Classification of integrable vector equations of geometric type
aa r X i v : . [ n li n . S I] A p r Classification of integrable vectorequations of geometric type
A.G. Meshkov a , V.V. Sokolov b,c a ) . Orel State University, 95, Komsomolskajastr., 302026, Orel, Russia b ) . Landau Institute for Theoretical Physics,142432, Chernogolovka, Russia c ) . Universidade Federal do ABC, 09210-580,Sao Paulo, Brazil
ABSTRACT. A complete classification of isotropic vector equations of the geometrictype that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type are found.
Consider evolution systems of the form u it = u ixxx + 3 A ijk ( u ) u jx u kxx + B ijkl ( u ) u jx u kx u lx , i, j, k, s = 1 , . . . , N, (1.1)where u = ( u , . . . , u N ). Here and below, we assume that the summation is carried out overrepeated indexes.Integrable systems of this type are connected with various geometric and algebraic structuresand are of interest by themselves. In addition, the most interesting of them play the role ofinfinitesimal symmetries for physically important hyperbolic systems of the form u ixy = C ijk ( u ) u jx u ky . (1.2)Having an efficient description of integrable systems (1.1), we can construct a class of integrablesystems of the form (1.2) following the approach from the papers [1, 2].An example of such type integrable system provides the following equation [3] U t = U xxx − U x U − U xx − U xx U − U x + 32 U x U − U x U − U x , where U ( x, t ) is an m × m matrix. In this case N = m . For any m this system has infinitelymany local symmetries and conservation laws.It is convenient to rewrite (1.1) in the following way u it = u i + 3 A ijk ( u ) u jx u kxx + ∂A ijk ∂u l + 2 A ils A sjk − A isl A sjk + β ijkl ! u jx u kx u lx . (1.3)he class of systems (1.3) is invariant under the arbitrary point transformations u → Φ ( u ). Itis easy to see that under such a change of coordinates, the functions A ijk and β ijkm are transformedjust as components of an affine connection Γ and of a tensor β , respectively. Example 1.
In the case N = 1 equation (1.3) has the form u t = u xxx + 3 A ( u ) u x u xx + (cid:16) A ′ ( u ) + A ( u ) + β ( u ) (cid:17) u x . Using the symmetry approach (see [4]), one can verify that this equation possesses higher sym-metries iff β ′ = 2 Aβ.
By a proper point transformation of the form u → Φ( u ) the function A can be reduced to zero (for N = 1 any affine connection is flat) and the function β becomes aconstant. The equation u t = u xxx + const u x is known to be integrable and it is related to themKdV equation by a potentiation.Without loss of generality we assume that the tensor β is symmetric: β ( X, Y, Z ) = β ( Y, X, Z ) = β ( X, Z, Y )for any vectors
X, Y, Z . The functions β ijkm are defined by the values of β ( X, X, X ) . Suppose a system of the form (1.3) has higher symmetries and/or non-degenerate conservationlaws and A ijk = A ikj , i.e. the torsion tensor T is equal to zero. Then the corresponding affineconnected space is symmetric [6] which means that ∇ X (cid:16) R ( Y, Z, V ) (cid:17) = 0 , (1.4)where R is the curvature tensor . Let σ ( X, Y, Z ) def = β ( X, Y, Z ) − (cid:16) R ( X, Y, Z ) + R ( Z, Y, X ) (cid:17) . Then σ ( X, Y, Z ) = σ ( Z, Y, X ) , (1.5) ∇ X (cid:16) σ ( Y, Z, V ) (cid:17) = 0 , (1.6) R ( X, Y, Z ) = σ ( X, Z, Y ) − σ ( X, Y, Z ) , (1.7)and σ ( X, σ ( Y, Z, V ) , W ) − σ ( W, V, σ ( X, Y, Z )) + σ ( Z, Y, σ ( X, V, W )) − σ ( X, V, σ ( Z, Y, W )) = 0 . (1.8) It was discovered by S. Svinilupov and V. Sokolov and was published without proof in the survey [5] dedicatedto Sergey Svinolupov. We use here the following formula for the curvature tensor: R mijk = ∂∂u j A mki − ∂∂u k A mji + A mjs A sk − A mks A sji . u the tensor σ ( u ) defines a triple Jordansystem [7, 8]. Conjecture 1.
If a symmetric ( T = 0) affine connection and a tensor σ satisfy identities(1.4) – (1.7) and (1.8) then the corresponding system (1.3) possesses infinitely many localsymmetries and conservation laws.Several integrable systems of the form (1.3) that correspond to symmetric connections canbe found in [9] but no integrable models corresponding to the case T = 0 are known. In thispaper we construct examples of integrable models (1.3) such that T = 0 and R = 0 . Our goal is to find all non-triangular integrable systems of the form (1.1), which belong to aspecial class of vector isotropic equations of the form u t = u xxx + f u xx + f u x + f u , (1.9)where u ( x, t ) is an N -dimensional vector and the coefficients f i are supposed to be functions ofthe following six independent scalar products:( u , u ) , ( u , u x ) , ( u x , u x ) , ( u , u xx ) , ( u x , u xx ) , ( u xx , u xx ) . (1.10)Equations (1.9) are invariant with respect to the orthogonal group O N . It is clear that any equation (1.9) whose component form belong to the class of equations(1.1) has the following structure: u t = u xxx + a u [0 , u xx + ( a u [0 , + a u [1 , + a u , ) u x + ( a u [1 , + a u [0 , u [0 , + a u [1 , u [0 , + a u , ) u , (1.11)where u [ i,j ] = ( ∂ ix u , ∂ jx u ) , i j (1.12)and the coefficients a i are functions in one variable: a i = a i ( u [0 , ). In this case the components,the torsion and the curvature tensors for the corresponding affine connection are given by A ( X, Y ) = 13 (cid:16) a ( u , X ) Y + a ( u , Y ) X + (cid:16) a ( X, Y ) + a ( u , X )( u , Y ) (cid:17) u (cid:17) ,T ( X, Y ) = 13 ( a − a ) (cid:16) ( u , X ) Y − ( u , Y ) X (cid:17) and R ( X, Y, Z ) = 19 (cid:16) q ( u , X )( u , Z ) + p ( X, Z ) (cid:17) Y − (cid:16) q ( u , X )( u , Y ) + p ( X, Y ) (cid:17) Z + r (cid:16) ( u , Y )( X, Z ) − ( u , Z )( X, Y ) (cid:17) u , It is clear that β ( X, X, X ) = σ ( X, X, X ). p = a a u − a + 3 a , q = a a u + a + 3 a − a ′ , r = a a u + a − a + 6 a ′ . To find all integrable equations (1.11), we use a version of the symmetry approach developedin [10] for vector equations.In Section 2 we discuss necessary conditions [10] of the existence of higher symmetries forvector equations of the form (1.9). In Section 3 we present lists of integrable equations (1.11),formulate and prove classification statements. For some of these equations written in componentsof the vector u the torsion T is not zero. To justify the real integrability of equations found inSection 3, we detect (see Section 4) auto-B¨acklund transformations for these equations. Each ofthem is a new integrable semi-discrete model. Acknowledgments.
The authors are grateful to E. Ferapontov and P. Leal da Silva foruseful discussions. VS was supported by the state assignment No 0033-2019-0006. He is thankfulto IHES for its support and hospitality.
It was shown in [10] that if an equation of the form (1.9) has infinitely many vector highersymmetries u τ = f n u n + f n − u n − + · · · + f u , where u k = ∂ k u ∂x k , (2.1)then an infinite series of special local conservation laws exists for equation (1.9). Their densities ρ n , n = 0 , , . . . are called canonical .The first two canonical densities are given by ρ = − f , (2.2) ρ = 19 f − f + 13 D x f . (2.3)Using a technique developed in the papers [11, 12], one can obtain the following recursion formulafor other canonical densities for equations of the form (1.9): ρ n +2 = 13 (cid:20) θ n − f δ n, − f ρ n +1 − f D x ρ n − f ρ n (cid:21) − (cid:20) f n X s =0 ρ s ρ n − s + X s + k n ρ s ρ k ρ n − s − k + 3 n +1 X s =0 ρ s ρ n − s +1 (cid:21) − D x (cid:20) ρ n +1 + 12 n X s =0 ρ s ρ n − s + 13 D x ρ n (cid:21) , n > . (2.4) A conservation law D t ( ρ ) = D x ( θ ) is called local if ρ and θ are functions of variables (1.12). δ i,j denotes the Kronecker delta and the functions θ i are fluxes of the canonicalconservation laws D t ρ n = D x θ n , n = 0 , , , . . . (2.5)In this formula D x and D t are the total derivatives of x and t , respectively. For brevity, we callrelation (2.5) ρ n -integrability condition.Using formulas (2.2)–(2.4), one can obtain the next density ρ = − f + 13 θ − f + 19 f f − D x (cid:18) f + 29 D x f − f (cid:19) (2.6)and so on. Notice that the density ρ n , n > { θ , θ , . . . θ n − } . These fluxes are to be calculated from the previous conditions (2.5).To eliminate the function θ n from (2.5) one can apply the variational derivative δδ u = X i j (cid:20) ( − D x ) i (cid:18) u j ∂∂u [ i,j ] (cid:19) + ( − D x ) j (cid:18) u i ∂∂u [ i,j ] (cid:19)(cid:21) . (2.7)to both sides of (2.5) and use the fact that δ ( D x g ) δ u = 0for any function g (see, for example [13], chapter 4) to obtain δδ u ( D t ρ n ) = 0 , n = 1 , , . . . . (2.8)Conditions (2.8) are most efficient for the cases n = 1 , θ i . We are searching for non-triangular integrable equations of the form (1.11). In this section, integrability means the existence of an infinite sequence of higher symmetries [13, 4] of theform (2.1).Some equations (1.9) become triangular in the spherical coordinates, which is defined by theformulas u = R v , | v | = 1 , where R = | u | . Let v [ i,j ] = ( ∂ ix v , ∂ jx v ) , i j. Since v [0 , = 1, we have D x ( v [0 , ) = 2 v [0 , = 0. Moreover, D x v [0 , = v [0 , + v [1 , = 0, i.e. v [0 , = − v [1 , and so on. It is clear that all variables v [0 ,k ] can be expressed in terms of thevariables v [ i,k ] , i k < ∞ . 5e call equation (1.9) triangular if it can be rewritten in the spherical coordinates as v t = v xxx + g v xx + g v x + g v ,R t = R xxx + S ( v [1 , , v [1 , , v [2 , , R, R x , R xx ) , where the coefficients g i depend on v [1 , , v [1 , , v [2 , only. The class of equations of the form (1.11) is invariant with respect to the point transformationsof the form u = v ϕ ( v [0 , ) . (3.1)Under such a transformation the coefficient a changes as follows:˜ a ( v [0 , ) = 2 ϕ − ϕ ′ (cid:0) v [0 , a ϕ + 3 (cid:1) + a ϕ , where a ( u [0 , ) = a (cid:0) v [0 , ϕ (cid:1) . It easy to see that if a = − u [0 , − , then we obtain ˜ a = − v [0 , − . For any function a different from − u [0 , − we can choose the function ϕ such that ˜ a vanishes. Thus, up to thepoint transformations we have two non-equivalent cases: . a = 0 and . a = − u [0 , . Theorem 1.
Any non-triangular integrable equation of the form (1.11) with a = 0 can bereduced to one of equations from the following List 1 by a scaling of the form u → λ u . List 1. u t = u xxx + 3 λz u x u , z + u [0 , − u [1 , ! + 3 z u F, where λ = 1 or λ = 12 , (3.2) u t = u xxx − z u x zu [0 , z + u [0 , − u [0 , u [1 , z + u [0 , + u [0 , u , ( z + u [0 , ) ! + 3 z u F, (3.3) u t = u xxx − z u x zu [0 , z + u [0 , + ( z − u [0 , ) u [1 , z + u [0 , ) − (2 z − u [0 , ) u , z + u [0 , ) ! + 3 z u F, (3.4)where F = u [0 , u [0 , + u [1 , z + u [0 , − u [1 , − u , ( z + u [0 , ) . Here z = 0 is an arbitrary parameter. Theorem 2.
Any non-triangular integrable equation of the form (1.11) with a = − u [0 , can be reduced to one of the equations from the following List 2 by a point transformation ofthe form (3.1). 6 ist 2. u t = u xxx − u xx u [0 , u [0 , − u x u [1 , u [0 , − u , u , ! , (3.5) u t = u xxx − u xx u [0 , u [0 , − u x u [1 , u [0 , − u , u , ! , (3.6) u t = u xxx − u xx u [0 , u [0 , − u x (cid:18) u [0 , u [0 , + u [1 , u [0 , (cid:19) + 3 u u [1 , u [0 , − u [0 , u [1 , u , + 43 u , u , ! . (3.7) Remark 1.
Equation (3.7) is equivalent to the equation u t = u xxx − u xx u [0 , u [0 , − u x u [0 , u [0 , + u [1 , u [0 , − u , u , ! + 3 u u [1 , u [0 , − u [1 , u [0 , u , ! (3.8)found in [5, formula (59)]. Remark 2.
Using the formulas from Introduction, one can verify that for equations (3.2)and (3.7) the torsion T is equal to zero while for equations (3.3)–(3.6) we have T = 0 , R = 0. The equation under consideration is the following: u t = u xxx + ( a u [0 , + a u [1 , + a u , ) u x +( a u [1 , + a u [0 , u [0 , + a u [1 , u [0 , + a u , ) u . (3.9)For such equations the canonical densities (2.2), (2.3), and (2.6) are given by ρ = 0 , ρ = − (cid:0) a u [0 , + a u [1 , + a u , (cid:1) ,ρ = − (cid:0) a u [1 , + a u [0 , u [0 , + a u [1 , u [0 , + a u , (cid:1) + 13 D x ( a u [0 , + a u [1 , + a u , ) . Consider the ρ -condition. The equality (2.8) with n = 1 has the form P i =0 q i u i = 0 , where q = 4 u [0 , ( a ′ − a ′ ) . Hence, a = a + c , where c is a constant. Eliminating a , we find that q vanishes, whichallows us to write the coefficient q as q = ( u [0 , + 3 u [1 , ) (cid:0) ( a u [0 , − a ′ − a ) − c ( a + a ) (cid:1) + u [0 , (cid:0) u [0 , F + u [1 , F + u , F (cid:1) , where F i = F i ( u [0 , ). The functions F i are too cumbersome to be shown explicitly here whilethe difference F − F is very short: F − F = c ( a ′ + 2 a − a ) . n = 1 we have obtained three simple relations a = a + c , ( a u [0 , − a ′ − a ) − c ( a + a ) = 0 , c ( a ′ + 2 a − a ) = 0 . (3.10)Consider now the ρ -condition. We obtain δδ u ( D t ρ ) = X i =0 p i u i = 0 , where p = (cid:16) ( u [0 , + u [1 , ) ( a ′ + a − a ) + 2 u , ( a ′′ + a ′′ − a ′ ) (cid:17) . Equating p to zero, we find a = a ′ + a and conclude that this implies p = 0. Substituting a into third of equations (3.10), we obtain that c a ′ = 0. Equating now p to zero, we find onemore simple relation a ′ − a = 0. So the ρ -condition implies a = a ′ + a , c ( a − a ) = 0 , a ′ − a = 0 . (3.11)Several more useful relations can be derived from the ρ -condition. The density ρ has thefollowing structure: ρ = 13 θ − D x ( θ ) + R, where R does not depend on θ and θ . The term with θ disappears when we apply thevariational derivative in the formula (2.8) with n = 4. So to use the ρ -condition, we have tospecify the form of the function θ only.Using (3.11), we obtain that ρ is trivial: ρ = D x ( S ), where S = 16 (cid:16) a u , − a u , + 2 a u [0 , + 2 a u [1 , − a u [1 , + 2 c u [1 , (cid:17) . Therefore, θ = D t ( S ). Taking into account this expression for θ , we find that δδ u ( D t ρ ) = X i =0 r i u i , where r = 2 (cid:16) a ′′ u , + a ′ ( u [0 , + u [1 , ) (cid:17) . This means that a = c , where c is a constant. Substituting a = c into the second equationof (3.10), we obtain( c u [0 , − a ′ − a ) + ( a + c ) c = 0 , or a = 2 a ′ − c ( a + c ) c u [0 , − . (3.12)8sing (3.10), (3.11), (3.12), we express all coefficients in (3.9) in terms of a , a , c , c . Thenthe coefficient r vanishes and r turns into r = (cid:0) u [0 , + 4 u [1 , (cid:1)(cid:0) a c u [0 , + c − a (cid:1) + 4 u [[0 , u [0 , c (cid:0) a ′ ( c u [0 , −
3) + a c (cid:1) . The equation r = 0 is then equivalent to relations a = − c c u [0 , − c (cid:0) a ′ ( c u [0 , −
3) + a c (cid:1) = 0 (3.13)and we have proved the following: Lemma 1.
Any integrable equation (3.9) has the form u t = u xxx + u x (cid:18) a u [0 , + ( a + c ) u [1 , + (cid:16) a ′ − c a + c c u [0 , − (cid:17) u , (cid:19) + u c u [1 , − c u [0 , u [0 , + u [1 , c u [0 , − c u , ( c u [0 , − ! , (3.14)where a = a ( u [0 , ) and c i are constants.Let us consider the following two branches A . c = 0 and B . c = 0 . In Case A the ρ -condition leads to c = 0 and a = 0 and the linear equation u t = u appears.Case B we separate into two following subcases: B . . a = 0 and B . . a = 0 . Consider Case
B.1. If c = 0, then we arrive at the equation u t = u + u c u [1 , . Thisequation is not integrable since the ρ -condition leads to a contradiction.If c = 0, then equation (3.14) coincides with equation (3.2), where c = − z , c = − λz . The ρ -condition gives rise to the following equation(2 λ − λ −
1) = 0 . Consider Case
B.2.
The coefficient r in the ρ -condition is given by r = − u [0 , u [1 , c (cid:0) a ′ − a (cid:1) − u [0 , u [0 , c u [0 , − a c (cid:0) a ( c u [0 , −
3) + 3 c (cid:1) . c (cid:0) a ′ − a (cid:1) = 0 , a c (cid:16) a ( c u [0 , −
3) + 3 c (cid:17) = 0and we have a = − c c u [0 , − . Then ρ -condition provides the following equation( c + c )(2 c + c ) = 0 . The two possibilities c = − c and c = − c c = − z . (cid:3) Remark 3.
We have verified that all equations of List 1 satisfy the ρ n -conditions with n ρ n , where n = 0 , , ,
6, are total x -derivatives. In accordance with a generalstatement from [10] this is an indication of the existence of infinite series of local conservationlaws. The canonical conservation laws, corresponding to n = 1 , , ,
7, have the orders 1 , , , , respectively. Moreover, each equation from List 1 possesses a fifth order symmetry. Consider equations of the form u t = u xxx − u [0 , u [0 , u xx + ( a u [0 , + a u [1 , + a u , ) u x +( a u [1 , + a u [0 , u [0 , + a u [1 , u [0 , + a u , ) u . (3.15)The simplest canonical densities (2.2), (2.3), and (2.6) are given by ρ = 12 D x ( u [0 , ) , ρ = u , u , − (cid:16) a u [0 , + a u [1 , + a u , (cid:17) − D x (cid:16) u [0 , u [0 , (cid:17) ,ρ = 13 D x u [0 , u [0 , − u , u , − u [1 , u [0 , + 2 D x (cid:16) u [0 , u [0 , (cid:17) + a u [0 , + a u [1 , + a u , ! . Using the same line of reasoning as in Section 3.2, we derive short relations from the ρ – ρ conditions. Namely, it follows from the ρ -condition that a = a + c u − , , (3.16)( a u [0 , − u , a ′ − a u , + 3) = c u [0 , ( a + a ) , (3.17)where c is a constant. The ρ and ρ -conditions implies c (cid:16) a ′ u [0 , − ( a u [0 , + 3)( a u [0 , + a ) − a ( a u [0 , − (cid:17) = 0 , (3.18)10 (cid:16) ( a − a ) u [0 , + a (cid:17) ( a u [0 , + 3) = 0 , (3.19)and the ρ -condition leads to the following relations: (cid:16) a ( a u [0 , + 3) − a (cid:17) ( a u [0 , + 3) = 0 , (3.20)2 c u [0 , ( a ( a u [0 , + 3) − a ) = ( c + 3)(2 c + 3) , (3.21)2 u , ( a u [0 , +3)( a ′ + a − a )+( a u [0 , − a u , +2 a u [0 , − c u [0 , ( a + a ) = 0 . (3.22)It follows from (3.21) that c = 0 and we may reduce (3.18) and (3.19) by the factor c .Let us simplify the equation (3.15) by an appropriate point transformation of the form (3.1).It is more convenient for computations to rewrite it as u = (cid:16) fv [0 , (cid:17) / v , u [0 , = f ( v [0 , ) . (3.23)One can verify that under this transformation the coefficient a transforms as˜ a = ∂f∂v [0 , a f + 3 f − v [0 , . (3.24)It follows from this formula that we can reduce a to zero with the exception of the case a ( u [0 , ) = − u [0 , . Case A: a = 0 . It follows from (3.20), (3.18) and (3.19) that a = a = a = 0. From(3.17) we obtain a = 3 u , . Moreover, relation (3.21) leads to ( c + 3)(2 c + 3) = 0. Substituting a i , i ρ -condition, we obtain that a ′ = − a u [0 , or a = k/u , . Finally, ρ -condition gives rise to k = 0.In the case c = − c = −
32 leads to equation (3.6). Itfollows from (3.24) that in Case A the only admissible point transformations are u → const u and therefore equations (3.5) and (3.6) are non-equivalent. Case B: a = − u [0 , . Taking into account (3.16), we find that the equation has thefollowing form u t = u xxx − u [0 , u [0 , u xx + u x (cid:18) a u , − u [0 , u [0 , + ( c − u [1 , u [0 , (cid:19) + u (cid:16) a u [1 , + a u [0 , u [0 , + a u [1 , u [0 , + a u , (cid:17) . (3.25)Relations (3.17) and (3.21) can be rewritten as( c − c −
3) = 0 , ( a u [0 , − a u , + c −
9) = 0 . (3.26)For equations of the form (3.25) the ρ -condition provides the following additional relations:( a u , + c − a u , − a u , −
3) = 0 , (3.27) a ′ = a u [0 , ( a u , + c − − a u [0 , . (3.28)11oreover it follows from the ρ -condition that( a u , + c − c −
3) = 0 , (3.29)( a u , + c − u , a + 6 a ′ u , − a u , − a u , − a u , + 36) = 0 . (3.30)Under transformations (3.23) the coefficient a changes as follows:˜ a = f ′ f (cid:16) a f + c − (cid:17) + 9 − c v , . The condition ˜ a = 0 is a differential equation for f , which has a non-constant solution exceptfor the case a u , + c − B . . a = 0 and B . . a = 9 − c u , . In the case
Case B.1. it follows from (3.26) – (3.30) that a = 0 , a = 3 u [0 , , a = 0 , a = − u , , a = 4 u , , c = 32 . Substituting all these coefficients into equation (3.25), we obtain equation (3.7).Consider the
Case B.2.
According to equations (3.26) and (3.28) we have the followingequation: u t = u − u u [0 , u [0 , + u (9 − c ) u , u , − u [0 , u [0 , + ( c − u [1 , u [0 , ! + u (cid:0) a u [1 , + a u [0 , u [0 , + a u [0 , u [1 , + a u , (cid:1) , (3.31)where ( c − c −
3) = 0. It can be verified that in the spherical coordinates equation (3.31)has the form v t = v + c v v [1 , + 3 v v [1 , , ( c − c −
3) = 0 , (3.32) R t = R + R R R ( R b + R b − − R v [1 , ( R b − R b − R b − c )+ R R ( R b + R b + 6) + Rv [1 , ( R b − , (3.33)where b i ( R ) = a i ( u [0 , ) ≡ a i ( R ). So, the system (3.31) is triangular. (cid:3) Remark 4.
Both equations (3.32) are integrable equations on the sphere [10]. They haveinfinitely many conservation laws depending on the variables v [ i,j ] . This is a reason why allconditions from Section 2 are satisfied for any functions a − a . However, we can use thegeometric integrability conditions (1.4) - (1.8) for the classification of triangular systems (3.31)(see Appendix 6). 12 emark 5. It turns out that the equations of List 1 can be simplified by the point trans-formation (3.23) with f = z v [0 , a − v [0 , , where a = 0 is an arbitrary constant. As a result, the coefficients of u vanish and the equations(3.2), (3.3) and (3.4) transform to equations u t = u − u u [0 , a + u [0 , − u u [0 , a + u [0 , − u [1 , ( λ − a + u [0 , + u , ( λ − a + u [0 , ) ! , (3.34)where λ = 1 or λ = 12 , u t = u − u u [0 , a + u [0 , − u u [1 , a + u [0 , − u , ( a + u [0 , ) ! , (3.35)and u t = u − u u [0 , a + u [0 , − u u [1 , a + u [0 , − u , ( a + u [0 , ) ! , (3.36)respectively. Equations written in this form appeared in [14] (see formulas (3.12), (3.13), (3.16)and (3.17)). An auto-B¨acklund transformation of the first order for a vector equation of the form (1.9) isdefined by the formula u x = h v x + f u + g v , where u and v are solutions of (1.9). The functions f, g and h are (scalar) functions of variables u [0 , def = ( u , u ) , v [ i,j ] def = ( v i , v j ) , w i,j def = ( u i , v j ) , i, j > . Remark 6.
If the auto-B¨acklund transformation depends on an arbitrary parameter µ , onecan construct exact multi-parameter solutions of equation (1.9) by applying the transformationseveral times to a trivial solution. Remark 7.
The existence of a vector auto-Backlund transformation with an arbitraryparameter is the most easily verifiable evidence for the integrability of a vector equation.For equations (3.2)–(3.4) we use the canonical forms (3.34)–(3.36) since the auto-B¨acklundtransformations for them look more elegant. 13he auto-B¨acklund transformations for equation (3.34) with λ = 1 and with λ = 1 / u x = pq v x + p ( q w [0 , − p v [0 , ) q ( a − p q + w [0 , ) ( u − v ) − a µ pq ( u − v ) , (4.1)and u x = pq v x + p ( q w [0 , − p v [0 , q ( a − p q + w [0 , ) ( u − v ) + µ p / ( u − v ) q / √ a − p q + w [0 , , (4.2)respectively. Here p = √ u [0 , + a , q = √ v [0 , + a and µ is an arbitrary parameter.The auto-B¨acklund transformations for equations (3.35) and (3.36) have the following form: u x = pq v x + µ p ( p − q ) ( u − v ) a − p q + w [0 , (4.3)and u x = pq v x + µ p ( u − v ) √ a − p q + w [0 , . (4.4)The auto-B¨acklund transformations for equation (3.5), (3.6) and (3.8) have the followingform: u x = √ u [0 , √ v [0 , v x + √ v [0 , w [0 , − √ u [0 , v [0 , v [0 , ( w [0 , − √ u [0 , √ v [0 , ) (cid:0) u √ v [0 , − v √ u [0 , (cid:1) + µ √ u [0 , √ v [0 , u , (4.5) u x = √ u [0 , √ v [0 , v x + √ v [0 , w [0 , − √ u [0 , v [0 , v [0 , ( w [0 , − √ u [0 , √ v [0 , ) (cid:0) u √ v [0 , − v √ u [0 , (cid:1) + µ | u [0 , | / | v [0 , | / u , (4.6)and u x = √ u [0 , √ v [0 , v x + µ u (cid:0) √ u [0 , √ v [0 , − w [0 , (cid:1) / + µ u [0 , v − w [0 , u (cid:0) √ u [0 , √ v [0 , − w [0 , (cid:1) / + (cid:0) u √ v [0 , − v √ u [0 , (cid:1)(cid:0) √ u [0 , v [0 , − √ v [0 , w [0 , (cid:1) v [0 , ( √ u [0 , √ v [0 , − w [0 , ) , (4.7)respectively. T =0 The affine connections that correspond to equation (3.8) and to two equations (3.34) have zerotorsion: T = 0. We verified that they satisfy the integrability conditions (1.4)-(1.8). In theappendix we present explicit formulas for these equations.14 xample 2. In the equations (3.34) we have A ( X, Y ) = − ( u , X ) Y + ( u , Y ) Xa + u [0 , , where u [0 , = ( u , u ) . One can check that this connection is the Levi-Civita affine connection ofthe metric g ( X, Y ) = (
X, Y ) a + u [0 , − ( u , X )( u , Y )( a + u [0 , ) . The tensors β , R and σ can be expressed in terms of g as follows: β ( X, Y, Z ) = 3 λ − (cid:16) g ( X, Y ) Z + g ( Y, Z ) X + g ( Z, X ) Y (cid:17) ,R ( X, Y, Z ) = g ( X, Z ) Y − g ( X, Y ) Z , and σ ( X, Y, Z ) = λ g ( X, Y ) Z + λ g ( Z, Y ) X + ( λ − g ( X, Z ) Y. (5.1)It can be easily verified that for any bi-linear form g formula (5.1) defines a triple Jordansystem iff λ = 1 or λ = 12 (cf. (3.34)). Both these triple systems are known to be simple [8]. Example 3.
An elegant description [9] of all geometric objects for equation (3.8) can bedone in terms of the simple triple Jordan system (cf. (5.1)) S ( X, Y, Z ) def = ( X, Y ) Z + ( Z, Y ) X − ( X, Z ) Y. We have A ( X, Y ) = S ( X, F, Y ) , where F def = − u u [0 , ,σ ( X, Y, Z ) = − S ( X, S ( F, Y, F ) , Z ) ,R ( X, Y, Z ) = σ ( X, Z, Y ) − σ ( X, Y, Z ) , β ( X, X, X ) = σ ( X, X, X ) . The tensor β ( X, Y, Z ) can be obtained from β ( X, X, X ) by the symmetrization.
Since T = 0 for triangular systems of the form (3.31), we may use the intgerability conditions(1.4) - (1.8) for the classification of triangular systems. Lemma 2.
Using a transformation of the form (3.23), we can reduce the coefficient a in(3.31) to • Case a : a = 0; This formula means that we are dealing with the space of the constant curvature k = 1. Case b : a ( x ) = 3 x . In the
Case a the conditions (1.4) - (1.8) are equivalent to a = a = a = a = 0 and we arriveat the systems u t = u − u u [0 , u [0 , + u (9 − c ) u , u , − u [0 , u [0 , + ( c − u [1 , u [0 , ! , c = 3 , . In the
Case b we may use transformations (3.23) to vanish a . Transformations (3.23) with f ( x ) = k x k , (6.1)where k i are arbitrary constants, preserve the normalization a = 0 . From conditions (1.4) -(1.8) it follows that
Case b : a ( x ) = − x or Case b : a ( x ) = − c + 3 x . In the Case b conditions (1.4) - (1.8) imply c = 32 and a ( x ) = − x and we obtain theequation u t = u − u u [0 , u [0 , + u u , u , − u [0 , u [0 , − u [1 , u [0 , ! + u u [1 , u [0 , − u [0 , u [1 , u , − u , u , ! . This equation is invariant with respect to the group of transformations (3.23), (6.1).In the Case b we get a = kx , where k is a constant. By a transformation (3.23), (6.1) wecan bring k to zero. As a result we obtain u t = u − u u [0 , u [0 , + u (9 − c ) u , u , − u [0 , u [0 , + ( c − u [1 , u [0 , ! + u u [1 , u [0 , − ( c + 3) u [0 , u [1 , u , ! c = 3 , . Both of these equations admit a total separation of variables in the spherical coordinates: theequation with c = 3 is converted to v t = v xxx + 3 v x v [1 , + 3 v v [1 , , R t = R xxx − R x R xx R while the equation with c = 32 turns into v t = v xxx + 32 v x v [1 , + 3 v v [1 , , R t = R xxx − R x R xx R + 32 R x R . In both cases the scalar equation for R is point equivalent to the integrable equation ˜ R t =˜ R xxx + ˜ R x .The equation from Case b and the equations from Case a admit a partial separation ofvariables in the spherical coordinates. 16 eferences [1] Meshkov A. G. and Sokolov V. V., Hyperbolic equations with symmetries of third order , Theoret. and Math. Phys. , 2011, (1), 43–57.[2] Meshkov A. G. and Sokolov V. V.,
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