On the discretization of Darboux Integrable Systems admitting the second-order integrals
aa r X i v : . [ n li n . S I] J u l On the discretization of Darboux Integrable Systems admitting thesecond-order integrals.
Kostyantyn Zheltukhin Department of Mathematics, Middle East Technical University, Ankara, TurkeyNatalya Zheltukhina Department of Mathematics, Faculty of Science, Bilkent University, Ankara, Turkey
Abstract
We study discretization of Darboux integrable systems. The discretization is doneby using x - or y -integrals of the considered systems. New examples of semi-discrete Darbouxintegrable systems are obtained.Keywords: semi-discrete system, Darboux integrability, x -integral, n -integral, discretization. In the present paper we study the problem of discretization of integrable equations so that theproperty of integrability is preserved. In particular, we consider hyperbolic type systems p ixy = f i ( x, y, p, p x , p y ) i = 1 , . . . , m , (1)where p = ( p , . . . , p m ), p x = ( p x , . . . , p mx ) and p y = ( p y , . . . , p my ).For such hyperbolic systems it is convenient to use Darboux integrability [1]. The abovesystem is said to be integrable if it admits m functionally independent non-trivial x − integrals and m functionally independent non-trivial y − integrals. A function I ( x, y, p, p y , p yy , ... ) is called an x − integral of the system (1) if D x I ( x, y, p, p y , p yy , ... ) = 0 on all solutions of (1) , (2)where D x is the total derivative with respect to x . One can define y − integrals in a similar way.The Darboux integrable systems are extensively studied, see [2]-[11] and a review paper [12].The extension of the notion of Darboux integrability to discrete and semi-discrete Darbouxintegrable systems was developed by Habibullin and Pekcan [13], see also [14]. In recent yearsthere is an interest in studying such systems, see [15]-[25]. A semi-discrete system q inx = f i ( x, n, q, q x , q ) i = 1 , . . . , m , (3) e-mail: [email protected] e-mail: [email protected] q = ( q , . . . , q m ), q x = ( q x , . . . , q mx ) and q = ( q ( x, n +1) , . . . , q m ( x, n +1)), is called Darbouxintegrable if it admits m functionally independent non-trivial x − integrals and m functionallyindependent non-trivial n − integrals. A function J ( x, n, q, q x , q xx , ... ) is called an n − integral ofthe system (3) if DJ ( x, y, q, q x , q xx , ... ) = J ( x, n, q, q x , q xx , ... ) on all solutions of (3) , (4)where D is the shift operator, that is Dq = q . Note that Dq k = q k +1 , k = 1 , , , ... . The x − integrals I ( x, n, q, q , q , ... ) for the system (3) are defined in the same way as for continuoussystem.A hypothesis states that any continuous Darboux integrable system can be discretized withrespect to one of the independent variables such that the resulting semi-discrete system is Darbouxintegrable and admits the set of x − or y − integrals of the original system as n − integrals [25]. Theresults of our work support the above hypothesis. We complete the discretization of continuousDarboux integrable equations derived by Zhiber and Kostrigina in [8]. In their paper Zhiberand Kostrigina considered the classification problem for continuous Darboux integrable systemsadmitting x − and y − integrals of the first and second orders. They found all such systems togetherwith their x - and y -integrals. Following [8] we have two types of systems. The first system is u xy = u x u y u + v + (cid:18) u + v + αu + α v (cid:19) u x v y ,v xy = α v x v y u + α v + (cid:18) α ( u + v ) + αu + α v (cid:19) u x v y . (5)For α = 1 it has y -integrals I = 2 v − u + v + cu x + 2 c ln u x u + v + c , (6) J = u xx u x − u x + v x ( u + v + c ) (7)and the x -integrals have the same form in u, u y , u yy and v, v y , v yy variables.For α = 1 it has y -integrals I = (cid:18) α (cid:19) v (cid:18) u x u + v (cid:19) − α − v x (cid:18) u x u + v (cid:19) − α , (8) J = u xx u x − ( α + 1) u x + αv x α ( u + v ) (9)and the x -integrals have the same form in u, u y , u yy and v, v y , v yy variables.The second system is u xy = vu x u y uv + d + (cid:18) uv + d + 1 α ( uv + c ) (cid:19) uu x v y ,v xy = uv x v y uv + c + (cid:18) αuv + d + 1 uv + c (cid:19) vu x v y . (10)2or α = 1 it has y -integrals I = ( d − c ) v u x uv + d ) − cu x v x uv + d (11)and J = u xx u x + ( d − c ) vu x − cuv x c ( uv + d ) , (12)where c and d are non-zero constants and the x -integrals have the same form in u, u y , u yy and v, v y , v yy variables.For α = 1 it has y -integrals I = u βx v x ( uv + d ) β + βv u β +1 x ( uv + d ) β +1 (13)and J = − u xx u x + 2 vu x + uv x uv + d , (14)where d and β = 1 are non-zero constants, and the x -integrals have the same form in u, u y , u yy and v, v y , v yy variables.To discretize the systems (5) and (10) we employ a method introduced by Habibullin et. all[21], (see also [24]-[26]). In this approach one takes x - or y -integrals of a system and looks fora semi-discrete system admitting such integrals as n -integrals. In general one gets a set of semi-discrete systems admitting these n -integrals. For all sets of y − integrals of systems (5) and (10) weobtained corresponding semi-discrete systems. In all cases we were able to choose a semi-discretesystem that gives the original system in the continuum limit. Also in examples where we canwrite a semi-discrete system explicitly we have shown that the system is Darboux integrable.The following theorems are formulated for a hyperbolic type semi-discrete system u xn = f ( x, n, u, v, u x , v x , u n , v n ) ,v xn = g ( x, n, u, v, u x , v x , u n , v n ) , (15)where variables u, v depend on a continuous variable x ∈ R and a discrete variable n ∈ N . Notethat the system (5) in the case α = 1 was discretized in [25]. Theorem 1
Let α = 1 . A system (15) admits n -integrals (8) and (9) if and only if it has theform u x = u + v u + v D α − u x ,v x = α +1 α v D α − − v D u + v u x + D v x . (16) The function D is equal to or given implicitly by H ( K , L ) = 0 , where H is any smoothfunction and K = αv D α − − αv D α − + (1 − D α − ) u ( D α − − α +1 , (17)3 = ( u − D α − u ) e D α − D ( D α − − a + ( − α e D α − ( αv D α − − αv D α − + (1 − D α − ) u )( D α − − α +1 . (18)Let us construct some examples. Example 1
In the case D = 1 the system (16) becomes u x = u + v u + v u x ,v x = (cid:18) α (cid:19) v − vu + v u x + v x . (19) This system is Darboux integrable. Indeed, it has two independent non-trivial n - integrals (8) , (9) and two independent non-trivial x -integrals F = v − v v − v and F = u − u ( v − v ) α α − α ( v − v ) α . (20) The x -integrals can be found by considering the x -algebra corresponding to the system. Example 2
Considering K = 0 and α = − we get D = u + v u + v . Using (16) we get thesystem u x = u + vu + v u x ,v x = u + v u + v v x . (21) This system is Darboux integrable. Indeed, it has two independent non-trivial n - integrals (8) , (9) and two independent non-trivial x -integrals F = ( u + v )( v − v )( u + v )( v − v ) and F = ( u − u )( u + v )( − u + u )( u + v ) . (22) Example 3
Considering K = 0 and α = − we get D = 4 u + 2 v v + p v + 16 u + 8 u v . Using (16) we get the system u x = u + v u + v v + p v + 16 u + 8 u v u + 2 v ! u x ,v x = − v u + v v + p v + 16 u + 8 u v u + 2 v ! − v (4 u + 2 v )( u + v )( v + p v + 16 u + 8 u v ) u x + 4 u + 2 v v + p v + 16 u + 8 u v v x (23) This system has two independent non-trivial n - integrals (8) and (9) . xample 4 Considering L = 0 and α = − we get D = v + p v + 16 u + 8 uv v + 4 u . Using (16) we get the system u x = u + v u + v v + 4 uv + p v + 16 u + 8 uv ! u x ,v x = − v u + v v + 4 uv + p v + 16 u + 8 uv ! − v ( v + p v + 16 u + 8 uv )( u + v )(2 v + 4 u ) u x + v + p v + 16 u + 8 uv v + 4 u v x (24) This system has two independent non-trivial n - integrals (8) and (9) . Remark 1
In both previous examples let us consider the corresponding x -rings. Denote by X = D x , Y = ∂∂u x , Y = ∂∂v x , E = [ Y , X ] , E = [ Y , X ] , E = [ E , E ] . Note that X = u x E + v x E + Y + Y . The following multiplication table [ E i , E j ] E E E E E − u + v ) − E E − E − u + v ) − E E u + v ) − E u + v ) − E shows that x -rings are finite-dimensional. Therefore systems (23) and (24) are Darboux integrable. Remark 2
Expansion of the function D α − given implicitly by H ( K , L ) = L = 0 , that is by αv D α − − αvD α − + (1 − D α − ) u = 0 , into a series of the form D α − ( u , v, v ) = ∞ X n =0 a n ( v − v ) n , where coefficients a n depend on vari-ables u and v only, yields D α − ( u , v, v ) = 1 + αu + α v ( v − v ) + ∞ X n =2 a n ( v − v ) n and D ( u , v, v ) = 1 + α u + α v ( v − v ) + ∞ X n =2 a n ( v − v ) n . By letting u = u + εu y , v = v + εv y and taking ε → one can see that the system (16) becomes(5). heorem 2 A system (15) admits n -integrals (11) and (12) if and only if it has the form u x = v ( u v + d ) D v ( uv + d ) u x ,v x = ( d − c ) vv ( D − c ( uv + d ) D u x + v v D v x . (25) The function D is given implicitly by H ( K , L ) = 0 , where H is any smooth function and K = v ( D − M − dc + d v D ( − cdu v + uv D M ) , L = v D M dc + d v , (26) where M = 2 cd + ( c + d )( D − u v D . Example 5
Considering K = 0 we can get D = (2 cd + ( c + d ) uv ) u v (2 cd + ( c + d ) u v ) uv . Using (25) we get thesystem u x = u ( u v + d )(2 cd + ( c + d ) uv ) u ( uv + d )(2 cd + ( c + d ) u v ) u x ,v x = ( d − c )2 c ( uv + d ) (cid:26) u v (2 cd + ( c + d ) uv ) u (2 cd + ( c + d ) u v ) − uv (2 cd + ( c + d ) u v ) u (2 cd + ( c + d ) uv ) (cid:27) u x + u (2 cd + ( c + d ) u v ) u (2 cd + ( c + d ) uv ) v x . (27) This system has two independent non-trivial n - integrals (11) and (12) . One can check that thissystem also has the following two n -integrals I ∗ = (2 cd + ( c + d ) uv ) u x u ( uv + d ) ,J ∗ = ( c − d ) uv u x c ( uv + d )(2 cd + ( c + d ) uv ) + uv x cd + ( c + d ) uv . Considering the corresponding x -algebra we can also find x -integrals given by F = u u (cid:18) cd + ( c + d ) uv cd + ( c + d ) u v (cid:19) c − dc + d , F = u v − uvu v − uv . (28) Example 6
Considering K = 0 we can also get D = ( c + d ) u v cd + ( c + d ) u v . Using (25) we get thesystem u x = ( c + d ) u v ( u v + d )( uv + d )(2 cd + ( c + d ) u v ) u x ,v x = ( d − c ) v c ( uv + d ) (cid:26) ( c + d ) u v cd + ( c + d ) u v − cd + ( c + d ) u v ( c + d ) u (cid:27) u x + 2 cd + ( c + d ) u v ( c + d ) u v v x . (29)6 his system has two independent non-trivial n - integrals (11) and (12) and two independent x -integrals F = 1 c + d (cid:18) cd + ( c + d ) u v vu (cid:19) c + d d + uu (cid:18) cd + ( c + d ) u v vu (cid:19) c − d d and F = v u d − c d (2 cd + ( c + d ) u v ) c + d d v c + d d (2 cd + ( c + d )( u v + u v )) . Remark 3
The expansion of the function D given implicitly by H ( K , L ) = L − (2 cd ) d/ ( c + d ) = 0 into a series of the form D ( u , v, v ) = ∞ X n =0 a n ( v − v ) n , where coefficients a n depend on vari-ables u and v only, yields D ( u , v, v ) = 1 + cv ( uv + c ) ( v − v ) + ∞ X n =2 a n ( v − v ) n . By letting u = u + εu y , v = v + εv y and taking ε → one can see that the system (25) becomes (10) with α = − . Theorem 3
A system (15) admits n -integrals (13) and (14) if and only if it has form u x = u v + d D ( uv + d ) u x ,v x = − βv D ( uv + d ) + βv D β uv + d ! u x + D β v x . (30) The function D is given implicitly by H ( K , L ) = 0 , where H is any smooth function and K = ( v − v D β ) β − ( d u − du D ) D , (31) L = ( v − v D β ) (1 − β ) β − ( d D β − − d + ( β − u ( v − v D β )) . (32) Example 7
Considering K = 0 we can get D = v /β v /β . Using (30) we get the system u x = ( u v + d ) v /β ( uv + d ) v /β u x ,v x = ( − βv v /β v /β ( uv + d ) + βv v v ( uv + d ) ) u x + v v v x . (33) This system has two independent non-trivial n - integrals (13) and (14) and two independent x -integrals F = (cid:18) − (cid:16) v v (cid:17) − ββ (cid:19) (cid:18) − d u + du (cid:16) v v (cid:17) β (cid:19) β − and F = v − ββ − v − ββ v − ββ − v − ββ . ne can check that this system also has the following two n -integrals I ∗ = v /β u x uv + d , J ∗ = v x v + βvu x uv + d . Example 8
Considering K = 0 we can also get D = d udu . Using (30) we get the system u x = ( u v + d ) du ( uv + d ) d u u x ,v x = ( − βdv u d u ( uv + d ) + βd β v u β d β u β ( uv + d ) ) u x + d β u β d β u β v x . (34) This system has two independent non-trivial n - integrals (13) and (14) and two independent x -integrals F = d β u β v − d β u β v d β u β v − d β u β v and F = ( d β u β v − d β u β v )( dd β u β u − d d β u β u + (1 − β ) uu ) dd uu . One can check that this system also has the following two n -integrals I ∗∗ = du x u ( uv + d ) , J ∗∗ = u β v x d β + βv u β u x d β ( uv + d ) . Example 9
Considering L = 0 with β = 2 we get D = d + R u v , where R = q d + 4 u v ( u v − d ) . Using (30) we get the system u x = ( u v + d )( d − R )2( uv + d )( d − u v ) u x ,v x = (cid:26) v ( R − d ) d − u v + d + 2 u v ( u v − d ) + d Ru (cid:27) u x uv + d + d + 2 u v ( u v − d ) + d R u v v x . (35) This system has two independent non-trivial n - integrals (13) and (14) . Example 10
Considering L = 0 with β = 1 / we get D / = 2 d + u v + R u v , where R = p (2 d + u v ) − d u v. sing (30) we get the system u x = ( u v + d )(2 d + u v − R ) d ( uv + d ) u x ,v x = (cid:26) − v (2 d + u v − R ) d + v (2 d + u v + R )4 u (cid:27) u x uv + d + 2 d + u v + R u v v x . (36) This system has two independent non-trivial n - integrals (13) and (14) . Remark 4
In both previous examples the corresponding x -rings have the following multiplicationtable [ E i , E j ] E E E E E − vd + uv E E − E − ud + uv E E vd + uv E ud + uv E where fields X , Y , Y , E , E and E are introduced in the same way as in Remark 1. It showsthat the x -rings are finite-dimensional and the corresponding systems are Darboux integrable. Remark 5
The expansion of the function D given implicitly by H ( K , L ) = L = 0 into aseries of the form D ( u , v, v ) = ∞ X n =0 a n ( v − v ) n , where coefficients a n depend on variables u and v only, yields D ( u , v, v ) = 1 + u βu v − d ( v − v ) + ∞ X n =2 a n ( v − v ) n . By letting u = u + εu y , v = v + εv y and taking ε → one can see that the system (30) becomes(10). Note that β = − α . It follows from the equality DJ = J that u xx u x − (cid:18) α (cid:19) u x u + v − v x u + v = u xx u x − ( α + 1) u x + αv x α ( u + v ) , (37)that is f x + f u u x + f v v x + f u f + f v g + f u x u xx + f v x v xx f − (cid:18) α (cid:19) fu + v − gu + v = u xx u x − ( α + 1) u x + αv x α ( u + v ) . (38)9y comparing the coefficients by v xx and u xx , we get f v x = 0 and f u x f = 1 u x . Hence f ( x, n, u, v, u , v , u x , v x ) = A ( x, n, u, v, u , v ) u x . (39)It follows from DI = I that (cid:18) α (cid:19) v (cid:18) Au x u + v (cid:19) − α − g (cid:18) Au x u + v (cid:19) − α = (cid:18) α (cid:19) v (cid:18) u x u + v (cid:19) − α − v x (cid:18) u x u + v (cid:19) − α , (40)that is g = (cid:18) α (cid:19) (cid:18) Av u + v − vA α u + v (cid:18) u + vu + v (cid:19) α (cid:19) u x + (cid:18) A u + vu + v (cid:19) α v x . (41)By substituting the expressions for f and g into (38) and comparing the coefficients by v x , u x andthe free term we get A x A = 0 , (42) A u A + A u + (cid:18) α (cid:19) (cid:20) A v v u + v − vA α A v A ( u + v ) (cid:18) u + vu + v (cid:19) α − Au + v − Av ( u + v ) + vA α ( u + v )( u + v ) (cid:18) u + vu + v (cid:19) α + 1 u + v (cid:21) = 0 , (43) A v A + A v A α A (cid:18) u + vu + v (cid:19) α − A α u + v (cid:18) u + vu + v (cid:19) α + 1 u + v = 0 . (44)Let D = (cid:18) u + vu + v (cid:19) α A α . (45)In terms of the function D the equations (43) and (44) become( u + v ) D u + ( u + v ) D α − D u + α + 1 α ( v D α − − v D ) D v − D ( D α − −
1) = 0 , (46) D v D + D v = 0 . (47)The set of solutions of the above system is not empty. For example, D = 1 is one singularsolution that leads to the Darboux integrable system (19). Let D = 1. For function W = W ( u, v, u , v , D ) equations (46) and (47) become( u + v ) W u + ( u + v ) D α − W u + α + 1 α ( v D α − − v D ) W v + D ( D α − − W D = 0 , (48) W v D + W v = 0 . (49)After the change of variables ˜ v = v , ˜ v = v − v D , ˜ u = u , ˜ u = u , ˜ D = D equations abovebecome (˜ u + ˜ v ) W ˜ u + (˜ u + ˜ v + ˜ v ˜ D ) ˜ D α − W ˜ u + ((1 + α )˜ v ˜ D α − + α ˜ v ( ˜ D α − − ˜ D )) W ˜ v +( ˜ D α − − ˜ D ) W ˜ D = 0 ,W ˜ v = 0 .
10e differentiate the first equation with respect to ˜ v , use W ˜ v = 0, and get two new equations W ˜ u + ˜ D α − W ˜ u + 1 α ( ˜ D α − − ˜ D ) W ˜ v = 0 , (50)˜ uW ˜ u + (˜ u + ˜ v ) ˜ D α − W ˜ u + α + 1 α ˜ v ˜ D α − W ˜ v + ( ˜ D α − − ˜ D ) W ˜ D = 0 . (51)After the change of variables u ∗ = ˜ u − ˜ D α − ˜ u , v ∗ = α ˜ D α − ˜ v + (1 − ˜ D α − )˜ u , u ∗ = ˜ u , v ∗ = ˜ v and D ∗ = ˜ D the last system becomes W u ∗ = 0 , (cid:16) ( D ∗ α − + α − (1 − D ∗ − α − )) u ∗ + α − v ∗ D ∗ − α − (cid:17) W u ∗ + α +1 α v ∗ D ∗ α − W v ∗ + D ∗ ( D ∗ α − − W D ∗ = 0 . The last equation has a general solution H ( K , L ) = 0, where K , L (rewritten in old variables)are given by (17), (18) and H is any smooth function. Now, using the equalities (45), (39) and(41) we obtain the system (16). (cid:3) The equality DJ = J implies f x + f u u x + f v v x + f u f + f v g + f u x u xx + f v x v xx f + ( d − c ) v f − c u gc ( u v + d )= u xx u x + ( d − c ) vu x − cuv x c ( uv + d ) . (52)By comparing the coefficients by u xx and v xx in the above equality we get f v x = 0 and f u x f = 1 u x .Hence f = A ( x, n, u, v, u , v ) u x . (53)Equality DI = I implies( d − c ) v A u x u v + d ) − cAgu v + d = ( d − c ) v u x uv + d ) − cv x uv + d . (54)It follows from (54) that g = (cid:18) ( d − c ) v A c ( u v + d ) − ( d − c ) v ( u v + d )2 c A ( uv + d ) (cid:19) u x + c ( u v + d ) c A ( uv + d ) v x . (55)11y substituting the expressions for f and g into (52) and comparing the coefficients by u x , v x andfree term we get A x A = 0 ,A u A + A u + (cid:18) A v A − u u v + d (cid:19) (cid:18) ( d − c ) v A c ( u v + d ) − ( d − c ) v ( u v + d )2 c A ( uv + d ) (cid:19) + ( d − c ) v Ac ( u v + d ) − ( d − c ) vc ( uv + d ) = 0 , (56) A v A + c ( u v + d ) c A ( uv + d ) (cid:18) A v A − u u v + d (cid:19) + uuv + d = 0 . One can check that A = v ( u v + d ) v ( uv + d ) is a particular solution provided d = d and c = c . Nowassuming that A = v ( u v + d ) v ( uv + d ) we introduce new function D = v ( uv + d ) v ( u v + d ) A. (57)In terms of D the system (56) becomes D x = 0 , ( uv + d ) D u + v ( u v + d ) D v D u + vv c (cid:0) ( d − c ) D − ( d − c ) D − (cid:1) D v − dvc D + ( d + c ) v c D + v ( d − c )2 c = 0 ,c v D D v + cv D v + ( − c D + c D ) = 0 . For function W = W ( u, v, u , v , D ) the last two equations become( uv + d ) W u + v ( u v + d ) v D W u + vv c (cid:0) ( d − c ) D − ( d − c ) D − (cid:1) W v + (cid:18) dvc D − ( d + c ) v c D − v ( d − c )2 c (cid:19) W D = 0 ,c v D W v + cv W v + ( c D − c D ) W D = 0 . In new variables ˜ u = u , ˜ u = u , ˜ v = v ( c D − c ), ˜ v = v ( c D − c ) D − , ˜ D = D the last systemcan be rewritten as (cid:16) ( c ˜ D − c )˜ u ˜ v + d ( c ˜ D − c ) (cid:17) W ˜ u + ˜ v ˜ v (cid:16) ˜ u ˜ v ˜ D ( c ˜ D − c ) + d ( c ˜ D − c ) (cid:17) W ˜ u +˜ v c d ˜ D c − ( d + c ) ˜ D c − d ! W ˜ v + ˜ v ˜ v ( d − c ) ˜ D − cd ˜ D c + c + d ! W ˜ v = 0 , (58) W ˜ D = 0 . D = c /c . We differentiate equation (58) withrespect to ˜ D three times and get the following system of three equations( dc − c ˜ u ˜ v ) W ˜ u + c d ˜ v ˜ v W ˜ u + ( c − d )˜ v W ˜ v + ( c + d )˜ v ˜ v W ˜ v = 0 , (59)( c ˜ u ˜ v − dc c ) W ˜ u − (2 d c c ˜ v ˜ v + c ˜ u ˜ v ) W ˜ u + c d ˜ v c W ˜ v − cd ˜ v ˜ v c W ˜ v = 0 , (60) dc W ˜ u + (cid:18) d c ˜ v ˜ v + c ˜ u ˜ v (cid:19) W ˜ u − ( d + c )˜ v W ˜ v + ( d − c )˜ v ˜ v W ˜ v = 0 , (61)that has no solutions if c = c or d = d . In case of c = c and d = d the system becomes W ˜ u − ˜ v (2 c d + ( c − d ) ˜ u ˜ v )2 c (˜ u ˜ u ˜ v ˜ v + cd (˜ u ˜ v − ˜ u ˜ v )) W ˜ v − ˜ v ˜ v (2 c d + ( c + d ) ˜ u ˜ v )2 c (˜ u ˜ u ˜ v ˜ v + cd (˜ u ˜ v − ˜ u ˜ v )) W ˜ v = 0 , (62) W ˜ u − ˜ v ˜ v ( − c d + ( c + d )˜ u ˜ v )2 c (˜ u ˜ u ˜ v ˜ v + cd (˜ u ˜ v − ˜ u ˜ v )) W ˜ v + ˜ v (2 c d + ( − c + d )˜ u ˜ v )2 c (˜ u ˜ u ˜ v ˜ v + cd (˜ u ˜ v − ˜ u ˜ v )) W ˜ v = 0 . (63)After the change of variables u ∗ = ˜ u , v ∗ = ˜ v , v ∗ = ˜ v ˜ v (2 c d + ( c + d ) ˜ u ˜ v ) dc + d , u ∗ = ˜ u ˜ v (2 c d + ( c + d ) ˜ u ˜ v ) c − dc + d − c d ˜ u ˜ v ˜ v − (2 c d + ( c + d ) ˜ u ˜ v ) − dc + d equations (62) and (63)become W v ∗ = 0 and W u ∗ = 0 respectively. We rewrite these first integrals in old variables and getthat the general solution is given implicitly by H ( K , L ) = 0, where H is any smooth functionand K , L are given by (26). The form of system (25) follows from (53), (55) and (57). (cid:3) Proof.
Equality DJ = J implies − f x + f u u x + f v v x + f u f + f v g + f u x u xx + f v x v xx f + 2 v f + u gu v + d = − u xx u x + 2 vu x + uv x uv + d . (64)By comparing the coefficients by u xx and v xx in the above equality we get f v x = 0 and f u x f = 1 u x .Hence f = A ( x, n, u, v, u , v ) u x . (65)Equality DI = I implies f β g ( u v + d ) β + βv f β +1 ( u v + d ) β +1 = u βx v x ( uv + d ) β + βv u β +1 x ( uv + d ) β +1 . (66)Let D = u v + d A ( uv + d ) . (67)13hen from (66) we get g = − βv D ( uv + d ) + βv D β ( uv + d ) ! u x + D β v x . (68)The equality (64) in terms of D takes the form D x D + D u D + u v + d D ( uv + d ) D u + β ( v D β − v D − ) D ( uv + d ) D v + v D ( uv + d ) − v ( uv + d ) ! u x + (cid:18) D v D + D β − D v (cid:19) v x = 0 . (69)By comparing the coefficients by u x , v x and free term we get D x = 0 , (70) uv + d D D u + u v + d D D u + βv D β − βv D − D D v + v D − v = 0 , (71) D v + D β D v = 0 . (72)In new variables ˜ v = v − v D β , ˜ v = v , ˜ u = u , ˜ u = u , ˜ D = D equations (72) and (71) can berewritten for function W = W (˜ u, ˜ v, ˜ u , ˜ v , ˜ D ) as follows W ˜ v = 0 , ˜ D (˜ u ˜ v + d ) W ˜ u + ( ˜ u ( ˜ v + ˜ v ˜ D β ) + d ) W ˜ u + ˜ D (˜ v ( ˜ D − ˜ D β ) − ˜ v ) W ˜ D − β ˜ v ( ˜ v + ˜ v ˜ D β ) W ˜ v = 0 . We differentiate the last equation with respect to ˜ v , use the fact that W ˜ v = 0 and get the newsystem of equations˜ u ˜ D W ˜ u + ˜ u ˜ D β W ˜ u + ( ˜ D − ˜ D β +1 ) W ˜ D − β ˜ v ˜ D β W ˜ v = 0 ,d ˜ D W ˜ u + ( ˜ u ˜ v + d ) W ˜ u − ˜ D ˜ v W ˜ D − β ˜ v W ˜ v = 0 , that can be rewritten as W ˜ u + d ˜ D − d ˜ D β + ˜ D ˜ u ˜ v d ˜ u − d ˜ D β ˜ u + ˜ u ˜ u ˜ v W ˜ D − βd ˜ v ˜ D β − d ˜ u − d ˜ D β ˜ u + ˜ u ˜ u ˜ v W ˜ v = 0 ,W ˜ u − ˜ D ( d ˜ D − d ˜ D β + ˜ u ˜ v ) d ˜ u − d ˜ D β ˜ u + ˜ u ˜ u ˜ v W ˜ D + β ˜ v ( d ˜ D β − ˜ u ˜ v ) d ˜ u − d ˜ D β ˜ u + ˜ u ˜ u ˜ v W ˜ v = 0 . After the change of variables u ∗ = ˜ u ˜ v /β d / (1 − β )1 ˜ D − − dd β/ (1 − β )1 ˜ u ˜ v /β , D ∗ = ˜ v − β ) /β ˜ D β − − ˜ v − β ) /β +( β − d − ˜ u ˜ v /β , u ∗ = ˜ u , v ∗ = ˜ v , v ∗ = ˜ v the last two equationsbecome respectively W v ∗ = 0 and W u ∗ = 0. We rewrite these first integrals in old variables andget that general solution is given implicitly by H ( K , L ) = 0, where H is any smooth functionand K , L are given by (31), (32). The form of system (30) follows from (65), (68) and (67). (cid:3) eferences [1] Darboux G. Leconsur la theory generale des surface et les applicationgeometriques du calculusinfinitesimal vol. (Paris: Gautier Villas) (1915)[2] Shabat A. B., Yamilov R. I. Exponential systems of type I and Cartan matrices (Russian)
Preprint
BBAS USSR Ufa (1981)[3] Leznov A.N., Smirnov V.G. and Shabat A.B.
Internal symmetry group and integrabilityconditions for two-dimensional dynamical systems (Russian) Teoret. Mat. Fiz. , 10-21(1982)[4] Sokolov V.V., Zhiber A.V. On the Darboux integrable hyperbolic equations
Phys. Lett. A , 303-308 (1995)[5] Zhiber A.V., Sokolov V.V. and Startsev S.Ya.
On nonlinear Darboux-integrable hyperbolicequations , (Russian) Dokl. Akad. Nauk , 746-748 (1995)[6] Zhiber A. V., Sokolov V. V.
Exactly integrable hyperbolic equations of Liouville type , RussianMath. Surveys , 61-101 (2001)[7] Zhiber A.V., Murtazina R.D. On the characteristic Lie algebras for the equations u xy = f ( u, u x ), J. Math. Sci. (N. Y.) , 3112-3122 (2008)[8] Kostrigina O.S., Zhiber A.V. Darboux-integrable two-component nonlinear hyperbolic sys-tems of equations , J. Math. Phys. , 033503 (2011)[9] Murtazina R. D. Nonlinear hyperbolic equations with characteristic ring of dimension 3 ,(Russian) Ufa Math. J. , 113-118 (2011)[10] Anderson I. M., Fels M. E. The Cauchy problem for Darboux integrable systems and non-linear d’Alembert formulas , SIGMA , Paper 017, (2013)[11] Anderson, I. M., Fels, M. E. and Vassiliou P. J. On Darboux integrability , SPT 2007Symmetryand perturbation theory, 1320, World Sci. Publ., Hackensack, NJ, (2008)[12] Zhiber A.B., Murtazina R.D., Habibullin I.T. and Shabat A.B.
Characteristic Lie rings andintegrable models in mathematical physics , Ufa Math. J. , 17-85 (2012)1513] Habibullin I.T., Pekcan A. Characteristic Lie algebra and the classification of semi-discretemodels , Theoret. and Math. Phys. , 781-790 (2007)[14] Adler V. E., Startsev S.Ya.
On discrete analogues of the Liouville equation , Theoret. andMath. Phys. , 1484-1495 (1999)[15] Habibullin I.T.
Characteristic algebras of fully discrete hyperbolic type equations , SIGMASymmetry Integrability Geom.: Methods Appl. , 9 (2005)[16] Habibullin I.T., Zheltukhin K. and Yangubaeva M. Cartan matrices and integrable latticeToda field equations
J. Phys. A , no. 46, 465202 (2011)[17] Habibullin I.T., Zheltukhina N. and Pekcan A. On some algebraic properties of semi-discretehyperbolic type equations , Turkish J. Math. , 277-292 (2008)[18] Habibullin I.T., Zheltukhina N. and Pekcan A. On the classification of Darboux integrablechains , J. Math. Phys. , 102702 (2008)[19] Habibullin I.T., Zheltukhina N. and Pekcan A. Complete list of Darboux integrable chainsof the form t x = t x + d ( t, t , 102710 (2009)[20] Habibullin I.T., Zheltukhina N. and Sakieva A. On Darboux-integrable semi-discrete chain ,J. Phys. A , 434017 (2010)[21] Habibullin I., Zheltukhina N. and Sakieva A. Discretization of hyperbolic type Darboux in-tegrable equations preserving integrability , J. Math. Phys. , 093507( 2011).[22] Habibullin I.T. and Gudkova E.V. Classification of integrable discrete Klein-Gordon models ,Physica Scripta bf 81, 045003 (2011)[23] Smirnov S V.
Semidiscrete Toda lattices , Theoret. and Math. Phys. , no. 3, 12171231(2012)[24] Habibullin I.T. and Zheltukhina N.