and ˆ D t . If pro longed equation (10) has a solution dep ending on nonlo cal v ariables, then this solution is called a nonlo cal symmetry . Differen tial equations that are in teresting for applications hav e low or ders. That is why to obtain in teresting nonev olution in tegrable systems one o ugh t to consider low order conserv ed densities and nonlo cal symmetries. W e restrict ourselv es with considering the lo cal v ariables u α 0 and u α 1 only . Moreo v er, the nonlo cal v ariables are computed till the second or der b ecause the symmetries dep endent on higher order nonlo cal v ariables a r e v ery cum b ersome. If the system tak es the following form u α t = D x K α ( t, x, u, u x , . . . , u n − 1 ) , (16) then it is called div ergen t. Setting here u α = U α x , w e obtain the system U α t = K α ( t, x, U x , . . . , U n ) (17) that is usually called a p oten tial v ersion of system (16), as U is the p otential for u . Belo w w e consider systems of the f o rm (17) with t w o functions u and v . Therefore in t he general formulas considered ab ov e one m ust change u 1 and u 2 to u and v resp ectiv ely . 4 3 Nonlo cal symmetries Some of the systems found in [1 ] do not p ossess any no nlo cal symmetries. O t her systems p os- sess multi-parametric nonlo cal symmetries . Arbitrary constants contained in the symmetries are denoted a s c i or k i . 1. The system u t = u 3 + 3 2 u 1 v 2 − 3 4 u 1 v 2 1 + 1 4 u 1 3 , v t = − 1 2 v 3 − 3 4 (2 u 1 u 2 + u 2 1 v 1 ) + 1 4 v 3 1 (18) admits of the follo wing nonlo cal symme try u τ = c 2 w 2 + c 3 w 3 + c 4 ( w 4 − w 1 w 2 ) + c 5 ( w 5 − w 1 w 3 ) + c 6 ( w 3 w 4 − w 2 w 5 ) + c 7 (2 w 7 − w 3 w 1 2 ) + c 8 ( w 2 w 7 + w 3 w 6 + w 4 w 5 − w 1 w 3 w 4 − w 1 w 2 w 5 ) + c 9 ( w 1 w 3 w 4 + w 1 w 2 w 5 − 2 w 3 w 6 − w 4 w 5 ) , v τ = c 1 w 1 − c 2 w 2 + c 3 w 3 + c 4 w 1 w 2 − c 5 w 1 w 3 + c 6 ( w 3 w 4 + w 2 w 5 − w 8 ) − c 7 (2 w 7 − 2 w 1 w 5 + w 3 w 1 2 ) + c 8 ( w 3 w 6 − w 2 w 7 − w 1 w 3 w 4 + w 1 w 2 w 5 ) + c 9 ( w 3 w 1 w 4 + w 1 w 8 − 2 w 3 w 6 − w 1 w 2 w 5 ) , (19) where w 1 = D − 1 x e v , w 2 = D − 1 x e u − v , w 3 = D − 1 x e − u − v , w 4 = D − 1 x w 2 e v , w 5 = D − 1 x w 3 e v , w 6 = D − 1 x w 1 w 2 e v , w 7 = D − 1 x w 1 w 3 e v , w 8 = D − 1 x w 2 w 3 e v . W e ha v e verifie d that flo w (19) comm utes with flo w (18) and with the next flow from the same hierarch y: u t = u 5 + 5 4 v 4 u 1 + 5 4 u 3 (2 v 2 − v 2 1 ) + 5 4 v 3 (2 u 2 − u 1 v 1 ) − 5 2 u 2 v 2 v 1 − 5 8 v 2 2 u 1 − 5 8 u 1 v 2 ( u 2 1 + v 2 1 ) + 1 32 u 1 (5 v 5 1 − 3 u 4 1 + 10 u 2 1 v 2 1 ) , v t = − 1 4 v 5 − 5 4 u 1 u 4 − 5 4 u 3 ( u 2 + u 1 v 1 ) − 5 8 v 3 ( u 2 1 − v 2 1 ) + 5 8 v 1 v 2 2 − 5 4 u 1 u 2 v 2 + 5 8 u 1 u 2 ( u 2 1 + v 2 1 ) + 1 32 v 1 (5 u 4 1 − 3 v 4 1 + 10 u 2 1 v 2 1 ) . So, there is a reason to b elieve that the exact in tegrabilit y of system (19) holds. F or all systems considered in the pap er we hav e a lso v erified commutativit y of the nonlo cal flo ws and of the higher members of the corresp onding hierarchie s. W e do no t men tion it b elo w a nd we do not write out the higher members of hierarc hies for brevit y . 1.a. Setting in (19) c i = 0 , i > 3, we obtain the T o da lattice: u τ x = c 2 e u − v + c 3 e − u − v , v τ x = c 1 e v − c 2 e u − v + c 3 e − u − v . (20) 5 In notation of paper [5] consider the system u i,tx = exp  P j u j A j i  , where A j i , i, j = 1 , 2 , 3 is the Cartan matrix of the affine algebra D (2) 3 with the following D ynkin diagram ◦ < = ◦ = > ◦ . One has explicitly: u 1 ,tx = exp(2 u 1 − 2 u 3 ) , u 2 ,tx = exp(2 u 2 − 2 u 3 ) , u 3 ,tx = exp(2 u 3 − u 1 − u 2 ) . It is ob vious that the functions p = u 1 − u 3 , q = u 2 − u 3 satisfy the system p tx = e 2 p − e − p − q , q tx = e 2 q − e − p − q . If c i 6 = 0, then the same system is obtained from (2 0) b y the substitution u − v = 2 p, u + v = − 2 q . The constan ts c i ma y v anish in (20). In particular, if c 1 = 0, then the sys tem decomp oses into a pair of t he Liouville equations. System (20) can b e r epresen ted in sev eral forms. F or example, choosing p = w 1 and q = w 2 as the new unknown functions, we obtain: p τ x = p x ( c 1 p − c 2 q + c 3 w ) , q τ x = q x (2 c 2 q − c 1 p ) , w x = p − 2 x q − 1 x . 1.b. If in (19) c 4 = 1 and c i = 0 , i > 4, then there are sev eral p ossibilities. Consider the follo wing examples. (1) Adopting p = w 1 and q = ln w 2 as the new unkno wn function and setting c 1 = c 3 = 0 w e obtain: p τ x = e q pp x , q τ x = − e q pq x + f ( τ ) q x , (21) where f ( τ ) is an integration “constan t”. (2) Substitution u = ln  U x ( V x /U x ) x  , v = ln U x , w 1 = U, w 2 = V x /U x , w 4 = V results in another system under condition c 3 = 0: U τ x = c 1 U U x − c 2 V x + U V x , V τ x = c 1 V U x + V V x + f ( τ ) U x . (22) If one considers in this p oint c 3 6 = 0, then the result is t he third order cumbersome system. Notice that if c 4 = 0 and c 5 6 = 0 in (19), then a sligh tly differen t substitution results in (22) again. (3) D ouble differen tiation of syste m (19) giv es the t hird order lo cal sys tem u τ x x = (2 u x + z x ) u τ x + cu x e z − e u , z τ x x = − ( z x + u x ) z τ x + c ( u x + 2 z x ) e z − e u for any c 1 , c 2 , c 3 . Here z = − u − v , c = − 2 c 3 . 1.c. Adopting c 7 = − 1 , c i = 0 , i > 3 in (19) one can obtain the follo wing system: u txx = − u tx ( p x + 2 u x ) + 2 c 2 u x e − p + 2 p u tx e p − c 2 − c 3 e − 2 u , p txx = p tx ( u x + p x ) + 2 c 2 ( u x + 2 p x ) e − p + 2 p u tx e p − c 2 − c 3 e − 2 u , (23) 6 where p = v − u . 2. The system u t = u 3 − 3 v 3 + 3 v 2 ( v 1 − 2 u 1 ) + 3 u 1 v 2 1 − 2 u 3 1 , v t = − 3 u 3 + 4 v 3 − 3 u 2 ( v 1 − 2 u 1 ) + 3 v 1 u 2 1 − 2 v 3 1 , (24) admits of the follo wing nonlo cal symme try u τ = c 2 w 2 + c 3 w 3 + c 4 w 4 + c 5 (2 w 2 w 3 − w 5 ) + c 6 w 6 + c 7 (2 w 3 w 4 − w 7 ) + c 8 (2 w 3 w 6 − w 9 ) + c 9 w 2 ( w 2 w 3 − w 5 ) − c 10 ( w 5 w 6 + w 2 w 9 − 2 w 4 w 7 + 2 w 3 w 2 4 − 2 w 2 w 3 w 6 ) , v τ = c 1 w 1 + c 2 w 2 + c 4 w 1 w 2 + c 5 w 5 + c 6 (2 w 1 w 4 − w 6 ) + c 7 w 1 w 5 + c 8 (2 w 1 w 7 − w 9 ) + c 9 (2 w 8 − w 2 w 5 ) + c 10 ( w 5 w 6 − w 2 w 9 − 2 w 1 w 4 w 5 + 2 w 1 w 2 w 7 ) , (25) where w 1 = D − 1 x e v , w 2 = D − 1 x e − u − v , w 3 = D − 1 x e 2 u , w 4 = D − 1 x w 1 e − u − v , w 5 = D − 1 x w 3 e − u − v , w 6 = D − 1 x w 2 1 e − u − v , w 7 = D − 1 x w 1 w 3 e − u − v , w 8 = D − 1 x w 2 w 3 e − u − v , w 9 = D − 1 x w 2 1 w 3 e − u − v . Let us presen t some simple lo cal syste ms that follow from (25). 2.a. Setting c i = 0 for i > 3 in (25) w e obtain the T o da lattice: u τ x = c 2 e − u − v + c 3 e 2 u , v τ x = c 1 e v + c 2 e − u − v . (26) Let us write the system u i,tx = exp  P j u j A j i  , where A j i is t he Cartan matrix for the affine algebra A (2) 4 with the following Dynkin diagram ◦ = > ◦ = > ◦ . Substitution u = 2 u 1 − u 2 , v = 2 u 3 − u 2 results in system (26) with c 1 = 2 , c 2 = − 1 , c 3 = 1. Sys tem (26) can b e rewritten in sev eral different forms. F or example, the functions p = w 1 , q = w 2 satisfy the sys tem: p τ x = p x ( c 1 p + c 2 q ) , q τ x = − q x ( c 1 p + 2 c 2 q + c 3 w ) , w x = ( p x q x ) − 2 . 2.b. If c i = 0 , i > 4 , c 4 = 1, t hen do uble differen tiation o f system (25) give s the fo llo wing lo cal system: u τ x x = − u τ x ( u x + v x ) + c ( 3 u x + q x ) e 2 u + e − u , ( c = c 3 ) , v τ x x = v x v τ x − u τ x ( u x + 2 v x ) + c ( u x + 2 v x ) e 2 u + 2 e − u . (27) It is obv ious that the o rder of the second equation can b e decreased b y the substitution v x → v . If, under the previous conditions, o ne c ho oses p = w 1 and q = w 2 as new unkno wn functions, then another system follows: p τ x = p x ( c 1 p + c 2 q + pq ) , (ln q x ) τ x = p x ( c 1 + q ) − c 3 p − 2 x q − 2 x . 7 2.c. If c 4 = 0 , c 5 = 1 , c i = 0 , i > 5, then double differen tiation of system (25) giv es the follo wing lo cal sys tem: u τ x x = 2 u x u τ x − v τ x (3 u x + v x ) + c ( 3 u x + v x ) e v + 3 e u − v , v τ x x = − v τ x ( u x + v x ) + c ( u x + 2 v x ) e v + e u − v , ( c = c 1 ) . (28) 2.d. If c 6 = 1 and c i = 0 , i > 4, then the follow ing lo cal system follows u τ x x = − 2 u τ x ( u x + q x ) + 2 c (2 u x + q x ) e 2 u + 2 q u τ x e 2 q + be − 2 u − ce 2( u + q ) , q τ x x = q τ x ( u x + 2 q x ) + 1 2 c (2 q x − u x ) e 2 u + q u τ x e 2 q + be − 2 u − ce 2( u + q ) , (29) where b = c 2 4 / 4 − c 2 , c = c 3 , q = ( v − u ) / 2. Notice that in the case b = c = 0 the order o f the first equation can b e decreased b y the substitution u x → u . 3. The next system u t = u 3 + u 1 v 2 − u 1 v 2 1 , v t = ( u 2 u 1 + u 2 1 v 1 ) , (30) admits of the follo wing nonlo cal symme try u τ = c 1 w 1 + c 2 w 2 + c 4 ( w 4 − w 1 w 3 ) + c 5 ( w 5 − w 1 w 4 ) + c 6 ( w 6 − w 2 w 3 ) + c 7 ( w 1 w 6 − w 2 w 4 ) + c 8 ( w 8 − w 2 w 6 ) , v τ = − c 1 w 1 + c 2 w 2 + c 3 w 3 + c 4 ( w 4 + w 1 w 3 ) + c 5 w 1 w 4 − c 6 ( w 6 + w 2 w 3 ) − c 7 ( w 1 w 6 + w 2 w 4 ) − c 8 w 2 w 6 , (31) where w 1 = D − 1 x e u − v , w 2 = D − 1 x e − u − v , w 3 = D − 1 x e 2 v , w 4 = D − 1 x w 1 e 2 v , w 5 = D − 1 x w 2 1 e 2 v , w 6 = D − 1 x w 2 e 2 v , w 7 = D − 1 x w 1 w 2 e 2 v , w 8 = D − 1 x w 2 2 e 2 v . 3.a. In the case c i = 0 , i > 3 the followin g T o da lattice is obta ined: u τ x = c 1 e u − v + c 2 e − u − v , v τ x = − c 1 e u − v + c 2 e − u − v + c 3 e 2 v . (32) In the new v ariables p = u − v , q = − u − v this system tak es the form p τ x = 2 c 1 e p − c 3 e − p − q , q τ x = − 2 c 2 e q − c 3 e − p − q . This allow s to connect sys tem (32) with the affine algebra C (1) 2 ha ving the following Dynkin diagram ◦ = > ◦ < = ◦ . In the case of c 3 = 0 this system decomp oses into a pair of t he Liouville equations ob viously . 3.b. If c 4 6 = 0 , c i = 0 , i > 4 , then in the terms of new v ariables p = w 1 , q = w 3 system (31) take s t he follo wing form: p τ x = p x (2 c 1 p − c 3 q − 2 c 4 pq ) , ( q − 1 x q τ x ) x = 2 c 3 q x − 2 c 1 p x + 4 c 4 pq x + 2 c 4 q p x + 2 c 2 p − 1 x q − 1 x . If one simply doubly differen tiates system (31), then the result is u τ x x = u τ x (2 u x − p x ) − 2 c 2 u x e − p − c 4 e p , p τ x x = 2 p τ x ( p x − u x ) + 2 c 2 e − p (2 u x − 3 p x ) + 2 c 4 e p , 8 where p = u + v . Here the order of the first equation can b e decreased b y the substitution u x → u . 3.c. If c 5 6 = 0 and the other constants c i = 0 , i > 3, then a do uble differen tiation of system (31) giv es u τ x x = u τ x (2 u x + q x ) − 2 c 2 u x e q + p aq τ x e 2 u + be − 2 q + 2 ac 2 e 2 u + q , q τ x x = − 2 q τ x ( u x + q x ) − 2 c 2 (2 u x + 3 q x ) e q + 2 p aq τ x e 2 u + be − 2 q + 2 ac 2 e 2 u + q , (33) where a = − c 5 , b = − c 3 c 5 , q = − u − v . Notice that systems (29) and (3 3 ) coincide when c 2 = c 3 = c 4 = 0. This is surprising, b ecause the symmetries of these systems, i.e. systems (24) and (30), are en tirely different. A p ossible explanation is as follo ws. The system u τ x x = u τ x (2 u x + q x ) + e u √ q τ x , q τ x x = − 2 q τ x ( u x + q x ) + 2 e u √ q τ x is Liouvillean and p ossesses a double sequence of symmetries that are constructed b y differen t in tegrals. 4. The next system u t = u 3 + v 1 v 2 − 1 2 u 3 1 + 1 2 u 1 v 2 1 + c 1 v 1 , v t = u 2 v 1 − 1 2 u 2 1 v 1 + 1 2 v 3 1 − c 1 u 1 + c 2 v 1 (34) con tains t w o essen tia l constan ts that affect the form and quan tit y of admissible symme tries. It b ecomes clear if one tak es in to accoun t that system (34) can b e obtained from the Ito system by a differen tial substitution dep ending on c 1 and c 2 (see [1]). If c 1 = 0, then the differen tial substitution is essen tially simplified, and the differen tial substitution v anishes when c 1 = c 2 = 0. 4.1. If c 1 = 0 , c 2 = 0, then system (34 ) is degenerate. In fact, the substitution u = ln U x , v = V x U − 1 x giv es U t = U xxx − 3 2 U 2 xx U x − V x V xx U xx U 2 x + 1 2 V 2 x U 2 xx U 3 x + 1 2 V 2 xx U x , V t = V x U t U x . Hence, V = F ( U ) and w e hav e: U t = U xxx − 3 2 U 2 xx U x + 1 2 ( F ′′ ) 2 U 3 x . This equation is exactly integrable iff F I V = 0 (see [7]). Hence, system (34) with c 1 = 0, c 2 = 0 is not in tegrable in the general case. 4.2. If c 1 = 0 , c 2 6 = 0, then system (34) admits of the following no nlo cal 4-parametric symmetry u τ = k 1 w 1 + k 2 w 2 + k 3 ( w 5 + 2 w 2 w 3 ) + k 4 ( w 1 w 5 − 4 w 1 w 2 + 2 w 2 w 4 ) , v τ = − 2 k 2 e − u v x − 4 k 3 e − u v x (2 + w 3 ) − 4 k 4 e − u v x w 4 , (35) 9 where w 1 = D − 1 x e u , w 2 = D − 1 x e − u ( v 2 x + c 2 ) , w 3 = D − 1 x e u w 2 , w 4 = D − 1 x e u w 2 w 1 , w 5 = D − 1 x (4 v 2 x e − u − e u w 2 2 ) . 4.2.a. If k i = 0 , i > 2, then differen tiation of system (35) give s: u τ x = k 1 e u + k 2 ( v 2 x + c 2 ) e − u , v τ = − 2 k 2 v x e − u . 4.2.b. If k 3 6 = 0 , k 4 = 0, then one can obtain k 3 = 1 / 4 b y a dilatation of τ . In this case differen tiation o f system (35) gives the following syste m: u τ x = 1 4 e u  v τ v x  x + v τ u x v x  2 − 1 2 v τ ( v x + c 2 v − 1 x ) + k 1 e u − c 2 e − u ,  ln v x  τ x =  v τ v 2 x ( v xx − u x v x )  x − ( v 2 x + c 2 ) e − u . 4.3. If c 1 6 = 0 in (34), then a dilatation of v , t and x g iv es c 1 = 1: u t = u 3 + v 1 v 2 − 1 2 u 3 1 + 1 2 u 1 v 2 1 + v 1 , v t = u 2 v 1 − 1 2 u 2 1 v 1 + 1 2 v 3 1 − u 1 + c 2 v 1 , (34 a ) There are three cases for three different v a lues of c 2 . 4.3.a. c 2 = − 2 ε , ε = ± 1. In this case system (34 a ) admits the following nonlo cal symmetry: u τ = k 1 w 1 − k 2 w 2 + k 3 w 3 + k 4 w 4 + k 5 w 1 w 3 + k 6 (6 w 4 w 5 − 6 w 3 w 6 − 3 w 1 + w 8 ) , v τ = ( k 2 − εk 1 ) w 1 + εk 2 w 2 + ε ( k 3 + k 5 w 1 )(2 v x e − u − εv − w 3 ) + k 4 ( w 3 + 2 εv v x e − u − εv − εw 4 − 2 v x e − u − εv ) + 4 k 6 e − u − εv (6 v v x + v x v 3 + 3 εv x v w 5 − 3 v x w 5 − 6 εv x v 2 − 3 εv x w 6 ) + 3 k 6 ( εw 1 + 2 w 3 w 5 + w 7 − 4 w 4 + 2 εw 3 w 6 − 2 εw 4 w 5 ) − k 6 εw 8 . (36) Here, w 1 = D − 1 x e u + εv , w 2 = D − 1 x v e u + εv , w 3 = D − 1 x e − u − εv (1 − εv 2 x ) , w 5 = D − 1 x w 3 e u + εv , w 4 = D − 1 x e − u − εv  v + v 2 x (1 − εv )  , w 6 = D − 1 x e u + εv ( v w 3 − w 4 ) , w 7 = D − 1 x  2 v e − u − εv ( εv + v 2 x (2 ε − v )) − e u + εv w 2 3  , w 8 = D − 1 x  2 v e − u − εv  v 2 ( ε − v 2 x ) − 3 v + 6 v 2 x ( εv − 1)  + 3 e u + εv w 3 ( v w 3 − 2 w 4 )  . If k 5 = k 6 = 0, then differen tiation of system (35) gives the followin g system: u τ x = ( k 1 − k 2 v ) e u + εv + k 3 e − u − εv (1 − εv 2 x ) + k 4 e − u − εv  v + v 2 x (1 − εv )  , v τ x = ( k 2 − εk 1 + εk 2 v ) e u + εv − εk 3 e − u − εv (2 u x v x − 2 v xx + εv 2 x + 1) + k 4 e − u − εv  (2 v xx − 2 u x v x − 1)( ε v − 1) + v 2 x (2 ε − v )  . If one set here k 3 = k 4 = 0, then the triangle system ( u + εv ) τ x = εk 2 e u + εv follo ws. 10 If k 5 = ε/ 2, and the other constan ts k i = 0, then the follo wing lo cal system follo ws p τ x = εp √ ε − εp x q x , q τ x = 2( q x p x − 1) p − 1 x + pp xx (2 − q x p x ) p − 3 x + pp − 1 x q xx + εq √ ε − εp x q x , (37) where p = w 1 , q = w 3 . Other com binations o f constan ts in (3 6) giv e v ery cum b ersome systems . 4.3.b. If c 2 = − 2 k , | k | < 1, t hen system (34 a ) p osse sses the following nonlo cal symmetry: u τ = k 1 w 1 − k 2 w 2 + k 3 w 3 + k 4 w 4 + k 5 ( w 4 w 1 + w 3 w 2 ) + k 6 ( − w 4 w 2 + w 1 w 3 ) − k 7 ( − w 7 + 2 c 2 w 3 w 6 − 2 c 2 w 4 w 5 ) + k 8 (2 c 2 w 4 w 6 + 2 c 2 w 3 w 5 + w 8 ) , v τ = k 1 ( cw 2 − k w 1 ) + k 2 ( k w 2 + cw 1 ) + k 3  2 v x sin( cv ) e − u − k v + w 4 c − k w 3  + k 4  2 v x cos( cv ) e − u − k v − w 3 c − k w 4  + 2 v x k 5  w 1 cos( cv ) + w 2 sin( cv )  e − u − k v − k 5 ( k w 3 w 2 + cw 3 w 1 − cw 4 w 2 + k w 4 w 1 ) + 2 v x k 6  w 1 sin( cv ) − w 2 cos( cv )  e − u − k v + k 6 ( cw 1 w 4 + cw 2 w 3 − k w 1 w 3 + k w 2 w 4 ) − 2 v x k 7  sin( cv )(1 − 2 c 2 + 2 c 2 w 6 ) − 2 c cos( cv ) ( v + k − cw 5 )  e − u − k v + k 7 (2 k c 2 ( w 3 w 6 − w 4 w 5 ) − 2 c 3 ( w 4 w 6 + w 3 w 5 ) + 2 cw 4 − k w 7 − cw 8 ) + 2 k 8 v x  cos( cv )(1 − 2 c 2 + 2 c 2 w 6 ) + 2 c sin( cv )( cw 5 − k + v )  e − u − k v + k 8  2 c 3 ( w 4 w 5 − w 3 w 6 ) − 2 k c 2 ( w 4 w 6 + w 3 w 5 ) + cw 7 + 2 cw 3 + k w 8  . (38) Here c = √ 1 − k 2 and w 1 = D − 1 x cos( cv ) e u + k v , w 3 = D − 1 x  cos( cv − α ) − v 2 x sin( cv )  e − u − k v , w 2 = D − 1 x sin( cv ) e u + k v , w 4 = D − 1 x  sin( α − cv ) − v 2 x cos( cv )  e − u − k v , w 5 = D − 1 x  w 4 sin( cv ) − w 3 cos( cv )  e u + k v , w 6 = D − 1 x  w 4 cos( cv ) + w 3 sin( cv )  e u + k v , w 7 = D − 1 x  c 2 e u + k v  sin( cv )( w 2 3 − w 2 4 ) + 2 w 3 w 4 cos( cv )  + + e − u − k v  c cos( cv )(2 k v 2 x − 2 v v 2 x + 2 k v − 1) − sin ( cv )( 2 c 2 v 2 x − v 2 x + 2 c 2 v + k )   , w 8 = D − 1 x  c 2 e u + k v  cos( cv )( w 2 3 − w 2 4 ) − 2 w 4 w 3 sin( cv )  + + e − u − k v  cos( cv )(2 c 2 v 2 x − v 2 x + 2 c 2 v + k ) + c sin( cv )(2 k v 2 x − 2 v v 2 x + 2 k v − 1)   , k = sin α, c = cos α, − π 2 < α < π 2 . Simple lo cal equations exist under conditions k i = 0 , i > 4 only: u τ x = ( k 1 cos( cv ) + k 2 sin( cv )) e u + k v + k 3 v 2 x sin( cv + α ) e − u − k v − k 4 v 2 x cos( cv + α ) e − u − k v −  k 3 cos( cv ) + k 4 sin( cv )  e − u − k v , v τ x =  k 1 sin( cv − α ) − k 2 cos( cv − α )  e u + k v + (2 v x u x − 2 v xx + 1)  k 3 sin( cv + α ) − k 4 cos( cv + α )  e − u − k v − v 2 x  k 3 cos( cv + 2 α ) + k 4 sin( cv + 2 α )  e − u − k v . (39) 11 If k 3 = k 4 = 0, then this system decomp oses into t w o Lio uville equations in the terms of v ariables p = u + ie − iα v , q = u − ie iα v . 4.3.c. If c 2 = − a − a − 1 , | a | 6 = 1, then system (34 a ) p ossesses the follow ing nonlo cal symmetry: u τ = − ak 1 w 1 + k 2 w 2 + ak 3 w 3 + k 4 w 4 + ak 5 w 1 w 3 + k 6 w 2 w 4 + k 7 ( w 7 + 2 w 4 w 5 ( a 2 − 1) 2 ) + ak 8  2 w 3 w 6 ( a 2 − 1) 2 + w 8  , v τ = k 1 w 1 − ak 2 w 2 − k 3 ( w 3 − 2 av x e − u − v /a ) − ak 4 ( w 4 − 2 v x e − u − av ) + k 5 w 1 ( − w 3 + 2 ae − u − v /a v x ) + k 6 w 2 a (2 v x e − u − av − w 4 ) − ak 7  w 7 + 2 w 4 ( a 2 − 1)( w 5 a 2 + 2 a − w 5 )  − 4 ak 7 e − u − av v x  2 a 2 v ( a 2 − 1) + 2 a − w 5 ( a 2 − 1) 2  + k 8  2 w 3 ( a 2 − 1)( w 6 − a 2 w 6 + 2 a 2 ) − w 8  + 4 ak 8 e − u − v /a v x  w 6 ( a 2 − 1) 2 + 2 av ( a 2 − 1) − 2 a 4 )  . (40) Here, w 1 = D − 1 x e u + v/a , w 2 = D − 1 x e u + av , w 3 = D − 1 x e − u − v /a ( a − v 2 x ) , w 4 = D − 1 x e − u − av (1 − av 2 x ) , w 5 = D − 1 x w 4 e u + av , w 6 = D − 1 x w 3 e u + av w 7 = D − 1 x  4 a 2 e − u − av  v − a 2 v − a + v 2 x ( a 3 v − av + 1)  − e u + av w 2 4 ( a 2 − 1) 2  , w 8 = D − 1 x  4 ae − u − v /a  a 3 v − a 2 − av + v 2 x ( − a 2 v + a 3 + v )  − e u + v/a w 2 3 ( a 2 − 1) 2  . If k i = 0 , i > 4, then the following lo cal system follows: u τ x = − ak 1 e u + v/a + k 2 e u + av + ak 3 e − u − v /a ( a − v 2 x ) + k 4 e − u − av (1 − av 2 x ) , v τ x = k 1 e u + v/a − ak 2 e u + av − k 3 e − u − v /a (2 au x v x − 2 av xx + v 2 x + a ) − ak 4 e − u − av (2 u x v x − 2 v xx + av 2 x + 1) . If k 3 = k 4 = 0 then this system decomp oses in to tw o Liouville equations in the terms of v ariables p = u + av , q = u + v /a . If k 5 = a and the other constants k i = 0, then the follo wing lo cal system follo ws p τ x = pq p x ( a 2 − 1) + 2 ap √ a + p x q x , q τ x = 4 a 2 ( a + p x q x ) p − 1 x + 2 a 2 pp − 1 x q xx − 2 ap − 1 x ( q p x + 2 pq x p xx ) √ a + p x q x − 2 a 2 pp xx ( p x q x + 2 a ) p − 3 x + (1 − a 2 ) pq q x , (41) where p = w 1 , q = w 3 . Other com binations of the constan ts in (40) giv e more cum b ersome systems . Notice that all for m ulas from p oints 4.3.b and 4.3.c are connected with eac h o ther b y the transformation a = k + ic, a − 1 = k − ic, c = √ 1 − k 2 . All formulas from p oint 4.3.a can b e obtained from corresp onding form ulas of p oint 4.3 .b as the limit k → ε = ± 1 , c → 0. 12 But these calculations are v ery cum bersome. In particular, syste m (41 ) is reduced in to (37) under the substitution a = ε = ± 1, q → − εq All remaining systems found in [1] ha v e no nonlo cal symmetries o r ha v e trivial nonlo cal symmetries that lead to the Lio uville equation. 4 Zero cu r v atur e represe ntations W e presen t here the matrices U and V realizing zero curv ature represen tations U τ − V x + [ U, V ] = 0 for some o f the systems connected with (24). Sp ectral parameter is denoted as k ev erywhere. System (2 4) can b e obta ined from the Drinfeld-Sok olo v system [5] m t = m 3 − 3 n 3 − 3 m x (4 m − 9 n ) + 3 n x (8 m − 15 n ) , n t = − 3 m 3 + 4 n 3 + 12 m x n + 6 n x ( m − 4 n ) (42) b y the follo wing differen tial substitution: m = u 2 x + 1 2 v 2 x − u 2 − v 2 , n = u 2 x − u 2 . (43) First, w e write t he matrices U 0 V 0 that for m the zero curv ature r epresen t a tion for system (42): U 0 =       0 1 n − m 0 0 0 0 0 1 0 − 1 0 0 0 m − n 0 n 0 0 k 0 0 1 0 0       , V 0 =       h 1 ,x h 2 f 1 0 − 5 k 0 h 3 ,x − 5 k − 2 h 3 0 − h 1 0 0 5 − f 1 5 k f 2 0 − h 3 ,x k h 2 0 5 h 1 0 − h 1 ,x       . (44) Here, h 1 = 7 n − 4 m, h 2 = m − 3 n, h 3 = 4 n − 3 m, f 1 = − 4 m 2 + 7 n 2 + 4 m 2 + 7 n 2 − 11 mn, f 2 = − 3 m 2 + 4 n 2 − 8 n 2 + 6 mn. Matrices (44) are em bedded in sl (5 , C ) . P erforming substitution (43) in matr ices (4 4) a nd excluding u 2 and v 2 from U 0 b y a gauge transformation U = S − 1 ( U 0 S − S x ) , V = S − 1 ( V 0 S − S t ), w e o btain the zero curv ature represen tation for system (24): U =       v x 1 0 0 0 0 − u x 0 1 0 − 1 0 0 0 0 0 0 0 u x k 0 0 1 0 − v x       , V =       ϕ 1 ϕ 2 0 − 5 v x − 5 k 0 ϕ 3 − 5 k ϕ 4 5 k v x − ϕ 5 5 h 0 5 0 5 k 0 5 kh − ϕ 3 k ϕ 2 0 5 ϕ 5 0 − ϕ 1       . (45) 13 Here, ϕ 1 = 4 v 3 − 3 u 3 + 3 u 2 (2 u x − v x ) + 3 v x u 2 x − 2 v 3 x , ϕ 2 = 2 u 2 − v 2 − 2 u 2 x − 2 v 2 x + 5 u x v x , ϕ 3 = 3 v 3 − u 3 + 3 v 2 (2 u x − v x ) − 3 u x v 2 x + 2 u 3 x , h = v x − u x , ϕ 4 = 2 u 2 − 6 v 2 − 2 u 2 x + 3 v 2 x , ϕ 5 = 4 v 2 − 3 u 2 + 3 u 2 x − 2 v 2 x . The sys tem Ψ x = U Ψ, where U takes the form (4 5), can b e reduced to t he fo llowing single equation ( ∂ x − u x )( ∂ x + u x )( ∂ x − v x ) ∂ x ( ∂ x − v x )Ψ 5 + k Ψ 5 = 0 . The sp ectral problem for this equation is obv iously no ntrivial. System (24) is presen ted in [5], but in another form (see table 5, A (2) 4 ). The zero curv a ture represen tations fo r this system a nd corresp onding T o da la t t ice are contained in the same pap er. But it w as simpler for us to compute these zero curv ature represen tations anew. Matrix U for t he T o da lattice (26) is sho wn in (45) and V take s t he follo wing form: V =       0 0 − c 1 e v 0 0 − c 2 e − u − v 0 0 0 0 0 0 0 0 c 1 e v 0 c 3 e 2 u 0 0 0 0 0 0 − k − 1 c 2 e − u − v 0       . W e hav e assumed that systems (27) – (29) b elong to the same hierarc h y as system (24). If this is true, the matrix U is common for all mentioned systems. The calculations hav e confirmed our assumption and w e presen t b elow o nly the matrices V for the mentioned systems . F or system (27): V =       0 0 u τ x − v τ x − ce 2 u 0 0 ce 2 u − u τ x 0 − e − u 0 0 0 0 0 − k − 1 e − u v τ x − u τ x + ce 2 u 0 ce 2 u 0 0 0 0 0 0 k − 1 ( ce 2 u − u τ x ) 0       . (46) F or system (28): V =       0 0 − ce v 0 0 ce v − v τ x 0 0 0 0 0 0 0 0 ce v − e u − v u τ x − v τ x + ce v 0 0 0 0 k − 1 e u − v 0 k − 1 ( ce v − v τ x ) 0       . (47) F or system (29): V =       0 0 − 2 q τ x − ce 2 u 2 k − 1 e 2 q 0 ce 2 u − u τ x 0 − 2 r 0 2 e 2 q 0 0 0 − 2 k − 1 r 2 q τ x + ce 2 u 0 ce 2 u 0 0 0 0 0 0 k − 1 ( ce 2 u − u τ x ) 0       . (48) 14 Here r = p u τ x e 2 q + be − 2 u − ce 2( u + q ) and the substitution v = u + 2 q mus t b e perfo r med in the matrix U (see (45)). Conclus ion As it w as mentioned ab o v e, eac h nonlo cal symmetry presen ted in this pap er is a symmetry for the system under consideration as w ell as for its hig her a na logue. This giv es grounds to b eliev e that a ll presen ted systems are exactly integrable. But this assumption mus t b e pro v ed, of course. Such pro of s hav e b een presen ted for systems (27) – (2 9 ). F or other systems this problem should b e further in v es tigated. References [1] A. G. Meshk o v, F undamentalnaya i Prikla dnaya Matematika, 12 :7, (2006), 141–161, (in Russian). [2] S. Kumei, J. Math. Phys. , 16 :12 (19 7 5), 2461–2468 . [3] Zhijun Qiao , arXiv:nlin/0201065 v1 [nlin.SI], 31.01.20 02. [4] V. G. Drinfeld, V. V. Sok olo v, Sov. Math. Dokl., 23 (19 8 1), 457.) [5] V. G. D rinfeld, V. V. Sok olov , Lie algebr as and e quations of Kortewe g-de V ries typ e , in Curr ent pr oblems in m athematics , 24 , Ito gi nauki i tehniki , VINITI, Mosco w, 1 9 84, 81-180, (in Russian), tr anslation in J. Sov. Math. , 30 ( 1985), 197 5 –2035. [6] P . J. Olver, Applic ations of Lie Gr oups to Differ ential Equations , Spring er- V erlag, New Y ork, 19 8 9; N. H. Ibragimo v, Applic ations of T r ansformation Gr oups to Mathematic al Physics, Nauk a , Mosk ow, 19 83. [7] A. V. Mikhailo v, A. B. Shabat and V. V. Sok olo v, “The symmetry approac h to classifi- cation of integrable equations”, What is Inte gr ability ? , Springer-V erla g (Springer Series in Nonlinear Dynamics), New Y ork, 19 91, 115–18 4. [8] V. V. Sok olov and S. I Svinolupov, Math. Notes , 48 :5-6 (1991), 12 3 4–1239; I.Sh. Akha- to v, R.K. Gazizov, N. H. Ibragimo v, “ Nonlo cal Symme tries. Heuristic Approac h”, Ito gi nauki i tehniki , 34 , VINITI, Mosco w , 1989, 3–83, (in Russian). [9] A. Sergye y ev, “On recursion op erato r s and nonlo cal symmetries of ev olution equations”, Hr o c. Sem. Diff. Ge om. , Math. Publications V. 2, D. Kr upk a, ed., Silesian Univers it y in Opa v a, Opa v a, 2000, 159–173. 15