Group classification of the general second-order evolution equation. Semi-simple invariance groups R. Zhdanov Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivska Street, 01601 Kyiv, Ukraine ∗ V. Lahno State Pedagogical University, 2 Ostrogradskogo Street, 36000 Poltava, Ukraine † Abstract In this paper we consider the problem of group classification of the generic secondorder evolution equation in one spatial variable. We construct all inequivalent evolution equations whose invariance groups are either semi-simple or semi-direct products of semisimple and solvable Lie groups. The obtained lists of invariant equations contain both already known equations and the broad classes of new evolution equations possessing non-trivial Lie symmetry. 1 Introduction Utilization of group properties of partial differential equations (PDEs) has already become a universal and convenient tool for analysis of these equations. Clearly for this technique to work, equations under study should actually have nontrivial group properties. Within this viewpoint the whole class of differential equations splits into two subclasses of equations with nontrivial symmetry, S, and without any symmetries. The whole history of exploration of symmetries of differential equations is the sequence of attempts to extend the class S by modifying somehow the classical concept of Lie symmetry. As the paper title implies, we restrict our analysis to equations from S. Moreover, we narrow the meaning of admitted symmetry by considering invariance with respect to Lie transformation groups only. The basic facts and all the necessary information about group analysis of differential equations can be found in [1]–[4]. In this paper we study the general evolution equation ut = F (t, x, u, ux , uxx ) ∗ † (1.1) Corresponding author, e-mail: renat.zhdanov@bio-key.com e-mail: lvi@pdpu.poltava.ua 1 in order to get an answer to the following seemingly simple question: Is it possible to describe all possible functions F such that Eq.(1.1) admits nontrivial Lie transformation group? By nontrivial symmetry group we mean a Lie transformation group which is at least one-parameter. 2 Hereafter, u = u(t, x), ut = ∂u , ux = ∂u , uxx = ∂∂xu2 , and F is an arbitrary sufficiently smooth ∂t ∂x function. The first paper on group classification of a subclass of linear equations from the class (1.1) was published by the creator of the theory of transformation groups Sophus Lie as early as in 1881 [5]. However, the real boom of interest in group classification of differential equations was initiated by the Ovsyannikov’s paper [6]. It was followed by numerous publications (see [9]–[20] and the references therein) analyzing various specific subclasses of the general class of evolution equations (1.1). A detailed account of group properties of the equations considered in the above papers can be found in [8, 21, 22]. Surprisingly, there is still no ultimate solution of the classification problem for the general evolution equation (1.1). The main reason is that the class of equations (1.1) is too general for the traditional Ovsyannikov’s classification method to be practical. This method is not very efficient when the class of equations under study involves functions of several variables. Recently, we developed an efficient approach to solving group classification problem for low-dimensional partial differential equations. It enabled us to classify the broad classes of heat conductivity [23, 21], Schrödinger [27], third-order evolution [28] and wave [29] equations admitting nontrivial Lie symmetry. Note that some elements of this approach were utilized earlier by Fushchych & Serov [24], Gagnon & Winternitz [25] and Zhdanov et al [26] in order to perform symmetry classification of the nonlinear d’Alembert, Schrödinger and multi-component wave equations, correspondingly. In the present paper we apply the method of [21] to obtain exhaustive classification of evolution equations that admit n-parameter Lie transformation group for all possible values of n >= 1. As group classification of linear PDEs of the form (1.1) has already been performed in [5], we consider essentially nonlinear evolution equations only. By ’essentially nonlinear’ we mean PDEs (1.1) that cannot be linearized by point transformations of the space of the variables t, x, u. 2 Preliminary group analysis of Eq.(1.1) It is a common knowledge, that the most general (in Lie’s sense) transformation group admitted by (1.1) is generated by the infinitesimal operators v = τ ∂t + ξ∂x + η∂u . (2.1) Here τ = τ (t, x, u), ξ = ξ(t, x, u), η = η(t, x, u) are arbitrary smooth functions defined in the space of V = R2 × R1 of two independent R2 = ht, xi and one dependent R1 = hui variables. 2 Constructing the second prolongation of the infinitesimal operator v we get where e v = v + ϕt ∂ ∂ ∂ ∂ ∂ + ϕx + ϕtt + ϕtx + ϕxx , ∂ut ∂ux ∂utt ∂utx ∂uxx ϕt = Dt (η) − ut Dt (τ ) − ux Dt (ξ), ϕx = Dx (η) − ut Dx (τ ) − ux Dx (ξ), ϕxx = Dx (ϕx ) − utx Dx (τ ) − uxx Dx (ξ), ∂ ∂ ∂ ∂ Dt = + ut + utt + utx + ..., ∂t ∂u ∂ut ∂ux ∂ ∂ ∂ ∂ Dx = + uxx + .... + ux + utx ∂x ∂u ∂ut ∂ux We do not give the formulas for the coefficients ϕtt , ϕtx since they are not used in the sequel. Acting by e v on Eq.(1.1) we arrive at the following invariance criterion: ¢¯ ¡ t (2.2) ϕ − τ Ft − ξFx − ηFu − ϕx Fux − ϕxx Fuxx ¯ut →F (t,x,u,ux ,uxx ) = 0. The subscript formula in (2.2) means that one needs to replace utx with Dx F , utt with Dt F and ut with F in the expression within the parentheses. If we construct the general solution of (2.2), then we obtain the most general (local) Lie transformation group admitted by Eq.(1.1). Note that this group is also called the classical Lie symmetry of nonlinear PDE (1.1). Analyzing relations (2.2) we prove the following technical assertion. Assertion 2.1 The most general invariance group of Eq.(1.1) is generated by the infinitesimal operators v = τ (t)∂t + ξ(t, x, u)∂x + η(t, x, u)∂u , (2.3) the functions τ, ξ, η and F satisfying the following equation: ηt − ux ξt + (ηu − τt − ux ξu )F = [ηx + ux (ηu − ξx ) − u2x ξu ]Fux +[ηxx + ux (2ηxu − ξxx ) + u2x (ηuu − 2ξxu ) − u3x ξuu (2.4) +uxx (ηu − 2ξx ) − 3ux uxx ξu ]Fuxx + τ Ft + ξFx + ηFu . As the forms of unknown functions τ, ξ, η depend essentially on the function F , it is customary to call (2.4) the classifying equation. Now the problem of group classification of Eq.(1.1) becomes entirely algorithmic. It reduces to constructing all possible solutions of a single partial differential equation (2.4). It is that simple. The difficulty, however, is that we have to deal with the under-determined system of partial differential equations that consists of one equation for three unknown functions τ, ξ, η. 3 To make things even more complicated, this equation contains unknown function F of variables t, x, u, ux , uxx , which is to be determined as well. To proceed any further we need extra information, either on the form of the function F or on the form of the functions τ, ξ, η which would narrow the class of invariant equations or the (u) 2 ux , split the resulting equation set of possible symmetries. Say, we may put F = f (u)uxx + dfdu by the variables ux , uxx and obtain an over-determined system of partial differential equations for τ, ξ, η. Solving the latter yields the classical Ovsyannikov’s classification result [6]. An alternative approach would be fixing a priori the symmetry group of Eq.(1.1) and solving (2.4) for this specific choice of coefficients of the infinitesimal group operator, τ, ξ, η. To this end one can also use the popular nowadays technique called conventionally ”the method of moving frames”. The latter can be used efficiently, if one succeeds in obtaining the explicit form of the finite group transformations of the infinitesimal symmetry group whose differential invariants are to be constructed (see, e.g., [7] and the references therein). The problem with the above approaches is that some functions F providing for extension of symmetry group of Eq.(1.1) might be lost. To prevent this from happening, the classifying equation should be the only source of constraints on the form of unknown functions τ, ξ, η, F . Another difficulty is that the method of moving frames is inefficient when the group under consideration is infinite-parameter and this is exactly the case we are dealing with. In [26] we suggested an elegant way to approach this problem based on the fact that the general solution (τ, ξ, η) of PDE (2.4) can always be represented as a linear combination of its basis solutions, va = (τa , ξa , ηa ), a = 1, . . . , n, forming a Lie algebra ℓn . Consequently, if we succeed in describing all possible subalgebras of the infinite-dimensional Lie algebra ℓ∞ generated by operators (2.3) and solve for each of them the classifying equation (2.4), then the problem of group classification of initial Eq.(1.1) will be completely solved. So that we can re-formulate the problem of group classification of Eq.(1.1) in a purely algebraic way. Namely, to solve the classification problem we need to • construct all subalgebras of the infinite-dimensional Lie algebra ℓ∞ , and • select those subalgebras whose basis elements satisfy the classifying equation (2.4). Saying it another way, we can replace (2.2) with the (possibly infinite) set of systems of PDEs   Equation (2.4)|v→va , a = 1, . . . , n,     P   Qi τj − Qj τi = n Cijk τk , k=1 (2.5) P n k   Q ξ − Q ξ = C ξ , i j j i k  k=1 ij     Q η − Q η = Pn C k η , i j j i k=1 ij k where Qi = τi ∂t + ξi ∂x + ηi ∂u , Cijk are structure constants of a Lie algebra ℓn , i, j = 1, . . . , n and n = 1, 2, 3, . . .. 4 If we solve over-determined system of PDEs (2.5) for all possible dimensions n ≥ 1 of all admissible Lie algebras ℓn , then the problem of group classification of Eq.(1.1) is completely solved. Consequently, group classification of the general evolution equation (1.1) reduces to integrating over-determined systems of PDEs (2.5) for all n = 1, 2, . . . , nmax , where nmax is the maximal dimension of the Lie algebra admitted by the equation under study. Note that nmax may be equal to ∞. According to the Magadeev theorem [30], we have either nmax <= 7 or nmax = ∞. And what is more, in the latter case the corresponding invariant equation is mapped to a linear PDE by a contact transformation. So to describe all essentially nonlinear evolution equations (i.e., those inequivalent to linear ones) one needs, in fact, to consider all possible Lie algebras of the dimension up to 7. However, combining our Lie algebraic classification approach with the Ovsyannikov method it suffices to consider Lie algebras of the dimension n < 5 only. At this point, the classical Mubarakzyanov results [37, 39] come into play. He described all inequivalent abstract Lie algebras of the dimension n < 6. This means that the structure constants Cijk in (2.5) are already known. Let us mention here the Reid’s procedure of calculating the Lie algebra admitted by PDE without integrating determining equations [39]. The basic idea of his approach is investigating compatibility of systems (2.5) thus deriving the admissible forms of the structure constants Cijk . In our mind a more natural approach is actually to integrate equations (2.5) so that the compatibility conditions come as a by-product. Summarizing we conclude that if we 1) construct all realizations of of all subalgebras of ℓ∞ by operators, coefficients of which satisfy Eq.(2.5), and 2) prove that these are maximal invariance algebras of (1.1), then the problem of group classification of Eq.(1.1) is completely solved. Thus it is clear what to do to achieve complete group classification of the generic class of PDEs (1.1). However, there is still a question how to do this. Saying it another way, what is the practical approach to classify all inequivalent subalgebras of the algebra ℓ∞ ? Before introducing the details of our approach to solving this problem we give a brief account of necessary notions and facts from the general theory of Lie algebras. Let the symbol L stand for a Lie algebra. Denote by the symbol [L, L] the Lie algebra spanned by all the possible commutators of basis elements of L. Then the Lie algebra N is called a subalgebra of L provided [N, N ] ⊂ N . Next, if the relation [L, N ] ⊂ N holds true, then the Lie algebra N is called the ideal of L. Given two ideals, N1 and N2 of the algebra L, the Lie algebra [N1 , N2 ] is also the ideal of L. Consider the following sequence of ideals: L[0] = L, L[1] = [L[0] , L], . . . , L[n] = [L[n−1] , L], . . . 5 If L[n] = 0 for some n > 1, then the Lie algebra L is called nilpotent. If, in particular, L[1] = 0, then L is commutative or Abelian Lie algebra. Now consider another sequence of ideals of L (the composition series of the Lie algebra L) L(0) = L, L(1) = [L(0) , L(0) ], . . . , L(n) = [L(n−1) , L(n−1) ], . . . The algebra L is called solvable if there is n > 0 such that L(n) = 0. The Lie algebra R is called the radical of L, provided it is the maximal solvable ideal of L containing any other solvable ideal of L. The Lie algebra L is called semi-simple, provided it does not contain non-zero solvable ideals. Finally, the algebra L is called simple if it contains no ideals different from 0 and L and L[1] 6= 0. Evidently, a simple algebra is semi-simple. On the other hand, every semi-simple Lie algebra can be decomposed into a direct sum of simple Lie algebras. The fundamental Levi-Maltsev theorem (see, e.g., [21, 32]) says that for any Lie algebra L with radical R there exists a semi-simple Lie algebra S such that L=S⊂ +R. This relation is called the Levi decomposition of the algebra L, and what is more, the algebra S is called the Levi factor of L. The further details can be found, for example, in [21, 32]. Due to the Levi-Maltsev theorem the problem of classification of subalgebras of the algebra ℓ∞ can be divided into three subproblems: 1. Classification of semi-simple subalgebras. 2. Classification of solvable subalgebras. 3. Classification of subalgebras which are semi-direct sums of semi-simple and solvable Lie algebras. Remarkably, it is possible to complete all of the above classifications for the case of the very general class of PDEs (1.1) starting with one-dimensional subalgebras and increasing step-bystep the dimension of the algebras involved. One of our principal results presented below is the analog of the Magadeev theorem for transformation group acting in the space of variables t, x, u. Namely, we prove that the dimension of the maximal invariance algebra of (1.1) is either less or equal to seven or infinite. In the latter case the invariant equation can be linearized by a point transformation of the space of variables t, x, u. Our algorithm for group classification in its present form has been suggested in [21, 23]. As the first step, we compute the maximal equivalence group, E, of the class of PDEs (1.1). The equivalence group consists of one-to-one transformations of the space V t̄ = α(t, x, u), x̄ = β(t, x, u), v = γ(t, x, u), 6 D(α, β, γ) 6= 0, D(t, x, u) (2.6) which preserve the differential structure of equation (1.1). Note that we use a concept of equivalence group which is different from that employed by Ovsyannikov. Indeed, the group E does not involve the function F as an extra variable, as it is customary within the Ovsyannikov’s classification scheme [2]. Making change of variables (2.6) in Eq.(1.1) and requiring for the transformed equation to belong to the same class of PDEs as the initial one, namely, vt̄ = Φ(t̄, x̄, v, vx̄ , vx̄x̄ ), one gets the following result (see, e.g., [33]) Assertion 2.2 The maximal equivalence group of the class of equations (1.1) is formed by the transformations t̄ = T (t), x̄ = X(t, x, u), v = U (t, x, u), T ′ = D(X, U ) dT 6= 0, 6= 0. dt D(x, u) (2.7) Note that the above group contains arbitrary functions, which means that it is the infiniteparameter Lie transformation group. The equivalence group splits the class of equations (1.1) into conjugacy subclasses, each of them being uniquely characterized by a single representative. So it suffices to provide a representative of each conjugacy class in order to get the complete description of invariant equations. This property is used to simplify the form of infinitesimal operators forming the basis of admissible realizations of Lie algebras. Classification of these realizations is thus reduced to constructing inequivalent realizations of abstract Lie algebras by differential operators in three real variables of the form (2.3). Note that the systematic analysis of realizations of Lie algebras by differential operators in one and two variables was performed by Sophus Lie itself [34]–[36]. Consider, as an example, one-dimensional Lie algebras generated by operators (2.3). Change of variables (2.7) reduces (2.3) to the form ṽ = τ T ′ ∂t̄ + (τ Xt + ξXx + ηXu )∂x̄ + (τ Ut + ξUx + ηUu )∂v . (2.8) If τ 6= 0, then choosing a solution of equation τ T ′ = 1 as T and taking as X, U the fundamental solution of the system of equations τ Xt + ξXx + ηXu = 0, τ Ut + ξUx + ηUu = 0, D(X, U ) 6= 0, D(x, u) we get ṽ = ∂t̄ . Now if τ = 0, then ξ 6= 0 or η 6= 0 (otherwise, operator (2.3) vanishes identically). In the case when ξ 6= 0, η = 0, making the change of variables t̄ = t, x̄ = u, 7 v = x, which belongs to the group E, we reduce (2.3) to the form ṽ = ξ(t̄, x̄, v)∂v . Consequently, we can suppose that η 6= 0 without any loss of generality. Choosing as X and U non-vanishing identically solutions of the system of PDEs ξXx + ηXu = 0, ξUx + ηUu = 1, we transform (2.8) to become ṽ = ∂v . We summarize the above reasonings in the form of lemma. Lemma 2.1 Operator (2.3) is equivalent to one of the canonical operators v1 = ∂t , v2 = ∂u . Thus there are two E-inequivalent classes of realizations of one-dimensional Lie algebras by operators (2.3). Consequently, there are only two inequivalent classes of PDEs (1.1) admitting one-dimensional Lie algebras. They are easily obtained by integration of the determining equations for the corresponding infinitesimal operators ∂t , ∂u . Theorem 2.1 There are two inequivalent classes of invariant equations of the form (1.1) admitting one-parameter Lie groups. Below we give the representatives of these classes and the corresponding one-dimensional invariance Lie algebras A1 ut = F (x, u, ux , uxx ) : A11 = h∂t i; ux = F (t, x, ux , uxx ) : A21 = h∂u i. What is more, if the function F is arbitrary, then the algebras A11 and A21 are maximal in Lie’s sense invariance algebras admitted by the corresponding PDEs. As we noted in [31] the set of invariant equations (1.1) is naturally split into two classes, C1 and C2 . Let Ak = hv1 , v2 , . . . , vk i, where vi = τ i ∂t + ξ i ∂x + η i ∂u , i = 1, 2, . . . , k (2.9) be the maximal invariance algebra of Eq.(1.1). We say that this equation belongs to the first class, C1 , if the functions τ i (i = 1, 2, . . . , k) are linearly independent. Otherwise, it belongs to C2 . According to [31] any nonlinear PDE from the second class C2 can be mapped into quasilinear evolution equations by a non-point transformation. So if one constructs all E-inequivalent equations belonging to the classes C1 , C2 , then the problem of group classification of the general evolution equation is solved. We provide full calculation details for the first class C1 . In the case of class C2 we will give outlines of the proofs of the corresponding theorems together with the lists of invariant equations and their maximal invariance algebras. 8 The obtained list of invariant equations is too large to fit into a single paper. That is why, in order to keep the exposition compact we split the material into two parts. The first part deals with invariant equations from the class C1 and those equations from C2 whose invariance algebras are either semi-simple or semi-direct sums of semi-simple and solvable Lie algebras. This paper contains exhaustive classification of these equations. Classification results for PDEs (1.1) invariant under the solvable Lie algebras will be the topic of our subsequent publication. 3 Group classification of equations from C1 It is the direct consequence of Lemma 2.1 that one of the basis operators of the invariance algebra of an equation from C1 can always be chosen in the from ∂t . Also according to Theorem 2.1 the most general equation from the class C1 that admits one-dimensional Lie algebra is equivalent to PDE ut = F (x, u, ux , uxx ), Fuxx 6= 0. (3.1) The maximal invariance algebra of the above equation reads as A11 = h∂t i. Now we turn to equations admitting two-dimensional Lie algebras. It so happens that equations from C1 cannot admit two-dimensional commutative algebras. Lemma 3.1 Class C1 does not contain PDEs invariant under two-dimensional commutative Lie algebras. Proof. Suppose that the assertion of the lemma does not hold. Then one of the basis operators of the symmetry algebra can be reduced to the form v1 = ∂t , while the second operator, v2 , is of the generic form (2.3). Inserting these operators into the commutation relation for the two-dimensional commutative Lie algebra, [v1 , v2 ] = 0, yields that τt = 0, ξt = 0, ηt = 0. Hence it follows that the equation under study belongs to the class C2 . We arrive at the contradiction proving the lemma. It follows from the Levi-Maltsev theorem (see, e.g., [32]) that the set of finite-dimensional real Lie algebras consists of solvable algebras and Lie algebras having nontrivial Levi ideal. That is why we proceed now to studying equations from C1 that admit solvable and semi-simple Lie algebras of symmetry operators (2.8). 3.1. Invariance under solvable Lie algebras There are two inequivalent two-dimensional solvable Lie algebras A2.1 = he1 , e2 i : [e1 , e2 ] = 0, (3.2) A2.2 = he1 , e2 i : [e1 , e2 ] = e2 . 9 The case of commutative algebra A2.1 has already been considered. Turn to the algebra A2.2 . Without loss of generality we may choose one of the basis operators, say, e2 to be equal to ∂t (Lemma 2.1). Taking as e1 an arbitrary operator of the form (2.3) with τ 6= 0 and inserting e1 into (3.2) yields τt = −1, ξt = ηt = 0. Consequently, the operator e1 has the form e1 = −t∂t + ξ(x, u)∂x + η(x, u)∂u , (3.3) where ξ, η are arbitrary smooth functions. If ξ = η = 0, then we get the realization h−t∂t , ∂t i. Next, provided |ξ| + |η| 6= 0, there is a transformation from the group E which does not alter e2 and reduce e1 (3.3) to the form ẽ1 = −t̄∂t̄ − v∂v . Thus there exist two inequivalent realizations of the Lie algebra A2.2 h−t∂t , ∂t i, h−t∂t − u∂u , ∂t i that can be admitted by equations from C1 . However, if we insert coefficients of the first basis operator of the first realization into the classifying equation (2.4), we get F = 0. Consequently, C1 contains no equations invariant under the first realization of A2.2 . A similar analysis of the second realization yields the following class of A2.2 -invariant PDEs: ω = u−1 ux , ut = F (x, ω, w), w = u−1 uxx . Consider now three-dimensional solvable Lie algebras. It is a common knowledge that any solvable three-dimensional Lie algebra, A3 = he1 , e2 , e3 i, is either equivalent to one of the two decomposable A3.1 = A2.1 ⊕ A1 : [ei , ej ] = 0, i, j = 1, 2, 3; A3.2 = A2.2 ⊕ A1 : [e1 , e2 ] = e2 , [e1 , e3 ] = [e2 , e3 ] = 0; or equivalent to one of the seven non-decomposable algebras A3.3 : [e2 , e3 ] = e1 , [e1 , e2 ] = [e1 , e3 ] = 0; A3.4 : [e1 , e3 ] = e1 , [e2 , e3 ] = e1 + e2 , A3.5 : [e1 , e3 ] = e1 , [e2 , e3 ] = e2 , A3.6 : [e1 , e3 ] = e1 , [e2 , e3 ] = −e2 , A3.7 : [e1 , e3 ] = e1 , [e2 , e3 ] = qe2 , A3.8 : [e1 , e3 ] = −e2 , [e2 , e3 ] = e1 , A3.9 : [e1 , e3 ] = qe1 − e2 , [e1 , e2 ] = 0; [e1 , e2 ] = 0; [e1 , e2 ] = 0; [e1 , e2 ] = 0 (0 < |q| < 1); [e1 , e2 ] = 0; [e2 , e3 ] = e1 + qe2 , [e1 , e2 ] = 0 (q > 0) (see, e.g., [37]). Each of the the above algebras contains a two-dimensional commutative subalgebra. By force of Lemma 3.1 there are no equations from C1 which admit one of the above threedimensional algebras. Since any n-dimensional solvable Lie algebra contains (n−1)-dimensional solvable subalgebra, we arrive at the following assertion. 10 Lemma 3.2 Any equation from C1 invariant under solvable Lie algebra of the dimension n >= 2 is equivalent to PDE ut = F (x, ω, w), ω = u−1 ux , w = u−1 uxx . If F is arbitrary, then the maximal invariance algebra of the above equation is the two-dimensional Lie algebra h−t∂t − u∂u , ∂t i. 3.2. Invariance under semi-simple Lie algebras The semi-simple Lie algebras of the lowest dimension read as [32] so(3) = he2 , e2 , e3 i : [e1 , e2 ] = e3 , [e1 , e3 ] = −e2 , sl(2, R) = he1 , e2 , e3 i : [e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e2 , e3 ] = e1 ; [e2 , e3 ] = e1 . The above algebras do not contain the commutative algebra A2.1 [40]. However, semi-simple algebras of higher dimension n > 3 do contain subalgebras equivalent to A2.1 (see, e.g., [32]) and, consequently, cannot be invariance algebras of PDEs from the class C1 . Thus the only semi-simple Lie algebras than may lead to new invariant equations are the algebras so(3) and sl(2, R). Consider first the algebra so(3). Inserting e1 = ∂t , e2 , e3 , where the the latter two are of generic form (2.3), into the first two commutation relations of so(3) and integrating the equations obtained yield e2 = C cos t∂t + (α cos t + β sin t)∂x + (γ cos t + θ sin t)∂u , e3 = [∂t , e2 ]. Here C is an arbitrary non-zero constant, α = α(x, u), β = β(x, u), γ = γ(x, u) and θ = θ(x, u) are arbitrary real-valued functions. Inserting the obtained expressions for e2 , e3 into the remaining commutation relation we get C 2 = −1. Since this equation has no real solutions, the class C1 contains no so(3)-invariant equations. Turn now to the algebra sl(2, R). Let the operator e3 be of the form ∂t and the operators e1 , e2 be of generic form (2.3). Inserting the expressions for e1 , e2 , e3 into the commutation relations and solving the obtained equations we get four E-inequivalent realizations of sl(2, R): h2t∂t , −t2 ∂t , ∂t i, h2t∂t + x∂x , −t2 ∂t − tx∂x , ∂t i, h2t∂t + x∂x , −t2 ∂t − tx∂x + x2 ∂u , ∂t i, h2t∂t + x∂x , −t2 ∂t + x(x2 − t)∂x , ∂t i. The first realization cannot be invariance algebra of an equation of the form (1.1). Requiring invariance of (1.1) under the second realization yields the following system of equations for F = F (x, u, ux , uxx ): 2F = −ux Fux − 2uxx Fuxx + xFx , xux − 2tF = tux Fux + 2tuxx Fuxx − txFx . 11 As the function F is independent of t, we have xux = 0. Consequently, this realization cannot be admitted by an equation of the form (1.1). The third realization gives rise to the following system of equations for the function F : 2Fuxx + 2xFux + x2 Fu = xux . 2uxx Fuxx + ux Fux − xFx = 2F, This system is compatible. Its general solution is F = x−1 uux − x−2 u2 + x−2 F̃ (ω, w), ω = x2 uxx − 2u, w = 2u − xux . Finally, the last realization yields the following system of determining equations for F = F (x, u, ux , uxx ): 2uxx Fuxx + ux Fux − xFx = 2F, 6x(ux + xuxx )Fuxx + 3x2 ux Fux − x3 Fx = −xux , whence 1 −1 −1 F = − x−1 ux + x−3 u−1 ω = u−2 x F̃ (u, ω), x uxx + 3x ux . 4 Summarizing the above reasonings, we conclude that the class C1 contains only two inequivalent classes of PDEs which are invariant under semi-simple Lie algebras of symmetry operators. In both cases the maximal invariance algebras are isomorphic to sl(2, R). 3.3. Finalizing group classification of equations from C1 To complete group classification we have to consider equations from C1 invariant under the Lie algebras ℓ having nontrivial Levi factor and non-zero radical. It is a common knowledge that these Lie algebras should be decomposable into semi-direct sums of semi-simple ℓ1 and solvable ℓ2 Lie algebras. Structure of Lie algebras having the Levi factor sl(2, R) is studied in [41]. Since the solvable algebra, ℓ2 , is either one-dimensional or isomorphic to A2.2 , it follows from the results of [41] that there are no algebras ℓ that are invariance algebras of equations (1.1) belonging to the first class C1 . Theorem 3.1 There are, at most, four inequivalent classes of PDEs from C1 that admit nontrivial invariance algebras. The representative of the first class is (3.1), its maximal symmetry algebra is h∂t i. The representative of the second class is given in Lemma 3.2, its maximal invariance algebra being isomorphic to A2.2 . Two other classes of PDEs are presented below ut = x−1 uux − x−2 u2 + x−2 F̃ (ω, w), ut ω = x2 uxx − 2u, w = 2u − xux : sl1 (2, R) = h2t∂t + x∂x , −t2 ∂t − tx∂x + x2 ∂u , ∂t i; 1 −1 −1 ω = u−2 = − x−1 ux + x−3 u−1 x F̃ (u, ω), x uxx + 3x ux : 4 sl2 (2, R) = h2t∂t + x∂x , −t2 ∂t + x(x2 − t)∂x , ∂t i. The algebras sl1 (2, R) and sl2 (2, R) are maximal invariance algebras of the corresponding PDEs, provided the functions F̃ are arbitrary. 12 4 Invariance of equations from C2 under Lie algebras having nontrivial Levi factor As we mentioned above, to describe equations (1.1) invariant under Lie algebras having nontrivial Levi factor we need to construct equations, which are invariant under semi-simple Lie algebras. That is why we begin group classification of equations from C2 by considering realizations of semi-simple Lie algebras by operators (2.3). 4.1. Invariance under semi-simple Lie algebras It follows from the definition of class C2 that one of the basis operators of an equation from C2 can be chosen as ∂u . Consequently, any invariant equation from C2 can always be transformed to become ut = F (t, x, ux , uxx ), Fuxx 6= 0. (4.1) Consider first the semi-simple algebras of the lowest dimension, namely, so(3) and sl(2, R). Algebra so(3). Let e1 = ∂u and let the operators e2 , e3 be of the form (2.3). Inserting these operators into the first two commutation relations of so(3) and solving the PDEs obtained we get within the equivalence relation E the following formulas for e2 and e3 : ³ ´ e2 = α(t, x) cos u∂x + β(t, x) cos u + γ(t, x) sin u ∂u , e3 = [∂t , e2 ]. The remaining commutation relation yields αβ = 0, αγx − β 2 − γ 2 = 1. (4.2) If α = 0, then β 2 + γ 2 = −1 and system (4.2) has no real solutions. In the case α 6= 0 we have β = 0. Making the change of variables from E t̄ = t, x̄ = X(t, x) (αXx = 1), v=u we get e2 = cos u∂x + γ(t, x) sin u∂u , e3 = [∂t , e2 ], where the function γ = γ(t, x) is a solution of PDE γx = 1 + γ 2 , i.e., γ = tan(x + ϕ(t)). Making the equivalence transformation t̄ = t, x̄ = x + ϕ(t), v=u we reduce γ to the form tan x. Consequently, there is only one realization of the algebra so(3) by operators (2.3), namely, so1 (3) = h∂u , cos u∂x + tan x sin u∂u , − sin u∂x + tan x cos u∂u i. 13 Inserting the coefficients of the above operators into the classifying equation (2.4) yields the following system of PDEs for the function F = F (t, x, ux , uxx ): ux F − (sec2 x + u2x )Fux − (2 tan x sec2 x − u2x tan x + 3ux uxx )Fuxx = 0, ux tan xFux + (2ux sec2 x + u3x + uxx tan x)Fuxx + Fx − tan xF = 0. The general solution of the above system reads as p sec2 x + u2x F̃ (t, ω), F = ³ ´ 2 2 ω = uxx cos x − (2 + ux cos x)ux sin x (1 + u2x cos2 x)−3/2 . (4.3) What is more, the algebra so(3) is the maximal invariance algebra of the corresponding equation, provided F̃ is arbitrary. Lemma 4.1 Any so(3)-invariant equation (4.1) from C2 is equivalent to PDE ³ ´ p ut = sec2 x + u2x F̃ (t, ω), ω = uxx cos x − (2 + u2x cos2 x)ux sin x (1 + u2x cos2 x)−3/2 . Its maximal symmetry algebra is so1 (3) = h∂u , cos u∂x + tan x sin u∂u , − sin u∂x + tan x cos u∂u i. Algebra sl(2, R). Let e3 = ∂u and e1 , e2 be of the form (2.3). In order to satisfy the commutation relations of the algebra sl(2, R) the operators e1 , e2 , e3 have to be of the form e2 = (αu + β)∂x + (−u2 + γu + θ)∂u , e1 = −[∂u , e2 ], e 3 = ∂u , where α = α(t, x), β = β(t, x), γ = γ(t, x), θ = θ(t, x) are solutions of the system of PDEs 2β = −αβx − αγ + βαx , 4θ = −αθx − γ 2 + βγx . (4.4) If the function α does not vanish identically, then using the transformation t̄ = t, x̄ = X(t, x), v = u + U (t, x), where X, U are solutions of PDEs αXx = X, XU = βXx , we simplify e1 , e2 e2 = xu∂x + (−u2 + γu + θ)∂u , e1 = −[∂u , e2 ], e 3 = ∂u . In this case, system (4.4) takes the form xγ = 0, 4θ = −xθx − γ 2 , whence it follows that γ = 0, θ = µ(t)x−4 . Provided µ = 0, we have the following realization of the algebra sl(2, R): h2u∂u − x∂x , −u2 ∂u + xu∂x , ∂u i. If µ 6= 0, then making the transformation t̄ = t, 1 x̄ = |µ|− 4 x, 14 v = u, we arrive at the two realizations of the algebra sl(2, R): h2u∂u − x∂x , (x−4 − u2 )∂u + xu∂x , ∂u i, h2u∂u − x∂x , −(x−4 + u2 ) + xu∂x , ∂u i. Given the condition α = 0, we get from (4.4) the following realization of sl(2, R): h2u∂u , −u2 ∂u , ∂u i. However, this realization cannot be invariance algebra of an equation of the form (4.1), while the preceding ones give rise to the three classes of sl(2, R)-invariant equations. Lemma 4.2 There are only three inequivalent classes of equations from C2 whose invariance algebras are isomorphic to sl(2, R). Below we present the canonical forms of these equations together with their maximal invariance algebras. ut = xux F̃ (t, ω), ut ut −6 −2 ω = x−5 u−3 x uxx + 2x ux : sl3 (2, R) = h2u∂u − x∂x , −u2 ∂u + xu∂x , ∂u i; ¶ µ p 1 9 3 −2 6 2 −3/2 4 3 6 2 = x 4 + x ux F̃ (t, ω), ω = (4 + x ux ) x uxx + 5x ux + x ux : 2 4 −4 2 sl (2, R) = h2u∂u − x∂x , (x − u )∂u + xu∂x , ∂u i; ¶ µ p 1 9 3 −2 6 2 −3/2 4 3 6 2 = x |x ux − 4| F̃ (t, ω), ω = (x ux − 4) x uxx + 5x ux − x ux : 2 5 −4 2 sl (2, R) = h2u∂u − x∂x , −(x + u )∂u + xu∂x , ∂u i. Semi-simple algebras of the dimension higher than 3. According to the general classification of semi-simple Lie algebras [32] the next possible dimension of a semi-simple Lie algebra is six. There are four non-isomorphic semi-simple algebras of the dimension six, so(4), so(3, 1), so(2, 2) and so∗ (4). It is well-known that so(4) = so(3) ⊕ so(3), so∗ (4) ∼ so(3) ⊕ sl(2, R). Consequently, we can utilize the results of classification of realizations of so(3) and sl(2, R). To classify so(4)-invariant equations (1.1) we need to construct all realizations of so(3) by operators (2.3), which commute with the basis operators of the realization so1 (3). It is straightforward to verify that such realizations do not exist. Similarly, to classify so∗ (4)-invariant equations (1.1) we have to construct all realizations of sl(2, R) by operators (2.3), which commute with the basis operators of the realization so1 (3). The only realization of sl(2, R) obeying the above constraint is equivalent to h2t∂t , −t2 ∂t , ∂t i. The latter cannot be invariance algebra of an equation from C2 . Consequently, there are no equations belonging to the class C2 , which are invariant under the algebras isomorphic to so(4) and so∗ (4). The same assertion holds for the algebra so(3, 1). Indeed, the algebra so(3, 1) has the CarP . tan decomposition he1 , e2 , e3 i + hN1 , N2 , N3 i, where he1 , e2 , e3 i = so(3), [ei , Nj ] = 3l=1 εijl Nl , P [Ni , Nj ] = − 3l=1 ǫijl el , i, j, l = 1, 2, 3, and εijl is the anti-symmetric third-order tensor with 15 ε123 = 1. Taking so(3) = so1 (3) after simple calculations we get the forms of operators N1 , N 2 , N 3 N1 = cos u∂u , N2 = − sec u cos x∂x + sin u sin x∂u , N3 = sec u sin x∂x + sin u cos x∂u . Inserting the coefficients of the operator N1 into the classifying equation (2.4) yields Fuxx = 0, whence it follows that it the realization obtained cannot be invariance algebra of PDE (1.1). In order to classify realizations of the algebra so(2, 2), we make use of the fact that so(2, 2) ∼ sl(2, R) ⊕ sl(2, R). So that we can choose hei , ēi |i = 1, 2, 3i as the basis of so(2, 2). Here he1 , e2 , e3 i = sl(2, R), hē1 , ē2 , ē3 i = sl(2, R) with [ei , ēj ] = 0, (i, j = 1, 2, 3). Choosing as ei (i = 1, 2, 3) the basis operators of one of the realizations slk (2, R) (k = 1, 2, . . . , 5) and analyzing the commutation relations of so(2, 2) we come to the conclusion that the class of operators (2.3) does not contain realizations of the algebra so(2, 2) which are admitted by a nonlinear equation of the form (1.1). The same assertion holds true for the semi-simple algebras sl(3, R), su(3) and su(2, 1), which are eight dimensional (there are no seven-dimensional semi-simple algebras). Actually, this is true for any semi-simple algebra of the dimension n > 3. Indeed, there are four basic types of classical simple Lie algebras over the field of real numbers • Type An−1 (n > 1) contains four real forms of the algebra sl(n, C) : su(n), sl(n, R), su(p, q) (p + q = n, p ≥ q), su∗ (2n). • Type Dn (n > 1) contains three real forms of the algebra so(2n, C) : so(2n), so(p, q) (p+q = 2n, p ≥ q), so∗ (2n). • Type Bn (n > 1) contains two real forms of the algebra so(2n + 1, C): so(2n + 1), so(p, q) (p + q = 2n + 1, p > q). • Type Cn (n > 1) contains three real forms of the algebra sp(n, C): sp(n), sp(n, R), sp(p, q) (p + q = n, p > q. As su∗ (4) ∼ so(5, 1), and furthermore the algebra so(5, 1) contains so(4) as a subalgebra, the class of operators (2.3) does not contain realizations of the algebras An−1 (n > 1) and Dn (n > 1), which differ from slk (2, R) (k = 1, 2, . . . , 5). The same assertion holds for the algebras Bn (n > 1) and Cn (n ≥ 1), as well. Indeed, algebras of the type Bn with n >= 2 contains so(4) and so(3, 1). What is more, sp(2, R) ∼ so(3, 2) and so(3, 1) is the subalgebra of so(3, 2), sp(1, 1) ∼ so(4, 1) and so(3, 1) is the subalgebra of so(4, 1), sp(2) ∼ so(5) and so(4) is the subalgebra of so(5). It remains to analyze the exceptional semi-simple Lie algebras G1 , G2 , F4 , E6 , E7 , E8 . We consider the case of the algebra G2 only, the remaining algebras are handled in the same way. A Lie algebra of the type G2 contains the compact real form g2 and the non-compact real form g2′ . Since g2 ∩ g2′ ∼ su(2) ⊕ su(2) ∼ so(4), the class of operators (2.3) does not contain 16 realizations of the algebras g2 and g2′ , which would be invariance algebras of Eq.(1.1). The same assertion holds true for the remaining exceptional semi-simple Lie algebras G1 , F4 , E6 , E7 , E8 . Let us summarize the above reasonings in the form of theorem. Theorem 4.1 Any equation from the class C2 , whose invariance algebra is semi-simple, is equivalent to one of the equations given in Lemmas 4.1, 4.2. 4.2 Invariance under the algebras having nontrivial Levi factor Now we utilize the results of classification of inequivalent equations (1.1) admitting semi-simple symmetry algebras to describe PDEs (1.1) whose symmetry algebras admit Levi decomposition. The class of Lie algebras that admit Levi decomposition splits into two non-intersecting subclasses • subclass of Lie algebras which are decomposable into direct sums of semi-simple and solvable Lie algebras, and • subclass of Lie algebras which are semi-direct sums of a Levi factor and non-zero radical. 4.2.1. Invariance under direct sum of semi-simple and solvable Lie algebras To describe equations whose symmetry algebras are direct sums semi-simple and solvable Lie algebras we can utilize the explicit forms of realizations of semi-simple algebras constructed in the previous subsections. What is more, we need to consider realizations of semi-simple algebras belonging both to the class C1 and to the class C2 . Consider the case of the sl1 (2, R)-invariant equation. We look for possible extensions of the realization sl1 (2, R) by operators (2.3) which commute with its basis operators. Analysis of the commutativity conditions yields the general form of the additional symmetry operators v = C1 x∂x + (C2 + 2C1 u)∂u , (4.5) where C1 , C2 are arbitrary constants. So we need to describe all possible solvable Lie algebras that have basis operators (4.5). Skipping intermediate calculations we formulate the final result. The list of isomorphic solvable Lie algebras realized by operators (4.5) is formed by the two onedimensional algebras h∂u i, hx∂x +2u∂u i and one two-dimensional algebra L2 = h∂u , x∂x +2u∂u i. Note that the latter is isomorphic to A2.2 . Inserting the coefficients of the above operators into the classifying equation and solving the resulting equations yield the forms of the unknown functions F in the corresponding invariant equations 1. Algebra sl1 (2, R) ⊕ h∂u i 1 ut = u2x + x−2 F̃ (ω), 4 17 ω = x2 uxx − xux ; 2. Algebra sl1 (2, R) ⊕ hx∂x + 2u∂u i ut = x−1 uux − x−2 u2 + x−2 (2u − xux )2 F̃ (ω), ω = (x2 uxx − 2u)(2u − xux )−1 ; 3. Algebra sl1 2, (R) ⊕ h∂u , x∂x + 2u∂u i ut = mx2 u2xx µ 1 − 2mxux uxx + m + 4 ¶ u2x , m 6= 0. Under arbitrary F̃ the given algebras are maximal in Lie’s sense invariance algebras of the corresponding equations. Analogously we prove that the basis operators of a solvable Lie algebra which commute with the basis operators of sl2 (2, R), are necessarily of the form v = η(u)∂u . (4.6) Further computations show that the maximal dimension of a solvable Lie algebra having basis operators (4.6) is two. There are two inequivalent realizations, one-dimensional L1 = h∂u i and two-dimensional L2 = h−u∂u , ∂u i with L2 ∼ A2.2 . Solving the corresponding classifying equations yields the forms of the rand-hand sides of invariant equations (1.1). 1. Algebra sl2 (2, R) ⊕ h∂u i 1 ut = − x−1 ux + x−3 u−1 x F̃ (ω), 4 −1 −1 ω = u−2 x uxx + 3x ux ; 2. Algebra sl2 (2, R) ⊕ h−u∂u , ∂u i 1 −2 −1 −1 −2 ut = − x−1 ux + mx−3 u−1 x (ux uxx + 3x ux ) , 4 m 6= 0. Under arbitrary F̃ and m the given algebras are maximal in Lie’s sense invariance algebras of the corresponding equations. The most general form of operators (2.3) commuting with the basis operators of sl3 (2, R) is v = τ (t)∂t + ξ(t)x∂x . (4.7) Our analysis shows that there exist only four E-inequivalent realizations of solvable Lie algebras by operators (4.7), which are invariance algebras of equations of the form (1.1). Namely, there are two one-dimensional realizations, h∂t i, htx∂x i, and two two-dimensional realizations, h−t∂t − mx∂x , ∂t i(m ∈ R), ht∂t , tx∂x i. Below we list the corresponding invariant equations. 1. Algebra sl3 (2, R) ⊕ h∂t i ut = xux F̃ (ω), −6 −2 ω = x−5 u−3 x uxx + 2x ux ; 18 2. Algebra sl3 (2, R) ⊕ htx∂x i ut = ¢ xux ¡ −5 −3 + xux F̃ (t); ln x ux uxx + 2x−6 u−2 x 4t 3. Algebra sl3 (2, R) ⊕ h−t∂t − mx∂x , ∂t i −6 −2 1/4m ut = λxux (x−5 u−3 , λ 6= 0, x uxx + 2x ux ) m 6= 0, 3 ± ; 4 4. Algebra sl3 (2, R) ⊕ ht∂t , tx∂x i ut = xux λxux −6 −2 ln(x−5 u−3 , x uxx + 2x ux ) + 4t t λ ∈ R. Under arbitrary F̃ , m and λ the given algebras are maximal in Lie’s sense invariance algebras of the corresponding equations. A similar analysis of extensions of the realizations algebras sl4 (2, R), sl5 (2, R) and so1 (3) yields three more invariant equations. Below we give the right-hand sides of invariant equations and their maximal invariance algebras. p sl4 (2, R) ⊕ h∂t i : ut = x−2 4 + x6 u2x F̃ (ω), ¶ µ 1 9 3 6 2 − 32 4 3 ω = (4 + x ux ) x uxx + 5x ux + x ux ; 2 p sl5 (2, R) ⊕ h∂t i : ut = x−2 |x6 u2x − 4|F̃ (ω), µ ¶ 1 9 3 4 3 6 2 − 23 x uxx + 5x ux − x ux ; ω = |x ux − 4| 2 p so1 (3) ⊕ h∂t i : ut = sec2 x + u2x F̃ (ω), ³ ´ ω = uxx cos x − (2 + u2x cos2 x)ux sin x (1 + u2x cos2 x)−3/2 . 4.2.2. Invariance under semi-direct sum of semi-simple and solvable Lie algebras To perform classification of equations from C2 whose invariance algebra are isomorphic to semidirect sum of semi-simple and solvable Lie algebras we need to apply a more sophisticated strategy. It is based on the well-known fact of the group analysis of differential equations, which is, the higher dimension of the invariance algebra admitted by PDE (1.1) the less arbitrary is the function F . So at some point, instead of an arbitrary function of five variables t, x, u, ux , uxx equation (1.1) would contain an arbitrary function of one variable or even arbitrary constants. When this is the case, we apply the Ovsyannikov classification method [2], since the determining equations split into over-determined systems of PDEs that can be effectively integrated. So our approach to classification of equations invariant under semi-direct sum of semisimple and solvable Lie algebras consist of the two major steps. Firstly, utilizing the results of classification of lower dimensional Lie algebras that can be decomposed into semi-direct sum of Levi factor and solvable radical [41] we describe all invariant equations containing arbitrary 19 functions of four, three, two, one arguments and/or arbitrary constants. The second step is utilizing the Ovsyannikov’s approach to classify those equations that have either arbitrary functions of one variable or arbitrary parameters. This will provide complete classification of Eq.(1.1) within the considered class of Lie algebras. We prove that in order to implement the first step of our method it suffices to consider 1. Algebras which are semi-simple sums of semi-simple and solvable Lie algebras of the dimension n ≤ 6, if the Levi factor is isomorphic to so(3). 2. Algebras which are semi-simple sums of semi-simple and solvable Lie algebras of the dimension n ≤ 5, if the Levi factor is isomorphic to sl(2, R). Further analysis shows that without any loss of generality we can restrict our considerations to the Lie algebras sl(2, R) ⊂+A2.1 , so(3) ⊂+A3.1 . After completing the first step, we apply the Ovsyannikov method to finalize the classification. In addition, we utilize this method to complete group classification of invariant equations having arbitrary functions of one variable or arbitrary constants obtained in the previous subsection. In a sequel, we use the list of non-isomorphic four-dimensional solvable Lie algebras obtained in [37]. This list is formed by by ten decomposable algebras A3.i ⊕ A1 (i = 1, 2, . . . , 9), 2A2.2 = A2.2 ⊕ A2.2 and ten non-decomposable algebras A4.i = he1 , e2 , e3 , e4 i (i = 1, 2, . . . , 10) (we give only non-zero commutation relations only) A4.1 : [e2 , e4 ] = e1 , A4.2 : [e1 , e4 ] = qe1 , [e3 , e4 ] = e2 ; [e2 , e4 ] = e2 , [e3 , e4 ] = e2 + e3 , q 6= 0; A4.3 : [e1 , e4 ] = e1 , [e3 , e4 ] = e2 ; A4.4 : [e1 , e4 ] = e1 , [e2 , e4 ] = e1 + e2 , A4.5 : [e1 , e4 ] = e1 , [e2 , e4 ] = qe2 , A4.6 : [e1 , e4 ] = qe1 , [e3 , e4 ] = e2 + e3 ; [e3 , e4 ] = pe3 , −1 ≤ p ≤ q ≤ 1, pq 6= 0; [e2 , e4 ] = pe2 − e3 , A4.7 : [e2 , e3 ] = e1 , [e1 , e4 ] = 2e1 , A4.8 : [e2 , e3 ] = e1 , [e1 , e4 ] = (1 + q)e1 , A4.9 : [e2 , e3 ] = e1 , [e1 , e4 ] = 2qe1 , A4.10 : [e1 , e3 ] = e1 , [e2 , e3 ] = e2 , [e3 , e4 ] = e2 + pe3 , q 6= 0, p ≥ 0; [e2 , e4 ] = e2 , [e3 , e4 ] = e2 + e3 ; [e2 , e4 ] = e2 , [e2 , e4 ] = qe2 − e3 , [e1 , e4 ] = −e2 , [e3 , e4 ] = qe3 , |q| ≤ 1; [e3 , e4 ] = e2 + qe3 , q ≥ 0; [e2 , e4 ] = e1 . We provide full calculation details for the case of the algebra sl1 (2, R) ⊂+A2.1 . The algebra so(3) ⊂+A3.1 is handled in the same way. Let sl(2, R) = he1 , e2 , e3 i, A2.1 = he4 , e5 i. Then the basis elements of sl1 (2, R) are Eequivalent to e1 = 2t∂t + x∂x , e2 = −t2 ∂t − tx∂x + x2 ∂u , e3 = ∂t . The remaining non-zero commutation relations of the algebra sl1 (2, R) ⊂+A2.1 read [e1 , e4 ] = e4 , [e1 , e5 ] = −e5 , [e2 , e5 ] = e4 , 20 [e3 , e4 ] = e5 . (4.8) Inserting operators e4 , e5 of the form (2.3) into (4.8) and solving the resulting equations we obtain the four inequivalent realizations of the algebra sl1 (2, R) ⊂+A2.1 , basis operators of A2.1 having the form 1) e4 = t∂x + 2tx−1 u∂u , e5 = ∂x + 2x−1 u∂u ; 2) e4 = t∂x + (tx−1 u − x)∂u , 3) e4 = tx−1 ∂u , 4) e4 = (tu + x2 )∂x + (2ux + 2tx−1 u2 )∂u , e5 = ∂x + x−1 u∂u ; e5 = x−1 ∂u ; e5 = u∂x + 2x−1 u2 ∂u . However, only the second and the third realization give rise to the symmetry algebras of equations of the form (1.1). The corresponding invariant equations are sl1 (2, R) ⊂+ ht∂x + (tx−1 u − x)∂u , ∂x + x−1 u∂u i : ut = λuxx + 2λx−2 u − 2λx−1 ux + x−1 uux − x−2 u2 , λ 6= 0; sl1 (2, R) ⊂+ h(tu + x2 )∂x + 2(xu + tx−1 u2 )∂u , u∂x + 2x−1 u2 ∂u i : ut = x−1 uux − x−2 u2 + λx−2 (2u − xux )(x2 uxx + 2u − 2xux )−1 , λ 6= 0. Note that the five-dimensional Lie algebras presented above are maximal in Lie’s sense. Analysis of the realizations sl2 (2, R), sl4 (2, R), sl5 (2, R) shows that they do cannot be extended up to a realization of the algebra sl(2, R) ⊂ +A2.1 . Similarly, the realization so1 (3) cannot be extended up to a realization of the algebra so1 (3) ⊂+A3.1 . The realization sl3 (2, R) do yield new realizations of the algebra sl(2, R) ⊂+A2.1 . We give these below together with the corresponding invariant equations. sl3 (2, R) ⊂+ h−∂x + x−1 u∂u , x−1 ∂u i : ut = x−1 (xuxx + 2ux )1/3 F (t). sl3 (2, R) ⊂+ hx2 u∂x , x2 ∂x i : ut = x3 u2x (xuxx + 2ux )−1/3 F (t). (4.9) (4.10) However these five-dimensional Lie algebras are not maximal. To find the most extensive symmetry algebras we apply the infinitesimal Lie algorithm directly. First of all, using the fact that F (t) 6= 0 we can make the change of variables Z t̄ = F (t)dt, x̄ = x, v = u and get F ≡ 1. Next, utilizing the Lie’s algorithm we obtain that the maximal invariance algebra of PDE (4.9) with F = 1 is the seven-dimensional Lie algebra 4 sl3 (2, R) ⊂+h∂t , x∂x + t∂t , x−1 ∂u , −∂x + x−1 u∂u i. 3 This algebra is isomorphic to the Lie algebra sl(2, R) ⊂+A4.5 with q = 1, p = 43 . The maximal invariance algebra admitted by (4.10) with F = 1 is the seven-dimensional Lie algebra 4 sl3 (2, R) ⊂+h∂t , t∂t − x∂x , x2 u∂x , x2 ∂x i. 3 21 It is isomorphic to the algebra sl(2, R) ⊂+A4.5 with q = 1, p = 34 . Now we proceed to finalizing group classification of the equations obtained in the previous subsection. Since these equations contain arbitrary functions of at most one variable, we can apply directly the Ovsyannikov classification method. Consider, as an example, the case of realization sl1 (2, R) ⊕ h∂u i. Inserting the function 1 F = u2x + x−2 F̃ (ω), 4 ω = x2 uxx − xux into the classifying equation (2.4) and splitting the obtained relation by the powers of the independent variable ux yields the following over-determined system of PDEs for τ, ξ, η and F̃ : ³ ´ −2 −1 −1 x (2x ξ + ηu − 2ξx )ω − x ηx + ηxx F̃ω = x−2 (ηu − τt + 2x−1 ξ)F̃ , 1 (3x−2 ξu ω + ξxx + x−1 ξx − x−2 ξ − 2ηxu )F̃ω = x−2 ξu F̃ + ηx + ξt , 2 1 1 1 1 (2ξxu − ηuu + 2x−1 ξu )F̃ω = ηu + τt − ξx , ξuu F̃ω = − ξu . 4 4 2 4 (4.11) If the function F̃ (4.11) is arbitrary, then the realization sl1 (2, R) ⊕ h∂u i is the maximal symmetry algebra of the corresponding equation. It follows from the last two equations from (4.11) that either F̃ is linear function of ω or ξu = ηuu = 0, 2ξx − ηu − τt = 0. (4.12) Provided F̃ = λω + C, λ 6= 0, C ∈ R, then the maximal invariance algebra is infinitedimensional. In the case when C 6= 3λ it is formed by the basis elements of the realization sl1 (2, R) ⊕ h∂u i and by the infinite set of operators ³u´ v∞ = α(t, x) exp ∂u , 4λ where α = α(t, x) is an arbitrary solution of the equation αt = λαxx − λx−1 αx − C −2 x α. 4λ In the case when C = 3λ, the corresponding PDE admits two more symmetry operators t∂x + 2(λx−1 t − x)∂u and ∂x + 2λx−1 ∂u . However, the change of variables from E t̄ = t, x̄ = x, u = 4λ ln |v|, v = v(t̄, x̄) reduce the equation under study to a linear heat conductivity equation vt̄ = λvx̄x̄ − λx−1 vx̄ + C −2 x̄ v. 4λ Consequently, the equation in question is equivalent to linear PDE and therefore is excluded from further consideration. 22 If the function F̃ is a non-linear function of ω, i.e., F̃ωω 6= 0, then taking into account (4.12) we derive from (4.11) the system of two equations for the functions τ, ξ = ξ(t, x), η = (2ξx − τx )u + θ(t, x) and F̃ ´ ³ −2 −1 −1 x (2x ξ + ηu − 2ξx )ω − x ηx + ηxx F̃ω = x−2 (ηu − τt + 2x−1 ξ)F̃ + ηt , (4.13) 1 −1 −2 (2ηxu − ξxx − x ξx + x ξ)F̃ω = − 2 ηx − ξt . Analysis of the first equation yields the following admissible forms of the function F̃ F̃ = λ exp(pω) + m, λp 6= 0, F̃ = λ ln |ω + b| + m, λ 6= 0, F̃ = λ|ω + b|p + m, λp 6= 0, m ∈ R; b, m ∈ R; p 6= 1, b, m ∈ R. Inserting these expressions into (4.13) shows that extension of symmetry algebra is only possible when F̃ = λω 2 . However, the maximal invariance algebra of this equation has already been obtained earlier in this subsection. A similar analysis of the remaining invariant equations from Subsection 4.2.1 yields the following results: • The only extension of the realization sl2 (2, R) ⊕ h∂u i, which is invariance algebra of an equation of the form (1.1), is the realization sl2 (2, R) ⊕ h−u∂u , ∂u i. • The only extension of the algebra sl3 (2, R) ⊕ htx∂x i, which is invariance algebra of an equation of the form (1.1), is the realization sl3 (2, R) ⊕ ht∂t , tx∂x i. • The list of possible extensions of the algebra sl3 (2, R)⊕h∂t i, which are invariance algebras of equations of the form (1.1), is exhausted by the following algebras: 1) sl3 (2, R) ⊕ h−t∂t − mx∂x , ∂t i, 2) sl3 (2, R) ⊕ ht∂t , tx∂x i, 3) sl3 (2, R) ⊂ +h∂t , x∂x + 43 t∂t , x−1 ∂u , −∂x + x−1 u∂u i, 4) sl3 (2, R) ⊂ +h∂t , 34 t∂t − x∂x , x2 u∂x , x2 ∂x i. • The realizations sl4 (2, R) ⊕ h∂t i, sl5 (2, R) ⊕ h∂t i and so1 (3) ⊕ h∂t i do not admit extensions to realizations admitted by equations of the form (1.1). Summarizing the above results we give the final list of inequivalent equations from the class C2 that have nontrivial Levi factor. 1 sl1 (2, R) ⊕ h∂u i : ut = u2x + x−2 F (ω), 4 2 ω = x uxx − xux ; sl1 (2, R) ⊕ hx∂x + 2u∂u i : ut = x−1 uux − x−2 u2 + x−2 (2u − xux )2 F (ω), 23 ω = (x2 uxx − 2u)(2u − xux )−1 ; 1 sl2 (2, R) ⊕ h∂u i : ut = − x−1 ux + x−3 u−1 x F (ω), 4 −1 −1 ω = u−2 x uxx + 3x ux ; sl3 (2, R) ⊕ h∂t i : ut = xux F (ω), sl3 (2, R) ⊕ sl4 (2, R) ⊕ sl5 (2, R) ⊕ so1 (3) ⊕ sl1 (2, R) ⊕ sl2 (2, R) ⊕ sl3 (2, R) ⊕ sl3 (2, R) ⊕ sl1 (2, R) ⊂ + −6 −2 ω = x−5 u−3 x uxx + 2x ux ; ¢ xux ¡ −5 −3 htx∂x i : ut = + xux F (t), ln x ux uxx + 2x−6 u−2 x 4t p h∂t i : ut = x−2 4 + x6 u2x F (ω), 1 ω = (4 + x6 u2x )−3/2 (x4 uxx + 5x3 ux + x9 u3x ); 2 p −2 6 2 x ux − 4F (ω), h∂t i : ut = x 1 ω = (x6 u2x − 4)−3/2 (x4 uxx + 5x3 ux − x9 u3x ); 2 p 2 2 h∂t i : ut = sec x + ux F (ω), ³ ´ ω = (1 + u2x cos2 x)−3/2 uxx cos x − (2 + u2x cos2 x)ux sin x ; µ ¶ 1 2 2 h∂u , x∂x + 2u∂u i : ut = λx uxx − 2λxux uxx + λ + u2x , λ 6= 0; 4 1 −2 −1 −1 −2 h−u∂u , ∂u i : ut = − x−1 ux + λx−3 u−1 x (ux uxx + 3x ux ) , λ 6= 0; 4 −6 −2 1/4m h−t∂t − mx∂x , ∂t i : ut = λxux |x−5 u−3 , λ 6= 0, x uxx + 2x ux | 3 m 6= 0, ± ; 4 ¢ λxux xux ¡ −5 −3 ht∂t , tx∂x i : ut = + ln x ux uxx + 2x−6 u−2 , λ ∈ R; x 4t t ht∂x + (tx−1 − x)∂u , ∂x + x−1 u∂u i : ut = λuxx + 2λx−2 u − 2λx−1 ux + x−1 uux − x−2 u2 , λ 6= 0; sl1 (2, R) ⊂ + h(tu + x2 )∂x + 2(xu + tx−1 u2 )∂u , u∂x + 2x−1 u2 ∂u i : ut = x−1 uux − x−2 u2 + λx−2 (2u − xux )(x2 uxx + 2u − 2xux )−1 , λ 6= 0; 4 sl3 (2, R) ⊂ + h∂t , x∂x + t∂t , x−1 ∂u , −∂x + x−1 u∂u i : ut = x−1 (xuxx + 2ux )1/3 ; 3 4 sl3 (2, R) ⊂ + h∂t , t∂t − x∂x , x2 u∂x , x2 ∂x i : ut = x3 u2x (xuxx + 2ux )−1/3 . 3 5 Concluding Remarks We present exhaustive description of invariant nonlinear evolution equations belonging to the class C1 . We remind that the latter is formed by PDEs (1.1) that are not E-equivalent to equations of the form ut = F (t, x, ux , uxx ). 24 The corresponding inequivalent classes of invariant equations and their maximal invariance algebras are described by Theorem 3.1. According to this theorem the list of inequivalent invariant equations from C1 consists of • equation admitting one-dimensional Lie algebra; • equation admitting two-dimensional solvable Lie algebra isomorphic to A2.2 ; • two equations admitting semi-simple Lie algebras isomorphic to sl(2, R). The invariant equations (1.1), which are equivalent to the above PDE, belong to the class C2 . This division comes naturally if we take into account that equations from the class C2 can be transformed into quasi-linear equations evolution equations [21]. This fact can also be utilized to construct quasi-local symmetries of nonlinear evolution equations [31]. We give complete classification of PDEs from the class C2 invariant under the Lie algebras which are either semi-simple or semi-direct sums of semi-simple and solvable Lie algebras. In Section 4 we provide the full list of invariant equations in question together with their maximal invariance algebras. It is comprised by Eq.(4.1), equations listed in Lemmas 4.1, 4.2 and equations presented at the end of Subsection 4.2. The algebraic properties of these equations can be summarized as follows. There are • one equation admitting one-dimensional Lie algebra; • four equations admitting semi-simple Lie algebras isomorphic to sl(2, R) and so(3); • sixteen equations admitting symmetry algebras having non-trivial Levi factor. To achieve a complete description of invariant equations of the form (1.1) we need to classify equations from C2 admitting solvable Lie algebras. This research is in progress now and will be reported in our future publications. 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