On realizations of the Virasoro algebra

On realizations of the Virasoro algebra∗ arXiv:1310.2846v1 [nlin.SI] 10 Oct 2013 Renat Zhdanov1 and Qing Huang2,3 1. BIO-key International, 55121 Eagan, MN, USA 2. Department of Mathematics, Northwest University, Xi’an 710069, China 3. Center for Nonlinear Studies, Northwest University, Xi’an 710069, China Abstract We obtain complete classification of inequivalent realizations of the Virasoro algebra by Lie vector fields over the three-dimensional field of real numbers. As an application we construct new classes of nonlinear second-order partial differential equations possessing infinite-dimensional Lie symmetries. 1 Introduction Since its introduction in 19th century Lie group analysis has become a very popular and powerful tool for solving nonlinear partial differential equations (PDEs). Given a PDE that possesses nontrivial symmetry, we can utilize symmetry reduction procedure to construct its exact solutions [9, 10]. Not surprisingly, the wider symmetry of an equation under study is, the better off we are when applying Lie approach to solve it. This is especially the case when symmetry group is infiniteparameter. If a nonlinear differential equation admits infinite Lie symmetry then it is often possible either to linearize it or construct its general solution [9]. The classical example is the hyperbolic type Liouville equation utx = exp(u). (1) It admits the infinite-parameter Lie group t′ = t + f (t), x′ = x + g(x), u′ = u − f˙(t) − ġ(x), (2) where f and g are arbitrary smooth functions. General solution of (1) can be obtained by the action of transformation group (2) on its particular traveling wave Supported by the NSF of China (Grant Nos. 11101332, 11201371), the Foundation of Shaanxi Educational Committee, China (Grant No. 11JK0482) and the NSF of Shaanxi Province, China (Grant No. 2012JQ1013). ∗ 1 solution of the form u(t, x) = ϕ(x + t) (see, e.g., [9]). An alternative way to solve the Liouville equation is to linearize it [9]. Note that the Lie algebra of Lie group (2) is the direct sum of two infinitedimensional Witt algebras, which are subalgebras of the Virasoro algebra. Unlike the finite-dimensional algebras, infinite-dimensional ones have not been systematically studied within the context of classical Lie group analysis of nonlinear PDEs. The situation is however drastically different for the case of generalized (higher) Lie symmetries which played critical role in success of the theory of integrable systems in (1 + 1) and (1 + 2) dimensions (see, e.g. [16]). The breakthrough in the analysis of integrable systems was nicely complemented by development of the theory of infinite-dimensional Lie algebras such as loop algebras [28], Kac-Moody algebras [19] and Virasoro algebras [17]. Virasoro algebra plays an increasingly important role in mathematical physics in general [4, 13] and in the theory of integrable systems in particular. Study of nonlinear evolution equations in (1+2)-dimensions arising in different areas of modern physics shows that in many cases the Virasoro algebra is their symmetry algebra. Let us mention among others the Kadomtsev-Petvishilvi (KP) [7, 8, 14], modified KP, cylindrical KP [22], the Davey-Stewartson [6, 15], Nizhnik-NovikovVeselov, stimulated Raman scattering, (1+2)-dimensional Sine-Gordon [30] and the KP hierarchy [26] equations. Note that there exist integrable equations which admit infinite-dimensional symmetry algebras that are not of Virasoro type. For instance, the breaking soliton and Zakharov-Strachan equations do not possess Virasoro type symmetry while being integrable [30]. It is a common belief that nonlinear PDEs admitting symmetry algebras of Virasoro type are prime candidates for the roles of integrable systems. Consequently, systematic classification of inequivalent realizations of the Virasoro algebra is a crucial step of symmetry approach to constructing integrable systems (see, e.g., [23, 24]). Classification of Lie algebras of vector fields of differential operators within action of local diffeomorphism group has been pioneered by Sophus Lie himself. It remains a very powerful method for group analysis of nonlinear differential equations. Some of more recent applications of this approach include geometric control theory [18], theory of systems of nonlinear ordinary differential equations possessing superposition principle [31], algebraic approach to molecular dynamics [2, 29] to mention only a few. Still the biggest bulk of results has been obtained in the area of classification of nonlinear PDEs possessing point and higher Lie symmetries (see [3] and references therein). Analysis of realizations of Lie algebras by first-order differential operators is in the core of almost every approach to group classification of partial differential equations (see, e.g., [1, 5, 10–12, 20, 21]) In this paper we concentrate on study of realizations of Virasoro algebras by first-order differential operators (Lie vector fields) in the space Rn with n ≤ 3. One of our motivations was to utilize these realizations for constructing (1 + 1)2 dimensional PDEs that possess infinite-dimensional Lie symmetry and in this sense are integrable. The paper is organized as follows. In Section 2 we give a brief account of necessary facts and definitions. In addition, the algorithmic procedure for classifying realizations of infinite-dimensional Virasoro and Witt algebras is described in detail. We construct all inequivalent realizations of the Witt algebra (a.k.a. centerless Virasoro algebra) in Section 3. The next section is devoted to analysis of realizations of the full Virasoro algebra. We prove that there are no central extensions of the Witt algebra in the space R3 that possess nonzero central element. In Section 5 we construct broad classes of nonlinear PDEs admitting infinite dimensional symmetry algebras which are realizations of the Witt algebra. The last section contains discussion of the obtained results and the outline of further work. 2 Notations and definitions The Virasoro algebra, V, is the infinite-dimensional Lie algebra with basis elements C, Ln , n = 0, ±1, ±2, . . . which satisfy the following commutation relations: [Lm , Ln ] = (m − n)Lm+n + 1 m(m2 − 1)δm,−n C, 12 [Lm , C] = 0, m, n ∈ Z. (3) Hereafter we denote the commutator of two Lie vector fields P and Q as [Q, P ], i.e., [Q, P ] = QP − P Q. The symbol δa,b stands for the Kronecker delta ( 1, a = b, δa,b = 0, otherwise. The operator C commuting with all other elements is called the central element of algebra (3). In the case when C equals to zero the algebra (3) is called the centerless Virasoro algebra or Witt algebra W. Consequently, the full Virasoro algebra is the universal central extension of the Witt algebra. The realization space of the Virasoro algebra is the infinite-dimensional Lie algebra L∞ of first-order differential operators of the form Q = τ (t, x, u)∂t + ξ(t, x, u)∂x + η(t, x, u)∂u (4) over the space R3 ∋ (t, x, u). One can readily verify that the set of operators (4) is invariant under the transformation of variables t, x, u t → t̃ = T (t, x, u), x → x̃ = X(t, x, u), u → ũ = U(t, x, u), (5) provided D(T, X, U)/D(t, x, u) 6= 0. Indeed, applying (5) to an arbitrary element of L∞ of the form (4) we get Q̃ = (τ Tt + ξTx + ηTu )∂t̃ + (τ Xt + ξXx + ηXu )∂x̃ + (τ Ut + ξUx + ηUu )∂ũ . 3 Evidently, Q̃ ∈ L∞ . It is a common knowledge that correspondence Q ∼ Q̃ is the equivalence relation and as such it splits the set of operators (4) into equivalence classes. Any two elements within the equivalence class are related through a transformation (5), while two elements belonging to different equivalence classless cannot be transformed one into another by a transformation of the form (5). Hence to describe all possible realizations of the Virasoro algebra within the class of Lie vector fields (4) one needs to construct a representative of each equivalence class. The remaining realizations are obtained by applying transformations (5) to the representatives in question. The procedure for constructing realizations of inequivalent algebra (3) consists of the following three steps: • Describe all inequivalent forms of L0 , L1 and L−1 such that the commutation relations of the Virasoro subalgebra, [L0 , L1 ] = −L1 , [L0 , L−1 ] = L−1 , [L1 , L−1 ] = 2L0 , (6) hold together with the relations [Li , C] = 0, (i = 0, 1, −1). Note that the algebra (6) is isomorphic to sl(2, R). • Construct all inequivalent realizations of the operators L2 and L−2 which satisfy the following commutation relations: [L0 , L2 ] = −2L2 , [L0 , L−2 ] = 2L−2 , [L−1 , L2 ] = −3L1 , [L1 , L−2 ] = 3L−1 , 1 [L2 , L−2 ] = 4L0 + C, 2 (7) [Li , C] = 0, (i = 2, −2). • Describe the remaining basis operators of the Virasoro algebra through the recursion relations Ln+1 = (1 − n)−1 [L1 , Ln ], L−n−1 = (n − 1)−1 [L−1 , L−n ] and [Ln+1 , L−n−1 ] = 2(n + 1)L0 + 1 n(n + 1)(n + 2)C, 12 [Li , C] = 0 with i = n + 1, −n − 1 and n = 2, 3, 4, . . .. In the Sections 3 and 4 we implement the above algorithm to construct all inequivalent realizations of the Witt and Virasoro algebras by operators (4). 4 3 Realizations of the Witt algebra Turn now to describing realizations of the Witt algebra. We remind that the Witt algebra is obtained from the Virasoro algebra by putting C = 0. We begin by letting the vector field L0 be of the general form (4), i.e., L0 = τ (t, x, u)∂t + ξ(t, x, u)∂x + η(t, x, u)∂u . Transformation (5) maps L0 into L̃0 = (τ Tt + ξTx + ηTu )∂t̃ + (τ Xt + ξXx + ηXu )∂x̃ + (τ Ut + ξUx + ηUu )∂ũ . We have τ 2 + ξ 2 + η 2 6= 0, since otherwise L0 is zero. Consequently, we can choose the solutions of equations τ Tt + ξTx + ηTu = 1, τ Xt + ξXx + ηXu = 0, τ Ut + ξUx + ηUu = 0. as T , X and U and reduce L0 to the form L0 = ∂t (hereafter we drop the tildes). Consequently, the vector field L0 is equivalent to the canonical operator ∂t . With L0 in hand we now proceed to constructing L1 and L−1 which obey the commutation relations (6). Let L1 be of the general form (4). Inserting it into [L0 , L1 ] = −L1 yields L1 = e−t f (x, u)∂t + e−t g(x, u)∂x + e−t h(x, u)∂u , where f, g, h are arbitrary smooth functions. To further simplify vector field L1 we utilize the equivalence transformation (5) preserving the form of L0 . Applying (5) to L0 gives L0 → L̃0 = Tt ∂t̃ + Xt ∂x̃ + Ut ∂ũ = ∂t̃ . Consequently, transformation t̃ = t + T (x, u), x̃ = X(x, u), ũ = U(x, u) is the most general transformation that does not alter the form of L0 . It maps the Lie vector field L1 into L̃1 = e−t (f + gTx + hTu )∂t̃ + e−t (gXx + hXu )∂x̃ + e−t (gUx + hUu )∂ũ . To further analyze the above class of realizations of L1 we need to differentiate between two separate cases g 2 + h2 = 0 and g 2 + h2 6= 0. Case 1. If g 2 + h2 = 0, then L˜1 = e−t f (x, u)∂t̃ . Choosing t̃ = t − ln |f (x, u)| we have L1 = e−t ∂t . Let L−1 be of the general form (4). Inserting L0 , L1 , L−1 into the commutation relations [L0 , L−1 ] = L−1 and [L1 , L−1 ] = 2L0 yields L−1 = et ∂t . Case 2. If g 2 + h2 6= 0 then we choose t̃ = t + T (x, u) where T (x, u) satisfies the equation e−T = f + gTx + hTu 5 and take solutions of the equations gXx + hXu = e−T , gUx + hUu = 0 as X and U thus mapping L1 into e−t (∂t +∂x ). Choosing L−1 in the general form (4) and taking into account commutation relations [L0 , L−1 ] = L−1 and [L1 , L−1 ] = 2L0 we get L−1 = et (1 − e−2x f1 (u))∂t + et (−1 − e−2x f1 (u) + e−x g1 (u))∂x + et−x h1 (u)∂u , where f1 , g1 , h1 are arbitrary smooth functions. The transformation t̃ = t, x̃ = x + X(u), ũ = U(u) (8) evidently does not alter the form of L0 , L1 . Applying (5) to L−1 yields L̃−1 = et (1−e−2x f1 (u))∂t̃ +et (−1−e−2x f1 (u)+e−x g1 (u)+e−x h1 Ẋ)∂x̃ +et−x h1 (u)U̇ ∂ũ . Here and after the dot over the symbol stands for derivative of the corresponding function of one variable. To complete the analysis we need to consider separately the cases f1 (u) 6= 0 and f1 (u) = 0. Provided f1 (u) 6= 0 we can choose p p X(u) = − ln |f1 (u)|, φ(u) = (g1 (u) + h1 (u)Ẋ(u))/ |f1 (u)|. p Selecting U t0 be a solution of h1 (u)U̇ = |f1 (u)| if h1 6= 0 or an arbitrary nonconstant function if h1 = 0, we have L−1 = et (1 + αe−2x )∂t + et (−1 + αe−2x + e−x φ(u))∂x + βet−x ∂u , where α = ±1 and β = 0, 1. The case f1 (u) = 0 leads to the realization L̃−1 = et ∂t̃ + et (−1 + e−x g1 (u) + e−x h1 (u)Ẋ)∂x̃ + et−x h1 (u)U̇∂ũ . Choosing X = 0 and U to be a solution of h1 (u)U̇ = 1 if h1 6= 0 or an arbitrary non-constant function if h1 = 0, we obtain L−1 = et ∂t + et (−1 + e−x g1 (u))∂x + βet−x ∂u with β = 0, 1. We summarize the results obtained above in the following lemma. 6 Lemma 1. There exist only two inequivalent realizations of the algebra (6) 1. L0 = ∂t , L1 = e−t ∂t , L−1 = et ∂t ; 2. L0 = ∂t , L1 = e−t (∂t + ∂x ), (9) L−1 = et (1 + αe−2x )∂t (10) +et (−1 + φ(u)e−x + αe−2x )∂x + βe(t−x) ∂u . Here α = 0, ±1, β = 0, 1 and φ(u) is an arbitrary smooth function. To get a complete description of inequivalent realizations of the Witt algebra, we need to extend algebras (9) and (10) by the operators L2 and L−2 and perform the last two steps of the classification procedure described in Section 2. We first present the final result and then give the detailed proof. Theorem 1. There exist at most eleven inequivalent realizations of the Witt algebra W over the space R3 . Below we give the list of representatives of each equivalence class Wi , i = 1, 2, . . . , 11. W1 : Ln = e−nt ∂t , W2 : −nt Ln = e ∂t + e W3 :  −nt   1 −x n + n(n − 1)αe ∂x , 2  1 2 e − (n + 1)γe + n(n + 1)γ (ex − γ)−n−1 ∂t Ln = e 2   1 x −nt+(n−1)x ne − n(n + 1)γ (ex − γ)−n ∂x , +e 2 −nt+(n−1)x 2x x W4 : L0 = ∂t , L1 = e−t ∂t + e−t ∂x , L−1 = et (1 + γe−2x )∂x + et (−1 + γe−2x + e−x φ)∂x , L2 = e−2t f (x, u)∂x + e−2t g(x, u)∂x ,  
1 −3x −2x 3 2 3/2 2t 6γφ + φ ± (4γ + φ ) ∂t − e L−2 = e 1 + 3γe 2  
1 −3x −x −2x 2t 3 2 3/2 − e + e −2 + 3e φ + 6γe 6γφ + φ ± (4γ + φ ) ∂x , 2 Ln+1 = (1 − n)−1 [L1 , Ln ], L−n−1 = (n − 1)−1 [L−1 , L−n ], 7 n ≥ 2, where 
f (x, u) = ex 4e4x − 10e3x φ − 36γe2x + 2ex 31γφ + 6φ3 ± 6(4γ + φ2 )3/2  −64γ 2 − 54γφ2 − 9φ4 ∓ 9φ(4γ + φ2 )3/2 r −1 
g(x, u) = ex 8e4x − 16e3x φ − 2e2x (44γ + 5φ2 ) + 2ex 44γφ + 9φ3 ± 9(4γ + φ2 )3/2  −64γ 2 − 54γφ2 − 9φ4 ∓ 9φ(4γ + φ2 )3/2 r −1 ,
r = 4e5x − 10e4x φ − 40γe3x + 10e2x 6γφ + φ3 ± (4γ + φ2 )3/2 − 10ex 6γ 2 + 6γφ2
+φ4 ± φ(4γ + φ2 )3/2 + 30γ 2 φ + 20γφ3 + 3φ5 ± (2γ + 3φ2 )(4γ + φ2 )3/2 , W5 : Ln = e−nt+(n−1)x (ex ± n)(ex ± 1)−n ∂t + ne−nt+(n−1)x (ex ± 1)1−n ∂x , W6 : Ln = e−nt ∂t + γe−nt [enx − (ex − γ)n ](ex − γ)1−n ∂x , W7 : L0 = ∂t , L1 = e−t ∂t + e−t ∂x , L−1 = et (1 + γe−2x )∂t + et (−1 + γe−2x + e−x φ)∂x , L2 = e−2t+x (ex − φ)(e2x − ex φ − γ)−1 ∂t + e−2t+x (2ex − φ)(e2x − ex φ − γ)−1 ∂x , L−2 = e2t−3x (e3x + 3γex − γφ)∂t + e2t−3x (2ex − φ)(−e2x + ex φ + γ)∂x , Ln+1 = (1 − n)−1 [L1 , Ln ], W8 : W9 : L−n−1 = (n − 1)−1 [L−1 , L−n ],  n ≥ 2,  X γ Ln = e−nt ∂t + e−nt n − sgn(n) j(j + 1)e−2x  ∂x , 2 j=1 |n|−1  |n|−1 X e  Ln = x (−1 + (2j + 1))n + (2n + 1)ex − (n + 2)e2x + e3x (e − 1)n+2 j=1 −nt+(n−1)x   |n|−1 −nt+(n−1)x X X e φ   j(j + 1) ∂t + x −(−1 + (2j + 1))n +sgn(n) 2 j=1 (e − 1)n+1 j=1 |n|−1  X φ j(j + 1) ∂x , −2nex + ne2x − sgn(n) 2 j=1 |n|−1 8 W10 : |n| −nt Ln = e −nt ∂t + ne W11 : −nt Ln = e −nt ∂t + e sgn(n) X j(j − 1)e−nt−2x ∂u , ∂x + 2 j=1   αn(n − 1) −x n(n − 1) −nt−x n+ ∂x + e e ∂u , 2 2 where n ∈ Z, α = 0, ±1, γ = ±1, sgn(·) is the standard sign function and  c, c ∈ R, φ(u) = u. Proof. To prove the theorem it suffices to consider the case when the operators L0 , L1 , L−1 are of the form (9) or (10). Case 1. If L0 , L1 , L−1 are given by (9), then it is straightforward to verify that due to (7) L2 and L−2 are of the forms L2 = e−2t ∂t , L−2 = e2t ∂t . The remaining basis elements of the Witt algebra are easily obtained through recursion, which yields Ln = e−nt ∂t , n ∈ Z. We arrive at the realization W1 from Theorem 1. Case 2. Turn now to the realization (10). Inserting L0 , L1 , L−1 into the commutation relations [L0 , L−2 ] = 2L−2 and [L1 , L−2 ] = 3L−1 and solving the latter for the coefficients of the operator L−2 yield L−2 = e2t (1 + 3αe−2x + ψ1 (u)e−3x )∂t + e2t (−2 + 3φ(u)e−x + ψ2 (u)e−2x + ψ1 (u)e−3x )∂x + e2t (3βe−x + ψ3 (u)e−2x )∂u , where ψ1 , ψ2 , ψ3 are arbitrary smooth functions of u. Utilizing the commutation relations [L0 , L2 ] = −2L2 and [L−1 , L2 ] = −3L1 in a similar fashion we derive the form of the basis element L2 L2 = e−2t f (x, u)∂t + e−2t g(x, u)∂x + e−2t h(x, u)∂u , where f, g, h are solutions of the system of three PDEs − 3(αe−2x + 1)f + 2αe−2x g + (φe−x + αe−2x − 1)fx + βe−x fu + 3 = 0, (11a) (1 − φe−x − αe−2x )f + (φe−x − 2)g − φu e−x h + (φe−x + αe−2x )gx (11b) + βe−x gu + 3 = 0, βe−x f − βe−x g + 2(1 + αe−2x )h − (φe−x + αe−2x − 1)hx − βe−x hu = 0. 9 (11c) Inserting L2 and L−2 into the commutation relation [L2 , L−2 ] = 4L0 yields three more PDEs (4e−3x ψ1 + 12αe−2x + 4)f + (−6αe−2x − 3e−3x ψ1 )g + he−3x ψ˙1 + (−e−3x ψ1 + 2 − 3φe−x − e−2x ψ2 )fx + (−e−2x ψ3 − 3e−x β)fu − 4 = 0, (2e−3x ψ1 + 6φe−x + 2e−2x ψ2 − 4)f + (6αe−2x + 2 − 3φe−x − e−3x ψ1 − 2e−2x ψ2 )g + (e−3x ψ˙1 + 3φ̇e−x + e−2x ψ˙2 )h + (−e−3x ψ1 + 2 − 3φe−x − e−2x ψ2 )gx + (−e−2x ψ3 − 3βe−x )gu = 0, (2e−2x ψ3 + 6βe−x )f + (−3βe−x − 2e−2x ψ3 )g + (6αe−2x + 2e−3x ψ1 + e−2x ψ˙3 + 2)h + (−e−3x ψ1 + 2 − 3φe−x − e−2x ψ2 )hx + (−e−2x ψ3 − 3βe−x )hu = 0. (12) To determine the forms of L2 and L−2 we have to solve Eqs. (11) and (12). It is straightforward to verify that the relation Q = e−t−4x [βe3x + ψ3 e2x + (βψ2 − φψ3 − 3αβ)ex + βψ1 − αψ3 ] 6= 0 is the necessary and sufficient condition for the system of equations (11) and (12) to have the unique solution in terms of fx fu , gx , gu , hx and hu . By this reason we need to differentiate between the cases Q = 0 and Q 6= 0. Case 2.1. Let Q = 0 or, equivalently, β = ψ3 = 0. Eqs. (11) and (12) do not contain derivatives of the functions f, g, h with respect to u. That is why the derivatives fx , gx , hx can be expressed in two different ways using (11) and (12). Equating the right-hand sides of the two expressions for hx yields hex e4x − 2φe3x − ψ2 e2x − 2ψ1 ex + 3α2 + φψ1 − αψ2 = 0. (e2x − φex − α)(2e3x − 3φe2x − ψ2 ex − ψ1 ) Hence h = 0. Similarly the compatibility conditions for the derivatives fx and hx give two linear equations for the functions f and g. The determinant of the obtained system of linear equations does not vanish. Hence the system in question has the unique solution f (x), g(x). Computing the derivatives of the so obtained f and g with respect to x and comparing the result with the previously obtained expressions for fx and gx yield (ψ2 − 6α)(φ3 + 2ψ1 + φψ2 )e11x + F10 [x, u] = 0, (13) and (10φ3 ψ1 − 3αφ2(3ψ2 − 8α) + 3φψ1 (2α + 3ψ2 ) + 2(5ψ12 −4α(2α2 − 3α2 + ψ22 )))e10x + F9 [x, u] = 0. 10 (14) Hereafter Fn [x, u], n ∈ N denotes a polynomial in exp(x) of the power less than or equal to n. To find f and g we need to construct the most general form of φ and ψi (i = 1, 2, 3) satisfying Eqs. (13) and (14). If (13) holds then at least one of the following equations ψ2 = 6α and ψ1 = −(φ3 + φψ2 )/2 should be satisfied. Case 2.1.1. If ψ2 = 6α then Eqs. (13) and (14) hold if and only if 16α3 + 3α2φ2 − 6αφψ1 − φ3 ψ1 − ψ12 = 0, 3 whence ψ1 = (−6αφ − φ3 ± (4α + φ2 ) 2 )/2. 3 Case 2.1.1.1. Suppose now that ψ1 = (−6αφ − φ3 − (4α + φ2 ) 2 ))/2. Provided α = 0 we have either ψ1 = 0 or ψ1 = −φ3 . The case α = ψ1 = 0 leads to L−1 = et ∂t +et (−1+e−x φ)∂x . Making the equivalence transformation x̃ = x+X(u) we can reduce φ to one of the forms a = 0, ±1 thus getting g = 2 + ae−x . f = 1, Utilizing the recurrence relations of the Witt algebra we arrive at the realization W2 . Provided α = 0 and ψ1 = −φ3 we can reduce the function φ to the form b = 0, ±1 by the equivalence transformation x̃ = x + X(u). Note that in the case when b = 0 we have ψ1 = 0 which leads to the realization W2 . The case b 6= 0 gives rise to the following forms of f and g: f= ex (e2x − 3bex + 3b2 ) , (ex − b)3 g= ex (2ex − 3b) . (ex − b)2 Hence we get the realization W3 . 3 Provided α = ±1 we have ψ1 = (−6αφ − φ3 − (4α + φ2 ) 2 )/2 and the realization W4 is obtained. Case 2.1.1.2. Let the function ψ1 be of the form ψ1 = (−6αφ − φ3 + (4α + 3 φ2 ) 2 ))/2. If α = 0 then we have ψ1 = 0 or −φ3 . This case has already been considered when we analyzed the Case 2.1.1.1. If the relation α = ±1 holds then we get the realization W4 . Case 2.1.2. If ψ1 = −(φ3 + φψ2 )/2 then Eq. (13) takes the form (4α + φ2 )(ψ2 − (4α − 5φ2 )/4)(ψ2 − (2α − φ2 ))e10x + F9 [x, u] = 0. To solve the equation above we need to analyze the following three sub-cases. Case 2.1.2.1. Provided ψ2 = (4α − 5φ2 )/4, Eqs. (13) and (14) are satisfied if and only if 4α + φ2 = 0. 1 Consequently α ≤ 0 and φ = 2b(−α) 2 with b = ±1. If α = −1 then φ = 2b, ψ1 = 2b, ψ2 = −6 and furthermore f= ex (ex − 2b) , (ex − b)2 11 g= 2ex . ex − b The realization W5 is obtained. If α = 0 and φ = ψ1 = ψ2 = 0 then we arrive at the realization W2 with α = 0. Case 2.1.2.2. Let ψ2 = 2α − φ2 and suppose that Eqs. (13) and (14) hold. Provided α = 0 we can transform φ to become b = ±1 (note that the case b = 0 has already been considered). Then f = 1, g= 2ex − b ex − b which yields the realization W6 . Given α = ±1 we have f= ex (ex − φ) , e2x − ex φ − b g= ex (2ex − φ) , e2x − ex φ − b where b = ±1. Since φ can be reduced to the form ũ by the equivalence transformation ũ = φ with φ̇ 6= 0, the realization W7 is obtained. Case 2.1.2.3. If 4α + φ2 = 0 and Eqs. (13) and (14) holds, then we get α ≤ 0, whence α = 0, −1. Given the relation α = 0 we can transform φ to become a = 0, ±1. Thus f = 1 and g = 2 − ae−x which give rise to the realization W8 . In the case when α = −1 we have f= ex (4 + 5ex − 4e2x + e3x + ψ2 ) , (ex − 1)4 g= ex (−4 − 4ex + 2e2x − ψ2 ) . (ex − 1)3 What is more, the function ψ2 can be reduced to the form ũ by the equivalence transformation ũ = ψ2 provided ψ2 is a nonconstant function. As a result we get the algebra W9 . Summarizing we conclude that the case Q = 0 leads to the realizations Wi , i = 2, 3, · · · , 9. Turn now to the case Q 6= 0. Case 2.2. If Q 6= 0 or, equivalently, β 2 + ψ32 6= 0 then Eqs. (11) and (12) can be solved with respect to fx , fu , gx , gu , hx and hu . The compatibility conditions fxu = fux , gxu = gux , hxu = hux can be rewritten in the form of the system of three linear equations for the functions f, g, h a1 f + a2 g + a3 h + d1 = 0, b1 f + b2 g + b3 h + d2 = 0, c1 f + c2 g + c3 h + d3 = 0. Here ai , bi , ci , di , (i = 1, 2, 3) are functions of t, x, φ, ψ1 , ψ2 , ψ3 . 12 It is straightforward to verify that the system above has the unique solution f, g, h when β 2 + ψ32 6= 0. We do not present here the explicit formulae for these functions as they are very cumbersome. Inserting f, g, h into Eq. (11a) yields αβ 6 e42x + F41 [x, u] = 0. Consequently we have α = 0 or β = 0. Case 2.2.1. If β = 0 then Eq. (11a) takes the form αψ36 e36x + F35 [x, u] = 0, which gives α = 0 and ψ3 6= 0 (since Q = 0 otherwise). Taking into account these relations we rewrite Eq. (12) as follows ψ1 ψ36 e36x + F35 [x, u] = 0, (15φ2 + 2ψ2 )ψ36 e37x + F36 [x, u] = 0, (57φ2 − 2ψ2 )ψ37 e35x + F34 [x, u] = 0. Hence we conclude that φ = ψ1 = ψ2 = 0. Inserting these formulae into the initial Eqs. (11) and (12) and solving the obtained system yield f = 1, g = 2, h = −e−2x ψ3 . The function ψ3 can be reduced to the form −1 by the equivalence transformation ũ = U(u), where U̇ = −1/ψ3 . As a result we have f = 1, g = 2, h = e−2x , which leads to the realization W10 . Case 2.2.2. Provided α = 0 Eq. (11c) takes the form β 5 (4βφψ3 − 6ψ 2 + β 2 ψ˙3 )e41x + 30β 5 φψ 2 e40x + F39 [x, u] = 0. 3 3 Note that the case α = β = 0 has been already studied while considering Case 2.2.1. Consequently, without any loss of generality we can restrict our considerations to the cases ψ3 = 0, β = 1 and φ = 0, β = 1. If ψ3 = 0 then it follows from (12) and (11c) that ψ1 = ψ2 = 0. Taking into account these relations we rewrite Eqs. (11) and (12) in the form f = 1, g = 2 + e−x φ, h = e−x . What is more, the function φ can be reduced to one the forms 0, ±1 by equivalence transformations x̃ = x + X(u) and ũ = U(u). Whence we get the realization W11 . If the relation φ = 0 holds Eqs. (11) and (12) are incompatible. It is straightforward to verify that Wi , (i = 1, 2, · · · , 11) cannot be transformed one into another with the transformation (5) and hence are inequivalent. This completes the proof of Theorem 1. While proving the above theorem we obtained exhaustive description of realizations of the Witt algebra over the spaces R1 and R2 . We present the corresponding results without proof. 13 Theorem 2. The realization W1 exhausts the list of inequivalent realizations of the Witt algebra W over the space R1 . Theorem 3. The realizations W1 –W9 with φ = C exhaust the list of inequivalent realizations of the Witt algebra W over the space R2 . 4 Realizations of the Virasoro algebra To construct all inequivalent realizations of the Virasoro algebra, V, we need to extend inequivalent realizations of the Witt algebra by all possible realizations of the nonzero central element of the Virasoro algebra. In this section we prove that there are no realizations of the Virasoro algebra with non-zero central element in the three-dimensional space R3 . Let us begin by constructing all possible central extensions of the subalgebra L = hL−1 , L0 , L1 i. In view of Lemma 1 we can restrict our considerations to realizations (9) and (10) of the algebra L. Case 1. Consider first the realization (9) L0 = ∂t , L1 = e−t ∂t , L−1 = et ∂t . Let the basis element C be of the general form (4). Inserting (4) into the commutation relations [Li , C] = 0, (i = 0, 1, −1) yields C = ξ(x, u)∂x + η(x, u)∂u , ξ 2 + η 2 6= 0. Applying the transformation t̃ = t, x̃ = X(x, u), ũ = U(x, u), which does not alter the forms of L0 , L1 and L−1 to the realization of C above, we get C → C̃ = (ξXx + ηXu )∂x̃ + (ξUx + ηUu )∂ũ . Choosing solutions of the equations ξXx + ηXu = 0, ξUx + ηUu = 1, as X and U yields C = ∂u . Proceed now to constructing L2 . It easily follows from the relations [L0 , L2 ] = −2L2 , [L−1 , L2 ] = −3L1 and [L2 , C] = 0 that L2 = e−2t ∂t . Next, let L−2 be of the general form (4). Then commutation relations (7) involving L−2 yield overdetermined system of PDEs for the unknown functions τ , ξ and η. This system turns out to be incompatible. Hence realization (9) cannot be extended to a realization of the Virasoro algebra with nonzero central element. 14 Case 2. We begin by utilizing commutation relations for the basis elements L0 , L1 and C thus getting C = f (u)e−x ∂t + (g(u) + f (u)e−x )∂x + h(u)∂u , where f , g and h are arbitrary smooth function of u. Applying transformation (8) that preserves the form of the basis elements L0 , L1 to C gives C̃ = f (u)e−x ∂t̃ + (g(u) + f (u)e−x + h(u)Ẋ(u))∂x̃ + h(u)U̇(u)∂ũ . If f (u) 6= 0 then choosing X(u) = − ln |f (u)| we have C̃ = e−x̃ ∂t̃ + (e−x̃ + β(g + hẊ))∂x̃ + βhU̇ ∂ũ , where β = ±1. Provided h = 0 and ġ 6= 0 we can make the transformation ũ = g(u) and thus get C1 = e−x ∂t +(e−x +u)∂x . The case h = ġ = 0 leads to C2 = e−x ∂t + (e−x + λ)∂x , where λ is an arbitrary constant. Next, if h 6= 0 we choose solutions of the equations g + hẊ = 0 and hU̇ = 1/β as X and U getting C3 = e−x ∂t + e−x ∂x + ∂u . Provided f (u) = 0 the generator C̃ = (g + hẊ)∂x̃ + hU̇ ∂ũ is obtained. If h 6= 0 then it is possible to choose X and U so that C4 = ∂u . Given the condition h = 0 we have C̃ = g∂x̃ . If g is nonconstant then selecting U = g(u) yields C5 = u∂x . Finally, the case of constant g leads to C6 = ∂x . Summing up we conclude that there exist six inequivalent nonzero realizations of the central element C for the case when L0 = ∂t and L1 = e−t ∂t + e−t ∂x . The next step is extending the algebras hL0 , L1 , Ci i, (i = 1, 2, · · · , 6) to the realizations of the full Virasoro algebra. We present the calculation details for the case when C1 = e−x ∂t + e−x ∂x + ∂u . The remaining five cases are handled in a similar fashion. In order to extend hL0 , L1 , C1i to the full Virasoro algebra, we construct al possible realizations of L−1 . Inserting L−1 of the general form (4) into the corresponding commutation relations from (6) gives L−1 = et−2x (u2 e2x − 1) et−2x (uex + 1)2 ∂ − ∂x . t u2 u2 With L−1 in hand we proceed to constructing L2 . Taking into account relations (7) we get uex (uex + 2) 2uex L2 = 2t x ∂ + ∂x . t e (ue + 1)2 e2t (uex + 1) Inserting the obtained expressions for L−1 and L2 into (7) yields incompatible system of equations for the coefficients of L−2 . Whence we conclude that a realization of the algebra hL0 , L1 , C1 i cannot be extended to a realization of the full Virasoro algebra. The same result holds for the remaining five realizations of the central elements C2 , C3 , . . . , C6 . Theorem 4. There are no realizations of the Virasoro algebra with nonzero central element C in three-dimensional space Rn , n = 1, 2, 3. 15 5 Some applications: PDEs admitting infinitedimensional symmetry groups In this section we construct several classes of second-order evolution equations in the space R3 of the variables t, x, u that admit realization of the Witt algebra listed in Theorem 1. Given a realization of the Witt algebra, we can apply the Lie infinitesimal approach to construct the corresponding invariant equation [25, 27]. Differential equation F (t, x, u, ut, ux , utt , utx , uxx ) = 0 is invariant with respect to the Witt algebra with basis elements L1 , L2 , . . . , Ln , . . . if and only if the condition pr(2) Ln (F )|F =0 = 0 holds for any n ∈ N, where pr(2) Ln is the second-order prolongation of the vector field Ln , that is pr(2) Ln = Ln + η t ∂ut + η x ∂ux + η tt ∂utt + η tx ∂utx + η xx ∂uxx with η t = Dt (η) − ut Dt (τ ) − ux Dt (ξ), η x = Dx (η) − ut Dx (τ ) − ux Dx (ξ), η tt = Dt (η t ) − utt Dt (τ ) − utx Dt (ξ), η tx = Dx (η t ) − utt Dx (τ ) − utx Dx (ξ), η xx = Dx (η x ) − uxt Dx (τ ) − uxx Dx (ξ). Here the symbols Dt and Dx stand for the total differentiation operators with respect to t and x, correspondingly Dt = ∂t + ut ∂u + utt ∂ut + uxt ∂ux + . . . , Dx = ∂x + ux ∂u + utx ∂ut + uxx ∂ux + . . . . As an example we consider the realization W1 = he−nt ∂t i. Making use of the formulas above we obtain pr(2) Ln = e−nt ∂t + ne−nt ut ∂ut + (2ne−nt utt − n2 e−nt ut )∂utt + ne−nt utx ∂utx . (15) Next step is computation of the full set of functionally-independent second-order differential invariants, Im (t, x, u, ut, ux , utt , utx , uxx ) (m = 1, 2, · · · , 7), associated with Ln . To get Im we need to solve the related system of characteristic equations dux dutx duxx dx du dut dutt dt = = = = = = = . e−nt 0 0 ne−nt ut 0 2ne−nt utt − n2 e−nt ut ne−nt utx 0 16 Integration of the equations above yields utx , I6 = e−nt ut , I7 = e−2nt utt − ne−2nt ut . ut I1 = x, I2 = u, I3 = ux , I4 = uxx , I5 = Hence the most general Ln -invariant equation is of the form F (I1 , I2 , · · · , I7 ) = 0. Since this equation has to be invariant under every basis element of the infinitedimensional Witt algebra W1 , it must be independent of n. To obey this restriction function F should be independent of I6 and I7 . Thus the final form of the most general second-order PDE invariant under W1 reads as F (I1 , I2 , I3 , I4 , I5 ) = 0, or, F  utx x, u, ux, uxx , ut  = 0. Below we list five more classes of second-order differential equations whose symmetry algebra is infinite-dimensional Witt algebra. W2 invariant PDEs   ut uxx − ux utx = 0, if α = 0 F u, ux , uxx, ex ux   uxx − ux ut ux − ut uxx + ux utx + u2x F u, , − 2αux = 0, if α = ±1, u2x ex ux W6 invariant PDEs F W8 invariant PDEs W10 invariant PDEs  γ(uxx + utx ) − ex (ux + uxx ) u, ux (γ(ut + ux ) − ex ux )  = 0,   uxx − 2ux F u, = 0, u2x F (ux + 2u, uxx − 4u) = 0. 6 Concluding Remarks The principal result of this paper is exhaustive classification of inequivalent realizations of the Virasoro algebra by Lie vector fields over the space Rn with n = 1, 2, 3. These realizations are listed in Theorems 1-4. According to Theorem 1 there exist eleven inequivalent realizations of the Witt algebra in the space R3 . What is more, 17 we proved that there exist only one realization of the Witt algebra over the space R1 and nine realizations of the Witt algebra over the space R2 . It has been established that realizations of the Virasoro algebra with nonzero central element do not exist in the space Rn with n ≤ 3. As an application of our algebraic classification we construct a number of nonlinear PDEs admitting infinite-dimensional symmetry algebras, which are realizations of the Witt algebra. An interesting application of the obtained results would be describing nonlinear PDEs whose symmetry algebras are direct sums of the Witt algebras. A nontrivial example is the Liouville equation (1). Since these equations would admit symmetry with two arbitrary functions they would automatically be classically integrable. Since Virasoro algebra is a subalgebra of the Kac-Moody algebra, the results of this paper can be directly applied to solving the problem of classification of integrable KP type PDEs in (1 + 2) dimensions. The starting point is a description of inequivalent realizations of the Kac-Moody algebras by differential operators over the space R4 . 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