Classification of local and nonlocal symmetries of fourth-order nonlinear evolution equations

REPORTS ON MATHEMATICAL PHYSICS

Vol. 65 (2010)

No.3

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES OF FOURTH-ORDER NONLINEAR EVOLUTION EQUATIONS* QING HUANG1,2, C. Z. QU1,2 and R. ZHDANOV 3 1Department of Mathematics, Northwest University, Xi'an 710069,

People's Republic of China 2Center for Nonlinear Studies, Northwest University, Xi'an 710069, People's Republic of China 3BIO-key International, 55121 Eagan, MN, USA (e-mail: [email protected]) (Received June 17, 2009 -

Revised February 23, 2010)

In this paper, we consider the group classification of local and quasi-local symmetries for a general fourth-order evolution equations in one spatial variable. Following the approach developed in [26), we construct all inequivalent evolution equations belonging to the class under study which admit either semi-simple Lie groups or solvable Lie groups. The obtained lists of invariant equations (up to a local change of variables) contain both the well-known equations and a variety of new ones possessing rich symmetry. Based on the results on the group classification for local symmetries, the group classification for quasi-local symmetries of the equations is also given. Keywords: evolution equation, quasi-local symmetry, nonlocal symmetry, Lie group, Lie algebra.

1. Introduction In the paper [26] we suggested an efficient approach to the group classification of partial differential equations containing arbitrary functions of several arguments. It has been successfully applied to classify Lie symmetries of heat conductivity [4], Schrodinger [28], KdV-type evolution [3, 11], nonlinear wave [15], general secondorder quasi-linear evolution [27], third-order nonlinear evolution [3] and fourth-order evolution [12] equations. Recently, we extended the approach in question in order to classify nonlocal symmetries of partial differential equations [23]. In the present paper we utilize the ideas of [4, 23] to classify local and nonlocal symmetries of the nonlinear fourth-order evolution equations of the form (1)

Here F and G are arbitrary smooth functions and F I- O. Hereafter we adopt the notation u = u(t, x), u, = aulat, U x = aulax, U xx = a2ulox 2 , U xxx = a3ulax 3 • "Supported by NSF-China grants 10671156 and 10771170. [337)

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Q. HUANG, C. Z. QU and R. ZHDANOV

The class of partial differential equations (1) contains a number of important mathematical physics equations. The Kuramoto-Sivashinsky equation

u, =

-U xxxx

-

U xx

1 2 - "2ux'

(2)

the extended Fisher-Kolmogorov equation Ut

=

-U xxxx

+ U xx -

U

3

+ u,

(3)

and the Swift-Hohenberg equation u, =

-U xxxx

- 2u xx - u 3 + (K

-

1)u,

K E

R

(4)

are obtained from Eq. (1) by specifying appropriately the arbitrary functions F and G. In addition, the class (1) includes the equation describing thin film flows u, = -(u 3u xxx + j(u, u x, uxx))x.

The latter is used for modelling fluid flows in physical situations such as coating, draining of foams, and movement of contact lenses [18]. One more important particular case of Eq. (1) reads as Ut

= -(f(u)uxxx)x + (g(u)ux)x.

(5)

This equation describes the motion of thin viscous films (for further details see [5] and the references therein). Provided j(u) = u", g(u) == 0, Eq. (5) turns into the generalized lubrication equation. Depending on the choice of the parameter n, the general lubrication equation describes [7]: • capillary-driven flow for n = 3, • slip models in the vicinity of u ---+ 0 if n < 3, • the thickness of a thin bridge between two masses of fluid in a Hele-Shaw cell if n = 1. In [12], we perform group classification of the particular case of the class of equations (1), namely, u, =

-U xxxx

+ G(t, x, u, u x, U xx , u xxx).

(6)

Note that the above equation is obtained from Eq. (1) by putting F = -1. The lists of invariant equations obtained in the present paper contain invariant equations of the form (6) as particular cases. Lie symmetries provide a researcher in the field of mathematical physics with powerful and versatile tools for the analysis of nonlinear differential equations. However, the scope of applicability of these tools is determined by how broad is the admitted symmetry group (if any!). That is why one of the primary questions of the theory of Lie symmetries of partial differential equations is whether a given equation admits nontrivial symmetry. Saying it another way, the group classification of Eqs. (1) is a necessary first step in utilizing the Lie and non-Lie symmetry methods and techniques. In the present paper, we perform preliminary group classification of

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

339

Eq. (1) and describe all possible forms of the functions F and G such that Eq. (1) admits symmetry groups of dimension n :s 4. The history of the group classification of differential equations goes back to Sophus Lie. In [16] he proved that a linear two-dimensional second-order partial differential equation (POE) can admit at most three-parameter invariance group (apart from the trivial infinite-parameter symmetry group , which is due to linearity). The modern formulation of the problem of group classification of POEs has been sugge sted by Ovsyannikov [20]. He introduced the regular method based on the concept of equivalence group (we will refer to it as the Lie-Ovsyannikov method). Ovsyannikov's approach works at its best in the case when the equivalence group of an equation under study is finite-dimensional and is not very efficient otherwise. As a result, partial differential equations with arbitrary elements depending on two and more arguments cannot be efficiently handled with the Lie-Ovsyannikov method. To overcome this difficulty, Zhdanov and Lahno developed a different approach enabling to classify classes of POEs that admit infinite-dimensional equivalence groups [26]. Here we follow the approach of [26] in order to describe fourth-order nonlinear evolution equations (1) having nontrivial symmetry properties. Our symmetry classification algorithm is a combination of the standard Lie infinitesimal algorithm, equivalence group techniques and the theory of abstract Lie algebras. It consists of three major steps [26]. 1. We compute the most general symmetry group of (1). As a by-product the classifying equations for the unknown functions F and G are obtained. In addition, we calculate the maximal local equivalence group admitted by Eq. (1) under consideration. 2. The second step is essentially based on the explicit forms of commutation relations of low-dimensional abstract Lie algebras [2, 21, 22]. Using these we construct all inequivalent realizations of symmetry algebras by basis infinitesimal operators admitted by Eq. (1). 3. At the third step we insert canonical forms of symmetry generators into the classifying equations. Solving the latter yields invariant equations. Finally we make sure that the so obtained symmetry algebras are maximal in Lie's sense. Since Lie symmetry is not always the best answer to all challenges of the modem theory of nonlinear differential equations, one has alway s been looking for ways to generalize it. One of the possible generalizations is allowing for infinitesimals to depend on integrals of the dependent variables , which is just the way the nonlocal symmetries arose. At present, nonlocal symmetries of linear POEs are well understood (see e.g. [9, 10]). However, much less is known about nonlocal symmetries of nonlinear differential equations. One of the possible approaches to constructing nonlocal symmetries has been suggested by Bluman [6, 8]. He derived nonlocal symmetries (called potential symmetries) admitted by a given differential equation by computing local symmetries of an associated auxiliary system .

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There is an alternative approach to constructing nonlocal symmetries of a differential equation admitting nontrivial Lie symmetries, which is performing a nonlocal transformation of the dependent and independent variables in the equation in question. As a result, some of the Lie symmetries of the initial equation remain Lie symmetries of the transformed equation, while the remaining ones become nonlocal symmetries. The so constructed symmetries are called quasi-local. The term 'quasi-local symmetry' has been introduced independently in [1] and [17] in order to distinguish nonlocal symmetries that are equivalent to local ones through a nonlocal transformation. It has been noted in [4] that the results of group classification for local symmetries can be utilized to derive quasi-local symmetries of PDEs under study. Zhdanov and Lahno obtained some nontrivial examples of second-order evolution equations with quasi-local symmetries in [25]. Recently, Zhdanov suggested a regular grouptheoretical approach to the problem of classification of evolution equations that admit quasi-local symmetries [23]. The principal motivation for writing the present paper is a need for a unified group classification approach enabling to obtain both local and quasi-local symmetries of equations of the form (1). This approach is essentially based on the methods developed in the papers mentioned above [23, 26]. The structure of this paper is as follows. In Section 2 we obtain the classifying equations for the functions F, G and compute the equivalence group of Eq. (1). In the next section we construct equations of the form (1) invariant with respect to semi-simple algebras. Section 4 is devoted to classification of evolution equations (1) admitting solvable Lie algebras of the dimension n .::: 4. In Section 5 we classify equations of the form (1) that are equivalent to PDEs (1) admitting quasi-local symmetries.

2. The preliminary group analysis of Eq, (1) We begin by calculating the most general invariance group admitted by (1). It is generated by the first-order differential operators of the form V = t tt; x, U)Ot

+ ~(t, x, u)ox + ry(t, x, u)ou,

(7)

where r, ~, ry are arbitrary, real-valued smooth functions defined in some subspace of the space V = ]R2 X ]R 1 of the independent (t, x) and dependent (u) variables. The necessary and sufficient condition for the operator (7) to generate a one-parameter invariance group of (1) reads as [14, 19]

+ ~ F; + ryFu + t]x Fux + ryXX Fuxx + t]xxx Fuxxx)uxxxx - ryXXXX F - tt G, + ~ F; + t]Gu + ryxGux + ryxxGuxx + ryxxxGuxxx») Ut--F Uxxxx +G

(ryt _ (r F.

Here r/ = Dt(t]) - utDt(r) - uxDt(~), ryX = Dx(t]) - utDxCr) - uxDx(~),

= O.

(8)

341

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

YJxx = DxCr() - UxtDxCr) - uxxDxC~), YJxxx = DxCYJ xX) - UxxtDxCr) - uxxxDx(~), YJxxxx = DxCYJ xXX) - uxxxtDxCr) - uxxxxDxC~),

and the symbols D, and D, stand for the total derivative operators with respect to the variables t and x, respectively,

= at + Utau + UttaUt + Uxtaux + , D x = ax + uxa u + UtxaUt + uxxaux + . Dt

In order to obtain the system of determining equations for coefficients of the symmetry operator V we need to • replace u, and its differential consequences with Fu xxxx + G and its differential consequences in the left-hand side of Eq. (8), and • split the so obtained relation by the independent variables U X , U xx , ... As a result, we get the over-determined system of linear PDEs for r, it yields the following assertion.

~,

YJ. Solving

PROPOSITION 1. The most general symmetry group of (1) is generated by the infinitesimal operator

V = r(t)at + ~(t, x, u)a x + YJ(t, x, u)a u, where r,

~

(4~uux

and YJ are real-valued functions satisfying the classifying equations:

+ 4~x -

i)F - r Ft - ~ Fx - YJFu + (ux~x - UxYJu

+ u;~u - YJx)Fux (-uxxYJu -YJxx + u;~uu - u;YJuu - 2u xYJxu + 2uxx~x + ux~xx + 2u;~xu + 3uxuxx~u)Fuxx + (-3u;YJx,u,u -YJxxx + u;~uuu - 3u xYJxxu - 3u xxYJxu + 3uxxx~x + 3u;x~u + 3uxx~xx + 3u;~xuu + ux~xxx + 6u;uxx~uu - u;YJuuu + 9uxuxx~xu - UxxxYJu + 4uxuxxx~u + 3u;~xxu - 3u xuxxYJuu) Fuxxx = 0, + ux~xxxx + u~~uuuu + 4u;~xuuu - 6u;YJxxuu + 6u;~xxuu - 4u xxxYJxu - 6uxxYJxxu - 4u xYJxxxu + 12u;x~xu - 4u;YJxuuu - 3u;xYJuu + 4u;~xxxu + 4uxx~xxx + 6uxxx~xx - YJxxxx - 6u;uxxYJuuu 16uxuxxx~xu - 4u xuxxxYJuu + lOu;uxxx~uu + 15uxu;x~uu + lOu;uxx~uuu + lOuxxuxxx~u + 24u;uxx~xuu + 18uxuxx~xxu)F + (YJu - i - ux~u)G - t G, - ~Gx - YJG u + (ux~x - UxYJu + u;~u - YJx)G ux + (-uxxYJu - YJxx + u;~uu - u;YJuu - 2uxYJxu + 2uxx~x + ux~xx + 2u;~xu + 3uxuxx~u)Guxx

(-12u xuxxYJxuu - u;YJuuuu

(9)

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Q. HUANG, C. Z. QU and R. ZHDANOV

(-3u;1Jxuu - llxxx + U;~uuu - 3U x1Jxxu - 3U xx1Jxu

+ 3uxxx~x + 3u;x~u + 3uxx~xx + 3u~~xuu + Ux~xxx + 6u;uxx~uu - U~1Juuu + 9uxuxx~xu - u xxx1Ju + 4uxuxxx~u + 3u;~xxu - 3UxUxx1Juu)Guxxx - Ux~t + 1Jt = O.

(10)

Hereafter the dot over a symbol stands for differentiation with respect to its argument.

The next thing to do is calculating the equivalence group of Eq. (1). To this end we have to construct all possible invertible changes of variables

t = T(t, x, u),

i = X(t, x, u),

u=

D(T, X, V)

----#0,

Vet, x, u),

Dtt . x, u)

which do not alter the form of Eq. (1). After a simple algebra we arrive at the following assertion. PROPOSITION

2 ([3]). The maximal equivalence group of Eq. (I) reads as

t=

T(t),

i = X(t, x, u),

U = Vet, x, u).

Here T, X, V are arbitrary sufficiently smooth functions and

r #0

(11) and ~rx:~;

# O.

Now we are ready to perform the second step of our algorithm. Namely, we are going to classify inequivalent realizations of Lie algebras of low dimension by operators (7) within equivalence transformation (11). We begin by classifying the realizations of one-dimensional Lie algebras. Then we proceed to classifying twodimensional algebras making use of the already known realizations of one-dimensional algebras. Next, we process the three-dimensional Lie algebras and so on. Performing such a classification we use the well known description of nonisomorphic low dimensional abstract Lie algebras (see [22] and the references therein). Inequivalent realizations of one-dimensional Lie algebras spanned by operators (9) are described by the following assertion. LEMMA 1. Within the point transformation (11), the vector field (9) is equivalent to one of the canonical operators

(12) Proof: Performing the change of variables (11) in (9) yields an operator of the form v = Ttor + (rX t + ~ Xx + 1JXu)ox + (TVt + ~Vx + T/Vu)Oj;. (13)

There are two inequivalent cases T # 0 and r = 0 which are to be considered separately. If r # 0, then choosing in (11) the function T satisfying t T = 1 and the functions X and V satisfying the equations rYt

yields the operator

at.

+ ~Yx + T/Yu = 0,

Y

=

yet, x, u)

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

343

Suppo se now that r = O. If this is the case, then ~2 + r;2 -10 (since otherwi se the operator (9) vanishes identically). With this constraint , (13) reads as

V=

(~Xx

+ r;Xu)ai + (~Ux + r;Uu)aii.

If ~ -I 0, then choosing in (11) a particular solution of ~ X, + r;X II = 1 as the function X and the fundamental solution of ~Ux + r;UII = 0 as U we transform (9) into ax. If ~ = 0, r; -10, then we make transformation (11) with t = t, X = u, U = x and get the case ~ -I 0, which has already been considered. It is straightforward to verify that operators at and ax are inequivalent. The assertion is proved. D Thus there exist only two inequivalent realizations of one-dimensional symmetry algebras, namely, (at) and (ax). Integrating the classifying equations for each symmetry operator, at and ax, yields the corresponding inequivalent equations from the class of POEs (1). In the sequel we adopt the notation A~ = (VI, V2 , • • • , Vk ) for a k-dimensional Lie algebra with basis elements Vj (j = 1,2, .. . , k), the index i standing for the number of the class to which the given algebra belong s. THEOREM 1. There are two inequivalent equations (1) invariant under the one-parameter symmetry groups:

Al = (at):

u,

AT = (ax):

u,

= F(x , u, ux , uxx, uxxx)uxxxx + G(x, u , ux , Uxx, Uxxx), = F(t , u , u x , Uxx , Uxxx)Uxxxx + Gtt , u , ux , Uxx , Uxxx).

Here F and G are arbitrary smooth functions. Furthermore, the associated one-dimensional invariance algebras are maximal in Lie's sense. 3.

Classification of equations invariant under semi-simple Lie algebras

In this section we construct all inequivalent equations (1) whose invariance algebras contain a semi-simple Lie algebra. It is a common knowledge that real semi-simple Lie algebras of the lowest dimension are isomorphic to one of the following two three-dimensional algebras:

so(3): sl(2 , lR):

= V3, [VI , V3] = -V2, [V2 , V3] = VI; [VI , V2] = 2V2, [VI, V3] = -2V3, [V2, V3] = VI. [Vj, V2]

We consider in some detail classification the operators (9). In the case of the algebra Taking into account the results of our -dimensional algebras we can chose one of the canonical forms at and ax.

of inequivalent realizations of so(3) by sl(2, lR) we give the final results only. classification of realizations of onethe basis operators, say Vi , in one of

344

Q. HUANG, C. Z. QU and R. ZHDANOV

Let VI = at and V2, V3 be of the form (9), Vi = t'i(t)at+~i(t, x, U)a x+77i(t, x, u)au , i = 2,3. It follows from the commutation relation [VI, V2] = V3 that i 2 = t'3. Next, taking into account that [VI, V3] = - V2, yields i 3 = -'[2. Hence, using the relation [V2, V3] = VI we obtain '[2 i 3 - t'3 i 2 = 1, whence + i 22 = -1. The obtained equation has no real solutions. Consequently, there are no realizations of the algebra so(3) with VI = at. Tum now to the case VI = ax. An analysis similar to the one above yields the unique realization of so(3)

ri

(ax, tanu sin r d,

+ cosxau , tanu cos xd, -

sinxau>.

The further check shows that the coefficients cannot satisfy the determining equations (8). Consequently, an equation of the form (1) cannot admit the obtained algebra. We summarize the above reasonings in the form of the assertion. THEOREM

2. There are no so(3)-invariant equations of the form (1).

A similar analysis of the admissible realizations of the algebra sl(2, 1R.) yields the following result. THEOREM 3. There are at most six inequivalent realizations of sl(2,1R.) by operators (9) which can be admitted by Eq. (1). These realizations and corresponding forms of the functions F, G are presented below: sll(2,1R.) = (2tat + xa x, -t 2at - txo; + x 2au , at>:

G

WI = xUx

-

=

G(Wl, W2, (3) X

W2 = x 2uxx - 2u,

2u,

2

xUU x - u 2

+ X2 W3 = x 3uxxx;

sz2(2, lR) = (2tat + xa x , -t 2at + (x 3 - tx)ax, at>: F(u, WI, (2)

F =

X

3 5

'

»:

Ux

X 6U5 x

W,=

xUxx + 3ux xu x2

sl\2,1R.) = (2xax - uau , -x 2ax + xuau , ax>: F(t, WI, F =

U

8

(2) '

4x'

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

G

= -8

2uxxxu z - 9uxuxxu + 9u; 11 F(t, u

F=

F(t, u, W) Ux4

-

WI, WZ)

+ uG(t, WI, WZ),

UxxxU Z - 9uxuxxu + 12u~

to: =

9

;

U

u xx z G = 6(uxx - uxuxxx)F(t, u, w) 6

,

345

Ux

+ uG(t, u, w),

_ 3u;x - 2uxu xxx .

W-

u4

F=

(U

,

Ux4

6

_

Z ZF(t,WI,WZ),

+ 4u x )

U Z 4 [146880u~ + 192u(641u5 - 167uxx)u~ - 896u2UxxxU~ 4(u + 4ux) + 96uZ(417u lO - 208uxxu5 + 28u;x)u; + 32u3uxxAu 5 + 20uxx)u; Z U5 - 160u3 )U Z + 6U3(855u15 - 1092uxxUIO - 176uxx xx x 9u 5 l\77u I O 5 - 162uxxu + 36u;x)]F(t, WI, WZ) + 32u xxA2u + 5uxx)ux + 3u

G= -

6

(U 6

+ WI

+ 4u Z )21 u

u

= (U 6

2 x

3

3

+ 4U~)2

_

G(t,

WI, WZ),

6

Z

(U - 2uuxx + lOuX),

3

Wz

=

U Z 3 [U xxxU 8 - 9uxuxxu7 + 12u;u 6 + 4uAu xu xxx - 3u;x)u Z (U + 4ux ) 6

+ 36u;u xxu u4 F=

6

60u~];

_

ZZF(t,WI,W2),

(U - 4ux)

5 6 U Z 4 [146880u~ - 192u(641u + 167uxx)u~ - 896uZuxxxU~ 4(u - 4ux) + 96uZ(417u lO + 208uxxu5 + 28u;x)u; - 32u3UXXX(U5 - 20uxx)u;

G= -

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Q. HUANG, C. Z. QU and R. ZHDANOV

2 3 5 - 6u 3(855u 15 + 1092uxxu 10 - 176uZ xxu + 160uxx)u x + 32u9u xxxC2u 5 - 5uxx)ux + 3u I4(77uI0 + 162uxxu 5 + 36u;J]F(t, WI, WZ)

+

1

(4u 2

- U6)2 _ x 2 G(t,

u

u

WI =

3 3

(u

u6 ) 2

(4ui -

6

WI, (2),

+ 2uu xx -

3

W2 =

U 2 3 [u xxxu 8 (u - 4u x) 6

-

2

lOux)'

9u xuxxu 7 + 12u;u 6 + 4uxC3u;x - UxUxxx)u2

- 36u;u xxu + 60u;J. The algebras sli (2, JR) (i = I, 2, ... ,6) are maximal invariance algebras of the corresponding PDEs provided the functions F and G are arbitrary. THEOREM 4. The invariant equations listed in Theorem 3 exhaust the list of all possible inequivalent PDEs (1) whose invariance algebras contain semi-simple subalgebras.

Proof: It is a common knowledge that semi-simple Lie algebras of the lowest dimension admit the following isomorphisms: so(3)

rv

su(2)

rv

sp(l),

sl(2, JR) "" su(l, 1)

rv

so(2, 1) "" sp(1, JR)

(see e.g. [2]). Hence it immediately follows that the realizations of the algebra sl(2, JR) exhaust the set of all possible inequivalent realizations of the three-dimensional semisimple Lie algebras admitted by (1). The next admissible dimension for classical semi-simple Lie algebras is six. There are four nonisomorphic semi-simple Lie algebras over the field of real numbers, so(4), 50*(4), so(3, 1), and so(2,2). As the relations so(4) rv so(3) $so(3), so*(4) rv so(3) E9 s/(2, JR) hold and the algebra so(3, 1) contains so(3), there are no realizations of these algebras by operator (9). Therefore the algebra so(2, 2) is the only candidate for a six-dimensional semi-simple symmetry algebra of (1). In view of so(2,2) rv s/(2, JR) E9 s/(2, JR) we can choose so(2,2) = (Qi, K, Ii = 1,2,3), where (Ql, Q2, Q3) and (K I , Kz, K 3) are all s/(2, JR) algebras with [Qi, K j ] = 0 (I, j = 1,2,3). Without any loss of generality we can choose QI, Q2, Q3 in the form of basis operators of sl(2, JR) listed in the formulation of Theorem 3. Taking the basis operators KI, K 2 , K 3 in the general form (9), after a simple algebra we establish that realizations of s/(2, JR) cannot be extended to a realization of so(2, 2). Consequently, no equation of the form (1) is invariant under a six-dimensional semi-simple Lie algebra. A similar reasoning yields that there are no realizations of eight-dimensional semi-simple Lie algebras s/(3, JR), su(3) and su(2, 1).

347

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

The same assertion holds true for the algebras An -I (n > 1), Dn (n > I), B; en (n 2: 1) and the exceptional semi-simple Lie algebras Gz, F4 , E6, E7 , Eg. The theorem is proved. D

(n 2: l ),

4. Classification of equations invariant under 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) iteratively. Namely, we start by constructing realizations of one-dimensional algebras. Next, we describe realizations of two-dimensional ones utilizing the fact that any twodimensional solvable algebra contains a one-dimensional algebra. Then we proceed to three-dimensional algebras an so on (for more details, see [4]). In this section we construct inequivalent equations of the form (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 nonisomorphic two-dimensional Lie algebras,

As both algebras Az.l and Az.z contain the subalgebra A I , we can assume that the basis operator of the latter is reduced to the canonical form. We present full calculation details for the case of the algebra Az.1 • The algebra A z.2 is handled in a similar way. Let VI = al and Vz be an operator of the most general form (9),

Vz = r(t)al

+ ~(t, x, u)a x + l1(t, x, u)a

ll •

Then the commutation relation of Az.1 implies that i = ~I = 111 = 0. Therefore r is a constant and ~ = ~(x, u), 11 = TJ(x, u). Consequently, without any loss of generality we can choose Vz = ~(x, u)a x + TJ(x, u)a ll • In order to simplify Vz with equivalence transformations (11) we need to select these transformations which do not alter the form of the basis operator VI ' Making general equivalence transformation yields

= tai + XI ax + ura u = ai . = UI = 0, so that T = t, X = X (x , u) VI

-4

VI

Hence, T = 1 and XI Transforming accordingly the operator Vz, we get

Vz -4

Vz =

(~Xx

and U

=

U(x , u).

+ I1 XII)a.I + ou, + I1UII )au'

If TJ = 0, then we choose U = U (u) and ~ satisfying ~Xx = I thus getting Provided 11 i- 0, we can take a solution of ~ Xx + T/X II = 1 as X and choose a solution of the equation ~ U, + I1UII = as U, thus getting the operator a,I

Vz = ax.

°

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Q. HUANG, C. Z. QU and R. ZHDANOV

again. Consequently, in the case under consideration there is only one inequivalent realization of the algebra A 2.1, namely (all ax). Consider now the case when VI = ax and V2 is an operator of the form (9). Inserting VI and V2 into the corresponding commutation relation yields V2 = r(t)at+~(t, u)a x+1)(t, u)a u ' The equivalence transformation, which leaves VI invariant, reads as x=x+X(t,u), u = Vet, u) with U;

1= 0.

This transformation reduces V2 to the form

= rTai + (rX t + ~ + 1)Xu)a; + (rVt + 1)Vu)au. The cases r = and r i= need to be handled separately. and 1) = 0, then V2 = ~ a;. Provided ~u i= 0, we can choose V = If r = V2

°ua;. ° ° ° ° 1= ° = + =°

~ which yields If ~u = and ~t = 0, then the relation V2 = a; holds. Next, given ~u = and ~t 0, we select T = ~ thus getting V2 = ta;. It is straightforward to verify that the obtained algebra (ax, tax) cannot be admitted by an equation of the form (1). Finally, if r = and 1) i= 0, then we choose V and X to satisfy PDEs 1)Vu 1 and ~ 1)X u and arrive at the canonical form au. Turn now to the remaining case r 1= 0. Choosing solutions of system of PDEs ci = 1, rXt+~+1)Xu =0, rVt+1)Vu =0 as T, X and V, respectively, we obtain the operator ai. So that we arrive at the already known realization (ax, at). This completes analysis of the realizations of the algebra A2.1. The case of A2.2 is treated similarly. Inserting the basis operators obtained above into the classifying equations and integrating the latter yields the corresponding invariant equations of the form (1). THEOREM 5. There exist three commutative and four noncommutative two-dimensional solvable Lie algebras admitted by (1). These algebras and the corresponding invariant equations are given below:

Algebra

F

G

Ai.1(at, ax)

F(u, u x, U xx , uxxx)

G(u, u x, U xx , u xxx)

A~.I (ax, au)

F(t, u x, Uxx, uxxx)

G(t, uX , u,«. u xxx)

A~.I (au, xa u)

F(t, x, U xx , u xxx) x 3 F(u, xUx, x 2uxx, x 3u xxx) t 3 F(u, tux, t 2uxx, t 3u xxx) u 4F(t, u x, uU xx, u 2U xxx) Ux-4F( i.u,u;-2 Uxx'U x-3 Uxxx)

G(t, x, uxx, u xxx) x -IG(u, xUx, x 2u xx, x 3 u xxx)

Ai.2(-tat - xa x, at) A~.2{-tOt - xa x, ax) A~.2(-xax - uau, ax)

Ai.2(-xox, ax)

t-IG(u, tux, t 2uxx, t 3u xxx) uG(t, ux, uU xx, u 2uxxx) G( t, u, U-2 u xx, U-3 u xxx) x x

4.2. Equations admitting three-dimensional solvable Lie algebras We split the set of three-dimensional solvable Lie algebras into the subsets of decomposable and nondecomposable algebras. The first subset consists of the

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

349

algebras which are direct sums of lower-dimensional algebras, while the second one includes the remaining three-dimensional solvable algebras. We consider the decomposable and nondecomposable Lie algebras separately. 4.2.1. Three-dimensional decomposable algebras

There exist two nonisomorphic three-dimensional decomposable Lie algebras, = Al EEl Al EB Al and A 3.2 = A 3.2 = A 2.2 EEl AI. with the commutation relations

A 3.1

i , j = 1,2,3,

and It is a common knowledge that any three-dimensional solvable Lie algebra contains a two-dimensional solvable algebra as a subalgebra. So 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 5 by vector fields V3 of the form (9). Then for each of the obtained realizations we simplify V3 utilizing equivalence transformations which preserve the operators VI and V2. After fulfilling these two steps we obtain the complete list of inequivalent equations (l) invariant under the three-dimensional solvable Lie algebras, A 3.1 and A 3.2 . A3.I-invariant equations:

AL =

(at, ax, au ) : F = F(u x , U xx , u xxx),

A ~. I = (at, s; x au) :

=

F

A ~.I

F (x, uxx, U xxx),

= (au, xa u, f(t, x)au), F = F (t, x,

W ),

f xx:l= 0 : I xxxx fxx

G = ---uxxF(t, x, w)

Ixxx fxx A3.2-invariant equations: W

=

U xxx

-

- - U xX'

Ai.2 = (-tat - xa x , ac. 8u ) F

=x

WI

=

3

:

F(WI , Wz, ( 3) ,

WJ =

XU x ,

A~.2 = (-tat - u8u, 8c. xu8u) : F =

u -IexO"\

F(x , W 1,

"~...~ )

,

X

3

uxxx ,

+ G(t, x, w) + -Itu xx, Ixx

350

Q. HUANG, C. Z. QU and R. ZHDANOV

= exer [ [(6a~a2 - 4aIa3 - 3a{)F(x, WI, (2) + G(x, WI, (2)), -1 -1 -1 al = U u x, a2 = U Uxx, a3 = U Uxxx, G

WI

A~.2

=

= a2 -

a~,

= a3 + 2a?

W2

-

3aI a2,

(-tat - xa x, ax, tuax) :

F = tu-;4F(u, WI, (2),

G

=

WI

= t -1 Ux-3 Uxx,

-5t-Iu-;6uxxC2uxuxxx - 3u;x)F(u, W2

WI,

= t -I Ux-s(UxU xxx -

+ uxG(u, WI, (2) -

W2)

3u 2

)

xx'

Aj.2 = (-tat - xax, ax, au) : F

= t 3 F(WI, W2, (3),

= t- I G(w[, W2, (3),

G

A~.2 = (-tat - xax, ax, tar) : F = t- Iu-;4F(u, WI, (2),

A~.2 = (-tat - xax, ax, tat

+ au) :

F = t 3 e- 4u F(WI, W2, (3),

G = rIG(WI, W2, (3), -2u W2=t e u xx, W3 = t 3e-3u u xxx, 2

A~.2 = (-xa x - ua u, ax, ua x) : F

= u4 Ux -4 F (t , WI, W2),

G = -5u4u-;6uxxC2uxuxxx - 3u;JF(t, WI, (2) WI

=

uU-3 U x

xx,

W2

=

U2U-s( uxu

A~.2 = (-xa x - ua u, ax, at) : F = u 4F(WI, W2, (3), G

A~.2

x

+ uuxG(t, WI, W2),

2 ) xxx - 3u xx'

= uG(WI, W2, (3),

=

(-xa x - ua u, ax, tuau) : F = u 4u-;4 F(t, WI, (2),

Aj?2 = (-xa x, ax, at) : F = u-;4F (t , WI, (2),

G = uG(t, WI, (2)

+ t- Iu In lux I,

r'»,

351

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

Aj.IZ = (-xa x, ax, au) : F = u;4F(u, WI, Wz),

4.2.2. . Three-dimensional nondecomposable algebras

The list of inequivalent three-dimensional nondecomposable Lie algebras contains seven algebras, A 3.3:

[Vz, V3] = VI;

A 3. 4

:

[VI, V3]

A 3.5

:

[VI, V3] = VI,

[Vz, V3] = Vz;

A 3. 6

:

[VI, V3] = VI,

[Vz, V3] = - Vz;

A 3.7

:

[VI, V3 ]

=

Vj ,

= VI,

[Vz, V3 ]

[Vz, V3 ]

A3.8:

[VI, V3] = - Vz,

A 3.9 :

[VI, V3 ] = qVI - V2,

=

VI

+ Vz;

= qVZ,

0<

Iql

< 1;

[V2, V3] = VI; [V2, V3] = VI

+ qV2,

q > O.

Note that we give the nonzero commutation relations only. All of the above algebras contain a two-dimensional Abelian ideal as a subalgebra. Consequently, we can use results of classification of AZ.1-invariant equations to construct equations of the form (1) which admit nondecomposable three-dimensional solvable Lie algebras. After some algebra we obtain the following list of inequivalent invariant equations. AD-invariant equations: At3 = (au,

all ax + tau) :

F = F(u x, U xx , u xxx),

A~.3 = (au, all (t + x)a u) : F = F(x, U xx , u xxx),

G = u;

+ G(x, U xx , U xxx ),

AL = (au, ax, tax + xa u) : Aj.3 = (au, ax, at + xa u) : F = Ftu ; - t, U xx , U xxx ),

G = G(u x - t , U xx , U xxx ),

A~.3 = (au, xa u, -ax) : F = F(t, U xx , U xxx ),

A~.3 = (au, xa u, at - ax) : F = F(x + t, U xx , U xxx ),

G = G(t, U xx , U xxx ), G = G(x

+ t, U xx , U xxx ),

352

Q. HUANG, C. Z. QU and R. ZHDANOV

2a A~.3 = (xa u, au, x x + xua u} : 6(3u + 2xu )F(t, WI, (2) G = 4x F = x 8F (t, wI, (2), xx xxx 4(3u. 3u wI = x xx, W2 = x u + xUxxx), A~.3 = (xa u, au, at + x 2ax + xuau} : F = x 8 F(WI, W2, (3), G = 4x 6(3u xx + 2xu xxx)F(WI, W2, (3) - xG(WI, W2, (3), 3U WI = t + x-I, W2 = x xx, W3 = x\3u xx + xUxxx). A3A-invariant equations: AjA = (au, at. tat + ax + (t + u)a u) : G =x F = e- x F(WI, W2, (3),

A~.4 = (au, at, tat

+ G(WI, W2, (3),

+ (t + u)a u) :

= u;I F(x, WI, (2), G = In IUxl + G(x, WI, (2), -1 -1 WI = U x u xx, W2 = U x u xxx, F

A~A = (au, ax, at + xa x + (x F = e4t F(WI, Wz, (3), WI

=

Ux -

+ u)a u) : G

t,

At4 = (au, ax, xax + (x + u)a u) : G F = e4ux F(t, WI, (2),

AL = (au, xa u, -ax F = F(t,

= etG(WI, W2, (3),

WI,

= eUxG(t, WI,

(2),

+ uau) :

Wz),

G

=

«
A~.4 = (au, xa u, at - ax + ua u) : t = F(WI, Wz, (3), G = e G(WI, W2, (3), WI = X + t , W2 = «i«: AjA = (xa u, au, x 2ax + (1 + x)ua u} :

F

F = x 8 F(t,

WI,

Wz),

+ xG (t, wI, (2),

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

3 1

4 1

w2 = X ex (3u xx + xu xxx),

WI = X exu xx,

AL =

2a (xa u, au, at + x x + (l F = x 8F(WI, W2, (3),

+ x)uau) :

G = 4x 6(3u xx + 2xu xxx)F(WI' W2, (3) - xe' G(WI, W2,

1 WI =t+-, X

(3),

) W2=X 3 e -t u xx, W3=X 4 e -t (3uxx+xuxxx'

A3.s-invariant equations:

A~.s = (at, au, tat

+ ax + uau) : x G F = e- F(Wl, W2, (3),

= G(WI, W2, (3), W3 = e -x U xxx ,

A~.s = (ax, au, xax + ua u) : F = u;; F(t, u«. u;;u xxx),

Ats = (ax, au, at + xax + uau ) : F = e4t F(WI' W2, (3), G = etG(WI, W2, (3), W3 = e2tuxxx, A~.s = (au, xa u, ua u) : G = uxxG(t, x, u;;u xxx),

F = F(t, x, u;;U xxx) ,

A~.5 = (ax, uax, at + xax) : F

=

G = etG(x, WI, (2),

F(x, WI, W2), -t

-t

WI = e u xx, W2 = e u xxx'

A3.6-invariant equations:

At6 = (at, au, tat - ua u) : F = uxF(x, U;l uxx, u;I Uxxx),

A~.6 = (at, au, tat

+ ax - uau ) x F = e- F(Wl, W2, (3),

G = Ux2G( X, Ux-I u xx, Ux-I u xxx) ,

:

G

= e- 2xG(WI, W2, (3),

353

354

Q. HUANG, C. Z. QU and R. ZHDANOV

A~.6 = (ax, au, xax - uOu) : I

G=

F = u;Z F(t, W], Wz),

u; G(t,

WI,

Wz),

to: = Ux-Z u xxx, Aj.6 = (ax, au, at + Ox - UOu) : F

= e41 F(w], Wz, (3),

G

= e-tG(wI, Wz, (3),

3t

Wz = e Uxx ,

W3

= e41u xxx,

A~.6 = (Ox, au, xOx - uOu) : I

F = x 4F (t , WI, Wz),

G = x'2 G(t,

WI,

Wz),

3

WI

= x'2uxx ,

A~.6 = (au, xa u, at + 2xox + uau) : F = e81 F(Wl, Wz, (V3),

G = etG(w], Wz, (3),

»: = e

3t

U xx ,

W3

= e51u xxx'

A3.7-invariant equations:

I_I

F = u; A~.7

=

(Of, au, tat

+ ax + quau)

F = e- x F(WI, Wz, (3), Wz =

G = e(q-l)xG(wI, Wz, (3), -qx -qx U e U xx , W3 = e xxx, -!L

WI,

wz),

q-I

Ux

G

= urI G(t, WI, wz),

~ q-I Wz = u; Uxxx,

?::::!I

=

G( x, U-I u U-I u ) , x xx, x xxx

:

4

F = u;-I F(t, WI

q

Uxx,

+ xOx + quau) : 4t F = e F(WI, wz, (3), G = eqtG(wI, wz, (3), r, e(l-q)t u x, WZ -e(Z-q)t u xx, ,.\ - e(3-q)lu xxx, LUI LU3 --

Aj.7 = (ax, au, Ot

A~.7 = {Ox, uOx, xox F

+ (l -

= x 4F(t, WI, wz),

q)uou) : I

G

= X l-q G(t, WI, wz),

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES 3q-Z

2q-1

WI

= X q-I u xx,

to: = X q-I u xxx,

A~.7 = (ax, uax, at + xa x + (l - q)uau) : F

= e4(l-q)t F(w], Wz, (3),

G = et G(WI, Wz, (3),

", - e(l-Zq)t u xx,

E.' - e(Z-3q)t u xxx'

U/Z -

U/3 -

A3.8-invariant equations:

Aj,8 = (ax, au, uax - xa u) : F

=

(l

+ u;)-z F(t, WI, wz),

G = -5u xuxx(l

+ u;)-\2u;u xxx -

3u xu;x

+ 2u xxx)F(t, WI, wz)

Z I + (1 + uxPG(t, WI, Wz),

= (l + u;)-i uxx, Wz = (1 + u;)~3(u;uxxx A~.8 = (ax, au, at + uo; - xa u) : WI

F = (l G= 5

A~.8

3u xu;x

+ U xxx ),

+ u;)-z F(WI, Wz, (3), U xx

Z 4[

6u Z arctan Z U + (2u Z Z x xx xu xxx + 12tu xx

+ ux ) Zu;x - 6u xu;x + 2u xxx) arctan u; + 6t - 6tu xu;x + 2tu;u xxx + 3u;u;x + 2tu xxx - 2u;u xxx - 2u xuxxx] Z I + F(WI, wz, (3) + (l + ux) 2: G(WI, Wz, (3), Z _3 WI = arctan a, + t, io: = (l + u x) 2:u xx, W3 = (1 + u;)-3(u;u xxx - 3uxu;x + u xxx + 3u;xWI), = (au, xa u, _(xz + l)ax - XUd u) : F = (l + x Z) 4 F(t, WI, Wz), G = 4x(1 + xZ)Z(2xZuxxx + 3xu xx + 2u xxx)F(t, WI, wz) I + (l +x 2 )'2G(t, WI, wz), z J. z:! Z WI = (1 +x )2u xx, W2 = (l +x )z(x U xxx + 3xu xx + u xxx), (l

Aj.8 = (au, Xd u, at - (x z + l)ax - xuau) : F = (l +x z)4F(WI,WZ,W3), G = 4x(l

+ xZ)Zuxx(2x2uxxx + 3xu xx + 2uxxx)]F(WI, Wz, (3) 2 I

+(l+x )2G(WI,WZ,W3),

355

356

Q. HUANG, C. Z. QU and R. ZHDANOV

WI

= t + arctan x,

W3 = (l

W2

5 + X2 )'lu xxx -

3 = (l + X2 )'lu xx,

3W2(WI - X).

A3.9-invariant equations: At9 = (ax, au, (qx

+ u)a x + (qu

- x)a u) :

+ u;)-2 e-4qarctanu x F(t, WI, (2), G = -5u xu xx (l + u;)-4e-4qarctanuX(2u;uxxx 1 + (l + u;Pe-qarctanuxG(t, WI, (2), F = (l

WI

3

= (l + u;)-zu xx,

W2

3u xu;x

= (l + u;)-3(u;U xxx -

+ 2u xxx)F(t, WI, (2)

3u xu;x

+ u xxx),

Aj.9 = (ax, au, at + (qx + u)a x + (qu - x)a u) : F = (l + u;)-2 e4qtF(WI, W2, (3), G = -5u xx (l

WI = W3 -_

+ u;)-4e4qt(2u~uxxx -

3u;u;x

+ 2u xuxxx + 3u;x)F(WI, W2, (3)

2 1 qt + (l + ux)"Ze G(WI, W2, (3), 3 t + arctanu., W2 = (l + u;)-"Zeqtu xx, (1 + U2)-3 e2q t (uxu 2 2 +u ) xxx - 3uxu xx xxx, x

A~.9 = (au, xa u, _(x 2 + l)ax + (q - x)uau) : F = (x 2 + 1)4F(t, WI, (2), G

= 4x(x 2 + 1)2[2(1 + x 2)u xxx + 3xu xxJF(t, wI, (2)

1 + (x 2 + 1)"Ze-qarctanxG(t, WI, (2), 3 WI = (x 2 + l)zeqarctanxuxx, ) W2 = (1 + U x2)J.2eqarctanx( X2Uxxx + 3XU xx + Uxxx, Aj.9 = (ax, uax, at + (q - u)xax - (u 2 + l)au) :

F = (u 2 + 1)4u;4 F(WI, W2, (3), G = [-5(u 2 + 1)4u;6uxA2uxuxxx - 3u;x) + 8u(u 2 + 1)3u;4uxx(uxuxxx - 3u;x) + 12(u 2 - l)(u 2 + 1)2u;2 uxxJF(wI' W2, (3)

+ (u 2 + l)!eqtuxG(WI, W2, (3), 3 WI = t + arctanu, W2 = (u 2 + Ipe-qtu;3uxx, _ 2 5 -qt -5 2 2 J. -qt -3 W3 - (u + 1)"Ze Ux (uxu xxx - 3u xx) + 3u(u + 1)2e U x u xx'

357

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

4.3. Equations invariant under four-dimensional solvable Lie algebras

Turn now to the problem of group classification of Eq. (1) admitting fourdimensional solvable Lie algebras. To this end we utilize the well-known classification of abstract four-dimensional Lie algebras as well as the results obtained in the previous sections. Skipping the cumbersome calculation details we give the final results, symmetry algebras and the corresponding invariant equations. Note that the classification results for the decomposable and nondecomposable algebras are presented separately. 4.3.1. Equations with four-dimensional decomposable algebras

The list of nonisomorphic four-dimensional decomposable Lie algebras contains the following ten algebras:

A2.2 EB A 2.2 = 2A 2.2, A 3.1 EB Al = 4A I , A 3.2 EB Al = A2.2 EB 2A I , A3.i EB AI, i = 3,4, ... ,9. The exhaustive list of equations (1) invariant under the above algebras is too long to fit into the present paper. It can be found in Appendix A of our online publication [13]. 4.3.2. Equations with four-dimensional nondecomposable algebras

There exist ten nonisomorphic four-dimensional nondecomposable Lie algebras, A4.i (i = 1,2, ... ,10): A 4. 1 :

[X 2 , X 4 ] = Xl,

[X3, X4] = X 2;

A 4.2 :

[Xl, X 4] = qX I ,

A 4.3

:

[Xl, X4] = Xl,

A 4.4 :

[Xl, X4] = Xl,

= X 2, [X 3, X 4 ] = X2; [X2, X4] = X I + X 2,

A 4.5 :

[Xl, X4] = Xl,

[~,~]=q~,

-l~p~q~l,

[X 2, X 4 ]

pq

q A 4.7 :

1= 0,

[X 2, X 3]

[X3, X 4]

= X 2 + X3;

~3,~]=P~,

1= 0; [X 2, X 4 ]

[Xl, X4] = qXI,

q

= p X2 -

X3,

p ~ 0;

= Xl,

[Xl, X 4]

= 2X I ,

[X2, X4] = X2,

[X 3, X4] = X 2 + X 3; A4.8 :

[X2, X 3] = Xl, [X 3, X4] = q X3,

[Xl, X 4 ]

= (1 + q)X I,

Iqj ~ 1;

[X2, X4] = X 2,

1= 0;

358

Q. HUANG, C. Z. QU and R. ZHDANOV

= x., [Xl, X 4 ] = 2qX I , [X 3 , X 4 ] = x, + qX 3 , q ;?: 0;

[Xl, X 3 ]

[Xl, X 3] = Xl,

[Xl, X3]

=

Xl,

Each of the above algebras can be decomposed into a semi-direct sum of a three-dimensional ideal N and a one-dimensional Lie algebra. Analysis of the commutation relations above shows that • N is of the type A 3. 1 for the algebras A 4 .i (i = 1,2, ... ,6), • N of the type A 3.3 for the algebras A 4. 7 , A 4. 8 , A4.9, and • N of the type A 3.5 for the algebra A 4 .1O . Consequently, we can use the already known realizations of three-dimensional solvable Lie algebras to obtain exhaustive descriptions of the four-dimensional nondecomposable solvable Lie algebras admitted by Eq. (1). The full list of inequivalent symmetry algebras and the corresponding invariant equations is given in Appendix B of the online publication [13].

5. Classification of equations admitting quasi-local symmetries Our method for the classification of nonlocal symmetries of fourth-order evolution equations (1) is based on the approach suggested recently in [23]. The approach in question makes use of Lie symmetries and nonlocal transformation techniques to derive nonlinear evolution equations admitting non-Lie symmetries. We give a brief description of the approach of [23] applied to the class of nonlinear evolution equations under study. As established in Section 2, the most general infinitesimal operator, V, admitted by the evolution equation (1) reads as V = r(t)ot

+ Ht, x, u)ox + r](t, x, u)ou.

And what is more, the maximal equivalence group of Eq. (1) takes the form

t= where

i

T(t),

x=

X(t, x, u),

u=

Utt ; x, u),

-I- O. r 0 and D(X,U) D(x,u) r Provided r = 0, the operator V takes the form V = ~(t, x, u)ox + r](t, x, u)ou' Consequently, there exists an equivalence transformation (t, x, u) -+ (t, x, u) that reduces V to the canonical form Ou (we drop the bars). So that Eq. (1) transforms to an equation of the form -I-

(14) Note that the right-hand side of Eq. (14) does not depend explicitly on u. Differentiating (14) with respect to x yields U tx

= Fuxxxxx

+ [(Fx + Fuxuxx + Fuxxuxxx + Fuxxxuxxxx)uxxxx + G, + Guxu xx + Guxxuxxx + Guxxxuxxxx].

359

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

Making the nonlocal change of variables t = t,

(15)

x =x,

and dropping the bars we finally get u,

=

Fu xxxx + [(Fx + r;«,

+ Fuxuxx + Fuxxuxxx)uxxx + G x + Guu x + Guxu xx + Guxxuxxx],

(16)

where F = Ftt , x, u, u x, u xx) and G = G(t, x, u, u x, u xx). Consequently, the nonlocal transformation (15) preserves the differential structure of the class of evolution equations (14). Assume now that Eq. (14) admits an r-parameter Lie transformation group ,

-+

= T(t, e),

t

,

-+

,

x = X(t, x, u, e),

-+

u = U(t,x,u,e)

(17)

e

with the vector of group parameters = (el , ... , er ) , r ::: 2. To obtain from (17) the symmetry group of the initial equation (16) we need to transform (17) according to (15). Computing the first prolongation of formulae (17) yields the transformation rule for the first derivative of u, ,

ux ' =

Uuu x + Ux

x,», + x, .

Consequently, the transformation group (17) takes the form ,

-+

t = T(t, e),

,

-+

(18)

x =X(t,x,v,e),

e).

with v = a-lu and U = U(t, x, v, Hence it follows that Eq. (16) admits the transformation group (18). If one of the right-hand sides of (18) depends explicitly on the nonlocal variable v, then (18) is a nonlocal (quasi-local) symmetry of Eq. (16). Evidently Eq. (16) admits a quasi-local symmetry if and only if transformation (18) satisfies one of the relations x) -a (UvU + U :f:0

or

Jv

Xvu

+ Xx

.

The latter can be rewritten to become or

Xv =0,

Expressing the above constraints via coefficients of the corresponding infinitesimal operator of the group (17) yields the following assertion. THEOREM 6 ([23]). The evolution equation (14) can be transformed to an equation admitting a quasi-local symmetry if it is invariant with respect to a Lie transformation group whose infinitesimal generator satisfies one of the following inequalities:

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Q. HUANG, C. Z. QU and R. ZHDANOV

(19) ~u

(20)

= 0,

Making the hodograph transformation

t = t,

x

= u,

u =x,

and dropping the bars we represent the evolution equation (21)

in the form (14). Hence we obtain that the following assertion holds. 1. Eq. (21) can be reduced to an equation with a quasi-local symmetry if Eq. (21) admits a Lie transformation group whose infinitesimal operator satisfies one of the inequalities COROLLARY

n, i= 0, n, = 0,

~;u

+

c. i=

0.

Summarizing the above reasonings we formulate the procedure for constructing evolution equations of the form (1) admitting quasi-local symmetries. 1. Select all invariant equations whose invariance algebras contain at least one operator of the form V = ~(t, x, u)ax + 'I(t, x, u)ou' 2. For each of these equations make a suitable local equivalence transformation reducing V to the canonical form au, the original equations being transformed to the evolution equations of the form (14). 3. For each Lie symmetry of the invariance algebra admitted by (14) check whether its infinitesimal generator satisfies one of conditions (19), (20) of Theorem 6. This analysis yields the list of evolution equations (1) that can be reduced to those admitting quasi-local symmetries. 4. Performing the nonlocal change of variables (15) transforms Eq. (14) to (16) which admits quasi-local symmetries (18). We process in this way all the invariant equations obtained in Sections 3 and 4 and thus obtain the list of fourth-order evolution equations (1) admitting quasi-local symmetries. Here we give the symmetry algebras of the corresponding invariant equations omitting the expressions for the functions F and G for brevity. Since we keep the same notation for the Lie algebra realizations as those used in the previous sections, it is straightforward to derive the explicit forms of invariant equations, given the form of its Lie symmetry algebra. Semi-simple Lie algebras: sz3(2, JR.) = (2xa x - uOu, -x 2ax + xuo u, ax),

sl4(2, JR.)

5

sl (2, JR.)

=

(2xox, -x2ox, ox),

=

(2xo x - uOu,

( u1

4

-

x

2) Ox + xuou, ox),

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

6

sl (2, JR) = (2xax - uau, -

(2x + u14) ax + xua u, aJ.

Three-dimensional solvable Lie algebras:

A~.3 = (xau, au, x 2ax + xua u), A~.3 = (xau, au, at + x 2ax + xua u), A~.4 = (xau, au, x 2ax + (l + x)uau), A~.4 = (xau, au, at + x 2ax + (l + x)uau), Aj.8 = (ax, au, uax - xa u), A~.8 = (ax, au, at + uax - xa u), A~.8 = (au, xa u, _(x 2 + l)a x - xua u), Aj.8 = (au, xa u, at - (x 2 + 1)ax - xuau),

+ u)ax + (qu - x)a u), (ax, au, at + (qx + u)ax + (qu - x)a u), (au, xa u, _(x 2 + l)a x + (q - x)ua u), (ax, uax, at + (q - u)xax - (u2 + l)a u).

Aj.9 = (ax, au, (qx A~.9 = A~.9 =

Aj.9 =

Four-dimensional solvable Lie algebras:

2Ai.2 = (-tat - uau, at, ax, ueXau), 2A~.2 = (-tat - xa x, at, au, eUau),

2A~.2

= (-tat - uau, at, ax, exax +aueXau), a i= 0, 2Ai~2 = (-tat - xa x, ax, au, eUau), 2Ai~2 = (-tat - uau, au, ax, aexa x + teXau), a i= 0, 2Ai 32 = (-tat - xa x, ax, au, teUa x), 2A~:2 = (-xa x, ax, au, teUau),

Aj.3 EB Al

= (ax, au, at + uax, at + atax + aa u), a

A~.3 EB Al = (ax, uax, -au, at), A~.3 EB Al = (xau, au, x 2ax + xua u, at),

Aj.4 EB Al = (ax, au, (x + u)ax + uau, at), AL EB Al = (uax, ax, x(l + u)ax + u2au, ue-~ ax),

E

R

361

362

Q. HUANG, C. Z. QU and R. ZHDANOV

+ u)ox + U20u, tue-~ ox), (uox, ox, x(l + u)ox + u20u, Ot),

A;.4 EB Al = (uox, ox, x(l A;.4 EB Al =

A~.4 EB Al = (uox, ox, Ot + x(l + u)ox + u20u, ue' I(t + ~ )ox), A1.8 EB Al

J" i= 0,

=

(ax, au, uax - xa u, at), A~.8 EB Al = (au, xa u, _(x 2 + l)ax - xuo u, Ot), 3

2

+ l)ax -

xuau, (x

3

2

+ l)ox -

xuo u, t(x

A 3.8 EB Al = (au, xa u, -(x A3.8 EB Al = (ou, xa u, -(x 4

A 3.8 EB Al = (ax, uax, at - xuax - (u A1.9 EB Al =

2

2

+ l)21 au), 2

+ 1)21 au),

+ l)ou, (1 + u2 )21 r« + arctanu)ax ) ,

I + I" i= 0, {ax, au, (qx + u)ox + (qu -

x)ou, at),

A~.9 EB Al = {au, xOu, _(x 2 + l)ax + (q - x)uau, at!, A~.9 EB Al = (ou, xa u, _(x 2 + l)a x + (q - x)uau, (x 2 + l)!e-qarctanxau), A~.9 EB Al = (au, xa u, _(x 2 + l)ax + (q - x)uou, t(x 2 + l)! e- qarctanxau) , At9 EB Al = {ax, uOx, at + (q - u)xax - (u 2 + l)ou, (u 2 + l)!eqtax!, At9 EB Al = {ax, uOx, at + (q - u)xox - (u 2 + l)ou, I(t +arctanu)(u 2 + l)!eqtax), A~.I =

I + I" i= 0, {xO u, au, Ot, x20x + (t + xu)au),

AJ.2 = (Ot, Ou, ax, qtat + xa x + (x + u)a u), A~.2 = {at, xa u, Ou, qtat + x 2ax + (x + l)UOu!, AJ.3 = (at, au, ax, tat + xa u), A1.3 = (of, xa u, au, tat + x 2ax + xuau), A~.4 = (au, ax, af, tat

AJ.6 =

+ (t + x)ax + (x + U)Ou), (xau, au, af, tat + x 2ax + [t + (x + l)u]au!, (at, ax, Ou, qtOt + (px + u)a x + (pu - x)a,,),

A~.6

(of, au, xa u, qtat - (x 2 + l)ox

A~.4 =

=

A1. 7 = (au, ax, tax + xa u, -at A17 = (au, ax, at + xau, tat

+ (p

- X)Uo,,),

+ xa x + 2uau!,

+ (t + x)ax + C~ + 2u)au),

CLASSIF1CATION OF LOCAL AND NONLOCAL SYMMETRIES

A~.7 = (aU,xaU, -ax,Xax + (u -

X;)aU)'

A1.7 = (au,xau, -ax, at +xax + (u - X;)au), 6

A 47 =

.

1 (xa u, au, x2 ax + xuau, -xax + (u - -2x )au!,

A~.7 = (xau, au, x 2a x + xuau, at A~.7 = (xau, au, at + x

2a

xax + (u -

2~ )au),

x + xuau, tat - xax + (t + u + t:x )au),

+ xau, (1 - q)tat + xax + (1 + q)uau), q 1= 1, A1.s = (au, ax, tax + xau, xax + 2uau!, q 1= 1, A~.s = (au, ax, at + xau, qtat + xax + (1 + q)uau), q 1= 0, A~.s = (au, ax, at + xau, xax + uau), A~.8 = (au, xau, -ax, -xax + tau!, A~.s = (au, xau, -ax, qxo; + (1 + q)uau), q 1= 1, Ats = (au, ax, tax

A~.s = (au, xau, -ax, xax + 2uau), A1~s = (au, xau, -ax, at AJ~

+ qxax + (1 + q)uau),

2a

= (xa u, au, x x + xuau, -qxax + Uau!,

q i= 1,

A1~s = (xa u, au, x 2ax + xuau, -xax + uau), A1~s = (xa u, au, x 2ax + xuau, xax + (tx + u)a u),

A2s = (xa u, au, x 2ax + xuau, at - qxo; + Ua u!, AJ~8 = (ua x, ax, at + xuo, + u2au, qtat + xax - quau),

AJ:8 = AJ.9

q i= 0,

2a

(ua x, ax, at + xuo, + u u, Xa x!,

= (au, ax, tax + xau, -(1 + t2)at + (q - t)xax + (_X; + 2qu )au),

ALo = (ax, au, xax + uau, uax - xau), A~.lO = (ax, au, xax + uau, at A~.lO = (ax, au, at

A1.10 = A~.lO =

+ uax - xau),

+ xax + uau, uax - xau!, (ax, au, at + xax + uau, at + uo, - xau), (au, xau, uau, -(1 + x 2)ax - xuau!,

363

364

Q. HUANG, C. Z. QU and R. ZHDANOV

+ x 2)ax - xuau), (au , x a,,, at + uau, -(1 + x 2)ax - x uau), (au , xa u, at + uau, at - (1 + x 2)ax - xua u)'

A ~.l0 = (au , x au, uau, at - (l A~. 10 =

A~.IO =

Making suitable equivalence transformations we reduce the basis elements V = + 1J(t , x ,u )ou of each of the algebras listed above to the canonical forms ou. Transforming accordingly the correspondin g invariant equations yield evolution equations (14). Next, differentiating the so obtained equations with respect to x and replacing u, with u yields an equation of the form (16) that admits a quasi-local symmetry. As an illustration we give two examples . ~(t,x, u)ox

EXAMPLE 1. Consider the algebra

sl\2, lR) = (2xo x, -x2ox, aJ. The hodograph

transformation

t = t,

u =x ,

x =u,

transforms the original algebra to become

(2uau, -u2ou, ou), (note that we have dropped the bars). The corresponding invariant equation reads as

u,

= F(t , x, w)uxxxx +

3u;x - 4ux uxxu xxx 2

Ux

F (t , x , w) + uxG(t , x, w).

Here F, G are arbitrary smooth functions and to = (2u xu xxx - 3u;x)u; 2. Differentiating the above equation with respect to x and replacing ux with u according to (15) we arrive at the evolution equation

u, = Fuxxxx + CUxxx + (3u;x - 4uxuxxuxxx)u; 2)(Fx + 0' Fw ) - (4u 2(u;x + uxu xxx ) - 13uu;u xx + 6u; )u- 3 F + uxG + uG x + ua Gi; with w = (2uu xx - 3u; )u- 2 and a = 2(u 2uxxx - 4uu xuxx + 3u;)u- 3 • This equation admits the nonlocal transformation group I

t = t, where

e is

x' = x,

I

U

u = ----

(ev+l) 2'

the group parameter and v = a-Iu.

EX AMPLE 2. Consider the Lie algebra

2Ai .2 = (- tOt - xOx, Or. ou, eUau) and the corresponding invariant equation

u, = x 3 F (w), Wz )u xxxx - x 3 (u; - 6u;u xx + 4u xuxxx + 3u;x) F(WI, (2) + uxG (WI , Wz), where W I = x (u; - uxx)u; l, W2 = x 2(u; - 3uxuxx + uxxx)u; ! and F, G are arbitrary smooth functions. Differentiating the above equation with respect to x and replacing U x by u according to (15) yields the evolution equation

CLASSIFICATION OF LOCAL AND NONLOCAL SYMMETRIES

365

u, = X3 Fu xxxx + 3x 2 Fu xxx + [X 3U xxx - X3(U; - 6u;u xx + 4uxuxxx + 3u;x)][Fw ] 0"1

+ FW2 0"2] with

-

x 2[3u 4 + 4xu 3u x - 6(xu xx + 3ux)u 2 + (-12xu ;

+ 12uxx

+ 4xu xxx )U + 9u; + lOxuxuxx]F + uxG + UO" I GW[ + U0"2GW2' x(u 2 - ux ) x 2(u 3 - 3uu x + uxx) WI

=

Wz =

u

u

and 0"1

=

u 3 + xu xu2

-

(xu xx + ux)u + xu; U

2

'

x[2u 4 + 2xu xu 3 - 3(xu xx + 2ux)u 2 + (xu xxx + 2uxx)u - XUxU xx] u2 The equation obtained admits the nonlocal symmetry u I U =--x' =x, t' = t , 0"2 =

Here () is a group parameter and v

= a-Iu .

6. Concluding remarks We have performed a preliminary group classification of a rather general class of fourth-order evolution equations (1). In particular, we have constructed all inequivalent PDEs of the form (1) whose invariance algebra contains a semi-simple algebra. Furthermore, an exhaustive description of the evolution equations (1), whose invariance algebra contains a solvable algebra of the dimension n :::: 4, has been obtained. The second part of the paper deals with the so-called quasi-local symmetries of nonlinear evolution equations. We have developed the regular procedure for constructing PDEs (1) admitting non-Lie symmetries. The procedure in question is essentially based on the results of group classification of the class of equations (1) and is purely algebraic by its nature. Our approach enabled to obtain families of three- and four-dimensional Lie algebras leading to nonlinear fourth-order evolution equations admitting nonlocal symmetries. We have given two examples of nonlinear equations of the form (1) which possess nonlocal symmetries. Since these symmetries include integral of the dependent variable they cannot be obtained within the framework of the traditional Lie approach and are new. One more important remark is about the potential symmetries of equations of the form (1). As we proved recently [24], any potential symmetry is a quasi-local one. Therefore, any potential symmetry of an equation of the form (1) can be obtained within the approach suggested in this paper. In one of our future publications we intend to utilize the nonlocal symmetries of nonlinear evolution equations derived in the present paper to construct their explicit solutions. Another interesting application of the obtained results is classification of nonlinear PDEs (1) that can be linearized via nonlocal transformations. These and related problems are under study now and will reported elsewhere.

366

Q. HUANG , C. Z. QU and R. ZHD ANOV

REFERENCES [ I] I. S. Akha tov, R. K. Gazizov and N. H. Ibragimov: Non-local symme tries: A heuristic appro ach, J. Sovie t Math. 55 (199 1), 1401-1 450. [2] A. O. Barut and R. Raczk a: Theory oj Group Representati ons and Applications, World Scientific, Singapore 1986. [3] P. Basarab-Horwath, F. Gun gor and V. Lahno: Symmet ry classi fication of third-order nonlinear evolution equations, arX iv:0802.0 367, 2008. [4] P. Basarab-Horwath, V. Lahno and R. Zhdanov: The structure of Lie algebras and the cla ssification problem for partial differ ential equation, Acta Appl. Math . 69 (200 1), 43-94. [5] A. L. Bertozzi: The mathem atics of moving contact line s in thin liquid films, No tices Am er. Ma th. Soc. 45 (1998), 689-697. [6] G. W. Bluman: Use and construct ion of potential symmetries, Math. Comput. Modelling 18 (199 3), 1-14. [7] P. Constantin, T. F. Dupont, R. E. Goldstein, L. P. Kadanoff, M. J. Shelley and S.-M. Zhou: Droplet breakup in a model of the Hele-Shaw cell, Phys. Rev. E 47 (1993), 4 169-4 181. [8] G. W. Bluman, G. 1. Reid and S. Kumei: New classes of symme tries' for partial differential equati ons, J. Math . Phys. 29 (1988) , 806-81 1. [9] W. I. Fushchych and A. G. Nikitin: Symmetries oj Maxwell's Equatio ns, Reidel, Dordrecht 1987. [10] W. I. Fushchych and A. G. Nikit in: Symmetries oj Equa tions oj Quantu m Mechanics , Allerton Press Inc., New York 1994. [II ] F. Gungor, V. I. Lahno and R. Z. Zhdanov: Symm etry c1sssification of KdV-type nonlin ear evolution equ ations, J. Ma th. Phys. 45 (2004) , 2280--2313. • [12] Q. Huang, V. Lahno, C. Z. Qu and R. Zhdanov: Prelimin ary gro up cla ssification of a class of fourth-order evolution equ ation s, J. Math. Phys. 50 (2009), 023503. [13] Q. Huang, C. Z. Qu, R. Zhd anov: Classification of classical and non-local symmetries of fourth- order nonlin ear evolution equations, LANL arXiv:090 5.2033v l, 2009. [ 14 ] N. H. Ibragimov: Transformation Groups App lied to Mathematica l Physics, Reidel, Dordre cht 1985. [15] V. Lahno and R. Zhd anov: Group classi fication of nonlinear wave equations, J. Math . Phys. 46 (2005), 330 1-3337. [16] S. Lie: Uber die Integration durch bestimmt e Integrale von einer Klasse linear partieller Different ialgleichungen, Arch. Math ., 6 (188 1), 328-368. [17] A. M. Meirrnanov, V. M. Pukhn achov and S. I. Shmarev: Evolut ion equations and Lagrangian coordinates , Walter de Gruyter, Berlin 1997 . [18] T. G. Myers: Th in films with high surface tension, SIAM Rev. 40 (1997), 441-462. [19] P. Ol ver: Applications oj Lie Groups to Differential Equati ons, Springer, New York 1987. [20] L. V. Ovsyann ikov: Group relations of the nonlinear heat equ ation, Dokl. Akad. Nauk SSSR 125 (1959), 492-495 (in Russian). [21] J. Patera , R. T. Sharp, P. Winte rnitz and H. Zassenhaus: Invariants of real low dimension Lie algebr as, J. Math. Phys. 17 (1976), 986-994. [22] P. Turkowski: Low-dimensional real Lie algebras, J. Math. Phys. 29 (1988), 2139-2144. [23] R. Zhd anov : Towards classification of quasi-local symm etries of evolution equations, LANL arXiv:0901.0578v1 , 2009. [24] R. Zhd anov: On relation betw een potential and contact symmetries of evolution equations, J. Math . Phys. 50 (2009), 053522. [25] R. Zhdano v and V. Lahno: Group class ification of the general evolution equation: local and quasilocal symmetries, SIGMA 1 (2005). [26] R. Z. Zhdanov and V. I. Lahno: Gro up classi fication of heat conductivity equations with a nonlin ear source, J. Phys. A: Math . Gen. 32 (1999), 7405-74 18. [27] R. Zhd anov and V. Lahno: Group classification of the general second-orde r evolution equation: semi-s imple invariance groups, J. Phys. A: Math . Theor. 40 (2007), 5083-51 03 . [28] R. Z. Zhd anov and O. Roman : On preliminary symmetry classification of nonlinear Schrodinger equ ation with some applications of Doebn er-Goldin model s, Rep. Math. Phys. 45 (2000 ). 273-29 1.