Integrable Anisotropic Evolution Equations on a Sphere

V.V. Sokolov's modifying symmetry approach is applied to anisotropic evolution equations of the third order on the n-dimensional sphere. The main result is a complete classification of such equations. Auto-B\"acklund transformations are also found for all equations.


Introduction
In this paper we are dealing with the problem of the symmetry classification of the integrable vector evolution equations of the third order. Completely integrable equations possess many remarkable properties and are often interesting for applications. As examples of such equations we may point out two well known modified Korteweg-de Vries equations u t = u xxx + (u, u)u x , u t = u xxx + (u, u)u x + (u, u x )u, where (· , ·) is a scalar product, u = (u 1 , . . . , u n ). These equations are integrable by the inverse scattering method for any vector dimension. Another example of integrable anisotropic evolution equation on the sphere is the higher order Landau-Lifshitz equation where R is a diagonal constant matrix. Complete integrability of this equation was proved in [1].
To investigate integrability of such equations a modification of the symmetry approach was proposed in [2]. Different examples of integrable vector equations can also be found in [2]. Sokolov's method can be applied to any vector equation u t = f n u n + f n−1 u n−1 + · · · + f 0 u (2) with scalar coefficients f i . Henceforth u = (u 1 , . . . , u n+1 ) denotes an unknown vector, u k = ∂ k u/∂x k and f i are some scalar functions depending on the scalar products (u i , u j ), i j. Moreover, dependence of f i on more than one scalar products (·, ·) i , i = 1, 2, . . . may be introduced. It is clear that any equation (2) with the Euclidean scalar product is invariant with respect to an arbitrary constant orthogonal transformation of the vector u. Therefore the equation (2) with a unique scalar product is called isotropic. When f i depend on two or more scalar products (u i , u j ) k , we call the equation (2) anisotropic. The scalar products may have a different nature. Only two properties of the scalar products are essential for us -bilinearity and continuity. Vector u may be both real and complex.
The symmetry approach [3,4,5,6,7] is based on the observation that all integrable evolution equations with one spatial variable possess local higher symmetries or, which is the same, higher commuting flows. Canonical conserved densities ρ i , i = 0, 1, . . . make the central notion of this approach. These densities can be expressed in terms of the coefficients of the equation under consideration. The evolutionary derivative of ρ i must be the total x-derivative of some local function θ i : It follows from (3) that the variational derivative δD t ρ i (u)/δu is zero. Both equations δD t ρ i /δu = 0, i = 0, 1, . . . and (3) are called the integrability conditions. The modified symmetry approach has been recently applied to the equations on S n [8] and on R n [9,10], where the third order equations in the form were classified under several restrictions: (u, u) = 1 in [8], [9] and f 2 = 0 in [10]. We have also tried to classify the equations (4) on R n but the attempt has failed because of great computational difficulties. Moreover, the integrable equations that we found proved to be too cumbersome to be applied to any scientific problem. That is why the restrictions were used in the above-mentioned articles. In contrast with R n the list of the anisotropic integrable equations on S n presented in Section 2 is short and contains at the least one interesting equation (10). Thus, the subject of the article may be defined as the symmetry classification of the equations (4) with the constraint (u, u) ≡ u 2 = 1. The coefficients f i are assumed to be depending on both isotropic variables u [i,j] and anisotropic variablesũ [i,j] with i j 2. The constraint u 2 = 1 implies (u, u t ) = 0, u [0,1] = 0, u [0,2] = −u [1,1] , u [0,3] = −3u [1,2] , . . . .
In this paper we shall consider the equations (4) that are integrable for arbitrary dimension n of the sphere. In addition, we assume that the coefficients f i do not depend on n. In virtue of the arbitrariness of n, the variables (5) will be regarded as independent. The functional independence of {u [i,j] ,ũ [i,j] , i j} is a crucial requirement in all our considerations.
It is easy to see that the stereographic projection maps any equation (4) on S n to some anisotropic equation on R n−1 .
In Section 2 we present a complete list of integrable anisotropic equations of the form (4) on the sphere S n . And a scheme of computations is presented in Section 3.
In order to prove that all equations from the list are really integrable, we find, in Section 4, an auto-Bäcklund transformation involving a "spectral" parameter for each of the equations.

Classification results
In this section we formulate some classification statements concerning integrable evolution equations of third order on the n-dimensional sphere. This classification problem is much simpler than the similar problem on R n . Indeed, the set of the independent variables (5) on S n is reduced because of the constraints (6). It is easy to see that we can express all variables of the form u [0,k] , k 1 in terms of the remaining independent scalar products. So, the complete set of dynamical variables on the sphere is Therefore the coefficients of the equation (4) on S n a priori depend on nine independent variables u [1,1] , u [1,2] , u [2,2] ,ũ [0,0] ,ũ [0,1] ,ũ [0,2] ,ũ [1,1] ,ũ [1,2] ,ũ [2,2] , whereas in the case of R n they are functions of twelve variables. Let g ij andg ij be the first and the second metric tensors, Then the equation (4) and the constraint u 2 = 1 are obviously invariant under the transformationg ij → αg ij + λg ij where α and λ are constants and α = 0. It is equivalent to the following transformation of the dependent variables We used this transformation to classify the integrable equations (4).

Remark 2.
Each of the equations (10)-(20) can contain the term cu 1 in its right-hand side. We removed these terms by the Galilean transformation as trivial. The constants a and b are arbitrary. One can set a = 0 in (13) and in (16). In (18) we may choose a = 0 or b = 0 but {a, b} = 0. If we set in (18) a = 0, then it takes the following form 1] . If we set in (19) a = 0 and then b = 0, then the equation is reduced to the following form [1,2] ξ (u 2 + u [1,1] u) + 3u [1,2] u Remark 3. In the classifying process we did not consider the isotropic equations because they were found earlier in [8]. Nevertheless, all equations (10)-(20) admit the reductionũ [i,j] = ku [i,j] that will be referred as the isotropic reduction. Each integrable isotropic equation can be obained from the list (10) We stress that the equation (19) with η = 0 is reduced to (11) from [8]. But its reduction under η = 0 -(19 ′ ), on the contrary, is reduced to the vector Schwartz-KdV equation (13) from [8]. Hence the properties of the equations (19) and (19 ′ ) are essentially different.

A sketchy proof of the main theorem
The equation (4) can be rewritten in the form This operator L is used for obtaining the canonical conserved densities by a technique proposed in [11]. Motivation and explanation of the technique, for vector equations, have been presented earlier in [8]. For more details, see also [12]. Let ρ and θ be the generating functions for the canonical densities and fluxes correspondingly: Then, from the equation L exp(D −1 x ρ) = 0, the recursion formula follows where δ i,j is the Kronecker delta and ρ 0 , ρ 1 are The corresponding functions θ i can be found from (3). The fact that the left-hand sides of (3) are total x-derivatives imposes rigid restrictions (see below) on the coefficients f i of (4).
Expressions for the next functions ρ i involve, besides f k , the functions θ j with j i − 2. For example and so on. It is shown in [8] that all even canonical densities ρ 2n are trivial and we have the following strengthened conditions of the integrability instead of (3).
Proof . To specify the form of the coefficient f 1 , we consider the condition D t ρ 1 = D x θ 1 , where ρ 1 is given by (22). To simplify the equation D t ρ 1 = D x θ 1 we use the equivalence relation that will be denoted as ∼. We say that F 1 and F 2 are equivalent ( Thus we may write D t ρ 1 ∼ 0, and this equivalence remains true after adding any x-derivative: D t ρ 1 + D x F ∼ 0, ∀ F . We call the transformation D t ρ 1 → D t ρ 1 + D x F the equivalence transformation. Using the equivalence transformation we can reduce the order of D t ρ 1 step by step.
Reducing the order of D t ρ 1 by the equivalence transformation we found that D t ρ 1 is equivalent to a third degree polynomial of the third order variables u [i,3] , i = 1, 2, 3 andũ [i, 3] , i = 0, 1, 2, 3. As this polynomial must be equivalent to zero, and it is obvious that any total derivative D x F is linear with respect to highest order variables, then the second and third degree terms must vanish. By equating the third degree terms to zero one can find that all third order derivatives of f 1 with respect to second order variables vanish, and, moreover, Integrating all these equations we obtain (25). Then we specified the functions g 0 , f 3 , . . . , f 12 in (24), (25) in the cases A, B, and C using the next integrability conditions (23). These computations are very cumbersome and we can not present it in a short article. The result of the computations is the Theorem 1.

Bäcklund transformations
To prove integrability of all equations from the list (10)-(20) we present in this section first order auto-Bäcklund transformations for all equations. Such transformations involving an arbitrary parameter allow us to build up both multi-soliton and finite-gap solutions even if the Lax representation is not known (see [7]). That is why the existence of an auto-Bäcklund transformation with additional "spectral" parameter λ is a convincing evidence of integrability.
For a scalar evolution equation, a first order auto-Bäcklund transformation is a relation between two solutions u and v of the same equation and their derivatives u x and v x . Writing this constraint as u x = φ(u, v, v x ), we can express all derivatives of u in terms of u, v, v x , . . . , v i , . . . . These variables are regarded as independent.
In the vector case, the independent variables are vectors and all their scalar products Following [8], we consider in this paper special vector auto-Bäcklund transformations of the form where f , g and h are scalar functions of the variables (27) with i, j 1.
Since (u, u) = 1, and (u, u 1 ) = 0, it follows from (28) that To find an auto-Bäcklund transformation for the equation (2), we differentiate (28) with respect to t in virtue of (2) and express all vector and scalar variables in terms of the independent variables (26) and (27). By definition of the Bäcklund transformation, the expression thus obtained must be identically equal to zero. Splitting this expression with respect to the vector variables (26) and the scalar variables (27) different from (29) we derive an overdetermined system of non-linear PDEs for the functions f and g. If the system has a solution depending on an essential parameter λ, this solution gives us the auto-Bäcklund transformation we are looking for.
We present below the result of our computations.
In the case of the equation (10) the auto-Bäcklund transformation reads as follows Next two equations, (11) and (12), are integrable on R n not only on S n . In fact, f 1 and f 2 do not depend on u [0,i] . Hence do not depend on u [0,i] for any i, j > 0. This implies that all ρ n and θ n for (11) and (12) do not depend on u [0,i] (see (22) and (21)). This means that the conditions D t ρ n = D x θ n are valid both on R n and on S n . The equations (11) and (12) on R n have the auto-Bäcklund transformations of the form The function F reads as for (11) and as If we set here a = 0, λ → bλ and then b = 0, we obtain the following expression for F : But it is a trivial solution because (30) is "auto-Bäcklund transformation" for (19 ′ ) when F = F (ũ [0,0] ,ṽ [0,0] , w 0 ,w 0 ) is an arbitrary solution of a quasilinear system of the three first order equations. (V. Sokolov and A. Meshkov found for the vector isotropic Schwartz-KdV equation that an "auto-Bäcklund transformation" containing an arbitrary function is equivalent to a point transformation. Unfortunately, formula (49) in [8] also gives a false auto-Bäcklund transformation.) The true auto-Bäcklund transformation for the anisotropic Schwartz-KdV equation (19 ′ ) has the following form This transformation is reduced in the isotropic limit to the true auto-Bäcklund transformation for isotropic Schwartz-KdV equation that was pesented in [9]. Finally, the auto-Bäcklund transformation for the equation (20) reads as follows

Concluding remarks
Each of the presented Bäcklund transformations is reduced to the true Bäcklund transformation under the isotropic reductionũ [i,j] = ku [i,j] in accordance with the diagram of the the isotropic reductions. It is convincing evidence that all Bäcklund transformations are real. We do not know any applied problems that lead to one of the equations from our list. If such problem emerge in future, it will be interesting to find the Lax representation for the corresponding equation.
The presented Bäcklund transformations can be used, first, for obtaining the soliton-like solutions (see [13]) and, secondly, for constructing the superposition formulas and new discrete integrable systems.