Preliminary group classification of a class of fourth-order evolution

Preliminary group classification of a class of fourth-order evolution equations∗ Qing Huang1,2 , V. Lahno3 , C. Z. Qu1,2 and R. Zhdanov4

†

1. Department of Mathematics, Northwest University, Xi’an 710069, PR China 2. Center for Nonlinear Studies, Northwest University, Xi’an 710069, PR China 3. State Pedagogical University, Poltava, Ukraine 4. BIO-key International, Eagan, USA

Abstract We perform preliminary group classification of a class of fourth-order evolution equations in one spatial variable. Following the approach developed in [1] we construct all inequivalent partial differential equations belonging to the class in question which admit semi-simple Lie groups. In addition, we describe all fourth-order evolution equations from the class under consideration which are invariant under solvable Lie groups of dimension n 0 and Y = ²T˙ 4 x + Y (t). The theorem is proved. Consider next the one-dimensional Lie algebras generated by (2). The transformation (6) reduce vector field (2) to the form ·

¸ 1 1 −1 ˜ = τ T˙ ∂t˜ + (τ T¨T˙ + τ˙ )(˜ X x − Y ) + τ Y˙ + ²ρT˙ 4 ∂x˜ 4 h ³² ´ i 1 + τ Ut + τ˙ T˙ − 4 (˜ x − Y ) + ρ Ux + ηUu ∂u˜ . 4 We consider the cases η = 0 and η 6= 0 separately. Case 1. η = 0. Choosing U = U (u), we have ¸ · 1 1 ¨ ˙ −1 ˜ ˙ ˙ ˙ X = τ T ∂t˜ + (τ T T + τ˙ )(˜ x − Y ) + τ Y + ²ρT 4 ∂x˜ . 4 If τ = 0, then inequality ρ 6= 0 holds, since otherwise operator (2) vanishes identi1 ˜ = ²∂x˜ . cally. Choosing as T a solution of equation |ρ|T˙ 4 = 1 yields the operator X ˜ Making use of the space reflection x → −x we get X = ∂x˜ . In the case when τ 6= 0, we can put ² = 1 in (6). Taking solutions of |τ |T˙ = 1,

1 τ Y˙ + ρT˙ 4 = 0,

T˙ > 0

˜ = ²∂t˜. as T and Y , yields the operator X Case 2. η 6= 0. Provided τ = ρ = 0, we choose U as a solution of ηUu = 1 and ˜ = ∂u˜ . Otherwise, we choose U to satisfy thus arrive at the operator X ¶ µ τ˙ x + ρ Ux + ηUu = 0, Uu 6= 0 τ Ut + 4 and we get the operator from Case 1. It is straightforward to verify that ²∂t , ∂x , ∂u cannot be transformed one into another with the transformation (6). We summarize the above results in the following lemma.

6

Lemma 1. Within the point transformation (6), the vector field (2) is equivalent to one of the following canonical operators ²∂t ,

∂x ,

∂u .

(7)

Thus there are three inequivalent realizations of one-dimensional Lie algebras by operators (2). Integrating the classifying equation for each symmetry operator yield the corresponding inequivalent equations from the class (1) which admit one-dimensional Lie algebras. In a sequel we adopt the notation Ak.i = hX1 , X2 , · · · , Xk i to denote an ith realization of a k-dimensional Lie algebra with Xi (i = 1, 2, · · · , k) being its basis elements. Theorem 1. There are three inequivalent equations of the form (1) invariant under one-dimensional Lie algebras: A11 = h∂t i : ut = −uxxxx + F (x, u, ux , uxx , uxxx ), A21 = h∂x i : ut = −uxxxx + F (t, u, ux , uxx , uxxx ), A31 = h∂u i : ut = −uxxxx + F (t, x, ux , uxx , uxxx ) where F is an arbitrary function. Furthermore, the corresponding symmetry algebra is maximal in Lie’s sense. 3. Classification of equations invariant under semi-simple Lie algebras

The lowest dimension semi-simple Lie algebras are isomorphic to one of the following two algebras so(3) :

[X1 , X2 ] = X3 , [X1 , X3 ] = −X2 , [X2 , X3 ] = X1 ;

sl(2, R) : [X1 , X2 ] = 2X2 , [X1 , X3 ] = −2X3 , [X2 , X3 ] = X1 . The following assertion holds. Theorem 2. There exists no realization of the algebra so(3) that can be a symmetry algebra of (1). Furthermore there are at most two inequivalent realizations of sl(2, R) by operators (2), which are admitted by Eq.(1). The realizations and the corresponding invariant equations read as 1 1 sl1 (2, R) = h−2t∂t − x∂x , ∂t , −t2 ∂t − tx∂x − x4 ∂u i : 2 2 1 1 1 ut = −uxxxx + 4 [u2 + (24 − xux )u] + 4 G(ω1 , ω2 , ω3 ), x 2 x ω1 = xux − 4u, ω2 = x2 uxx − 12u, ω3 = x3 uxxx − 24u, sl2 (2, R) = h−2u∂u , ∂u , −u2 ∂u i : 7

ut = −uxxxx + 4

uxx uxxx u3 uxxx u2xx − 3 xx + u G(t, x, 2 − 3 ). x ux u2x ux u2x

The algebras sl1 (2, R) and sl2 (2, R) are the maximal invariance algebras of the corresponding PDEs, provided that the functions G are arbitrary. Proof. Consider first inequivalent realizations of the algebra so(3). Taking operators (2) as its basis Xi (i=1,2,3), inserting the latter into the commutation relations of so(3) and solving equations obtained we obtain all possible realizations of the algebra under study. In view of Lemma 1, we can assume without any loss of generality that one of the basis operators, say X1 , is of the one of three canonical forms ²∂t , ∂x and ∂u . Let X1 = ∂t and X2 , X3 be of the general form (2). Inserting them into the first two commutation relations of so(3) yields X2 = 4a cos t∂t + [−ax sin t + b cos(t + c)]∂x + φ(x, u) cos(t + ψ(x, u))∂u , X3 = −4a sin t∂t − [ax cos t + b sin(t + c)]∂x − φ(x, u) sin(t + ψ(x, u))∂u . Here a, b, c are arbitrary real constants and φ, ψ are arbitrary real-valued smooth functions. Substituting X2 , X3 into the last commutation relation implies that −16a2 = 1 which has no real solution a. Thus, there are no realizations of so(3) in the case when X1 = ∂t . The same assertion holds for X1 = −∂t , ∂x or ∂u . Consequently, there are no realizations of algebra so(3) within the class of operators (2) and, therefore, Eq.(1) does not admit so(3). The case of the algebra sl(2, R) is treated in a similar way which yields two inequivalent realizations given in the formulation of the theorem. Solving the determining equations for each of the above obtained realizations we arrive at the invariant equations presented in the formulation of the theorem. The assertion is proved. Corollary 1. The invariant equations listed in Theorem 2 exhaust the list of all possible inequivalent PDEs (1), whose invariance algebras are semi-simple. Proof. It is a common knowledge that there exist four types of classical semisimple Lie algebras, An−1 , Bn , Cn , Dn , and five exceptional semi-simple Lie algebras, G2 , F4 , E6 , E7 , E8 [31]. • An−1 (n > 1) has four real forms of the algebra sl(n, C): su(n), sl(n, R), su(p, q) (p + q = n, p ≥ q), su∗ (2n). • 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). • 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). • Dn (n > 1) has three real forms of the algebra so(2n, C): so(2n), so(p, q) (p + q = 2n, p ≥ q), so∗ (2n). 8

The lowest dimensional semi-simple Lie algebras admit the following isomorphisms so(3) ∼ su(2) ∼ sp(1),

sl(2, R) ∼ su(1, 1) ∼ so(2, 1) ∼ sp(1, R).

Hence, realizations of the algebra sl(2, R) exhaust the set of all possible realizations of three-dimensional semi-simple Lie algebras admitted by (1). The next admissible dimension for classical semi-simple Lie algebras is six. There are four non-isomorphic semi-simple Lie algebras, namely, so(4), so∗ (4), so(3, 1), and so(2, 2). As so(4) ∼ so(3) ⊕ so(3), so∗ (4) ∼ so(3) ⊕ sl(2, R), and so(3, 1) contains so(3) as a subalgebra, these algebras can not be invariance algebras of Eq.(1). Thus so(2, 2) is the only possible six-dimensional semi-simple algebra that can be admitted by (1). As so(2, 2) ∼ sl(2, R) ⊕ sl(2, R), we can choose so(2, 2) = hQi , Ki |i = 1, 2, 3i, where hQ1 , Q2 , Q3 i = sl(2, R), hK1 , K2 , K3 i = sl(2, R) and [Qi , Kj ] = 0, (i, j = 1, 2, 3). Taking Q1 , Q2 , Q3 to be the basis operators of the realizations of sl(2, R) given in the formulation of the theorem, and K1 , K2 , K3 be of the general form (2), after some algebra we prove that realizations of sl(2, R) cannot be extended to a realization of so(2, 2). Consequently, Eq.(1) does not admit six-dimensional semi-simple Lie algebras. The same assertion holds true for eight-dimensional semi-simple Lie algebras sl(3, R), su(3) and su(2, 1). As su∗ (4) ∼ so(5, 1) ⊃ so(4), the algebra An−1 (n > 1) has no realizations by operators (2) except for those given in the formulation of the theorem. There are also no realizations of the algebra Dn (n > 1), since the lowest dimensional algebras of this type so(4), so(2, 2), so∗ (4) have no realizations within the class of differential operators (2). The similar reasoning yields that there are no new realizations of the algebras Bn (n ≥ 1) and Cn (n ≥ 1), that can be symmetry algebras of Eq.(1). This follows from the fact that the algebra B2 contains subalgebras isomorphic to so(4) and so(1, 3), and from the following properties of the algebra C2 sp(2) ∼ so(5) ⊃ so(4), sp(2, R) ∼ so(3, 2) ⊃ so(3, 1), sp(1, 1) ∼ so(4, 1) ⊃ so(4). To complete the proof, we need to consider the exceptional semi-simple Lie algebras G2 , F4 , E6 , E7 , E8 . Here we only consider the first two algebras and others are handled in the same way. The algebra G2 has a compact real form g2 and one noncompact real form g˜2 , where g2 ∩ g˜2 ∼ su(2) ⊕ su(2) ∼ so(4). Since Eq.(1) cannot admit so(4) as an invariance algebra, G2 has no realization by operators (2). F4 has a compact real form f4 and two noncompact real forms f˜4 , fˆ4 with f4 ∩ f˜4 ∼ sp(3) ⊕ su(2) ⊃ so(3), f4 ∩ fˆ4 ∼ sp(9). Thus Eq.(2) cannot admit a realization of an algebra of the type F4 . The same assertion holds for the remaining exceptional semi-simple Lie algebras E6 , E7 , E8 . The corollary is proved. 9

4. Classification of equations invariant under low-dimensional solvable Lie algebras

Using the concept of compositional series for a solvable algebra we can construct all possible realizations of solvable Lie algebras admitted by Eq.(1) starting from the one-dimensional ones and proceeding to the solvable Lie algebras of the dimension 2, 3, 4, . . . (for more details, see [1]). In this section we describe inequivalent Eqs.(1) which are invariant under solvable Lie algebras of the dimension up to four. Since equations invariant with respect to one-dimensional algebras have already been constructed, we start by analyzing two-dimensional solvable algebras. 4.1. Equations with two-dimensional Lie algebras

There are two non-isomorphic two-dimensional Lie algebras, A2.1 : [X1 , X2 ] = 0,

A2.2 : [X1 , X2 ] = X2 .

As both A2.1 and A2.2 contain the algebra A1 , we can assume that the basis operator of the latter is reduced to a canonical form. We consider in detail the case of A2.1 , while for the case of algebra A2.2 we present the final results only. Let X1 = ∂t and X2 be the operator of the most generic form (2) X2 = τ (t)∂t + [τ˙ /4x + ρ(t)]∂x + η(t, x, u)∂u . Then the commutation relation implies that τ˙ = ρ˙ = ηt = 0. Therefore τ , ρ are constants and η = η(x, u). So without any loss of generality we can put X2 = c∂x + η(x, u)∂u , where and hereafter c = const. Before simplifying X2 with equivalence transformations (6), we seek those equivalent transformations which preserve the basis operator of the algebra A1.1 ¸ · 1 −1 ˜ 1 = T˙ ∂t˜ + T˙ T¨(˜ X1 → X x − Y ) + Y˙ ∂x˜ + Ut ∂u˜ = ∂t˜. 4 Hence, T˙ = 1 and Y˙ = Ut = 0. Consequently, we can take T = t, U = U (x, u) and Y = const. Performing this transformation yields X2 → X˜2 = ²c∂x˜ + (cUx + ηUu )∂u˜ . If η = 0, we choose U = U (u) thus getting X˜2 = ∂x˜ . Provided η 6= 0 and c = 0, we can take U as a solution of ηUu = 1 and get ∂u˜ . When η 6= 0 and c 6= 0, we select U to satisfy cUx + ηUu = 0, Uu 6= 0, whence we conclude that the operator ∂x˜ is admitted by Eq.(1). Thus within the action of equivalence group of Eq.(1), we get the following two-dimensional invariance algebras, h∂t , ∂x i, h∂t , ∂u i. 10

Consider now the case when X1 = ∂x and X2 is an operator of the form (2). Inserting X1 , X2 into the commutation relation yields that X2 = c∂t + ρ(t)∂x + η(t, u)∂u . The equivalence transformation, which leaves X1 invariant, reads as t˜ = t, x˜ = x + Y (t), u˜ = U (t, u) with Uu 6= 0. This transformation reduces X2 to the form ˜ 2 = c∂t˜ + [cY˙ + ρ(t)]∂x˜ + (cUt + ηUu )∂u˜ . X We consider the cases η = 0 and η 6= 0 separately. If η = 0 and c = 0, then ˜ X2 = ρ(t)∂x˜ . It is straightforward to verify that the so obtained two-dimensional Lie algebra, h∂x , g(t)∂x i, cannot be invariance algebras of an equation of the form (1). Provided η = 0 and c 6= 0, we can choose solutions of PDEs Ut = 0 and cY˙ + ρ = 0 as U and Y and thus get X˜2 = ²∂t˜. ˜ 2 = ρ(t)∂x˜ + ∂u˜ . Similarly, the case η 6= 0 and c = 0 leads to the operator X ˜ Next, if η 6= 0 and c 6= 0, then the operator X2 = ²∂t˜ is obtained. Turn now to the remaining case X1 = ∂u . With the choice of X2 of the form (2), the commutation relation implies X2 = τ (t)∂t + [τ˙ /4x + ρ(t)]∂x + η(t, x)∂u . Note that the transformation t˜ = T (t),

1 x˜ = ²T˙ 4 x + Y (t),

u˜ = u + U (t, x)

belongs to the equivalence group of (1), which preserves X1 and converts X2 into ¸ · 1 1 ˙ −1 ¨ ˙ ˙ ˙ ˙ 4 τ T ∂t˜ + (τ T T + τ˜)(˜ x − Y ) + τ Y + ²ρT ∂x˜ 4 µ ¶ 1 [τ Ut + τ˙ x + ρ Ux + η]∂u˜ . 4 Choosing T , Y and U suitably, we can reduce the operator X2 to one of the forms, ∂x˜ , ²∂t˜ and η(t, x)∂u˜ . This completes the analysis of the realizations of the algebra A2.1 . The case of A2.2 is treated similarly. Substituting the so obtained basis operators into the classifying equation and solving the latter yields the corresponding invariant equations. Theorem 3. There exist four Abelian and five non-Abelian two-dimensional symmetry algebras admitted by (1). These algebras and the corresponding invariant equations are given below A12.1 = h∂t , ∂x i :

ut = −uxxxx + G(u, ux , uxx , uxxx ),

A22.1 = h∂t , ∂u i :

ut = −uxxxx + G(x, ux , uxx , uxxx ),

A32.1 = h∂x , g(t)∂x + ∂u i :

ut = −uxxxx + g 0 uux + G(t, ux , uxx , uxxx ),

A42.1 = h∂u , g(t, x)∂u i, gx 6= 0 : ut = −uxxxx + (gxxxx − gt )gx−1 ux + G(t, x, ω1 , ω2 ), 11

ω1 = (gx uxx − gxx ux )gx−1 , ω2 = (gx uxxx − gxxx ux )gx−1 , x A12.2 = h−t∂t − ∂x , ∂t i : 4

ut = −uxxxx + x−4 G(u, xux , x2 uxx , x3 uxxx ),

A22.2 = h−4t∂t − x∂x , ∂x i :

ut = −uxxxx + t−1 G(u, t 4 ux , t 2 uxx , t 4 uxxx ),

A32.2 = h−u∂u , ∂u i :

−1 ut = −uxxxx + ux G(t, x, u−1 x uxx , ux uxxx ),

A42.2 = h∂x − u∂u , ∂u i :

ut = −uxxxx + e−x G(t, ex ux , ex uxx , ex uxxx ),

A52.2 = h²∂t − u∂u , ∂u i :

ut = −uxxxx + ux G(x, e²t ux , e²t uxx , e²t uxxx ).

1

1

3

4.2. Equations admitting three-dimensional solvable Lie algebras

We consider the cases of decomposable and non-decomposable Lie algebras separately. 4.2.1 Three-dimensional decomposable algebras

There exist two non-isomorphic three-dimensional decomposable Lie algebras, A3.1 and A3.2 , A3.1 : [Xi , Xj ] = 0 (i, j = 1, 2, 3) (A3.1 = A1 ⊕ A1 ⊕ A1 ), A3.2 : [X1 , X2 ] = X2 , [X1 , X3 ] = [X2 , X3 ] = 0,

(A3.2 = A2.2 ⊕ A1 ).

It is a common knowledge that any three-dimensional solvable Lie algebra contains two-dimensional solvable algebra. So that to describe all possible realizations of three-dimensional solvable algebras admitted by Eq.(1) it suffices to consider all possible extensions of two dimensional algebras listed in Theorem 3 by vector fields, X3 , of the form (2). Then for each of the so obtained realization we simplify X3 using equivalence transformations which preserve the operators X1 and X2 . Having performed these two steps we obtain the following list of invariant equations (1): A3.1 − invariant equations A13.1 = h∂t , ∂x , ∂u i : F = G(ux , uxx , uxxx ), A23.1 = h∂t , ∂u , g(x)∂u i, g 0 6= 0 : F =

g (4) g 00 g 000 u + G(x, u − u , ux − uxxx ). x x xx g0 g0 g0

A3.2 − invariant equations x A13.2 = h−t∂t − ∂x , ∂t , ∂u i : 4 12

F =

1 G(xux , x2 uxx , x3 uxxx ), x4

A23.2 = h−4t∂t − x∂x , ∂x , ∂u i : 1 1 3 1 F = G(|t| 4 ux , |t| 2 uxx , |t| 4 uxxx ), t 1

A33.2 = h−4t∂t − x∂x , ∂x , |t| 4 ∂x + ∂u i : F =−

uux

1 1 3 1 + G(|t| 4 ux , |t| 2 uxx , |t| 4 uxxx ), t 4|t| 3 4

A43.2 = h−u∂u , ∂u , ∂x i : uxx uxxx , ), ux ux

F = ux G(t,

A53.2 = h−u∂u , ∂u , ∂t i : F = ux G(x,

uxx uxxx , ), ux ux

A63.2 = h∂x − u∂u , ∂u , e−x g(t)∂u i, g 6= 0 : F = −(1 +

g0 )ux + e−x G(t, ex (uxx + ux ), ex (uxxx − ux )), g

A73.2 = h∂x − u∂u , ∂u , g(t)∂x i, g 6= 0 : g0 uxx uxxx F = − (x + ln |ux |)ux + ux G(t, , ), g ux ux A83.2 = h∂x − u∂u , ∂u , ∂t i : F = e−x G(ex ux , ex uxx , ex uxxx ), A93.2 = h²∂t − u∂u , ∂u , e−²t g(x)∂u i, g 0 6= 0 : F =

00 000 g (4) − ²g −²x ²t g ²t g u + e G(x, e ( u − u ), e ( ux − uxxx )), x x xx g0 g0 g0

A10 3.2 = h²∂t − u∂u , ∂u , ∂x i : F = e−²t G(e²t uxx , e²t ux , e²t uxxx ), A11 3.2 = h²∂t − u∂u , ∂u , ∂t + λ∂x i, λ 6= 0 : x

x

x

x

F = e²( λ −t) G(e²(t− λ ) ux , e²(t− λ ) uxx , e²(t− λ ) uxxx ), A12 3.2 = h²∂t − u∂u , ∂u , ∂t i :

13

F = ux G(x,

uxx uxxx , ). ux ux

4.2.2 Three-dimensional non-decomposable algebras

There are seven non-isomorphic non-decomposable three-dimensional real solvable Lie algebras. The list of these algebras is exhausted by one nilpotent Lie algebra, A3.3 : [X2 , X3 ] = X1 ,

[X1 , X2 ] = [X1 , X3 ] = 0,

and six solvable Lie algebras (only nonzero commutation relations are given), A3.4 : [X1 , X3 ] = X1 ,

[X2 , X3 ] = X1 + X2 ,

A3.5 : [X1 , X3 ] = X1 ,

[X2 , X3 ] = X2 ,

A3.6 : [X1 , X3 ] = X1 ,

[X2 , X3 ] = −X2 ,

A3.7 : [X1 , X3 ] = X1 ,

[X2 , X3 ] = qX2 , (0 < |q| < 1),

A3.8 : [X1 , X3 ] = −X2 ,

[X2 , X3 ] = X1 ,

A3.9 : [X1 , X3 ] = qX1 − X2 ,

[X2 , X3 ] = X1 + qX2 , (q > 0).

All these algebras contain a two-dimensional Abelian ideal as a subalgebra. Thus we can use our classification of A2.1 − invariant equations to construct Eqs.(1), which admit non-decomposable three-dimensional solvable Lie algebras. We skip intermediate calculations and present the final result, the list of invariant equations and the corresponding symmetry algebras. A3.3 − invariant equations A13.3 = h∂x , ∂t , t∂x + ∂u i : F = −uux + G(ux , uxx , uxxx ), A23.3 = h∂u , ∂t , λ∂x + t∂u i, λ > 0 : F =

x + G(ux , uxx , uxxx ), λ

A33.3 = h∂u , ∂x , g(t)∂x + x∂u i : 1 F = − g 0 u2x + G(t, uxx , uxxx ), 2 A43.3 = h∂u , ∂x , λ∂t + x∂u i, λ 6= 0 : F = G(λux − t, uxx , uxxx ), 1

4

1

1

A53.3 = hλt 3 ∂x + ∂u , ∂x , 3λt 3 ∂t + λt 3 x∂x + (x − λt 3 u)∂u i, λ 6= 0 : 14

5 2 1 4 λ 2 F = − t− 3 uux + t− 3 G(λt 3 ux − t 3 , tuxx , t 3 uxxx ), 3

A63.3 = h∂u , [g(x) − t]∂u , ∂t i, g 0 6= 0 : F =

g (4) − 1 g 00 g 000 u + G(x, u − u , ux − uxxx ), x x xx g0 g0 g0

A73.3 = h∂u , −x∂u , ∂x i : F = G(t, uxx , uxxx ), A83.3 = h∂u , (²t − x)∂u , ∂x i : F = −²ux + G(t, uxx , uxxx ), 1 u A93.3 = h− ∂u , ∂u , ∂x − ∂u i : x x 4 1 F = − uxxx + G(t, xuxx + 2ux , xuxxx + 3uxx ). x x A3.4 − invariant equations 1 A13.4 = h∂u , ∂t , t∂t + x∂x + (u + t)∂u i : 4 ux uxx uxxx F = 4 ln |ux | + G( 3 , 2 , ), x x x A23.4 = h∂u , ∂x , 4t∂t + x∂x + (u + x)∂u i : 3

F = t− 4 G(ux − A33.4

1 1 1 ln |t|, |t| 4 uxx , |t| 2 uxxx ), 4

α2 = hα∂x + ∂u , ∂x , 0 ∂t + (1 + α)x∂x + [x + (1 − α)u]∂u i : α 3

1

2

F = −α0 uux + α−5 e α G(α2 ux − α, α3 e− α uxx , α4 e− α uxxx ), where α = α(t), α0 6= 0 and α2 α00 + 2(2 + α)α02 = 0, 1 A43.4 = h∂x , − ln |t|∂x + ∂u , 4t∂t + x∂x + u∂u i : 4 1 1 3 1 F = t−1 uux + |t|− 4 G(ux , |t| 4 uxx , |t| 2 uxxx ), 4 A53.4 = h∂u , −x∂u , ∂x + u∂u i : F = ex G(t, e−x uxx , e−x uxxx ), A63.4 = h∂u , [g(x) − ²t]∂u , ²∂t + u∂u i, g 0 6= 0 : 15

00 000 g (4) − ² ²t −²t g −²t g u + e G(x, e ( u − u ), e ( ux − uxxx )). x x xx g0 g0 g0

F =

A3.5 − invariant equations 1 A13.5 = h∂u , ∂t , t∂t + x∂x + u∂u i : 4 ux uxx uxxx F = G( 3 , 2 , ), x x x A23.5 = h∂u , ∂x , 4t∂t + x∂x + u∂u i : 3

1

1

F = t− 4 G(ux , |t| 4 uxx , |t| 2 uxxx ), A33.5 = h∂u , g(x)∂u , ²∂t + u∂u i, g 0 6= 0 : F =

00 000 g (4) ²t −²t g −²t g u + e G(x, e ( u − u ), e ( ux − uxxx )). x x xx g0 g0 g0

A3.6 − invariant equations 1 A13.6 = h∂t , ∂u , t∂t + x∂x − u∂u i : 4 F = x−8 G(x5 ux , x6 uxx , x7 uxxx ), 1

A23.6 = h∂x , λ|t| 2 ∂x + ∂u , 4t∂t + x∂x − u∂u i, λ ∈ R : 5 1 3 λ 1 F = − t− 2 uux + t− 4 G(|t| 2 ux , |t| 4 uxx , tuxxx ), 2

A33.6 = h∂u , e2x g(t)∂u , ∂x + u∂u i, g 6= 0 : F =(

g0 + 8)ux + ex G(t, e−x (uxx − 2ux ), e−x (uxxx − 4ux )), 2g

A43.6 = h∂u , e2²t g(x)∂u , ²∂t + u∂u i, g 0 6= 0 : F =

00 000 g (4) + 2²g ²t −²t g −²t g u + e G(x, e ( u − u ), e ( ux − uxxx )). x x xx g0 g0 g0

A3.7 − invariant equations 1 A13.7 = h∂t , ∂x , t∂t + x∂x i : 4 uxx uxxx F = u4x G(u, 2 , 3 ), ux ux 1 A23.7 = h∂t , ∂x , t∂t + x∂x + u∂u i : 4 16

u4x u4xx u4xxx , , ), u3 u2 u 1 = h∂t , ∂u , t∂t + x∂x + qu∂u i, q 6= 0, ±1 : 4 F = G(

A33.7

F = x4(q−1) G(x(1−4q) ux , x(2−4q) uxx , x(3−4q) uxxx ), A43.7 = h∂x , λt

1−q 4

∂x + ∂u , 4t∂t + x∂x + qu∂u i, q 6= 0, ±1, λ ∈ R :

q+3 1−q 2−q 3−q 1 1 F = λ(q − 1)|t|− 4 uux + |t| 4 (q−4) G(|t| 4 ux , |t| 4 uxx , |t| 4 uxxx ), 4

A53.7 = h∂u , e(1−q)x g(t)∂u , ∂x + u∂u i, g 6= 0, q 6= 0, ±1 : F =[

g0 + (1 − q)3 ]ux + ex G(t, ω1 , ω2 ), (1 − q)g

where ω1 = e−x [uxx − (1 − q)ux ], ω2 = e−x [uxxx − (1 − q)2 ux ], A63.7 = h∂u , e²(1−q)t g(x)∂u , ²∂t + u∂u i, g 0 6= 0, q 6= 0, ±1 : 00 000 g (4) + ²(1 − q)g ²t −²t g −²t g F = ux + e G(x, e ( 0 ux − uxx ), e ( 0 ux − uxxx )). g0 g g

A3.8 − invariant equations A13.8 = h∂x , α∂x + ∂u , −

1 + α2 ∂t − αx∂x + (αu − x)∂u i : α0 5

F = −α0 uux + (1 + α2 )− 2 G(ω1 , ω2 , ω3 ), where α = α(t) with α0 6= 0 satisfies 2αα02 + (1 + α2 )α00 = 0, 3

ω1 = (1 + α2 )ux − α, ω2 = (1 + α2 ) 2 uxx , ω3 = (1 + α2 )2 uxxx . A3.9 − invariant equations A13.9 = h∂x , α∂x + ∂u , −

1 + α2 ∂t + (q − α)x∂x + [(α + q)u − x]∂u i, q > 0 : α0 5

F = −α0 uux + (1 + α2 )− 2 e3q arctan α G(ω1 , ω2 , ω3 ), where α = α(t) with α0 6= 0 satisfies (2α − 4q)α02 + (1 + α2 )α00 = 0, 3

ω1 = (1 + α2 )ux − α, ω2 = (1 + α2 ) 2 e−q arctan α uxx , ω3 = (1 + α2 )2 e−2q arctan α uxxx . Note that the nonlinear ordinary differential equation (2α − 4q)α02 + (1 + α2 )α00 = 0 17

can be solved by quadratures. Its general solution is given by the following implicit formula: Z α(t) e−4q arctan z (1 + z 2 )dz + C1 t + C2 = 0, {C1 , C2 } ⊂ R. 5. Classification of equations invariant under four-dimensional solvable Lie algebras

Now we perform group classification of Eqs.(1) admitting four-dimensional solvable Lie algebras. To this end we use the realizations of three-dimensional solvable algebras obtained in the previous section and the fact that any four-dimensional solvable Lie algebra contains a three-dimensional solvable algebra. 5.1. Equations with four-dimensional decomposable algebras

The list of non-isomorphic four-dimensional decomposable algebras contains following ten algebras: A2.2 ⊕ A2.2 = 2A2.2 , A3.1 ⊕ A1 = 4A1 , A3.2 ⊕ A1 = A2.2 ⊕ 2A1 , A3.i ⊕ A1 (i = 3, 4, · · · , 9). The complete list of Eqs.(1) invariant with respect to the above algebras is given below. 2A2.2 − invariant equations x 2A12.2 = h−t∂t − ∂x , ∂t , ∂u , eu ∂u i : 4 F = u4x − 6u2x uxx + 4ux uxxx + 3u2xx + ω1 =

ux G(ω1 , ω2 ), x3

x 2 x2 (ux − uxx ), ω2 = (u3x + uxxx − 3ux uxx ), ux ux 1

2A22.2 = h−4t∂t − x∂x , ∂x , λ|t| 4 ∂x − u∂u , ∂u i F =

1 uxx 1 uxxx λ ω ω ln |ω| + G(|t| 4 , |t| 2 ), 4|t| |t| ux ux 1

ω = |t| 4 ux , x 2A32.2 = h−u∂u , ∂u , ²∂t , e²t ∂t + ²e²t ( + 1)∂x i : 4 18

F =−

x+4 1 uxx 2 uxxx ux + u G((x + 4) , (x + 4) ), x 4 (x + 4)3 ux ux t

2A42.2 = h∂x − u∂u , ∂u , λ∂t , e λ ∂x i, λ 6= 0 : F =−

x + ln |ux | uxx uxxx ux + ux G( , ), λ ux u x t

2A42.2 = h∂x − u∂u , ∂u , λ∂t , e λ −x ∂u i, λ 6= 0 : F =−

λ+1 ux + e−x G(ex (uxx + ux ), ex (uxxx − ux )), λ

2A52.2 = h²∂t − u∂u , ∂u , β∂t + γ∂x , e

²β+1 x−²t γ

∂u i, β 6= −², γ 6= 0 :

(²β + 1)4 − ²γ 4 x−t) ²( β γ F = G(ω1 , ω2 ), u + e x 3 γ (²β + 1) β

β

ω1 = e²(t− γ x) [γuxx − (²β + 1)ux ], ω2 = e²(t− γ x) [γ 2 uxx − (²β + 1)2 ux ]. A3.2 ⊕ A1 − invariant equations A43.2 ⊕ A1 = h−u∂u , ∂u , ∂x , ∂t i : F = ux G(

uxx uxxx , ), ux ux

A83.2 ⊕ A1 = h∂x − u∂u , ∂u , ∂t , e−x ∂u i : F = −ux + e−x G(ex (uxx + ux ), ex (uxxx − ux ), A93.2 (g = eλx ) ⊕ A1 = h²∂t − u∂u , ∂u , eλx−²t ∂u , λ∂t + ∂x i, λ 6= 0 : F =

λ4 − ² ux + e²(λx−t) G(e²(t−λx) (uxx − λux ), e²(t−λx) (uxxx − λ2 ux )). λ

A3.3 ⊕ A1 − invariant equations, x A23.3 ⊕ A1 = h∂u , ∂t , λ∂x + t∂u , ∂t + β∂x + ∂u i, λ > 0, β ∈ R : λ x F = −βux + + G(uxx , uxxx ), λ t A43.3 ⊕ A1 = h∂u , ∂x , λ∂t + x∂u , β∂t + ∂x + ∂u i, λβ 6= 0 : λ 1 t F = − ux + + G(uxx , uxxx ), β βλ A63.3 (g = λx) ⊕ A1 = h∂u , (λx − t)∂u , ∂t , ∂x + λ∂t i, λ 6= 0 : 19

1 F = − ux + G(uxx , uxxx ), λ A73.3 ⊕ A1 = h∂u , −x∂u , ∂x , ∂t i : F = G(uxx , uxxx ), 1 u g(t) A93.3 ⊕ A1 = h− ∂u , ∂u , ∂x − ∂u , ∂t + ∂u i : x x x g − 4uxxx 1 F = + G(xuxx + 2ux , xuxxx + 3uxx ). x x A3.4 ⊕ A1 − invariant equations, 1 A13.4 ⊕ A1 = h∂u , ∂t , t∂t + x∂x + (u + t)∂u , x4 ∂u i : 4 F = 4 ln |ux | +

xuxx − 3ux x2 uxxx − 6ux 6 u + G( , ), x x3 x3 x3

A53.4 ⊕ A1 = h∂u , −x∂u , ∂x + u∂u , ex g(t)∂u i, g 6= 0 : F =(

g0 + 1)uxx + ex G(t, e−x (uxxx − uxx )), g

A53.4 ⊕ A1 = h∂u , −x∂u , ∂x + u∂u , ∂t i : F = ex G(e−x uxx , e−x uxxx ), A63.4 (g = ²λx) ⊕ A1 = h∂u , ²(λx − t)∂u , ²∂t + u∂u , λ∂t + ∂x i, λ 6= 0 : 1 F = − ux + e²(t−λx) G(e²(λx−t) uxx , e²(λx−t) uxxx ). λ A3.5 ⊕ A1 − invariant equations 1 A13.5 ⊕ A1 = h∂u , ∂t , t∂t + x∂x + u∂u , x4 ∂u i : 4 F =6

ux xuxx − 3ux x2 uxxx − 6ux + G( , ), x3 x3 x3

A33.5 ⊕ A1 = h∂u , g(x)∂u , ²∂t + u∂u , ∂t i, g 0 6= 0 : F =

g (4) g 0 uxxx − g 000 ux 00 0 u + (g u − g u )G(x, ). x x xx g0 g 0 (g 0 uxx − g 00 ux )

A3.6 ⊕ A1 − invariant equations, 1 1 A13.6 ⊕ A1 = h∂t , ∂u , t∂t + x∂x − u∂u , 4 ∂u i : 4 x F = −210x−3 ux + x−8 G(x5 (xuxx + 5ux ), x5 (x2 uxxx − 30ux )), 20

A33.6 (g = 1) ⊕ A1 = h∂u , e2x ∂u , ∂x + u∂u , ∂t i : F = 8ux + ex G(e−x (uxx − 2ux ), e−x (uxxx − 4ux )), 2²

x

A43.6 (g = e− β x ) ⊕ A1 = h∂u , e2²(t− β ) ∂u , ²∂t + u∂u , ∂t + β∂x i, β 6= 0 : x

x

x

F = +e²(t− β ) G(e²( β −t) (βuxx + 2²ux ), e²( β −t) (β 2 uxxx − 4ux ))− (

8² + β)ux . β3

A3.7 ⊕ A1 − invariant equations 1 A13.7 ⊕ A1 = h∂t , ∂x , t∂t + x∂x , ∂u i : 4 u u xx xxx F = u4x G( 2 , 3 ), ux ux 1 A23.7 ⊕ A1 = h∂t , ∂x , t∂t + x∂x + u∂u , u∂u i : 4 F =

u4x uuxx u2 uxxx G( 2 , ), u3 ux u3x

1 A33.7 ⊕ A1 = h∂t , ∂u , t∂t + x∂x + qu∂u , x4q ∂u i, q 6= 0, ±1 : 4 F =

(4q − 3)(4q − 2)(4q − 1) ux + x4(q−1) G(ω1 , ω2 ), x3

ω1 = x(1−4q) [xuxx − (4q − 1)ux ], ω2 = x(1−4q) [x2 uxxx − (4q − 2)(4q − 1)ux ], A53.7 (g = 1) ⊕ A1 = h∂u , e(1−q)x ∂u , ∂x + u∂u , ∂t i, q 6= 0, ±1 : F = (1 − q)3 ux + ex G(e−x (uxx − (1 − q)ux ), e−x (uxxx − (1 − q)2 ux )), A63.7 (g = e

²(q−1) x β

F =[

x

) ⊕ A1 = h∂u , e²(1−q)(t− β ) ∂u , ²∂t + u∂u , ∂t + β∂x i, q 6= 0, ±1 :

x ²(q − 1)3 ²(t− β ) − β]u + e G(ω1 , ω2 ), x 3 β x

x

ω1 = e²( β −t) [βuxx − ²(q − 1)ux ], ω2 = e²( β −t) [β 2 uxxx − (q − 1)2 ux ]. 5.2. Equations with four-dimensional non-decomposable algebras

There exist ten non-isomorphic four-dimensional non-decomposable Lie algebras, A4.i (i = 1, 2, · · · , 10): A4.1 :

[X2 , X4 ] = X1 ,

[X3 , X4 ] = X2 ; 21

A4.2 :

[X1 , X4 ] = qX1 ,

[X2 , X4 ] = X2 ,

[X3 , X4 ] = X2 + X3 , q 6= 0;

A4.3 :

[X1 , X4 ] = X1 ,

[X3 , X4 ] = X2 ;

A4.4 :

[X1 , X4 ] = X1 ,

[X2 , X4 ] = X1 + X2 ,

A4.5 :

[X1 , X4 ] = X1 ,

[X2 , X4 ] = qX2 ,

[X3 , X4 ] = X2 + X3 ;

[X3 , X4 ] = pX3 ,

− 1 6 p 6 q 6 1, pq 6= 0; A4.6 :

[X1 , X4 ] = qX1 ,

[X2 , X4 ] = pX2 − X3 ,

[X3 , X4 ] = X2 + pX3 ,

q 6= 0, p > 0; A4.7 :

[X2 , X3 ] = X1 ,

[X1 , X4 ] = 2X1 ,

[X2 , X4 ] = X2 ,

[X3 , X4 ] = X2 + X3 ; A4.8 :

[X2 , X3 ] = X1 ,

[X1 , X4 ] = (1 + q)X1 ,

[X2 , X4 ] = X2 ,

[X3 , X4 ] = qX3 , |q| 6 1; A4.9 :

[X2 , X3 ] = X1 ,

[X1 , X4 ] = 2qX1 ,

[X2 , X4 ] = qX2 − X3 ,

[X3 , X4 ] = X2 + qX3 , q > 0; A4.10 :

[X1 , X3 ] = X1 ,

[X2 , X3 ] = X2 ,

[X1 , X4 ] = −X2 ,

[X2 , X4 ] = X1 .

Each of the above algebras can be decomposed into a semi-direct sum of a threedimensional ideal N and a one-dimensional Lie algebra. Analysis of the commutation relations above shows that N is of the type A3.1 for the algebras A4.i (i = 1, 2, · · · , 6), of the type A3.3 for the algebras A4.7 , A4.8 , A4.9 , and of the type A3.5 for the algebra A4.10 . Thus we can extend the already known realizations of threedimensional Lie algebras to obtain exhaustive description of the four-dimensional non-decomposable solvable Lie algebras admitted by Eq.(1). Below we present the final result, the lists of invariant equations together with the corresponding symmetry algebras. A4.1 − invariant equations A14.1 = h∂u , ∂x , ∂t , t∂x + x∂u i : 1 F = − u2x + G(uxx , uxxx ), 2 A24.1 = h∂u , x∂u , ∂t , ∂x + tx∂u i : 1 F = x2 + G(uxx , uxxx ). 2 A4.2 − invariant equations A14.2 = h∂t , ∂u , ∂x , 4t∂t + x∂x + (x + u)∂u i : 22

F = u3xx G(eux uxx , e2ux uxxx ), 1 A24.2 = h∂x , ∂u , ∂t , t∂t + x∂x + (t + u)∂u i : 4 F =

4 u3 u3xxx ln |ux | + G( xx , ), 3 u2x ux

4 q A34.2 = h∂t , ∂u , − ln |x|∂u , qt∂t + x∂x + u∂u i, q 6= 0 : q 4 1 ux F = −6 3 + x4( q −1) G(ω1 , ω2 ), x 4

4

ω1 = x1− q (xuxx + ux ), ω2 = x1− q (x2 uxxx − 2ux ), 1 A44.2 = hx4(1−q) ∂u , ∂u , ∂t , t∂t + x∂x + (t + u)∂u i, q 6= 0, 1 : 4 F =− ω1 =

(4q − 3)(4q − 2)(4q − 1) ux + 4 ln |x| + G(ω1 , ω2 ), x3

xuxx − (3 − 4q)ux x2 uxxx − (3 − 4q)(2 − 4q)ux , ω = . 2 x3 x3

A4.3 − invariant equations A14.3 = h∂u , ∂x , ∂t , t∂x + u∂u i : F = −ux ln |ux | + ux G(

uxx uxxx , ), ux ux

1 A24.3 = h∂t , ∂u , −4 ln |x|∂u , t∂t + x∂x i : 4 6 1 F = − 3 ux + 4 G(x2 uxx + xux , x3 uxxx − 2xux ), x x A34.3 = h∂u , ex ∂u , ∂t , ∂x + (tex + u)∂u i : F = ux + xex + ex G(e−x (uxx − ux ), e−x (uxxx − uxx )). A4.4 − invariant equations 1 A14.4 = h∂u , −4 ln |x|∂u , ∂t , t∂t + x∂x + (u − 4t ln |x|)∂u i : 4 F = −8 ln2 |x| − 6

ux xuxx + ux x2 uxxx − 2ux + G( , ). x3 x3 x3

A4.5 − invariant equations, 1 1 A14.5 = h∂t , ∂x , ∂u , t∂t + x∂x + pu∂u i, −1 6 p < , p 6= 0 : 4 4 23

4

1−p

2−4p

3−4p

F = ux 1−4p G(ux4p−1 uxx , ux4p−1 uxxx ), 1 1 A24.5 = h∂t , ∂x , ∂u , t∂t + x∂x + u∂u i : 4 4 uxxx F = u3xx G(ux , 2 ), uxx 1 A34.5 = h∂t , ∂u , x4(q−p) ∂u , t∂t + x∂x + qu∂u i, −1 6 p < q 6 1, pq 6= 0 : 4 F =

(4q − 4p − 1)(4q − 4p − 2)(4q − 4p − 3) ux + x4(q−1) G(ω1 , ω2 ), 3 x

ω1 = x1−4q [xuxx − (4q − 4p − 1)ux ], ω2 = x1−4q [x2 uxxx − (4q − 4p − 2)(4q − 4p − 1)ux ]. A4.7 − invariant equations 1 A14.7 = h∂u , ∂x , − ln |t|∂x + x∂u , 4t∂t + x∂x + 2u∂u i : 4 F =

1 u2x 1 + 1 G(uxx , |t| 4 uxxx ), 8t |t| 2

A24.7 = h∂u , α∂x + x∂u , −∂x , −

α2 1 ∂t + (1 − α)x∂x + (2u − x2 )∂u i : 0 α 2

2 1 F = − α0 u2x + α−4 e− α G(ω1 , ω2 ), 2 1

ω1 = α2 uxx − α, ω2 = e α α3 uxxx , where α = α(t) with α0 6= 0 satisfies α2 α00 + 2(α − 2)α02 = 0. 1 1 A34.7 = h∂u , (λx4 − t)∂u , ∂t , t∂t + x∂x + (2u + λtx4 − t2 )∂u i, λ 6= 0 : 4 2 24λ − 1 xuxx − 3ux x2 uxxx − 6ux 4 u + x G( , ), x 4λx3 x7 x7 1 = h∂u , (²t − x)∂u , ∂x , 4t∂t + (x + 3²t)∂x + (²xt + 2u − x2 )∂u i : 2 1 1 1 1 F = −²ux + t + 1 G(uxx + ln |t|, |t| 4 uxxx ), 6 4 |t| 2 F = 4λx4 ln |x| +

A44.7

1 A54.7 = h∂u , −x∂u , ∂x , 4t∂t + x∂x + (2u − x2 )∂u i : 2 1 1 1 F = 1 G(uxx + ln |t|, |t| 4 uxxx ), 4 |t| 2 24

1 u 1 A64.7 = h− ∂u , ∂u , ∂x − ∂u , 4t∂t + x∂x + (u + x)∂u i : x x 2 4 1 F = − uxxx + 1 G(ω1 , ω2 ), x x|t| 2 ω1 = xuxx + 2ux −

1 1 ln |t|, ω2 = |t| 4 (xuxxx + 3uxx ). 4

A4.8 − invariant equations 1 3 A14.8 = h∂x , ∂t , t∂x + ∂u , t∂t + x∂x − u∂u i : 4 4 7 uxx uxxx F = −uux + |ux | 4 G( 5 , 3 ), |ux | 4 |ux | 2 1 5 A24.8 = h∂u , ∂t , λ∂x + t∂u , t∂t + x∂x + u∂u i, λ > 0 : 4 4 1 1 uxx uxxx F = x + |ux | 4 G( 3 , 1 ), λ |ux | 4 |ux | 2 A34.8 = h∂u , ∂x , λt F =

1−q 4

∂x + x∂u , 4t∂t + x∂x + (1 + q)u∂u i, λ 6= 0, |q| 6 1 :

q−3 1−q 2−q λ(q − 1) − q+3 2 t 4 ux + t 4 G(t 4 uxx , t 4 uxxx ), 8

A44.8 = h∂u , ∂x , λ∂t + x∂u , 4t∂t + x∂x + 5u∂u i, λ 6= 0 : 1

F = |λux − t| 4 G(

A54.8

uxx

F =

,

uxxx

1 ), |λux − t| |λux − t| 2 q = h∂u , (λx4 − t)∂u , ∂t , qt∂t + x∂x + (1 + q)u∂u i, λq 6= 0, |q| 6 1 : 4 3 4

4 24λ − 1 −3− 4q −3− 4q q G(x (xu − 3u ), x (x2 uxxx − 6ux )), u + x xx x x 4λx3

A64.8 = h∂u , (λx4 − t)∂u , ∂t , u∂u i, λ 6= 0 : F =

24λ − 1 x2 uxxx − 6ux u + (xu − 3u )G(x, ), x xx x 4λx3 xuxx − 3ux

A74.8 = h∂u , −x∂u , ∂x , 4qt∂t + qx∂x + (1 + q)u∂u i, |q| 6 1, q 6= 0 : F =t

1−3q 4q

G(t

q−1 4q

uxx , t

2q−1 4q

uxxx ),

A84.8 = h∂u , −x∂u , ∂x , u∂u i : F = uxx G(t,

uxxx ), uxx 25

A94.8 = h∂u , (²t − x)∂u , ∂x , 4qt∂t + q(x + 3²t)∂x + (1 + q)u∂u i, |q| 6 1, q 6= 0 : F = −²ux + t

1−3q 4q

G(t

q−1 4q

uxx , t

2q−1 4q

uxxx ),

A10 4.8 = h∂u , (²t − x)∂u , ∂x , u∂u i : F = −²ux + uxx G(t,

uxxx ), uxx

1 u A11 4.8 = h− ∂u , ∂u , ∂x − ∂u , 4qt∂t + qx∂x + u∂u i, |q| 6 1, q 6= 0 : x x 1−3q

A12 4.8

2q−1 q−1 4 t 4q F = − uxxx + G(t 4q (xuxx + 2ux ), t 4q (xuxxx + 3uxx )), x x 1 u = h− ∂u , ∂u , ∂x − ∂u , u∂u i : x x 4 2 xuxxx + 3uxx F = − uxxx + (uxx + ux )G(t, ). x x xuxx + 2ux

A4.9 − invariant equations A14.9 = h∂u , ∂x , α∂x + x∂u , −

1 + α2 1 ∂t + (q − α)x∂x + (2qu − x2 )∂u i, q > 0 : 0 α 2

1 0 2 e2q arctan α F = − α ux + G(ω1 , ω2 ), 2 (1 + α2 )2 3

ω1 = (1 + α2 )uxx − α, ω2 = (1 + α2 ) 2 e−q arctan α uxxx , where α = α(t) with α0 6= 0 satisfies (1 + α2 )α00 + 2(α − 2q)α02 = 0. A4.10 − invariant equations A14.10 = h∂u , − tan x∂u , ∂t + u∂u , λ∂t + ∂x + u tan x∂u i, λ ∈ R : F = 8 tan xux + 4 tan xuxxx + et−λx sec xG(ω1 , ω2 ), ω1 = eλx−t (cos xuxx − 2 sin xux ), ω2 = eλx−t (cos xuxxx − 3 sin xuxx − 2 cos xux ). 6. Galilei-invariant equations

One of the important requirements in the classical mechanics is the requirement for the motion equation to satisfy the Galilei relativity principle. From the mathematical standpoint this requirement means that the motion equation has to admit Galilei group. Our approach enables to handle the problem of mathematical description of all possible PDEs of the form (1) that admit Galilei transformation 26

group or, equivalently, the Galilei algebra of first-order differential operators (vector fields). The Lie algebra g(1) = hT, P, Gi is called the classical Galilei algebra if its basis operators satisfy the following commutation relations: [T, P ] = 0, [P, G] = 0.

[T, G] = −P,

(8) (9)

The Lie algebra g˜(1) = hT, P, G, M i is called the extended classical Galilei algebra if the commutation relations (9) hold and what is more the basis operator M obeys the following commutation relations: Let an operator M satisfy the following commutation relations: [M, T ] = [M, P ] = [M, G] = 0,

[G, P ] = M.

Below we give without derivation expressions for the most general PDEs of the form (1) which are invariant under the Galilei algebras. The list of inequivalent Eqs.(1) invariant under the classical Galilei algebra is exhausted by the following five classes of PDEs: 1. ut = −uxxxx + uux + f (ux , uxx , uxxx ), T = ∂t , P = ∂x , G = −t∂x + ∂u ; 2. ut = −uxxxx − ux + f (x, uxx , uxxx ), T = ∂t , P = ∂u , G = (x − t)∂u ; 3. ut = −uxxxx + f (t + ux , uxx , uxxx ), T = ∂x , P = ∂u , G = ∂t − x∂u ; 1 4. ut = −uxxxx − u2x + f (t, uxx , uxxx ), 2 T = ∂x , P = ∂u , G = −t∂x − x∂u ; 5. ut = −uxxxx + f (t, uxx , uxxx ), T = ∂x , P = ∂u , G = −x∂u . Here f is an arbitrary real-valued smooth function. There are three inequivalent equations of the form (1) invariant under the extended classical Galilei algebra g˜(1): 1. ut = −uxxxx − u2x + f (uxx , uxxx ), 1 1 T = ∂t , P = −∂x , M = ∂u , G = t∂x + x∂u ; 2 2 1 uxxx 1 2. ut = −uxxxx + x − 4 + f (xuxx + 2ux , xuxxx + 3uxx ), 2λ x x u λ T = ∂t , X = −∂u , M = − ∂u , G = λ∂x + (t − λ )∂u i λ 6= 0; x x 27

1 2 x + f (uxx , uxxx ); 4λ2 1 1 1 P = x∂u , M = ∂u , G = λ∂x − xt∂u , 2λ 2 2λ

3. ut = −uxxxx − T = ∂t ,

λ 6= 0.

where f is an arbitrary smooth function. Note that the corresponding symmetry algebras are maximal invariance algebras provided f is arbitrary. 7. Concluding remarks

In this paper, we perform symmetry classification of the fourth-order parabolic equation of the very general form (1). As a result, the broad classes of invariant equations (1) are constructed together with their maximal symmetry algebras. Symmetry properties of these equations can be briefly summarized as follows, • there are three inequivalent equations admitting one-dimensional Lie algebra. • two equations which admit semi-simple Lie algebras isomorphic to sl(2, R). • there are nine equations admitting two-dimensional Lie algebras, among them, four equations admitting Abelian algebras and five admitting non-Abelian algebras. • there are forty-four equations admitting three-dimensional solvable Lie algebras. • there are fifty-seven equations admitting four-dimensional solvable Lie algebras. We have also established that the class of evolution equations (1) contains at most eight inequivalent classes of PDEs obeying the Galilei relativity principle. The corresponding equations are presented in the previous section. Finally we mention that in order to complete group classification of the general class of PDEs one needs to consider solvable Lie algebras of the dimension higher than 4. This work is in progress now and will be reported elsewhere.

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