JOURNAL OF MATHEMATICAL PHYSICS
VOLUME 45, NUMBER 6
JUNE 2004
Symmetry classification of KdV-type nonlinear
evolution equations
F. Güngöra)
Department of Mathematics, Faculty of Science and Letters, Istanbul Technical University,
34469, Istanbul, Turkey
V. I. Lahnob)
Pedagogical Institute, 2 Ostrogradskogo Street, 36003 Poltava, Ukraine
R. Z. Zhdanovc)
Institute of Mathematics, 3 Tereshchenkivska Street, 252004 Kyiv, Ukraine
共Received 11 August 2003; accepted 2 March 2004;
published online 11 May 2004兲
Group classification of a class of third-order nonlinear evolution equations generalizing KdV and mKdV equations is performed. It is shown that there are two
equations admitting simple Lie algebras of dimension three. Next, we prove that
there exist only four equations invariant with respect to Lie algebras having nontrivial Levi factors of dimension four and six. Our analysis shows that there are no
equations invariant under algebras which are semi-direct sums of Levi factor and
radical. Making use of these results we prove that there are three, nine, thirty-eight,
fifty-two inequivalent KdV-type nonlinear evolution equations admitting one-,
two-, three-, and four-dimensional solvable Lie algebras, respectively. Finally, we
perform a complete group classification of the most general linear third-order evolution equation. © 2004 American Institute of Physics.
关DOI: 10.1063/1.1737811兴
I. INTRODUCTION
The purpose of this article is classifying equations of the form
u t ⫽u xxx ⫹F 共 x,t,u,u x ,u xx 兲 ,
共1.1兲
which admit nontrivial Lie 共point兲 symmetries. The standard Korteweg–de Vries 共KdV兲 equation,
u t ⫽u xxx ⫹uu x ,
belongs to the family of evolution equations 共1.1兲. Classification of the KdV equation with variable coefficients 共vcKdV兲,
u t ⫽ f 共 x,t 兲 uu x ⫹g 共 x,t 兲 u xxx ,
f •g⫽0,
共1.2兲
by their symmetries is done in Ref. 1, where it is shown that the vcKdV can admit at most
four-dimensional Lie point symmetry group and those having four-dimensional symmetry group
can be transformed into the ordinary KdV equation by local point transformations. In Ref. 2, Eq.
共1.2兲 is investigated from the point of view of its integrability. It is shown, in particular, that
equations of the form 共1.2兲 with a three-dimensional Lie point symmetry group have a property of
‘‘partially integrability.’’
a兲
Electronic mail: [email protected]
Electronic mail: [email protected]
c兲
Electronic mail: [email protected]
b兲
0022-2488/2004/45(6)/2280/34/$22.00
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© 2004 American Institute of Physics
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
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Our motivation is the same as for classifying vcKdV equations. We start with a rather general
class of nonlinear equations generalizing 共1.2兲 for g x ⫽0. Note that any t dependent coefficient of
u xxx in 共1.1兲 can be normalized by a reparametrization of time. The main advantage of this
classification is that, if we know that the equation under study admits a nontrivial symmetry group,
then it is usually possible to apply the whole spectrum of the methods and algorithms of Lie group
analysis. This enables us to derive exact analytical solutions of equations that, under study, reveal
their integrability properties, find linearizing transformations, etc. Note that the connection between Lie point symmetries and integrability was discussed in Refs. 2, 3.
Recently, a novel generic approach to group classification of low-dimensional partial differential equations 共PDEs兲 has been developed in Ref. 4. The full account of ideas and algorithms
applied can be found in the review paper5 where the approach in question has been applied to
classify the most general second-order quasi-linear heat-conductivity equations admitting nontrivial Lie point symmetries. Here we adopt the same approach which basically consists of three
steps. We first construct the equivalence group, namely, the most general group of point transformations that transform any equation of the form 共1.1兲 to a 共possibly different兲 equation belonging
to the same class. Also, we find the most general element of the symmetry group together with a
determining equation for F. As a second step, we realize low-dimensional Lie algebras by vector
fields of the above form up to equivalence transformations. To this end, we use various results on
the structure of abstract Lie algebras.6 –9 A review of the classification results of nonisomorphic
finite-dimensional Lie algebras can be found in Ref. 5. In the last step, after transforming symmetry generators to canonical forms, we proceed to classifying equations that admit nontrivial
symmetries. We do this by inserting these generators into the symmetry condition and solving for
F.
Let us mention that similar ideas have been used by Winternitz and co-workers for the group
classification of several nonlinear partial differential equations1,10,11 and of discrete dynamical
systems.12–14 Note also that group classification of the nonlinear wave and Schrödinger equations
in the same spirit has been done in Refs. 15, 16.
The paper is organized as follows. In Sec. II we present the determining equations for the
symmetries and the equivalence group. Section III is devoted to the classification of the equations
invariant under low-dimensional symmetry groups. In Sec. IV we perform a classification of linear
equations in the class 共1.1兲. A discussion of results and some conclusions are presented in the final
section.
II. DETERMINING EQUATIONS AND EQUIVALENCE TRANSFORMATIONS
The Lie algebra of the symmetry group of Eq. 共1.1兲 is realized by vector fields of the form
X⫽ ␶ 共 x,t,u 兲 ⳵ t ⫹ ␰ 共 x,t,u 兲 ⳵ x ⫹ ␾ 共 x,t,u 兲 ⳵ u .
共2.1兲
In order to implement the symmetry algorithm we need to calculate the third order prolongation of the field vector field 共2.1兲,17–19
pr(3) X⫽X⫹ ␾ t ⳵ u t ⫹ ␾ x ⳵ u x ⫹ ␾ xx ⳵ u xx ⫹ ␾ xxx ⳵ u xxx ,
共2.2兲
where
␾ t ⫽D t ␾ ⫺u t D t ␶ ⫺u x D t ␰ ,
␾ x ⫽D x ␾ ⫺u t D x ␶ ⫺u x D x ␰ ,
␾ xx ⫽D x ␾ x ⫺u xt D x ␶ ⫺u xx D x ␰ ,
␾ xxx ⫽D x ␾ xx ⫺u xxt D x ␶ ⫺u xxx D x ␰ .
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J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
Here D x and D t denote the total space and time derivatives. In order to find the coefficients of the
vector field we require that the prolonged vector field 共2.2兲 annihilate Eq. 共1.1兲 on its solution
manifold,
pr(3) X 共 ⌬ 兲 兩 ⌬⫽0 ⫽0,
⌬⫽u t ⫺u xxx ⫺F.
共2.3兲
Equating coefficients of linearly independent terms of invariance criterion 共2.3兲 to zero yields an
overdetermined system of linear PDEs 共called determining equations兲. Solving this system we
obtain the following assertion.
Proposition 2.1: The symmetry group of the nonlinear equation (1.1) for an arbitrary (fixed)
function F is generated by the vector field
X⫽ ␶ 共 t 兲 ⳵ t ⫹
冉
冊
␶˙
x⫹ ␳ 共 t 兲 ⳵ x ⫹ ␾ 共 x,t,u 兲 ⳵ u ,
3
共2.4兲
where the functions ␶ (t), ␳ (t) and ␾ (x,t,u) satisfy the determining equation
⫺3 u x ˙␳ ⫺x u x ¨␶ ⫺9 u x u xx ␾ uu ⫺3 u 3x ␾ uuu ⫹3 ␾ t ⫺9 u xx ␾ xu ⫺9 u 2x ␾ xuu ⫺9 u x ␾ xxu ⫺3 ␾ xxx
⫹3 共 ␾ u ⫺ ␶˙ 兲 F⫹ 共 2 u xx ˙␶ ⫺3 u xx ␾ u ⫺3 u 2x ␾ uu ⫺6 u x ␾ xu ⫺3 ␾ xx 兲 F u xx
⫹ 共 u x ˙␶ ⫺3 u x ␾ u ⫺3 ␾ x 兲 F u x ⫺3 ␾ F u ⫺3 ␶ F t ⫺ 共 3 ␳ ⫹x ␶˙ 兲 F x ⫽0.
共2.5兲
Here the dot over a symbol stands for the time derivative.
If there are no restrictions on F, then 共2.5兲 should be satisfied identically, which is possible
only when the symmetry group is a trivial group of identity transformations. Here we shall be
concerned with the identification of all specific forms of F for which nontrivial symmetry groups
occur. The basic idea is to utilize the fact that for an arbitrarily fixed function F all admissible
vector fields form a Lie algebra. This immediately implies the idea of using the classical results on
the classification of low-dimensional Lie algebras obtained mostly in the late 1960s.6 – 8 Saying it
another way, we need to construct a kind of representation theory on low-dimensional Lie algebras
generated by Lie vector fields preserving the manifold 共2.5兲.
Our classification is up to equivalence under a group of locally invertible point transformations,
t̃ ⫽T 共 x,t,u 兲 ,
x̃⫽Y 共 x,t,u 兲 ,
ũ⫽U 共 x,t,u 兲 ,
共2.6兲
that preserve the form of the equation 共1.1兲, but 共possibly兲 change function F into a new one,
namely, we have
ũ˜t ⫽ũ x̃x̃x̃ ⫹F̃ 共 x̃, t̃ ,ũ,ũ x̃ ,ũ x̃x̃ 兲 .
共2.7兲
Inserting 共2.6兲 into 共1.1兲 and requiring that the form of the equation be preserved, we arrive at the
following assertion.
Proposition 2.2: The maximal equivalence group E has the form
t̃ ⫽T 共 t 兲 ,
x̃⫽Ṫ 1/3x⫹Y 共 t 兲 ,
ũ⫽U 共 x,t,u 兲 ,
共2.8兲
where Ṫ⫽0, U u ⫽0.
We note that the Lie infinitesimal technique can also be used to obtain the equivalence group
共2.8兲. It is straightforward to prove that both approaches produce the same results.
We make use of equivalence transformations 共2.8兲 to transform vector field X into a convenient 共canonical兲 form.
Proposition 2.3: Vector field 共2.4兲 is equivalent within a point transformation of the form 共2.6兲
to one of the following vector fields:
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
X⫽ ⳵ t ,
X⫽ ⳵ x ,
X⫽ ⳵ u .
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共2.9兲
Proof: Transformation 共2.8兲 transforms vector field 共2.4兲 into
X→X̃⫽ ␶ 共 t 兲 Ṫ 共 t 兲 ⳵˜t ⫹ 关 31 共 ␶ Ṫ ⫺1 T̈⫹ ␶˙ 兲共 x̃⫺Y 兲 ⫹ ␶ Ẏ ⫹ ␳ Ṫ 1/3兴 ⳵ x̃
⫹ 关 ␶ U t ⫹ 共 31 ␶˙ x⫹ ␳ 兲 U x ⫹ ␾ U u 兴 ⳵ ũ .
共2.10兲
There are two cases to consider.
共I兲 ␾ ⫽0. Choose U⫽U(u) so that we have
X̃⫽ ␶ 共 t 兲 Ṫ 共 t 兲 ⳵˜t ⫹ 关 31 共 ␶ Ṫ ⫺1 T̈⫹ ␶˙ 兲共 x̃⫺Y 兲 ⫹ ␶ Ẏ ⫹ ␳ Ṫ 1/3兴 ⳵ x̃ ⫹ ␾ U u ⳵ ũ .
共2.11兲
Now if ␶ ⫽0, then ␳ ⫽0 共otherwise X would be zero兲, and we choose T(t) to satisfy
Ṫ⫽ ␳ ⫺3 .
In this case X̃ is transformed into ⳵ x̃ .
If ␶ ⫽0, then we choose T and Y to satisfy
Ṫ⫽ ␶ ⫺1 ,
␶ Ẏ ⫹ ␳ Ṫ 1/3⫽0.
With this choice of T and Y vector field X̃ is transformed into ⳵˜t .
共II兲 ␾ ⫽0. If ␶ ⫽ ␳ ⫽0 then we can choose U to satisfy ␾ U u ⫽1 so that we have X̃⫽ ⳵ ũ .
Otherwise, U can be chosen to satisfy
␶ U t ⫹ 共 31 ␶˙ x⫹ ␳ 兲 U x ⫹ ␾ U u ⫽0.
Hence we recover Case I.
Summing up, the vector field 共2.4兲 is equivalent, up to equivalence under E, to one of the three
standard vector fields ⳵ x , ⳵ t , ⳵ u . This completes the proof.
III. GROUP CLASSIFICATION OF LINEAR EQUATIONS
To the best of our knowledge no group classification of the most general linear third-order
PDE appears in the literature. So we devote this section to the group classification of third-order
PDEs:
u t ⫽ f 1 共 x,t 兲 u xxx ⫹ f 2 共 x,t 兲 u xx ⫹ f 3 共 x,t 兲 u x ⫹ f 4 共 x,t 兲 u⫹ f 5 共 x,t 兲 .
共3.1兲
If we perform the local change of variables (x,t,u)→(x̃, t̃ ,ũ) preserving the form of 共3.1兲,
t̃ ⫽t,
x̃⫽F 共 x,t 兲 ,
u⫽V 共 x,t 兲v共 x̃, t̃ 兲 ⫹G 共 x,t 兲 ,
V⫽0, F x ⫽0,
共3.2兲
we obtain
v˜t ⫽ f 1 F 3x v x̃x̃x̃ ⫹ 兵 3 f 1 V ⫺1 关 V x F 2x ⫹VF x F xx 兴 ⫹ f 2 F 2x 其 v x̃x̃ ⫹ 兵 f 1 V ⫺1 关 3V xx F x ⫹3V x F xx ⫹VF xxx 兴
⫹ f 2 V ⫺1 关 2V x F x ⫹V f xx 兴 ⫹ f 3 F x ⫺F t 其 v x̃ ⫹ 兵 f 1 V ⫺1 V xxx ⫹ f 2 V ⫺1 V xx ⫹ f 3 V ⫺1 V x ⫹ f 4 ⫺V ⫺1 V t 其 v
⫹V ⫺1 关 f 1 G xxx ⫹ f 2 G xx ⫹ f 3 G x ⫹ f 4 G⫹ f 5 ⫺G t 兴 .
Now we choose the functions F,V, and G in 共3.2兲 to satisfy constraints,
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J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
f 1 F 3x ⫽1,
G t ⫽ f 1 G xxx ⫹ f 2 G xx ⫹ f 3 G x ⫹ f 4 G⫹ f 5 ,
3 f 1 F 2x V x ⫹ 关 3 f 1 F x F xx ⫹ f 2 F 2x 兴 V⫽0,
and thus normalize f 1 (x,t)→1, and set f 2 (x,t)→0, f 5 (x,t)→0.
Thus 共3.1兲 reduces to the following particular form:
u t ⫽u xxx ⫹A 共 x,t 兲 u x ⫹B 共 x,t 兲 u.
共3.3兲
Here A,B are arbitrary smooth functions of x and t.
The most general equivalence transformation preserving the class of equations 共3.3兲, which is
a subset of 共2.8兲, reads as
t̃ ⫽T 共 t 兲 ,
x̃⫽Ṫ 1/3x⫹Y 共 t 兲 ,
ũ⫽V 共 t 兲 u,
共3.4兲
with Ṫ⫽0, V⫽0.
Performing change of variables 共3.4兲 transforms Eq. 共3.3兲 to become
ũ˜t ⫽ũ x̃x̃x̃ ⫹Ãũ x̃ ⫹B̃ũ,
共3.5兲
where the coefficients Ã,B̃ are expressed in terms of the functions A,B and their derivatives as
follows:
Ã⫽Ṫ ⫺1 共 AṪ 1/3⫺ 31 T̈Ṫ ⫺ 2/3x⫺Ẏ 兲 ,
B̃⫽Ṫ ⫺1 共 B⫹V ⫺1 V̇ 兲 .
共3.6兲
As Eq. 共3.3兲 is linear, it admits trivial infinite-parameter group having the generator
X 共 ␤ 兲 ⫽ ␤ 共 x,t 兲 ⳵ u ,
␤ t ⫽ ␤ xxx ⫹A ␤ x ⫹B ␤ ,
and the one-parameter group generated by the operator u ⳵ u . These symmetries give no nontrivial
information about the solution structure of the equation under study and therefore are neglected in
the sequel.
The nontrivial invariance group of Eq. 共3.3兲 is generated by operators of the form
X⫽ ␶ 共 t 兲 ⳵ t ⫹ 共 31 ˙␶ x⫹ ␳ 共 t 兲兲 ⳵ x ⫹ ␣ 共 t 兲 u ⳵ u ,
共3.7兲
functions ␶ , ␳ , ␣ ,A and B satisfying equations
3 ␣˙ ⫺3B ␶˙ ⫺3 ␶ B t ⫺B x 共 3 ␳ ⫹x ␶˙ 兲 ⫽0,
⫺3 ␳˙ ⫺x ␶¨ ⫺2A ␶˙ ⫺3 ␶ A t ⫺A x 共 3 ␳ ⫹x ␶˙ 兲 ⫽0.
共3.8兲
Provided A⫽A(x,t), B⫽B(x,t) are arbitrary functions, ␶ ⫽ ␳ ⫽0, ␣˙ ⫽0. So in this case Eq.
共3.3兲 admits trivial symmetries only.
Transformation 共3.4兲 leaves operator X 1 ⫽u ⳵ u invariant while transforming operator 共3.5兲 to
become
(3.4)
X → X̃⫽ ␶ Ṫ ⳵˜t ⫹ 关 ␶ 共 31 T̈Ṫ ⫺ 2/3x⫹Ẏ 兲 ⫹Ṫ 1/3共 31 ␶˙ x⫹ ␳ 兲兴 ⳵ x̃ ⫹ 共 ␶ V̇⫹ ␣ V 兲 u ⳵ ũ .
共3.9兲
That is why, if ␶ ⫽0 in 共3.7兲, then putting
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
Ṫ⫽ ␶ ⫺1 ,
Y ⫽⫺
冕␳ ␰ ␶
t
共 兲
2285
⫺ 4/3
共 ␰ 兲d␰,
and taking V as a nonzero solution of the equation
␶ V̇⫹ ␣ V⫽0,
in 共3.4兲 transforms 共3.9兲 to the canonical form of the generator of time displacements
X̃⫽ ⳵˜t .
Next, if ␶ ⫽0, ␳ ⫽0 in 共3.7兲, then putting Ṫ⫽ ␳ ⫺3 in 共3.4兲 yields the operator
X̃⫽ ⳵ x̃ ⫹ ␣ ũ ⳵ ũ .
Finally, if ␶ ⫽ ␳ ⫽0, ␣˙ ⫽0 in 共3.7兲, we put T⫽ ␣ in 共3.4兲 thus getting the operator
X̃⫽ t̃ ũ ⳵ ũ .
Taking into account the above considerations, we see that there are transformations 共3.4兲, that
transform operator 共3.7兲 to one of the following inequivalent forms:
⳵t ,
⳵x ,
⳵ x ⫹ f 共 t 兲 u ⳵ u 共 ḟ ⫽0 兲 ,
tu ⳵ u .
In what follows, we analyze each of the above operators separately.
Operator X 1 ⫽ ⳵ t . The system of determining Eqs. 共3.8兲 for this operator reads as
B t ⫽A t ⫽0,
whence it follows that A⫽A(x), B⫽B(x). Inserting these functions into 共3.8兲 yields
3 ␣˙ ⫺3B ␶˙ ⫺B x 共 3 ␳ ⫹x ␶˙ 兲 ⫽0,
⫺3 ␳˙ ⫺x ␶¨ ⫺2A ␶˙ ⫺A x 共 3 ␳ ⫹x ␶˙ 兲 ⫽0.
Analyzing the above system of ordinary differential equations shows that for the case under
consideration Eq. 共3.3兲 admits an invariance group whose dimension is higher than one if and only
if the following occurs.
共1兲 A⫽mx ⫺2 , B⫽nx ⫺3 , 兩 m 兩 ⫹ 兩 n 兩 ⫽0 with the additional symmetry operator t ⳵ t ⫹ 31 x ⳵ x ;
共2兲 A⫽0, B⫽␧x (␧⫽⫾1) with the additional symmetry operator ⳵ t ⫹␧tu ⳵ u ;
共3兲 A⫽B⬅0 with the additional symmetry operators ⳵ x ,t ⳵ t ⫹ 31 x ⳵ x .
Operator X 2 ⫽ ⳵ x . If Eq. 共3.3兲 is invariant under X 2 , then A⫽A(t), B⫽B(t). What is more,
it follows from 共3.6兲 that there are transformations 共3.4兲, which reduce equation 共3.3兲 to the form
共3.5兲 with Ã⫽B̃⬅0. So we arrive at the already known case.
Operator X 3 ⫽ ⳵ x ⫹ f (t)u ⳵ u ( ḟ ⫽0). If Eq. 共3.3兲 admits operator X 3 , then we have A⫽0, B
⫽ ḟ x. Inserting these expressions into 共3.8兲 yields
␳˙ ⫽0,
␶¨ ⫽0,
␣˙ ⫽ ␳ ḟ ,
3 ␶ f̈ ⫹4 ␶˙ ḟ ⫽0.
共3.10兲
From the first three equations it follows that ␳ ⫽C 1 , ␶ ⫽C 2 t⫹C 3 , ␣ ⫽C 1 f ⫹C 4 ,
C 1 ,C 2 ,C 3 ,C 4 苸R. Hence we conclude that the last equation of system 共3.10兲 takes the form
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J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
TABLE I. Symmetry classification of 3.3.
N
A
B
Symmetry operators
1
A共x兲
0
B共x兲
⳵t
2
mx⫺2, m苸R
4
0
ḟ 共t兲x
nx ⫺3 , n苸R,
兩 m 兩 ⫹ 兩 n 兩 ⫽0
␧x, ␧⫽⫾1
⳵x⫹f 共t兲u⳵u , ḟ⫽0
3
5
0
⫺mt
⳵ t ,t ⳵ t ⫹
a苸R
⳵ x ⫹3mt ⫺1/3u ⳵ u ,
x,
0
x⳵x
⳵ t , ⳵ x ⫹␧tu ⳵ u
⫺4/3
m苸R, m⫽0
6
1
3
t ⳵ t⫹
⳵ t , ⳵ x ,t ⳵ t ⫹
1
3 x⳵x
1
3 共 x⫺2at 兲 ⳵ x
3 共 C 2 t⫹C 3 兲 f̈ ⫹4C 2 ḟ ⫽0.
Analyzing this equation we see that extension of the symmetry algebra of Eq. 共3.3兲 with A
⫽0, B⫽ ḟ x is only possible when
f ⫽3mt ⫺ 1/3, m⫽0;
f ⫽␧t, ␧⫽⫾1.
The second case has already been considered. In the first case the basis of nontrivial invariance
algebra is formed by the operators ⳵ x ⫹3mt ⫺1/3u ⳵ u , t ⳵ t ⫹ 31 x ⳵ x .
Operator X 4 ⫽tu ⳵ u . Inserting the coefficients of this operator into 共3.8兲 leads to the contradiction 3⫽0, whence it follows that the operator X 4 cannot be a symmetry operator of Eq. 共3.3兲.
We summarize the above classification results of in Table I, where we give the forms of the
functions A and B and basis operators of the nontrivial symmetry algebras of the corresponding
equations 共3.3兲.
So the equation u t ⫽u xxx has the highest symmetry within the class of equations 共3.3兲. Its
maximal finite-dimensional symmetry algebra is four-dimensional.
Note that according to Ref. 5 the class of nonlinear equations of the form
u t ⫽F 共 t,x,u,u x 兲 u xx ⫹G 共 t,x,u,u x 兲 ,
F⫽0,
共3.11兲
contains five nonlinear equations admitting five-dimensional symmetry algebras. Furthermore, an
equation admitting six-dimensional symmetry algebra is equivalent to the heat equation. It is the
linear heat conductivity equation u t ⫽u xx that possess the largest symmetry group within the class
of second-order equations 共3.11兲.
This is not the case for the class of third-order PDEs under consideration in the present paper.
We shall see that there are examples of nonlinear equations that admits higher symmetry algebras
than does the linear equation. For instance, the nonlinear Schwarzian KdV equation 共4.14兲 admits
a six-dimensional symmetry algebra.
IV. CLASSIFICATION OF EQUATIONS INVARIANT UNDER SEMI-SIMPLE ALGEBRAS
AND ALGEBRAS HAVING NONTRIVIAL LEVI DECOMPOSITIONS
In order to describe equations 共1.1兲 that admit Lie algebras isomorphic to the Lie algebras
having nontrivial Levi decomposition, we need, first of all, to describe equations whose invariance
algebras are semi-simple.
The lowest order semi-simple Lie algebras are isomorphic to one of the following threedimensional algebras:
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
sl共 2,R兲 : 关 X 1 ,X 3 兴 ⫽⫺2X 2 ,
so共 3 兲 : 关 X 1 ,X 2 兴 ⫽X 3 ,
关 X 1 ,X 2 兴 ⫽X 1 ,
关 X 2 ,X 3 兴 ⫽X 1 ,
2287
关 X 2 ,X 3 兴 ⫽X 3 ;
关 X 3 ,X 1 兴 ⫽X 2 .
Taking into account our preliminary classification we conclude that one of the basis operators
reduces to one of the canonical forms ⳵ t , ⳵ x , ⳵ u .
First, we study realizations of the algebra so共3兲 within the class of operators 共2.4兲.
Let X 1 ⫽ ⳵ t and let the operators X 2 ,X 3 be of the form 共2.4兲. Checking commutation relations
关 X 1 ,X 2 兴 ⫽X 3 , 关 X 3 ,X 1 兴 ⫽X 2 we see that
X 2 ⫽3 ␣ cos t ⳵ t ⫹ 关 ⫺ ␣ x sin t⫹ ␤ cos共 t⫹ ␥ 兲兴 ⳵ x ⫹ ␸ 共 x,u 兲 cos„t⫹ ␺ 共 x,u 兲 …⳵ u ,
X 3 ⫽⫺3 ␣ sin t ⳵ t ⫺ 关 ␣ x cos t⫹ ␤ sin 共 t⫹ ␥ 兲兴 ⳵ x ⫺ ␸ 共 x,u 兲 sin„t⫹ ␺ 共 x,u 兲 …⳵ u .
Here ␣, ␤, ␥ are arbitrary real constants and ␸, ␺ are arbitrary real-valued smooth functions.
The third commutation relation 关 X 2 ,X 3 兴 ⫽X 1 implies that 9 ␣ 2 ⫽⫺1. As this equation has no
real solutions, there are no realizations of so共3兲 with X 1 ⫽ ⳵ t .
The same assertion holds for the cases when X 1 ⫽ ⳵ x and X 1 ⫽ ⳵ u . So the class of operators
共2.4兲 contains no realizations of the algebra so共3兲. This means that there are no so共3兲-invariant
equations of the form 共1.1兲.
Theorem 4.1: There exist no realizations of the algebra so共3兲 in terms of vector fields 共2.4兲.
Hence no equation of the form 共1.1兲 is invariant under so共3兲 algebra.
Similar reasoning yields that there are three inequivalent realizations of the algebra sl共2,R兲 by
operators of the form 共2.4兲,
兵 ⳵ t ,t ⳵ t ⫹ 31 x ⳵ x ,⫺t 2 ⳵ t ⫺ 32 tx ⳵ x 其 ,
兵 ⳵ t ,t ⳵ t ⫹ 31 x ⳵ x ,⫺t 2 ⳵ t ⫺ 32 tx ⳵ x ⫺x 3 ⳵ u 其 ,
兵 ⳵ u ,u ⳵ u ,⫺u 2 ⳵ u 其 .
Inserting the coefficients of basis operators of the first realization of the algebra sl共2,R兲 into
invariance criterion yields the following classifying equations:
2u xx F u xx ⫹u x F u x ⫺xF x ⫺3F⫽0,
t 共 2u xx F u xx ⫹u x F u x ⫺xF x ⫺3F 兲 ⫺xu x ⫽0,
from which we get the equation xu x ⫽0. Consequently, the realization in question cannot be
invariance algebra of the equation under study.
The two remaining realizations of sl共2,R兲 do yield invariance algebras of equation 共1.1兲. The
forms of the function F in the corresponding invariant equations read as
兵 ⳵ t ,t ⳵ t ⫹ 31 x ⳵ x ,⫺t 2 ⳵ t ⫺ 32 tx ⳵ x ⫺x 3 ⳵ u 其
␻ 1 ⫽3u⫺xu x ,
兵 ⳵ u ,u ⳵ u ,⫺u 2 ⳵ u 其
: F⫽⫺x ⫺3 关 2xu x ⫹ 91 x 2 u 2x ⫺G 共 ␻ 1 , ␻ 2 兲兴 ,
␻ 2 ⫽6u⫺x 2 u xx ;
2
: F⫽⫺ 23 u ⫺1
x u xx ⫹u x G 共 x,t 兲 .
As any semi-simple or simple algebra contains either so共3兲 or sl共2,R兲 共or both兲 as
subalgebra共s兲,20 the above result can be utilized to perform the classification of equations 共1.1兲
admitting invariance algebras isomorphic to one having a nontrivial Levi decomposition.
First we turn to the equation
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J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
2
u t ⫽u xxx ⫺ 23 u ⫺1
x u xx ⫹u x G 共 x,t 兲 .
共4.1兲
Applying the Lie infinitesimal algorithm we see that the maximal invariance algebra of Eq.
共4.1兲 is spanned by the operators X 1 ⫽ ⳵ u , X 2 ⫽u ⳵ u , X 3 ⫽⫺u 2 ⳵ u , and
X 4 ⫽ ␶ 共 t 兲 ⳵ t ⫹ 共 31 ␶˙ x⫹ ␳ 共 t 兲兲 ⳵ x ,
共4.2兲
functions ␶, ␳ and G satisfying the equation
共 x ␶˙ ⫹3 ␳ 兲 G x ⫹3 ␶ G t ⫹2 ␶˙ G⫹x ␶¨ ⫹3 ␳˙ ⫽0.
共4.3兲
By direct verification we ensure that the form of basis operators of the realization of sl共2,R兲
under study is not altered by the transformations
t̃ ⫽T 共 t 兲 , x̃⫽Ṫ 1/3x⫹Y 共 t 兲 , ũ⫽ ␥ u, Ṫ⫽0, ␥ ⫽0.
共4.4兲
As transformation 共4.4兲 reduces 共4.2兲 to the form
(4.4)
X 4 → X̃ 4 ⫽ ␶ 共 t 兲 Ṫ 共 t 兲 ⳵˜t ⫹ 关 31 共 ␶ Ṫ ⫺1 T̈⫹ ␶ 兲共 x̃⫺Y 兲 ⫹ ␶ Ẏ ⫹ ␳ Ṫ 1/3兴 ⳵ x̃ ,
we can put X 4 ⫽ ⳵ t or X 4 ⫽ ⳵ x within the equivalence relation.
Provided X 4 ⫽ ⳵ t , it follows from 共4.3兲 that G⫽G̃(x) in 共4.1兲. Next, if X 4 ⫽ ⳵ x , then necessarily G⫽G̃(t). Consequently, the class of Eqs. 共4.1兲 contains two inequivalent equations:
2
u t ⫽u xxx ⫺ 23 u ⫺1
x u xx ⫹u x G̃ 共 x 兲
共4.5兲
2
u t ⫽u xxx ⫺ 23 u ⫺1
x u xx ⫹u x G̃ 共 t 兲 ,
共4.6兲
and
which are invariant under extensions of the algebra sl共2,R兲. Namely, they admit algebras
sl(2,R) 丣 兵 ⳵ t 其 and sl(2,R) 丣 兵 ⳵ x 其 , correspondingly. What is more, if the function G̃⫽G̃(x) in 共4.5兲
is arbitrary, the given algebra is maximal 共in Lie sense兲 invariance algebra of Eq. 共4.5兲.
Equation 共4.6兲 is reduced to PDE 共4.5兲 with G̃(x)⫽0 with the help of the change of variables,
t̃ ⫽t, x̃⫽x⫹
冕
t
G̃ 共 ␰ 兲 d ␰ , u⫽ v共 x̃, t̃ 兲 .
Therefore, we can restrict our further considerations to Eq. 共4.5兲, where we need to differentiate
between the cases G̃⫽0 and G̃⫽0.
Classifying Eq. 共4.3兲 with G⫽G̃(x) reads as
共 x ␶˙ ⫹3 ␳ 兲 G̃ x ⫹2 ␶˙ G̃⫹x ␶¨ ⫹3 ␳˙ ⫽0.
Hence it follows that there are two cases providing for extension of the symmetry algebra.
Namely, the case when G̃⫽0, which gives rise to two additional symmetry operators X 5 ⫽t ⳵ t
⫹ 13 x ⳵ x and X 6 ⫽ ⳵ x . Another case of the extension of symmetry of Eq. 共4.5兲 is when G̃
⫽␭x ⫺2 (␭⫽0). If this is the case, 共4.5兲 admits the additional operator X 5 ⫽t ⳵ t ⫹ 31 x ⳵ x .
Now we turn to the equation
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2289
u t ⫽u xxx ⫺2x ⫺2 u x ⫺ 91 x ⫺1 u 2x ⫹x ⫺3 G 共 ␻ 1 , ␻ 2 兲 ,
␻ 1 ⫽3u⫺xu x ,
共4.7兲
␻ 2 ⫽6u⫺x 2 u xx .
First of all, we ensure that the class of PDEs 共4.7兲 does not contain equations whose invariance algebras possess semi-simple subalgebras of the dimension n⬎3.
It is common knowledge20 that there are four types of abstract simple Lie algebras over the
field of real numbers:
• The type A n⫺1 (n⬎1) contains four real forms of the algebras sl(n,C): su(n), sl(n,R),
su(p,q) (p⫹q⫽n, p⭓q),su* (2n).
• The type B n (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).
• The type C n (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).
• The type D n (n⬎1) contains three real forms of the algebra so(2n,C): so(2n),so(p,q) ( p
⫹q⫽2n, p⭓q),so* (2n).
The lowest order classical semi-simple Lie algebras are three-dimensional. The next admissible dimension for classical semi-simple Lie algebras is six. There are four nonisomorphic semisimple Lie algebras: so共4兲, so共3,1兲, so共2,2兲 and so* (4). As so(4)⫽so(3) 丣 so(3), so* (4)
⬃so(3) 丣 sl(2,R), and the algebra so共3,1兲 contains so共3兲 as a subalgebra, the algebra so共2,2兲 is the
only possible six-dimensional semi-simple algebra that might be invariance algebra of Eq. 共4.7兲.
Taking into account that so(2,2)⬃sl(2,R) 丣 sl(2,R) and choosing so(2,2)⫽ 兵 X 1 ,X 2 ,X 3 其
丣 兵 X̃ 1 ,X̃ 2 ,X̃ 3 其 , where X 1 ,X 2 ,X 3 form a basis of sl共2,R兲, which is invariance algebra of 共4.7兲 and
X̃ 1 ,X̃ 2 ,X̃ 3 are of the form 共2.4兲, we require the commutation relations
关 X i ,X̃ j 兴 ⫽0 共 i, j⫽1,2,3 兲
to hold, whence
X̃ j ⫽␭ j ⳵ u
共 j⫽1,2,3 兲 ,
where ␭ j are arbitrary real constants. Hence we conclude that the class of operators 共2.4兲 does not
contain a realization of so共2,2兲.
The same result holds for eight-dimensional semi-simple Lie algebras sl共3,R兲, su共3兲, su共2,1兲.
As su* (4)⬃so(5,1) and the algebra so共5,1兲 contains so共4兲 as a subalgebra, the class of
operators 共2.4兲 contains no realizations of A n and D n (n⬎1) type algebras that are inequivalent to
the algebra sl共2,R兲.
The same assertion holds true for B n (n⬎1) and C n (n⭓1) type Lie algebras. Indeed, B 2
type algebras contain so共4兲 and so共3,1兲 and what is more,
sp共 2,R兲 ⬃so共 3,2兲 傻so共 3,1兲 ,
sp共 1,1兲 ⬃so共 4,1兲 傻so共 4 兲 ,
sp共 2 兲 ⬃so共 5 兲 傻so共 4 兲 .
What remains to be done is to consider the exceptional semi-simple Lie algebras that belong
to one of the following five types:20 G 1 ,F 4 ,E 6 ,E 7 ,E 8 . We consider in some detail G 1 type Lie
algebras.
The type G 1 contains one compact real form g 2 and one noncompact real form g 2⬘ . As
g 2 艚g ⬘2 ⬃su(2) 丣 su(2)⬃so(4) and the algebra so共4兲 has no realization within the class of operators 共2.4兲, the latter contains no realizations of type G 1 .
Summing up, we conclude that class of PDEs 共4.7兲 contains no equations, whose invariance
algebras are isomorphic to n-dimensional semi-simple Lie algebras 共or contains the latter as
subalgebras兲 under n⬎3.
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J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
Consider now Eqs. 共4.7兲, whose invariance algebras has nontrivial Levi factor. First, we turn
to equations which are invariant with respect to the Lie algebras that can be decomposed into a
direct sum of semi-simple Levi factor and radical, sl(2,R) 丣 L, L being a radical. To this end, we
will study possible extensions of the algebra sl共2,R兲 by operators 共2.4兲.
Let sl(2,R)⫽ 兵 X 1 ,X 2 ,X 3 其 , where X 1 ,X 2 ,X 3 , form a basis of the invariance algebra of Eq.
共4.7兲. Then it follows from
关 X i ,Y 兴 ⫽0
共 i⫽1,2,3 兲 ,
Y being an operator of the form 共2.4兲, that Y ⫽␭ ⳵ u , ␭⫽const. Hence L is the one-dimensional Lie
algebra spanned by the operator ⳵ u . For Eq. 共4.7兲 to admit the algebra sl(2,R) 丣 兵 ⳵ u 其 , the equation
G ␻ 1 ⫹2G ␻ 2 ⫽0,
has to be satisfied, whence
G⫽G̃ 共 ␴ 兲 ,
␴ ⫽x 2 u xx ⫺2xu x .
Consequently, an equation of the form 共4.7兲 admits invariance algebra which is the direct sum of
semi-simple Levi factor and radical iff it reads as
u t ⫽u xxx ⫺2x ⫺2 u x ⫺ 91 x ⫺1 u 2x ⫹x ⫺3 G̃ 共 ␴ 兲 ,
␴ ⫽x 2 u xx ⫺2xu x .
共4.8兲
As Eq. 共4.8兲 contains an arbitrary function of one variable, we can perform direct group
classification by a straightforward application of the Lie infinitesimal algorithm. The determining
equation for coefficients of the infinitesimal symmetry operator are of the form
共a兲
␾ uuu ⫽0;
共b兲
3 ␾ uu G̃ ␴ ⫹18␾ uu ⫹9x ␾ xuu ⫹ 31 共 x ⫺1 ␳ ⫺ ␾ u 兲 ⫽0;
共c兲
6x ⫺1 共 ␾ xu ⫹x ⫺2 ␳ 兲 G̃ ␴ ⫹9x ⫺2 ␴ ␾ uu ⫹3 ␳ t ⫹x ␶ tt ⫹6x ⫺1 共 3 ␾ xu ⫹2x ⫺2 ␳ 兲
⫹9 ␾ xxu ⫺ 32 x ⫺1 ␾ x ⫽0;
共d兲
关 ⫺3x ⫺3 共 ␾ u ⫹2x ⫺1 ␳ 兲 ␴ ⫹6x ⫺2 ␾ x ⫺3x ⫺1 ␾ xx 兴 G̃ ␴ ⫹3x ⫺3 共 ␾ u ⫹3x ⫺1 ␳ 兲 G̃⫺9x ⫺2 ␾ xu ␴
⫹3 关 ␾ t ⫺ ␾ xxx ⫹2x ⫺2 ␾ x 兴 ⫽0.
It follows from 共a兲 that
␾ ⫽ f 共 x,t 兲 u 2 ⫹g 共 x,t 兲 u⫹h 共 x,t 兲 ,
共4.9兲
where f ,g,h are arbitrary smooth functions. Inserting 共4.9兲 into 共b兲 yields
6 f G̃ ␴ ⫹36f ⫹18x f x ⫹ 31 共 x ⫺1 ␳ ⫺g 兲 ⫺ 32 f u⫽0.
Taking into account that functions f , ␳ ,g,G̃ do not depend on u, we get
f ⫽0, g⫽x ⫺1 ␳ .
So that equation 共c兲 reduces to
3 ␳ t ⫹x ␶ tt ⫹12x ⫺3 ␳ ⫹ 32 x ⫺3 ␳ u⫺ 32 x ⫺1 h x ⫽0.
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2291
Hence it follows that
␳ ⫽0, h⫽ 12 x 3 ␶ tt ⫹h̃ 共 t 兲 .
Finally, inserting the obtained expression for ␸ into equation 共d兲 gives
␶ ttt ⫽0, h̃ t ⫽0,
whence
␶ ⫽C 1 t 2 ⫹C 2 t⫹C 3 ,
h̃⫽C 4 .
Here C 1 ,C 2 ,C 3 ,C 4 are arbitrary 共integration兲 constants.
Summing up, we conclude that the algebra sl(2,R) 丣 兵 ⳵ u 其 is the maximal invariance algebra
admitted by Eq. 共4.8兲. It cannot be extended by specifying the form of an arbitrary function
G̃( ␴ ), ␴ ⫽x 2 u xx ⫺2xu x .
What remains to be done is classifying Eqs. 共4.7兲, whose invariance algebras are isomorphic
to semi-direct sums of a semi-simple Levi factor and radical, i.e., whose invariance algebras have
the following structure: sl(2,R)L. To perform this classification we utilize the classification of
these type of Lie algebras obtained by Turkowski.21
We choose sl(2,R)⫽ 兵 v 1 , v 2 , v 3 其 with
v 1 ⫽⫺2t ⳵ t ⫺ 23 x ⳵ x , v 2 ⫽ ⳵ t , v 3 ⫽⫺t 2 ⳵ t ⫺ 23 tx ⳵ x ⫺x 3 ⳵ u .
According to Ref. 21, there is only one five-dimensional Lie algebra of the desired form
sl(2,R)L with L⫽ 兵 e 1 ,e 2 其 , operators e 1 ,e 2 satisfying the commutation relations:
关 e 1 ,e 2 兴 ⫽0, 关v 1 ,e 1 兴 ⫽e 1 , 关v 1 ,e 2 兴 ⫽⫺e 2 ,
关v 2 ,e 1 兴 ⫽0, 关v 2 ,e 2 兴 ⫽e 1 ,
关v 3 ,e 1 兴 ⫽e 2 ,
关v 3 ,e 2 兴 ⫽0.
As operators e 1 ,e 2 are necessarily of the form 共2.4兲, we easily get that
e 1 ⫽ 兩 x 兩 ⫺3/2⳵ u ,
e 2 ⫽t 兩 x 兩 ⫺3/2⳵ u .
However, checking the invariance criterion for the above realization we find that the algebra in
question cannot be invariance algebra of an equation of the form 共4.7兲.
According to Ref. 21, there exist three six-dimensional Lie algebras that are semi-direct sums
of semi-simple Levi factor and radical, algebra L being of the form L⫽ 兵 e 1 ,e 2 ,e 3 其 . Nonzero
commutation relations for e 1 ,e 2 ,e 3 read as
共1兲
关v 1 ,e 1 兴 ⫽2e 1 , 关v 2 ,e 2 兴 ⫽2e 1 , 关v 3 ,e 1 兴 ⫽e 2 ,
关v 1 ,e 3 兴 ⫽⫺2e 3 , 关v 2 ,e 3 兴 ⫽e 2 , 关v 3 ,e 2 兴 ⫽2e 3 ;
共2兲
关v 1 ,e 1 兴 ⫽e 1 , 关v 2 ,e 2 兴 ⫽e 1 , 关v 3 ,e 1 兴 ⫽e 2 ,
关v 1 ,e 2 兴 ⫽⫺e 2 , 关 e 1 ,e 2 兴 ⫽e 3 ;
共3兲
关v 1 ,e 1 兴 ⫽e 1 , 关v 2 ,e 2 兴 ⫽e 1 ,
关v 3 ,e 1 兴 ⫽e 2 ,
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J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
[ v 1 ,e 2 ]⫽⫺e 2 , [e 1 ,e 3 ]⫽e 1 , [e 2 ,e 3 ]⫽e 2 .
Solving the above relations we see that the corresponding realizations cannot be invariance
algebras of Eq. 共4.7兲.
Next, we consider seven-dimensional algebras from the Turkowski’s classification. According
to Ref. 21 there are five inequivalent algebras of the targeted dimension. Four of them contain the
above five- and six-dimensional algebras as subalgebras. So we need to consider only the fifth
algebra sl(2,R)L, where L⫽ 兵 e 1 ,e 2 ,e 3 ,e 4 其 and the following commutation relations hold:
关v 1 ,e 1 兴 ⫽3e 1 ,
关v 2 ,e 2 兴 ⫽3e 1 ,
关v 3 ,e 1 兴 ⫽e 2 ,
关v 1 ,e 2 兴 ⫽e 2 ,
关v 2 ,e 3 兴 ⫽2e 2 ,
关v 3 ,e 2 兴 ⫽2e 3 ,
关v 1 ,e 3 兴 ⫽⫺e 3 ,
关v 2 ,e 4 兴 ⫽e 3 ,
关v 3 ,e 3 兴 ⫽3e 4 ,
关v 1 ,e 4 兴 ⫽⫺3e 4 .
The most general form of operators e 1 ,e 2 ,e 3 ,e 4 satisfying the above relations is as follows:
e 1 ⫽ 兩 x 兩 ⫺ 9/2⳵ u ,
e 3 ⫽3t 2 兩 x 兩 ⫺ 9/2⳵ u ,
e 2 ⫽3t 兩 x 兩 ⫺ 9/2⳵ u ,
e 4 ⫽t 3 兩 x 兩 ⫺ 9/2⳵ u .
However, verifying the invariance criterion yields that this algebra cannot be the symmetry algebra of Eq. 共4.7兲.
Thus we proved that the class of PDEs 共4.7兲 contains no equations admitting symmetry
algebras of the dimension n⭐7, which are semi-direct sums of the Levi factor and radical. It is
natural to conjecture that the same assertion holds for an arbitrary n. To prove this fact we need
to consider in full details classification of nonlinear equations 共1.1兲, whose invariance algebras are
solvable.
Let us sum up the above results as theorems.
Theorem 4.2: The class of PDEs (1.1) contains two inequivalent equations whose invariance
algebra are semi-simple „sl(2,R)…,
2
u t ⫽u xxx ⫺ 23 u ⫺1
x u xx ⫹u x G 共 x,t 兲 ;
u t ⫽u xxx ⫺x ⫺3 关 2xu x ⫹ 91 x 2 u 2x ⫺G 共 ␻ 1 , ␻ 2 兲兴 ,
␻ 1 ⫽3u⫺xu x ,
␻ 2 ⫽6u⫺x 2 u xx .
The maximal invariance algebras of the above equations under arbitrary G read as
sl1 共 2,R兲 ⫽ 兵 ⳵ u ,u ⳵ u ,⫺u 2 ⳵ u 其 ;
sl2 共 2,R兲 ⫽ 兵 ⳵ t ,t ⳵ t ⫹ 31 x ⳵ x ,⫺t 2 ⳵ t ⫺ 32 tx ⳵ x ⫺x 3 ⳵ u 其 .
Theorem 4.3: Nonlinear equation (1.1) whose invariance algebra is isomorphic to a Lie
algebra having nontrivial Levi decomposition is represented by one of the following equations:
2
u t ⫽u xxx ⫺ 23 u ⫺1
x u xx ⫹u x G̃ 共 x 兲 ,
2
⫺2
u t ⫽u xxx ⫺ 23 u ⫺1
ux ,
x u xx ⫹␭x
␭⫽0,
sl1 共 2,R 兲 丣 兵 ⳵ t 其 ;
sl1 共 2,R兲 丣 兵 ⳵ t ,t ⳵ t ⫹ 31 x ⳵ x 其 ;
共4.10兲
共4.11兲
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2
u t ⫽u xxx ⫺ 23 u ⫺1
x u xx ,
sl1 共 2,R兲 丣 兵 ⳵ t , ⳵ x ,t ⳵ t ⫹ 31 x ⳵ x 其 ;
u t ⫽u xxx ⫺2x ⫺2 u x ⫺ 91 x ⫺1 u 2x ⫹x ⫺3 G̃ 共 ␴ 兲 , ␴ ⫽x 2 u xx ⫺2xu x ,
sl2 共 2,R兲 丣 兵 ⳵ u 其 ,
2293
共4.12兲
共4.13兲
where G̃ is an arbitrary function of x or ␴. Moreover, the associated symmetry algebras are
maximal.
Note that Eq. 共4.12兲 can be expressed in the form
ut
⫽ 兵 u;x 其 ,
ux
共4.14兲
where 兵 u;x 其 denotes the Schwarzian derivative of u with respect to x. It is known that a nonpoint
transformation taking this equation into the usual KdV exists.
V. CLASSIFICATION OF EQUATIONS INVARIANT UNDER LOW-DIMENSIONAL
SOLVABLE SYMMETRY ALGEBRAS
In this section we apply the strategy summarized in the Introduction to identify representative
classes of equations of the form 共1.1兲 invariant under one-, two-, and three-dimensional solvable
symmetry algebras. In order to approach this task in a systematic manner we realize all possible
inequivalent algebras in terms of vector fields 共2.4兲 under the action of the equivalence group E.
A. Equations with one-dimensional symmetry algebras
We assume that for a given F, Eq. 共1.1兲 is invariant under a one-parameter symmetry group,
generated by the vector field 共2.4兲 with coefficients subject to the constraint 共2.5兲. We make use of
Proposition 2.3 which characterizes the canonical forms of the vector field X of 共2.4兲. We then
substitute the coefficients of the canonical vector field into the determining equation 共2.5兲, which
is a first order linear homogeneous PDE for F, and solve the latter in order to construct invariant
equations.
According to Proposition 2.3 we have three types of one-dimensional symmetry algebras:
A 1,1 :
X 1⫽ ⳵ t ,
A 1,2 :X 1 ⫽ ⳵ x ,
A 1,3 :X 1 ⫽ ⳵ u .
共5.1兲
The corresponding invariant equations will have the form
A 1,1 :
u t ⫽u xxx ⫹F 共 x,u,u x ,u xx 兲 ,
共5.2a兲
A 1,2 :
u t ⫽u xxx ⫹F 共 t,u,u x ,u xx 兲 ,
共5.2b兲
A 1,3 :
u t ⫽u xxx ⫹F 共 x,t,u x ,u xx 兲 .
共5.2c兲
Theorem 5.1: There are three inequivalent classes of Eqs. (1.1) invariant under oneparameter symmetry group. Their representatives are given by (5.2).
B. Equations with two-dimensional symmetry algebras
There are two isomorphy classes of two-dimensional Lie algebras, Abelian and non-Abelian,
satisfying the commutation relations 关 X 1 ,X 2 兴 ⫽ ␬ X 2 , ␬ ⫽0,1. We denote them by A 2,1 and A 2,2 .
1. Abelian
We start from each of the one-dimensional cases obtained in 共5.1兲 and add to it vector fields
X 2 of the form 共2.4兲 commuting with X 1 . We then simplify X 2 by equivalence transformations
leaving the vector field X 1 invariant. For further details we refer the reader to Ref. 4. The standardized X 2 and the restricted form of F in 共5.2兲 are then substituted into 共2.5兲. Solving this
equation will further restrict the form of the function F. The number of variables of F reduces by
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J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
one, three variables in this case. Thus, we find that there exist precisely four classes of twodimensional Abelian symmetry algebras represented by the following ones:
3
:
A 2,1
1
:
A 2,1
X 1⫽ ⳵ t ,
X 2⫽ ⳵ x ,
F⫽F 共 u,u x ,u xx 兲 ;
共5.3兲
2
A 2,1
:
X 1⫽ ⳵ t ,
X 2⫽ ⳵ u ,
F⫽F 共 x,u x ,u xx 兲 ;
共5.4兲
F⫽⫺ ␣˙ uu x ⫹F̃ 共 t,u x ,u xx 兲 ;
共5.5兲
X 1⫽ ⳵ x ,
4
:
A 2,1
X 2⫽ ␣共 t 兲⳵ x⫹ ⳵ u ,
X 1⫽ ⳵ u ,
X 2 ⫽g 共 x,t 兲 ⳵ u ,
F⫽ 共 g t ⫺g xxx 兲 g ⫺1
x u x ⫹F̃ 共 x,t, ␻ 兲 ,
g x ⫽const;
共5.6兲
␻ ⫽g xx u x ⫺g x u xx .
2. Non-Abelian
Imposing that X 1 reads as 共5.1兲 and X 2 is in generic form and that they satisfy 关 X 1 ,X 2 兴
⫽X 2 , we find that five classes of symmetry algebras exist. Those algebras and nonlinear functions
F are represented by
1
A 2,2
:
X 1⫽ ⳵ t ,
F⫽x ⫺3 F̃ 共 u, ␻ 1 , ␻ 2 兲 ,
2
:
A 2,2
X 1 ⫽⫺u ⳵ u ,
4
:
A 2,2
X 2⫽ ⳵ u ,
␻ 2 ⫽t 2/3u xx ;
F⫽u x F̃ 共 x,t, ␻ 兲 ,
␻ ⫽u ⫺1
x u xx ;
X 2⫽ ⳵ u ;
␻ 1 ⫽e x u x ,
X 1 ⫽ ⳵ t ⫺u ⳵ u ,
F⫽u x F̃ 共 x, ␻ 1 , ␻ 2 兲 ,
␻ 2 ⫽x 2 u xx ;
X 2⫽ ⳵ x ,
␻ 1 ⫽t 1/3u x ,
X 1 ⫽ ⳵ x ⫺u ⳵ u ,
F⫽e ⫺x F̃ 共 t, ␻ 1 , ␻ 2 兲 ,
5
A 2,2
:
␻ 1 ⫽xu x ,
X 1 ⫽⫺3t ⳵ t ⫺x ⳵ x ,
F⫽t ⫺1 F̃ 共 u, ␻ 1 , ␻ 2 兲 ,
3
A 2,2
:
x
X 2 ⫽⫺t ⳵ t ⫺ ⳵ x ,
3
␻ 2 ⫽e x u xx ;
X 2⫽ ⳵ u ,
␻ 1 ⫽e t u x ,
␻ 2 ⫽e t u xx .
Theorem 5.2: There exist nine classes of two-dimensional symmetry algebras admitted by Eq.
1
4
1
5
,...,A 2,1
and A 2,2
,...,A 2,2
.
(1.1). They are represented by the algebras A 2,1
C. Equations with three-dimensional symmetry algebras
1. Decomposable algebras
A Lie algebra is decomposable if it can be written as a direct sum of two or more Lie algebras
L⫽L 1 丣 L 2 with 关 L 1 ,L 2 兴 ⫽0. There are two types of 3-dimensional decomposable Lie algebras:
A 3,1⫽3A 1 ⫽A 1 丣 A 2 丣 A 3 with 关 X i ,X j 兴 ⫽0 for i, j⫽1,2,3 and A 3,2⫽A 2,2 丣 A 1 with 关 X 1 ,X 2 兴
⫽X 2 , 关 X 1 ,X 3 兴 ⫽0, 关 X 2 ,X 3 兴 ⫽0.
We start from the two-dimensional algebras in 共5.3兲 and add a further linearly independent
vector field X 3 in the form 共2.4兲 and impose the above commutation relations. We simplify X 3
using equivalence transformations leaving the space 兵 X 1 ,X 2 其 invariant. We present the following
result without proof. We emphasize that there exist several realizations that do not produce invariant equations of the form 共1.1兲:
Downloaded 17 May 2004 to 160.75.21.17. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp
J. Math. Phys., Vol. 45, No. 6, June 2004
X 1⫽ ⳵ t ,
1
A 3,1
:
KdV-type nonlinear evolution equations
X 2⫽ ⳵ x ,
X 1⫽ ⳵ t ,
2
:
A 3,1
F⫽⫺
X 3⫽ ⳵ u ,
X 2⫽ ⳵ u ,
f⵮
u ⫹F̃ 共 x, ␻ 兲 ,
f⬘ x
F⫽x ⫺3 F̃ 共 ␻ 1 , ␻ 2 兲 ,
␻ 1 ⫽xu x ,
F⫽t ⫺1 F̃ 共 ␻ 1 , ␻ 2 兲 ,
X 1 ⫽ ⳵ x ⫺u ⳵ u ,
5
A 3,2
:
F⫽⫺
X 2⫽ ⳵ u ,
X 1 ⫽ ⳵ x ⫺u ⳵ u ,
F⫽e ⫺x F̃ 共 ␻ 1 , ␻ 2 兲 ,
7
:
A 3,2
X 1 ⫽ ⳵ t ⫺u ⳵ u ,
F⫽⫺
␻ 1 ⫽e x u x ,
X 1 ⫽ ⳵ t ⫺u ⳵ u ,
X 1 ⫽ ⳵ t ⫺u ⳵ u ,
f ⫽0,
␣ ⫽0,
␻ ⫽u ⫺1
x u xx ;
X 3⫽ ⳵ t ,
␻ 2 ⫽e x u xx ;
X 3 ⫽e ⫺t f 共 x 兲 ⳵ u ,
X 2⫽ ⳵ u ,
F⫽e ⫺t F̃ 共 ␻ 1 , ␻ 2 兲 ,
9
:
A 3,2
X 3⫽ ␣共 t 兲⳵ x ,
X 2⫽ ⳵ u ,
f ⵮⫹ f
u x ⫹e ⫺t F̃ 共 x, ␻ 兲 ,
f⬘
8
:
A 3,2
3
␻ 2 ⫽t 2 u xx
;
␻ ⫽e x 共 u x ⫹u xx 兲 ;
␣˙
u ln共 e x u x 兲 ⫹u x F̃ 共 t, ␻ 兲 ,
␣ x
6
A 3,2
:
X 3 ⫽t 1/3⳵ x ⫹ ⳵ u ,
X 3 ⫽e ⫺x f 共 t 兲 ⳵ u ,
ḟ
u ⫹e ⫺x F̃ 共 t, ␻ 兲 ,
f x
X 1 ⫽ ⳵ x ⫺u ⳵ u ,
X 3⫽ ⳵ u ,
␻ 1 ⫽tu 3x ,
X 2⫽ ⳵ u ,
冉 冊
F⫽⫺ 1⫹
␻ 2 ⫽x 2 u xx ;
3
␻ 2 ⫽t 2 u xx
;
X 2⫽ ⳵ x ,
F⫽⫺ 13 t ⫺2/3uu x ⫹t ⫺1 F̃ 共 ␻ 1 , ␻ 2 兲 ,
4
:
A 3,2
X 3⫽ ⳵ u ,
X 2⫽ ⳵ x ,
␻ 1 ⫽tu 3x ,
X 1 ⫽⫺3t ⳵ t ⫺x ⳵ x ,
3
:
A 3,2
X 3⫽ f 共 x 兲⳵ u ,
X 2⫽ ⳵ t ,
X 1 ⫽⫺3t ⳵ t ⫺x ⳵ x ,
2
:
A 3,2
F⫽F 共 u x ,u xx 兲 ;
␻ ⫽ f ⬙ u x ⫺ f ⬘ u xx ;
x
X 1 ⫽⫺t ⳵ t ⫺ ⳵ x ,
3
1
:
A 3,2
2295
␻ ⫽e t 共 f ⬙ u x ⫺ f ⬘ u xx 兲 ;
X 2⫽ ⳵ u ,
␻ 1 ⫽e t u x ,
X 2⫽ ⳵ u ,
f ⬘ ⫽0,
X 3⫽ ⳵ x ,
␻ 2 ⫽e t u xx ;
X 3 ⫽ ⳵ t ⫹␭ ⳵ x ,
␭⫽0,
F⫽exp共 x/t⫺␭ 兲 F̃ 共 ␻ 1 , ␻ 2 兲 ,
␻ 1 ⫽exp共 t⫺x/␭ 兲 u x ,
10
:
A 3,2
X 1 ⫽ ⳵ t ⫺u ⳵ u ,
␻ 2 ⫽exp共 t⫺x/␭ 兲 u xx ;
X 2⫽ ⳵ u ,
X 3⫽ ⳵ t ,
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J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
␻ ⫽u ⫺1
x u xx .
F⫽u x F̃ 共 x, ␻ 兲 ,
2. Nondecomposable algebras
The isomorphy classes of these algebras are represented by the following list:
关 X 2 ,X 3 兴 ⫽X 1 ,
A 3,3 :
关 X 1 ,X 3 兴 ⫽X 1 ,
A 3,4 :
A 3,6 :
A 3,9 :
关 X 2 ,X 3 兴 ⫽X 2 ;
关 X 1 ,X 3 兴 ⫽X 1 ,
关 X 1 ,X 3 兴 ⫽X 1 ,
A 3,8 :
关 X 2 ,X 3 兴 ⫽X 1 ⫹X 2 ;
关 X 1 ,X 3 兴 ⫽X 1 ,
A 3,5 :
A 3,7 :
关 X 1 ,X 2 兴 ⫽ 关 X 1 ,X 3 兴 ⫽0;
关 X 2 ,X 3 兴 ⫽⫺X 2 ;
关 X 2 ,X 3 兴 ⫽qX 2 共 0⬍ 兩 q 兩 ⬍1 兲 ;
关 X 1 ,X 3 兴 ⫽⫺X 2 ,
关 X 1 ,X 3 兴 ⫽qX 1 ⫺X 2 ,
关 X 2 ,X 3 兴 ⫽X 1 ;
关 X 2 ,X 3 兴 ⫽X 1 ⫹qX 2 , q⬎0.
Remark: Solvable nondecomposable algebras can be written as semidirect sums of a onedimensional subalgebra 兵 X 3 其 and an Abelian ideal 兵 X 1 ,X 2 其 . Note that the algebras A 3,6 and A 3,8
are isomorphic to e共1,1兲, and e共2兲, respectively. The algebra A 3,3 is a non-Abelian nilpotent algebra
共Heisenberg algebra兲.
The commutation relations of the algebras in question can be represented in the matrix notation
冉
冊 冉 冊
X1
关 X 1 ,X 3 兴
⫽J
,
X2
关 X 2 ,X 3 兴
关 X 1 ,X 2 兴 ⫽0,
where J is a 2⫻2 real matrix that can be taken in Jordan canonical form.
A solvable three-dimensional Lie algebra always possesses a two-dimensional Abelian ideal.
We assume that the ideal 兵 X 1 ,X 2 其 is already of the form 共5.3兲 and add a third element X 3 in the
form 共2.4兲 acting on the ideal. Imposing commutation relations and simplifying with equivalence
transformations 共2.6兲 共we consider each canonical form of the matrix individually兲 yield the
realizations of solvable Lie algebras together with the corresponding invariant equations.
There exist nine classes of realizations of nilpotent algebras which give rise to invariant
equations:
A 3,3 :
1
:
A 3,3
X 1⫽ ⳵ t ,
X 2⫽ ⳵ u ,
F⫽
2
:
A 3,3
X 1⫽ ⳵ u ,
J⫽
冉 冊
0
0
1
0
共5.7兲
.
X 3 ⫽t ⳵ u ⫹␭ ⳵ x ,
␭⬎0,
x
⫹F̃ 共 u x ,u xx 兲 ;
␭
X 2⫽ ⳵ x ,
X 3 ⫽x ⳵ u ⫹b 共 t 兲 ⳵ x ,
ḃ⫽0,
ḃ
F⫽⫺ u 2x ⫹F̃ 共 t,u xx 兲 ;
2
3
:
A 3,3
X 1⫽ ⳵ u ,
X 2⫽ ⳵ x ,
X 3 ⫽x ⳵ u ⫹␭ ⳵ t ,
␭⫽0,
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2297
F⫽F̃ 共 t⫺3␭u x ,u xx 兲 ;
X 1 ⫽ ⳵ u ⫹3␭t 1/2⳵ x ,
4
A 3,3
:
X 2⫽ ⳵ x ,
X 3 ⫽6␭t 3/2⳵ t ⫹3␭t 1/2x ⳵ x ⫹ 共 x⫺3␭t 1/2u 兲 ⳵ u ,
F⫽⫺ 32 ␭t ⫺1/2uu x ⫹t ⫺2 F̃ 共 ␻ 1 , ␻ 2 兲 ,
X 1⫽ ⳵ x ,
5
:
A 3,3
␻ 1 ⫽tu x ⫺
X 2⫽ ⳵ t ,
␭⫽0,
1 1/2
t ,
3␭
␻ 2 ⫽t 3/2u xx ;
X 3 ⫽t ⳵ x ⫹ ⳵ u ,
F⫽⫺uu x ⫹F̃ 共 u x ,u xx 兲 ;
X 1⫽ ⳵ u ,
6
A 3,3
:
X 3 ⫽ ⳵ t , 共 f ⬘ ⫽0 兲 ,
X 2 ⫽ 共 f 共 x 兲 ⫺t 兲 ⳵ u ,
F⫽⫺ 共 1⫹ f ⵮ 兲共 f ⬘ 兲 ⫺1 u x ⫹F̃ 共 x, ␻ 兲 ,
␻ ⫽ f ⬙ u x ⫺ f ⬘ u xx ;
X 1⫽ ⳵ u ,
X 3⫽ ⳵ x ,
7
:
A 3,3
X 2 ⫽ 共 t⫺x 兲 ⳵ u ,
F⫽u x ⫹F̃ 共 t,u xx 兲 ;
8
:
A 3,3
X 1⫽ ⳵ u ,
X 2 ⫽⫺x ⳵ u ,
X 3⫽ ⳵ x ,
F⫽⫺F̃ 共 t,u xx 兲 ;
9
:
A 3,3
X 1 ⫽⫺x ⫺1 ⳵ u ,
F⫽3x ⫺1 u xx ⫹x ⫺1 F̃ 共 t, ␻ 兲 ,
A 3,4 :
1
A 3,4
:
X 1⫽ ⳵ u ,
J⫽
X 2⫽ ⳵ t ,
X 1⫽ ⳵ x ,
F⫽
3
A 3,4
:
冉 冊
1
0
1
1
X 2 ⫽ ⳵ u ⫺ 13 ln t ⳵ x ,
X 2⫽ ⳵ x ,
F⫽t ⫺2/3F̃ 共 ␻ 1 , ␻ 2 兲 ,
4
:
A 3,4
共5.8兲
;
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫹ 共 u⫹t 兲 ⳵ u ,
3
1
uu ⫹t ⫺2/3F̃ 共 u x , ␻ 兲 ,
3t x
X 1⫽ ⳵ u ,
␻ ⫽2u x ⫹xu xx ;
␻ 1 ⫽x ⫺2 u x ,
F⫽3 ln x⫹F̃ 共 ␻ 1 , ␻ 2 兲 ,
2
:
A 3,4
X 3 ⫽ ⳵ x ⫺x ⫺1 u ⳵ u ,
X 2⫽ ⳵ u ,
␻ 2 ⫽x ⫺1 u xx ;
X 3 ⫽3t ⳵ t ⫹x ⳵ x ⫹u ⳵ u ,
␻ ⫽t 1/3u xx ;
X 3 ⫽3t ⳵ t ⫹x ⳵ x ⫹ 共 u⫹x 兲 ⳵ u ,
␻ 1 ⫽u x ⫺ 31 ln t, ␻ 2 ⫽t 1/3u xx ;
X 1⫽ ␣共 t 兲⳵ x⫹ ⳵ u , X 2⫽ ⳵ x ,
X 3 ⫽ 共 ␣ ⬘ 兲 ⫺1 ␣ 2 ⳵ t ⫹ 共 1⫹ ␣ 兲 x ⳵ x ⫹ 关 x⫹ 共 1⫺ ␣ 兲 u 兴 ⳵ u ,
␣ ⬘ ⫽0,
␣ 2 ␣ ⬙ ⫹ 共 3⫹ ␣ 兲共 ␣ ⬘ 兲 2 ⫽0,
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Güngör, Lahno, and Zhdanov
F⫽⫺ ␣ ⬘ uu x ⫹ ␣ ⫺4 exp共 2 ␣ ⫺1 兲 F̃ 共 ␻ 1 , ␻ 2 兲 ,
␻ 1 ⫽ ␣ 3 exp共 ⫺ ␣ ⫺1 兲 u xx ,
X 2 ⫽„⫺t⫹ f 共 x 兲 …⳵ u ,
X 1⫽ ⳵ u ,
5
A 3,4
:
␻ 2⫽ ␣ 2u x⫺ ␣ ;
F⫽⫺ 共 1⫹ f ⵮ 兲共 f ⬘ 兲 ⫺1 u x ⫹e t F̃ 共 x, ␻ 兲 ,
6
A 3,4
:
X 1⫽ ⳵ u ,
X 1⫽ ⳵ t ,
X 1⫽ ⳵ x ,
J⫽
冉 冊
1
0
0
1
共5.9兲
;
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫹u ⳵ u ,
3
X 2⫽ ⳵ u ,
␻ 1 ⫽x ⫺2 u x ,
F⫽F̃ 共 ␻ 1 , ␻ 2 兲 ,
2
A 3,5
:
X 3 ⫽ ⳵ x ⫹u ⳵ u ,
␻ ⫽e ⫺x u xx ;
F⫽e x F̃ 共 t, ␻ 兲 ,
1
:
A 3,5
␻ ⫽e ⫺t 共 f ⬙ u x ⫺ f ⬘ u xx 兲 ;
X 2 ⫽⫺x ⳵ u ,
A 3,5 :
f ⬘ ⫽0,
X 3 ⫽ ⳵ t ⫹u ⳵ u ,
X 2⫽ ⳵ u ,
␻ 2 ⫽x ⫺1 u xx ;
X 3 ⫽3t ⳵ t ⫹x ⳵ x ⫹u ⳵ u ,
F⫽t ⫺2/3F̃ 共 u x ,t 1/3u xx 兲 ;
X 3 ⫽ ⳵ t ⫹u ⳵ u , f ⬘ ⫽0,
X 1⫽ ⳵ u , X 2⫽ f 共 x 兲⳵ u ,
3
A 3,5
:
F⫽⫺ f ⵮ 共 f ⬘ 兲 ⫺1 u x ⫹e t F̃ 共 x, ␻ 兲 ,
␻ ⫽e ⫺t 关 f ⬙ u x ⫺ f ⬘ u xx 兴 ;
A 3,6 :
X 1⫽ ⳵ t ,
1
:
A 3,6
J⫽
冉
1
0
0
⫺1
X 2⫽ ⳵ u ,
冊
共5.10兲
;
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫺u ⳵ u ,
3
F⫽x ⫺6 F̃ 共 x 4 u x ,x 5 u xx 兲 ;
2
A 3,6
:
X 1⫽ ⳵ x ,
F⫽⫺
3
:
A 3,6
X 1⫽ ⳵ u ,
X 2 ⫽ ⳵ u ⫹␭t 2/3⳵ x ,
2␭ ⫺1/3
t
uu x ⫹t ⫺4/3F̃ 共 t 2/3u x ,tu xx 兲 ;
3
X 2 ⫽e 2t f 共 x 兲 ⳵ u ,
F⫽ 共 2 f ⫺ f ⵮ 兲共 f ⬘ 兲 ⫺1 u x ⫹e t F̃ 共 x, ␻ 兲 ,
4
:
A 3,6
X 1⫽ ⳵ u ,
X 3 ⫽3t ⳵ t ⫹x ⳵ x ⫺u ⳵ u ,
X 2 ⫽e 2 f
⫺1 x
h共 t 兲⳵u ,
X 3 ⫽ ⳵ t ⫹u ⳵ u ,
f ⬘ ⫽0,
␻ ⫽e ⫺t 共 f ⬙ u x ⫺ f ⬘ u xx 兲 ;
X 3 ⫽ f 共 t 兲 ⳵ x ⫹u ⳵ u ,
F⫽⫺ 关 4 f ⫺2 ⫺ 21 hh ⫺1 f ⫹ f ⫺1 f ⬘ x 兴 u x ⫹e f
⫺1 x
f h⫽0,
F̃ 共 t, ␻ 兲 ,
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
␻ ⫽e ⫺ f
A 3,7 :
1
A 3,7
:
J⫽
X 1⫽ ⳵ t ,
⫺1 x
2299
共 2u x ⫺ f u xx 兲 ;
冉 冊
1
0
0
q
,
0⬍ 兩 q 兩 ⬍1;
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ,
3
X 2⫽ ⳵ x ,
共5.11兲
q⫽1/3,
F⫽u 3x F̃ 共 u,u ⫺2
x u xx 兲 ;
2
A 3,7
:
X 1⫽ ⳵ t ,
X 1⫽ ⳵ t ,
q⫽1/3,
␻ 1 ⫽u ⫺2/3u x , ␻ 2 ⫽u ⫺1/3u xx ;
F⫽F̃ 共 ␻ 1 , ␻ 2 兲 ,
3
A 3,7
:
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫹u ⳵ u ,
3
X 2⫽ ⳵ x ,
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫹qu ⳵ u ,
3
X 2⫽ ⳵ u ,
q⫽0,⫾1,
F⫽x 3(q⫺1) F̃ 共 ␻ 1 , ␻ 2 兲 , ␻ 1 ⫽x 1⫺3q u x , ␻ 2 ⫽x 2⫺3q u xx ;
4
:
A 3,7
X 1⫽ ⳵ x ,
X 2 ⫽ ⳵ u ⫹␭t (1⫺q)/3⳵ x ,
X 3 ⫽3t ⳵ t ⫹x ⳵ x ⫹qu ⳵ u ,
q⫽0,⫾1, ␭苸R,
F⫽
␭
共 q⫺1 兲 t ⫺(q⫹2)/3uu x ⫹F̃ 共 ␻ 1 , ␻ 2 兲 ,
3
␻ 1 ⫽t ⫺(q⫺1)/3u x ,
5
:
A 3,7
X 1⫽ ⳵ u ,
␻ 2 ⫽t ⫺(q⫺2)/3u xx ;
X 2 ⫽e (1⫺q)t f 共 x 兲 ⳵ u ,
X 3 ⫽ ⳵ t ⫹u ⳵ u ,
f ⬘ ⫽0,
q⫽0,⫾1,
F⫽ 关共 1⫺q 兲 f ⫺ f ⵮ 兴共 f ⬘ 兲 ⫺1 u x ⫹e t F̃ 共 x, ␻ 兲 ,
␻ ⫽e ⫺t 关 f ⬙ u x ⫺ f ⬘ u xx 兴 ;
6
:
A 3,7
X 1⫽ ⳵ u ,
X 2 ⫽e (1⫺q) f
X 3 ⫽ f 共 t 兲 ⳵ x ⫹u ⳵ u ,
f •h⫽0,
⫺1 (t)x
h共 t 兲⳵u ,
q⫽0, ⫾1,
F⫽⫺ 关共 1⫺q 兲 2 f 2 ⫹ f ⫺1 f ⬘ x⫺ 共 1⫺q 兲 ⫺1 f h ⫺1 h ⬘ 兴 u x ⫹e f
␻ ⫽e ⫺ f
⫺1 x
⫺1 x
F̃ 共 t, ␻ 兲 ,
关共 1⫺q 兲 u x ⫺ f u xx 兴 .
Remark: The algebra A 3,7 has another realization,
兵 X 1 ⫽ ⳵ t ,X 2 ⫽ ⳵ x ,X 3 ⫽t ⳵ t ⫹ 31 共 x⫹b 0 t 兲 ⳵ x ⫹u ⳵ u 其 ,
2
that is isomorphic to A 3,7
under the change of basis
X 1 →X 1 ⫹
b0
X ,
2 2
X 2 →X 2 ,
X 3 →X 3 .
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2300
J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
Note that its equivalence to the latter is established by the change of variables,
t̃ ⫽t,
x̃⫽x⫺ 21 b 0 t,
ũ⫽u.
That is why we have excluded it from the above list:
A 3,8 :
1
:
A 3,8
X 3 ⫽⫺
J⫽
X 1⫽ ⳵ x ,
冉
1
⫺1
1
0
冊
;
共5.12兲
X 2⫽ ␣共 t 兲⳵ x⫹ ⳵ u ,
1
共 1⫹ ␣ 2 兲 ⳵ t ⫺ ␣ x ⳵ x ⫹ 共 ␣ u⫺x 兲 ⳵ u ,
␣˙
F⫽⫺ ␣˙ uu x ⫹ 共 1⫹ ␣ 2 兲 ⫺2 F̃ 共 ␻ 1 , ␻ 2 兲 ,
␻ 1 ⫽ 共 1⫹ ␣ 2 兲 u x ⫺ ␣ , ␻ 2 ⫽ 共 1⫹ ␣ 2 兲 3/2u xx ,
where ␣ (t), ␣˙ ⫽0 satisfies
共 1⫹ ␣ 2 兲 ␣¨ ⫹ ␣␣˙ 2 ⫽0;
A 3,9 :
1
A 3,9
:
X 3 ⫽⫺
J⫽
冉
q
⫺1
1
q
X 1⫽ ⳵ x ,
冊
,
q⬎0;
共5.13兲
共5.14兲
X 2⫽ ␣共 t 兲⳵ x⫹ ⳵ u ,
1
共 1⫹ ␣ 2 兲 ⳵ t ⫹ 共 q⫺ ␣ 兲 x ⳵ x ⫹ 关共 q⫹ ␣ 兲 u⫺x 兴 ⳵ u ,
␣˙
F⫽⫺ ␣˙ uu x ⫹exp兵 2q arctan ␣ 其 共 1⫹ ␣ 2 兲 ⫺2 F̃ 共 ␻ 1 , ␻ 2 兲 ,
␻ 1 ⫽ 共 1⫹ ␣ 2 兲 u x ⫺ ␣ , ␻ 2 ⫽ 共 1⫹ ␣ 2 兲 3/2 exp兵 ⫺q arctan ␣ 其 u xx ,
where ␣ (t), ␣˙ ⫽0 satisfies
共 1⫹ ␣ 2 兲 ␣¨ ⫹ 共 ␣ ⫺3q 兲 ␣˙ 2 ⫽0.
共5.15兲
Remark: ␣ (t) can be obtained implicitly by quadratures as
冕
␣
exp共 ⫺3q arctan ␰ 兲共 1⫹ ␰ 2 兲 1/2d ␰ ⫽c 1 t⫹c 0 .
Theorem 5.3: There are thirty-eight inequivalent three-dimensional solvable symmetry algebras admitted by Eq. (1.1).
VI. EQUATIONS WITH FOUR-DIMENSIONAL SOLVABLE ALGEBRAS
For dimL⫽4, we proceed exactly in the same manner as above. We start from the already
standardized three-dimensional algebras, and add a further linearly independent element X 4 , and
require that they form a Lie algebra.
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2301
A. Decomposable algebras
The list of decomposable four-dimensional Lie algebras consists of the twelve algebras:
4A 1 ⫽A 3,1 丣 A 1 , A 2,2 丣 2A 1 ⫽A 3,2 丣 A 1 , 2A 2,2⫽A 2,2 丣 A 2,2 , A 3,i 丣 A 1 (i⫽3,4,...,9). We preserve
the notations of the previous section.
There are four inequivalent realizations of the algebra 2A 2,2 which are invariance algebras of
PDEs of the form 共1.1兲. We give these realizations together with the corresponding invariant
equations:
x
X 1 ⫽⫺t ⳵ t ⫺ ⳵ x ,
3
1
:
2A 2,2
X 2⫽ ⳵ t ,
X 1 ⫽⫺3t ⳵ t ⫺x ⳵ x ,
F⫽
3
:
2A 2,2
X 2⫽ ⳵ x ,
␭
␻1
F̃ 共 ␻ 兲 ,
␻ 1 ln兩 ␻ 1 兩 ⫹
3t
t
X 1 ⫽ ⳵ x ⫺u ⳵ u ,
X 3 ⫽⫺u ⳵ u ⫹␭t 1/3⳵ x ,
1
X 3⫽ ⳵ t ,
␭
X 1 ⫽ ⳵ x ⫺u ⳵ u ,
X 2⫽ ⳵ u ,
5
:
2A 2,2
X 1 ⫽ ⳵ t ⫺u ⳵ u ,
⫺1 )x⫺t
␻ ⫽e t⫹( ␤
⫺1 t⫺x
⳵u ,
␭⫽0,
␻ ⫽e x 共 u x ⫹u xx 兲 ;
X 2⫽ ⳵ u ,
X 4 ⫽e ␥ x⫺t ⳵ u ,
F⫽e ( ␥ ⫺ ␤
X 4 ⫽e ␭
X 3 ⫽␭ ⳵ t ,
F⫽ 共 1⫹␭ ⫺1 兲 u x ⫹e ⫺x F̃ 共 ␻ 兲 ,
X 4 ⫽exp共 ␭t 兲 ⳵ x ,
␻ ⫽u ⫺1
x u xx ;
F⫽⫺␭xu x ⫺␭u x ln兩 u x 兩 ⫹u x F̃ 共 ␻ 兲 ,
4
2A 2,2
:
X 4⫽ ⳵ u ,
␻ ⫽t 1/3u ⫺1
x u xx ;
␻ 1 ⫽t 1/3u x ,
X 2⫽ ⳵ u ,
X 4 ⫽e u ⳵ u ,
␻ ⫽x 共 u ⫺1
x u xx ⫺u x 兲 ;
F⫽u 3x ⫺3u x u xx ⫹x ⫺2 u x F̃ 共 ␻ 兲 ,
2
:
2A 2,2
X 3⫽ ⳵ u ,
X 3⫽ ␤共 ⳵ x⫹ ␥ ⳵ t 兲⫺ ⳵ t ,
␤␥ ⫽0,
F̃ 共 ␻ 兲 ⫺ ␥ ⫺1 共 1⫹ ␥ 3 兲 u x ,
⫺1 ⫺ ␥ )x
共 ␥ u x ⫺u xx 兲 .
Equations invariant under the algebra A 2,2 丣 2A 1 ⫽A 3.2 丣 A 1 :
X 1 ⫽ ⳵ x ⫺u ⳵ u ,
6
丣 兵 X 4其 :
A 3,2
X 2⫽ ⳵ u ,
F⫽⫺u x ⫹e ⫺x F̃ 共 ␻ 兲 ,
6
A 3,2
丣 兵 X 4其 :
X 1 ⫽ ⳵ x ⫺u ⳵ u ,
F⫽u x F̃ 共 ␻ 兲 ,
7
A 3,2
共 f ⫽e ␭x , ␭⫽0 兲 丣 兵 X 4 其 :
X 3 ⫽e ␭x⫺t ⳵ u ,
X 4 ⫽e ⫺x ⳵ u ,
X 3⫽ ⳵ t ,
␻ ⫽e x 共 u x ⫹u xx 兲 ;
X 2⫽ ⳵ u ,
X 3⫽ ⳵ t ,
X 4⫽ ⳵ x ,
␻ ⫽u xx u ⫺1
x ;
X 1 ⫽ ⳵ t ⫺u ⳵ u ,
X 4 ⫽ ⳵ x ⫹␭ ⳵ t ,
X 2⫽ ⳵ u ,
␭⫽0,
F⫽⫺ 共 ␭ 3 ⫹1 兲 ␭ ⫺1 u x ⫹e ⫺t⫹␭x F̃ 共 ␻ 兲 ,
␻ ⫽e t⫺␭x 共 ␭u x ⫺u xx 兲 ;
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2302
J. Math. Phys., Vol. 45, No. 6, June 2004
1
A 3.3
丣 兵 X 4其 :
Güngör, Lahno, and Zhdanov
X 1⫽ ⳵ t ,
X 2⫽ ⳵ u ,
X 4 ⫽ ⳵ t ⫹␭ ⫺1 x ⳵ u ⫹ ␤ ⳵ x ,
X 3 ⫽t ⳵ u ⫹␭ ⳵ x ,
␤ 苸R,
␭⬎0,
F⫽␭ ⫺1 x⫺ ␤ u x ⫹F̃ 共 u xx 兲 ;
3
A 3,5
丣 兵 X 4其 :
X 1⫽ ⳵ u ,
␭⫽0,
X 2⫽ ⳵ x ,
␤ 苸R,
F⫽ ␤ 共 ␭ ⫺1 t⫺3u x 兲 ⫹F̃ 共 u xx 兲 ;
6
A 3,5
共 f ⫽␭ ⫺1 x, ␭⫽0 兲 丣 兵 X 4 其 :
X 3⫽ ⳵ t ,
X 4 ⫽ ⳵ t ⫹ ␤ 共 ⳵ x ⫹␭ ⫺1 t ⳵ u 兲 ,
X 3 ⫽x ⳵ u ⫹␭ ⳵ t ,
X 2 ⫽ 共 ␭ ⫺1 x⫺t 兲 ⳵ u ,
X 1⫽ ⳵ u ,
X 4 ⫽ ⳵ t ⫹␭ ⳵ x ,
␭⫽0,
F⫽⫺u x ⫹F̃ 共 u xx 兲 ;
9
丣 兵 X 4其 :
A 3,5
X 1 ⫽⫺x ⫺1 ⳵ u ,
X 3 ⫽ ⳵ x ⫺x ⫺1 u ⳵ u ,
X 2⫽ ⳵ u ,
F⫽3x ⫺1 u xx ⫹x ⫺1 F̃ 共 ␻ 兲 ,
1
A 3,4
丣 兵 X 4其 :
X 4⫽ ⳵ t ,
␻ ⫽2u x ⫹xu xx ;
X 1⫽ ⳵ u , X 2⫽ ⳵ t ,
X 3 ⫽t ⳵ t ⫹ 13 x ⳵ x ⫹ 共 u⫹t 兲 ⳵ u ,
X 4 ⫽x 3 ⳵ u ,
F⫽3 ln x⫺2x ⫺2 u x ⫹F̃ 共 ␻ 兲 , ␻ ⫽x ⫺1 u xx ⫺2x ⫺2 u x ;
5
A 3,4
共 f ⫽␭x, ␭⫽0 兲 丣 兵 X 4 其 :
X 2 ⫽ 共 ⫺t⫹␭x 兲 ⳵ u ,
X 1⫽ ⳵ u ,
X 3 ⫽ ⳵ t ⫹u ⳵ u ,
X 4 ⫽ ⳵ x ⫹␭ ⳵ t ,
␭⫽0,
F⫽⫺␭ ⫺1 u x ⫹e t⫺␭x F̃ 共 ␻ 兲 , ␻ ⫽e ⫺t⫹␭x u xx ;
6
A 3,4
丣 兵 X 4其 :
X 1⫽ ⳵ u ,
X 3 ⫽ ⳵ x ⫹u ⳵ u ,
F⫽e x F̃ 共 ␻ 兲 ,
1
A 3,5
丣 兵 X 4其 :
X 2 ⫽⫺x ⳵ u ,
X 4⫽ ⳵ t ,
␻ ⫽e ⫺x u xx ;
X 1⫽ ⳵ t ,
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫹u ⳵ u ,
3
F⫽⫺2x ⫺2 u x ⫹F̃ 共 ␻ 兲 ,
1
A 3,6
丣 兵 X 4其 :
X 2⫽ ⳵ u ,
X 4 ⫽x 3 ⳵ u ,
␻ ⫽x ⫺1 u xx ⫺2x ⫺2 u x ;
X 1⫽ ⳵ t ,
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫺u ⳵ u ,
3
X 2⫽ ⳵ u ,
X 4 ⫽x ⫺3 ⳵ u ,
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
F⫽⫺20x ⫺2 u x ⫹x ⫺6 F̃ 共 ␻ 兲 ,
3
A 3,6
共 f ⫽e ⫺2 ␤
⫺1 x
, ␤ ⫽0 兲 丣 兵 X 4 其 :
␻ ⫽4x 4 u x ⫺x 5 u xx ,
X 2 ⫽e 2(t⫺ ␤
X 1⫽ ⳵ u ,
X 4⫽ ⳵ t⫹ ␤ ⳵ x ,
X 3 ⫽ ⳵ t ⫹u ⳵ u ,
F⫽⫺ 共 ␤ ⫹4 ␤ ⫺2 兲 u x ⫹e t⫺ ␤
␻ ⫽e ⫺t⫹ ␤
⫺1 x
⳵u ,
␤ ⫽0,
F̃ 共 ␻ 兲 ,
X 1⫽ ⳵ u ,
X 3 ⫽ ⳵ x ⫹u ⳵ u ,
X 2 ⫽e 2x ⳵ u ,
X 4⫽ ⳵ t ,
␻ ⫽e ⫺x 共 2u x ⫺u xx 兲 ;
F⫽⫺4u x ⫹e x F̃ 共 ␻ 兲 ,
X 1⫽ ⳵ t ,
X 3 ⫽t ⳵ t ⫹ 31 x ⳵ x ,
F⫽u ⫺2 u 3x F̃ 共 ␻ 兲 ,
3
A 3,7
丣 兵 X 4其 :
⫺1 x
⫺1 x)
共 2u x ⫹ ␤ u xx 兲 ;
4
A 3,6
共 f ⫽h⫽1 兲 丣 兵 X 4 其 :
1
丣 兵 X 4其 :
A 3,7
2303
X 2⫽ ⳵ x ,
X 4 ⫽u ⳵ u ,
␻ ⫽u ⫺2
x uu xx ;
X 1⫽ ⳵ t ,
X 2⫽ ⳵ u ,
x
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫹qu ⳵ u ,
3
X 4 ⫽x 3q ⳵ u ,
q⫽0, ⫾1,
F⫽⫺ 共 3q⫺1 兲共 3q⫺2 兲 x ⫺2 u x ⫹x 3(q⫺1) F̃ 共 ␻ 兲 ,
␻ ⫽x 1⫺3q 关共 3q⫺1 兲 u x ⫺xu xx 兴 ;
5
A 3,7
共 f ⫽e ⫺1(1⫺q) ␤
⫺1 x
, ␤ ⫽0 兲 丣 兵 X 4 其 :
X 1⫽ ⳵ u ,
X 4⫽ ⳵ t⫹ ␤ ⳵ x ,
X 2 ⫽e (1⫺q)(t⫺ ␤
⫺1 x
6
A 3,7
共 f ⫽h⫽1 兲 丣 兵 X 4 其 :
X 3 ⫽ ⳵ x ⫹u ⳵ u ,
⳵u ,
X 3 ⫽ ⳵ t ⫹u ⳵ u ,
␤ ⫽0, q⫽0, ⫾1,
F⫽⫺ 关 ␤ ⫹ 共 1⫺q 兲 2 ␤ ⫺2 兴 u x ⫹e t⫺ ␤
␻ ⫽e ⫺t⫹ ␤
⫺1 x)
⫺1 x
F̃ 共 ␻ 兲 ,
关共 1⫺q 兲 u x ⫹ ␤ u xx 兴 ;
X 1⫽ ⳵ u ,
X 4⫽ ⳵ t ,
X 2 ⫽e (1⫺q)x ⳵ u ,
q⫽0, ⫾1,
F⫽⫺ 共 1⫺q 兲 2 u x ⫹e x F̃ 共 ␻ 兲 ,
␻ ⫽e ⫺x 关共 1⫺q 兲 u x ⫺u xx 兴 .
1
丣 A 1 invariant equation is
Remark: The A 3,7
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2304
J. Math. Phys., Vol. 45, No. 6, June 2004
u t ⫽u xxx ⫹
Güngör, Lahno, and Zhdanov
u 3x
u
2
F̃ 共 ␻ 兲 ,
␻⫽
uu xx
u 2x
共6.1兲
.
If, in particular, F̃⫽c ␻ 2 , c⫽const, namely,
2
F⫽cu ⫺1
x u xx ,
共6.2兲
the symmetry algebra is further extended by X 5 ⫽ ⳵ u to a five-dimensional one.
B. Nondecomposable algebras
The set of inequivalent abstract four-dimensional Lie algebras contains ten real nondecomposable Lie algebras A 4,i ⫽ 兵 X 1 ,X 2 ,X 3 ,X 4 其 (i⫽1,...,10). 4,5 They are all solvable and therefore
can be written as semidirect sums of a one-dimensional Lie algebra 兵 X 4 其 and a three-dimensional
ideal N⫽ 兵 X 1 ,X 2 ,X 3 其 . For A 4,i (i⫽1,...,6), N is Abelian, for A 4,7 ,A 4,8 ,A 4,9 it is of type A 3,3
共nilpotent兲, and for A 4,10 it is of the type A 3,5 . The nonzero commutation relations read as
A 4,1 :
关 X 2 ,X 4 兴 ⫽X 1 ,
关 X 3 ,X 4 兴 ⫽X 2 ;
A 4,2 :
关 X 1 ,X 4 兴 ⫽qX 1 ,
关 X 2 ,X 4 兴 ⫽X 2 ,
关 X 3 ,X 4 兴 ⫽X 2 ⫹X 3 ,
关 X 1 ,X 4 兴 ⫽X 1 ,
A 4,3 :
关 X 1 ,X 4 兴 ⫽X 1 ,
A 4,4 :
q⫽0;
关 X 3 ,X 4 兴 ⫽X 2 ;
关 X 2 ,X 4 兴 ⫽X 1 ⫹X 2 ,
关 X 3 ,X 4 兴 ⫽X 2 ⫹X 3 ;
A 4,5 :
关 X 1 ,X 4 兴 ⫽X 1 ,
关 X 3 ,X 4 兴 ⫽pX 3 ,
⫺1⭐ p⭐q⭐1,
关 X 1 ,X 4 兴 ⫽qX 1 ,
A 4,6 :
关 X 2 ,X 4 兴 ⫽qX 2 ,
关 X 2 ,X 4 兴 ⫽ pX 2 ⫺X 3 ,
关 X 3 ,X 4 兴 ⫽X 2 ⫹ pX 3 ,
A 4,7 :
关 X 2 ,X 4 兴 ⫽X 2 ,
A 4,9 :
A 4,10 :
关 X 1 ,X 4 兴 ⫽2X 1 ,
关 X 1 ,X 4 兴 ⫽ 共 1⫹q 兲 X 1 ,
关 X 3 ,X 4 兴 ⫽qX 3 ,
关 X 2 ,X 3 兴 ⫽X 1 ,
关 X 2 ,X 4 兴 ⫽qX 2 ⫺X 3 ,
p⭓0;
关 X 3 ,X 4 兴 ⫽X 2 ⫹X 3 ;
关 X 2 ,X 3 兴 ⫽X 1 ,
A 4,8 :
q⫽0,
关 X 2 ,X 3 兴 ⫽X 1 ,
关 X 2 ,X 4 兴 ⫽X 2 ,
pq⫽0;
兩 q 兩 ⭐1;
关 X 1 ,X 4 兴 ⫽2qX 1 ,
关 X 3 ,X 4 兴 ⫽X 2 ⫹qX 3 ,
关 X 1 ,X 3 兴 ⫽X 1 ,
关 X 1 ,X 4 兴 ⫽⫺X 2 ,
q⭓0;
关 X 2 ,X 3 兴 ⫽X 2 ,
关 X 2 ,X 4 兴 ⫽X 1 .
In order to obtain realizations of solvable four-dimensional symmetry algebras of PDEs that
belong to the class 共1.1兲, we add X 4 in the generic form 共2.4兲 to the already constructed three-
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2305
dimensional symmetry algebras and impose the above commutation relations. Once the algebra is
found we insert X 4 into Eq. 共2.5兲 and solve it for the function F. The form of F which is invariant
under a three-dimensional algebra is further restricted:
1
:
A 4,1
X 1⫽ ⳵ u ,
X 2⫽ ⳵ x ,
X 3⫽ ⳵ t ,
X 4 ⫽t ⳵ x ⫹x ⳵ u ,
F⫽⫺ 12 u 2x ⫹F̃ 共 u xx 兲 ;
X 1⫽ ⳵ u ,
2
A 4,1
:
X 2 ⫽x ⳵ u ,
X 3⫽ ⳵ t ,
X 4 ⫽ ⳵ x ⫹tx ⳵ u ,
F⫽ 12 x 2 ⫹F̃ 共 u xx 兲 ;
1
A 4,2
:
X 1⫽ ⳵ t ,
X 2⫽ ⳵ u ,
X 3⫽ ⳵ x ,
X 4 ⫽3t ⳵ t ⫹x ⳵ x ⫹ 共 x⫹u 兲 ⳵ u ,
2
F̃ 共 e u x u xx 兲 ;
F⫽u xx
2
:
A 4,2
X 1⫽ ⳵ x ,
X 2⫽ ⳵ u ,
x
X 4 ⫽t ⳵ t ⫹ ⳵ x ⫹ 共 t⫹u 兲 ⳵ u ,
3
X 3⫽ ⳵ t ,
3
F⫽ ln兩 u x 兩 ⫹F̃ 共 ␻ 兲 ,
2
X 1⫽ ⳵ t ,
3
A 4,2
:
␻⫽
4
:
A 4,2
⫺1 ⫺1)
F̃ 共 ␻ 兲 ,
⫺1
X 2⫽ ⳵ u ,
F⫽⫺ 共 2⫺3q 兲共 1⫺3q 兲 x ⫺2 u x ⫹3 ln x⫹F̃ 共 ␻ 兲 ,
X 2⫽ ⳵ x ,
X 1⫽ ⳵ t ,
X 3⫽ ⳵ t ,
q⫽0,1,
␻ ⫽ 共 2⫺3q 兲 x ⫺2 u x ⫺x ⫺1 u xx ;
X 3⫽ ⳵ t ,
F⫽⫺u x ln兩 u x 兩 ⫹u x F̃
2
:
A 4,3
⫺1
␻ ⫽x 1⫺3q u x ⫹x 2⫺3q u xx ;
X 1 ⫽x 3(1⫺q) ⳵ u ,
X 1⫽ ⳵ u ,
;
q⫽0,
X 4 ⫽t ⳵ t ⫹ 13 x ⳵ x ⫹ 共 u⫹t 兲 ⳵ u ,
1
:
A 4,3
ux
X 3 ⫽⫺3q ⫺1 ln x ⳵ u ,
X 2⫽ ⳵ u ,
X 4 ⫽qt ⳵ t ⫹ 13 qx ⳵ x ⫹u ⳵ u ,
F⫽⫺2x ⫺2 u x ⫹x 3(q
2
u xx
X 2⫽ ⳵ u ,
X 4 ⫽t ⳵ x ⫹u ⳵ u ,
冉 冊
u xx
;
ux
X 3 ⫽⫺3 ln x ⳵ u ,
X 4 ⫽t ⳵ t ⫹ 13 x ⳵ x ,
F⫽⫺2x ⫺2 u x ⫹x ⫺3 F̃ 共 ␻ 兲 ,
3
:
A 4,3
X 1⫽ ⳵ u ,
␻ ⫽xu x ⫹x 2 u xx ;
X 2 ⫽e x ⳵ u ,
X 3⫽ ⳵ t ,
X 4 ⫽ ⳵ x ⫹ 共 u⫹te x 兲 ⳵ u ;
F⫽⫺u x ⫹xe x ⫹e x F̃ 共 ␻ 兲 ,
␻ ⫽e ⫺x 共 u x ⫺u xx 兲 ;
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2306
J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
X 1 ⫽ ⳵ u , X 2 ⫽⫺3 ln x ⳵ u , X 3 ⫽ ⳵ t ,
1
A 4,4
:
X 4 ⫽t ⳵ t ⫹ 13 x ⳵ x ⫹ 共 u⫺3t ln x 兲 ⳵ u ;
1
:
A 4,5
X 1⫽ ⳵ t ,
X 2⫽ ⳵ x ,
X 3⫽ ⳵ u ,
F̃ 共 ␻ 兲 ,
F⫽u 3(1⫺k)/(1⫺3k)
x
2
:
A 4,5
X 1⫽ ⳵ t ,
x
X 4 ⫽t ⳵ t ⫹ ⳵ x ⫹ku ⳵ u ,
3
k⫽0,
1
,
3
␻ ⫽u (3k⫺2)/(1⫺3k)
u xx ;
x
X 2⫽ ⳵ x ,
X 3⫽ ⳵ u ,
x
u
X 4 ⫽t ⳵ t ⫹ ⳵ x ⫹ ⳵ u ,
3
3
2
F⫽u xx
F̃ 共 u xx 兲 ;
3
:
A 4,5
X 1⫽ ⳵ t ,
X 2 ⫽u, X 3 ⫽x 3(q⫺p) ⳵ u ,
X 4 ⫽t ⳵ t ⫹ 13 x ⳵ x ⫹qu ⳵ u ,
q⫽ p,
q•p⫽0,
F⫽⫺ 关 3 共 q⫺ p 兲 ⫺1 兴关 3 共 q⫺ p 兲 ⫺2 兴 x ⫺2 u x ⫹x 3(q⫺1) F̃ 共 ␻ 兲 ,
␻ ⫽ 关 3 共 q⫺p 兲 ⫺1 兴 x 1⫺3q u x ⫺x 2⫺3q u xx ;
1
A 4,7
:
X 1⫽ ⳵ u ,
X 2⫽ ⳵ x ,
F⫽
2
:
A 4,7
X 3 ⫽x ⳵ u ⫺ 13 ln t ⳵ x ,
X 4 ⫽3t ⳵ t ⫹x ⳵ x ⫹2u ⳵ u ,
1 2 ⫺1/3
u ⫹t
F̃ 共 u xx 兲 ;
6t x
X 1⫽ ⳵ u ,
X 2 ⫽x ⳵ u ⫹b ⳵ x ,
X 3 ⫽⫺ ⳵ x ,
X 4 ⫽⫺b 2 共 b ⬘ 兲 ⫺1 ⳵ t ⫹ 共 1⫺b 兲 x ⳵ x ⫹ 共 2u⫺ 21 x 2 兲 ⳵ u ,
b⫽b 共 t 兲 , b ⬘ ⫽0,
b 2 b ⬙ ⫹ 共 b⫺3 兲共 b ⬘ 兲 2 ⫽0,
⫺1
F⫽⫺ 12 b ⬘ u 2x ⫹b ⫺3 e ⫺b F̃ 共 ␻ 兲 ,
3
:
A 4,7
X 1⫽ ⳵ u ,
␻ ⫽b 2 u xx ⫺b;
X 2 ⫽ 共 ␭x 3 ⫺t 兲 ⳵ u ,
X 3⫽ ⳵ t ,
X 4 ⫽t ⳵ t ⫹ 13 x ⳵ x ⫹ 共 2u⫺ 12 t 2 ⫹␭tx 3 兲 ⳵ u ,
F⫽⫺ 13 ␭ ⫺1 共 1⫹6␭ 兲 x ⫺2 u x ⫹3␭x 3 ln x⫹x 3 F̃ 共 ␻ 兲 ,
4
:
A 4,7
X 1⫽ ⳵ u ,
X 2 ⫽ 共 t⫺x 兲 ⳵ u ,
冉
X 4 ⫽3t ⳵ t ⫹ 共 x⫹2t 兲 ⳵ x ⫹ xt⫺
t
F⫽⫺u x ⫹t ⫺1/3F̃ 共 ␻ 兲 ⫹ ,
4
␭⫽0,
␻ ⫽2x ⫺5 u x ⫺x ⫺4 u xx ;
X 3⫽ ⳵ x ,
冊
x2
⫹2u ⳵ u ,
2
1
␻ ⫽u xx ⫹ ln t;
3
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J. Math. Phys., Vol. 45, No. 6, June 2004
5
A 4,7
:
KdV-type nonlinear evolution equations
X 1⫽ ⳵ u ,
X 2 ⫽⫺x ⳵ u ,
冉
X 4 ⫽3t ⳵ t ⫹x ⳵ x ⫹ 2u⫺
F⫽t ⫺1/3F̃ 共 ␻ 兲 ,
X 1 ⫽⫺x ⫺1 ⳵ u ,
6
A 4,7
:
2307
X 3⫽ ⳵ x ,
冊
x2
⳵ ,
2 u
␻ ⫽u xx ⫹ 31 ln t;
X 3 ⫽ ⳵ x ⫺x ⫺1 u ⳵ u ,
X 2⫽ ⳵ u ,
X 4 ⫽3t ⳵ t ⫹x ⳵ x ⫹ 共 u⫹ 21 x 兲 ⳵ u ,
F⫽3x ⫺1 u xx ⫹x ⫺1 t ⫺ 1/3F̃ 共 ␻ 兲 ,
1
A 4,8
:
X 1⫽ ⳵ x ,
X 2⫽ ⳵ t ,
␻ ⫽2u x ⫹xu xx ⫺ 31 ln t;
x
2
X 4 ⫽t ⳵ t ⫹ ⳵ x ⫺ u ⳵ u ,
3
3
X 3 ⫽t ⳵ x ⫹ ⳵ u ,
⫺4/3
F⫽⫺uu x ⫹u 5/3
u xx 兲 ;
x F̃ 共 u x
2
:
A 4,8
X 1⫽ ⳵ u ,
X 2⫽ ⳵ t ,
F⫽
x
⫹u 1/3
x F̃ 共 ␻ 兲 ,
␭
X 1⫽ ⳵ u ,
3
:
A 4,8
x
4
X 4 ⫽t ⳵ t ⫹ ⳵ x ⫹ u ⳵ u ,
3
3
X 3 ⫽t ⳵ u ⫹␭ ⳵ x ,
␻ ⫽u ⫺2/3
u xx ,
x
X 2⫽ ⳵ x ,
X 3 ⫽x ⳵ u ⫹␭t (1⫺q)/3⳵ x ,
X 4 ⫽3t ⳵ t ⫹x ⳵ x ⫹ 共 1⫹q 兲 u ⳵ u ,
F⫽
␭ 共 q⫺1 兲 ⫺(2⫹q)/3 2 (q⫺2)/3
u x ⫹t
F̃ 共 ␻ 兲 ,
t
6
4
A 4,8
:
X 1⫽ ⳵ u ,
X 1⫽ ⳵ u ,
3
␻ ⫽u xx
共 t⫺3␭u x 兲 2 ;
X 2 ⫽ 共 ␭x 3 ⫺t 兲 ⳵ u ,
⫺1
F⫽⫺ 13 ␭ ⫺1 共 1⫹6␭ 兲 x ⫺2 u x ⫹x 3q F̃ 共 ␻ 兲 ,
X 1⫽ ⳵ u ,
兩 q 兩 ⫽1;
␭⫽0,
X 4 ⫽qt ⳵ t ⫹ 13 qx ⳵ x ⫹ 共 1⫹q 兲 u ⳵ u ,
6
:
A 4,8
␭⫽0,
X 3 ⫽x ⳵ u ⫹␭ ⳵ t ,
X 4 ⫽3t ⳵ t ⫹x ⳵ x ⫹4u ⳵ u ,
5
:
A 4,8
q苸R,
␻ ⫽t (1⫺q)/3u xx ,
X 2⫽ ⳵ x ,
F⫽ 共 t⫺3␭u x 兲 1/3F̃ 共 ␻ 兲 ,
␭⬎0;
X 3⫽ ⳵ t ,
␭•q⫽0,
␻ ⫽2x ⫺(2⫹3q
X 2 ⫽ 共 t⫺x 兲 ⳵ u ,
⫺1 )
u x ⫺x ⫺(1⫹3q
⫺1 )
u xx ;
X 3⫽ ⳵ x ,
X 4 ⫽3qt ⳵ t ⫹q 共 x⫹2t 兲 ⳵ x ⫹ 共 1⫹q 兲 u ⳵ u ,
⫺1
F⫽⫺u x ⫹t 共 1/3兲 (1⫺2q)q F̃ 共 ␻ 兲 ,
7
A 4,8
:
X 1 ⫽⫺x ⫺1 ⳵ u ,
X 2⫽ ⳵ u ,
⫺1
␻ ⫽t 共 1/3兲 (q⫺1)q u xx ;
X 3 ⫽ ⳵ x ⫺x ⫺1 u ⳵ u ,
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2308
J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
X 4 ⫽3qt⫹qx ⳵ x ⫹u ⳵ u ,
⫺1
⫺1
F⫽3x ⫺1 u xx ⫹x ⫺1 t 共 1/3兲 (q⫺1)q F̃ 共 ␻ 兲 ,
8
:
A 4,8
X 1⫽ ⳵ u ,
X 2 ⫽⫺x ⳵ u ,
␻ ⫽t 共 1/3兲 (q⫺1)q 共 2u x ⫹xu xx 兲 ;
X 3 ⫽x ⳵ x ,
⫺1
q⫽0,
X 4 ⫽3qt ⳵ t ⫹qx ⳵ x ⫹ 共 1⫹q 兲 u ⳵ u ,
⫺1
F⫽t 共 1/3兲 (1⫺2q)q F̃ 共 ␻ 兲 ,
␻ ⫽t 共 1/3兲 (q⫺1)q u xx .
Remarks:
• There exists a realization of A 4,6 :
1
A4,6
: X 1⫽ ⳵ t ,
X 2 ⫽tan ␺⳵u ,
X3⫽⳵u ,
X 4 ⫽2t ⳵ t ⫹ 23 x ⳵ x ⫹ 关 p⫹tan ␺ 兴 u ⳵ u ,
␺ ⫽ 32 ln x,
p苸R.
However there are no equations that can be invariant under this algebra.
1
is isomorphic to the KdV algebra which is the semidirect sum of the
• The algebra A 4,8
nilradical 共maximal nilpotent ideal兲 h(2)⫽ 兵 X 1 ,X 2 ,X 3 其 and the dilation D⫽ 兵 X 4 其 :
1
:
A 4,9
X 4 ⫽⫺
X 1⫽ ⳵ u ,
X 2⫽ ⳵ x ,
X 3 ⫽ ␣ 共 t 兲 ⳵ x ⫹x ⳵ u ,
冉
冊
x2
共 1⫹ ␣ 2 兲
⳵ t ⫹ 共 q⫺ ␣ 兲 x ⳵ x ⫹ 2qu⫺
⳵ ,
␣˙
2 u
F⫽⫺ 12 ␣˙ u 2x ⫹ 共 1⫹ ␣ 2 兲 ⫺3/2 exp共 q arctan ␣ 兲 F̃ 共 ␻ 兲 ,
q苸R,
␻ ⫽ 共 1⫹ ␣ 2 兲 u xx ⫺ ␣ ,
共 1⫹ ␣ 2 兲 ␣¨ ⫹ 共 ␣ ⫺3q 兲 ␣˙ 2 ⫽0.
The function ␣ (t), ␣˙ ⫽0 is a solution of the ordinary differential equation 共5.15兲:
1
:
A 4,10
X 1⫽ ⳵ u ,
X 2 ⫽⫺tan x ⳵ u ,
X 4 ⫽ ␤ ⳵ t ⫹ ⳵ x ⫹u tan x ⳵ u ,
X 3 ⫽ ⳵ t ⫹u ⳵ u ,
␤ 苸R,
F⫽⫺2u x ⫺3 tan xu xx ⫹e t⫺ ␤ x sec xF̃ 共 ␻ 兲 ,
␻ ⫽e ␤ x⫺t 共 cos xu xx ⫺2 sin xu x 兲 .
We sum up the above results as a theorem.
Theorem 6.1: There exist fifty-two inequivalent four-dimensional symmetry algebras admitted
by Eq. (1.1). The explicit forms of those algebras as well as the associated invariant equations are
given above.
VII. DISCUSSION AND CONCLUSIONS
In this paper we provide a symmetry classification of the KdV type equations involving an
arbitrary function of five arguments. We find that the equivalence classes of invariant equations
involve an arbitrary function of four, three, two variables and one variable as soon as the symmetry algebra is one-, two-, three- and four-dimensional, respectively. In particular, we studied
symmetries of the most general third order linear evolution equation. What came out from this, to
our surprise, is that the symmetry group allowed is four-dimensional at most, while there are
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2309
nonlinear equations with symmetry algebras greater than four. This result is in contrast to the
second-order evolution equations. It is exactly the linear heat equation that allows for the maximal
symmetry algebra.
To complete the classification list, it only remains to obtain the inequivalent equations invariant under solvable algebras of the dimension dimL⭓5. But this would require to going through a
large number of isomorphism classes. To give an idea of the complexity of this task let us recall
that there are sixty-six classes of nonisomorphic real, solvable Lie algebras of dimension five. For
dimension six, there exist ninety-nine classes of them with a nilpotent element. We plan to devote
a separate article to study equations admitting higher-dimensional symmetry algebras.
Whenever F is an arbitrary function of its arguments, the symmetry algebras given in the
paper are maximal. In particular, if we impose the requirement that function F is be independent
of u xx then we find that ␾ ⫽R(t)u⫹S(x,t) in 共2.4兲. In this case, invariance under fourdimensional algebras will force F to depend on an arbitrary constant rather than on an arbitrary
function. Then, they may admit symmetry groups of the dimension higher than four. We have
analyzed this restricted class of equations and obtained that the only equation whose symmetry
1
for F̃⫽const. On the
algebra is higher than four is the one corresponding to the realization A 4,1
other hand, for the specific choices of F̃ involving one variable, the equations with fourdimensional symmetry algebras may be invariant under larger symmetry groups. For instance, the
1
4/3
obtained by setting F̃⫽cu xx
, c⫽const admits
particular case of the equation invariant under A 4,1
an additional symmetry group generated by the dilation operator X 5 ⫽3t ⳵ t ⫹x ⳵ x ⫺u ⳵ u .
We only presented representative lists of equivalence classes of invariant equations. All other
invariant equations can be recovered from these lists by applying the point transformations 共2.6兲.
In other words, an equation in the class 共1.1兲 will have a symmetry group with dimension satisfying dimL⭐6 if and only if it can be transformed to one in the 共canonical兲 equations from the
list.
As we mentioned, our classification is performed within point transformations of coordinates.
Two equations are equivalent if one can be obtained from the other by a change of variables. On
the other hand, consider a special case of 共6.2兲 for c⫽⫺3/4, 24
u t ⫽u xxx ⫺
2
3 u xx
,
4 ux
which additionally allows a symmetry group generated by 兵 ⳵ u 其 . Though this equation is equivalent to the third-order linear equation v t ⫽ v xxx under the 共no-point兲 transformation v ⫽ 冑u x , we
treat them as inequivalent.
To give a reader an insight into possible applications of the results of this article, we consider
a subclass of Eqs. 共1.1兲,
u t ⫽u xxx ⫹uu x ⫹ f 共 t 兲 u,
共7.1兲
which arises in several physical applications such as the propagation of waves in shallow water of
variable depth.
When f (t) is arbitrary, 共7.1兲 admits a two-dimensional Abelian symmetry algebra generated
by
X 1⫽ ⳵ x ,
X 2 ⫽ ␰ 共 t 兲 ⳵ x ⫺ ␰˙ 共 t 兲 ⳵ u ,
␰⫽
冕 再冕
exp
f 共 t 兲 dt
冎
dt.
共7.2兲
3
By the change of dependent variable ũ⫽u/ ␰˙ , the generators are transformed to the realization A 2,1
with ␣ ⫽⫺ ␰ . The corresponding invariant equation takes the form
ũ t ⫽ũ xxx ⫹ ␰˙ ũũ x ,
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2310
J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
which is a particular case of 共1.2兲.
For the special case f (t)⫽at k (a⫽0), the algebra is larger and we have the following
possibilities for the algebra to be either three- or four-dimensional.
共1兲 (a,k)⫽(a,⫺1), a⫽⫺1: The equation admits the three-dimensional indecomposable
solvable symmetry algebra spanned by
X 1⫽ ⳵ x ,
X 2 ⫽t 1⫹a ⳵ x ⫺ 共 1⫹a 兲 t a ⳵ u ,
x
2
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫺ u ⳵ u ,
3
3
共7.3兲
with nonzero commutation relations
1
关 X 1 ,X 3 兴 ⫽⫺ X 1 ,
3
关 X 2 ,X 3 兴 ⫽⫺
3a⫹2
X2 .
3
For a⫽⫺1/3, the algebra is isomorphic, up to the scaling of basis elements, to A 3,5 , for
⫺1⬍a⬍⫺ 13 , to A 3,7 .
For a⫽⫺2/3 it is isomorphic to the decomposable solvable algebra A 3,2 and a suitable basis
is
1
X 2 ⫽t 1/3⳵ x ⫺ t ⫺2/3⳵ u ,
3
X 1⫽ ⳵ x ,
x
2
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫺ u ⳵ u .
3
3
With the equivalence transformation
t̃ ⫽t,
ũ⫽⫺3t 2/3u,
x̃⫽x,
3
. The transformed equation
the basis elements are transformed, up to scaling, to the realization A 3,2
is
ũ t ⫽ũ xxx ⫺ 31 t ⫺2/3ũũ x .
3
This equation belongs to the class corresponding to the realization A 3,2
.
2
We note that a member of 共1.2兲 for f ⫽1, g⫽t 共see Ref. 1兲 is equivalent, under appropriate
point transformation, to the above equation. Similarly, the particular case a⫽⫺ ␣ /(1⫹ ␣ ), ␣
⫽0,1,2 is equivalent to f ⫽1, g⫽t ␣ of 共1.2兲. In this case, the symmetry algebra is indecomposable
and solvable.
共2兲 (a,k)⫽(⫺1,⫺1): the spherical KdV 共sKdV兲 equation.
In this case the equation is invariant with respect to a three-dimensional symmetry algebra.
We choose its basis to be
X 1⫽ ⳵ x ,
1
X 2 ⫽ln t ⳵ x ⫺ ⳵ u ,
t
x
2
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫺ u ⳵ u ,
3
3
共7.4兲
with nonzero commutation relations
关 X 3 ,X 1 兴 ⫽⫺ 31 X 1 ,
关 X 3 ,X 2 兴 ⫽X 1 ⫺ 13 X 2 .
It is easy to see that this algebra is isomorphic to A 3,4 . Under the transformation ũ⫽3tu, the
2
. The sKdV equation takes the form
generators are transformed to the realization A 3,4
ũ t ⫽ũ xxx ⫹
1
ũũ ,
3t x
2
.
which is a particular case of the equation invariant under the algebra A 3,4
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2311
We note that a member of 共1.2兲 for f ⫽1, g⫽e 3t1 is equivalent to the case 共2兲, i.e. the sKdV
equation.
共3兲 (a,k)⫽(a,0): The basis of the symmetry algebra reads as
X 1⫽ ⳵ x ,
X 2 ⫽e at 共 ⳵ x ⫺a ⳵ u 兲 ,
X 3⫽ ⳵ t ,
共7.5兲
with nonzero commutation relation 关 X 3 ,X 2 兴 ⫽aX 2 . The algebra is isomorphic to A 3,2 . With the
transformation ũ⫽e ⫺at u the equation is transformed to a special case of 共1.2兲 for f ⫽1,g⫽e at .
共4兲 (a,k)⫽(⫺1/2,⫺1): the cylindrical KdV 共cKdV兲 equation.
In this case the symmetry algebra is four-dimensional. In a convenient basis we have
X 1 ⫽2 冑t ⳵ x ⫺
X 2 ⫽4t 3/2⳵ t ⫹2x 冑t ⳵ x ⫺
X 3⫽ ⳵ x ,
1
冑t
冉
⳵u ,
x
冑t
冊
⫹4 冑tu ⳵ u ,
共7.6兲
X 4 ⫽3t ⳵ t ⫹x ⳵ x ⫺2u ⳵ u ,
with nonzero commutation relations
关 X 2 ,X 3 兴 ⫽⫺X 1 ,
关 X 1 ,X 4 兴 ⫽⫺ 21 X 1 ,
关 X 2 ,X 4 兴 ⫽⫺ 32 X 2 ,
关 X 3 ,X 4 兴 ⫽X 3 .
We see that the symmetry algebra of the cKdV equation is isomorphic to the algebra A 4,8 with
q⫽1. The existence of such an isomorphism is a necessary, but not sufficient condition for a local
point transformation to exist, transforming the two equations into each other. Comparing these
generators with 共2.10兲 and choosing 共2.8兲 suitably, for example, first transforming the commuting
elements 兵 X 1 ,X 3 其 into 兵 ⳵ x̃ , t̃ ⳵ x̃ ⫹ ⳵ ũ 其 and then transforming the remaining ones with the aid of the
freedom left in equivalence transformations we arrive at
t̃ ⫽2t ⫺1/2,
x̃⫽t ⫺1/2x,
x
ũ⫽tu⫹ ,
2
which establishes the equivalence of the Lie algebra with basis 共7.6兲 and the cKdV equation to the
1
) and KdV equation. This connection between the KdV and cKdV equations is
KdV algebra (A 4,8
well-known in the literature.19
As a further comparison of the results obtained in the article we consider
u t ⫹u xxx ⫹ f 共 u 兲 u kx ⫽0,
k⬎0,
共7.7兲
which is clearly a special case of 共1.1兲. Group classification of this equation is given in a table 共see
Table II兲.22 These results can immediately be derived from those obtained in this paper either
directly or performing a change of independent or dependent variables.
Note that the equations that do not appear in the classification list can be recovered from those
by suitable point transformations.
A number of integrable KdV type equations can be reproduced by restricting the arbitrary
2
is
functions contained in invariant equations of this article. For example, the realization A 3,7
⫺1/3
equivalent to 兵 ⳵ t , ⳵ x ,t ⳵ t ⫹x/3⳵ x ⫺u/3⳵ u 其 under the transformation u→u
. We have the invariant function
F⫽u 4 F̃ 共 ␻ 1 , ␻ 2 兲 ,
␻ 1 ⫽u ⫺2 u x ,
␻ 2 ⫽u ⫺3 u xx .
Setting F̃⫽ ␻ 1 produces the modified KdV 共mKdV兲 equation
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2312
J. Math. Phys., Vol. 45, No. 6, June 2004
Güngör, Lahno, and Zhdanov
TABLE II. Symmetry classification of 共7.7兲.
N
k
f (u)
Symmetry generators
Symmetry algebra
1
arb.
arb.
⳵t ,⳵x
A 12,1
2
k
un
A 23,7
3
k
eu
4
k
1
5
3
arb.
6
3
u ⫺2
7
3
1
8
1
u n ⫹c
9
1
u
x
k⫺3
u⳵ , k⫹n⫽1
⳵t ,⳵x ,t⳵t⫹ ⳵x⫹
3
k⫹n⫺1 u
x
⳵t ,⳵x ,t⳵t⫹ ⳵x⫹共k⫺3兲⳵u
3
x
k⫺3
⳵t ,⳵x ,t⳵t⫹ ⳵x⫹
u⳵ ,⳵ , k⫽1
3
3共k⫺1兲 u u
x
⳵t ,⳵x ,t⳵t⫹ ⳵x
3
x
⳵t ,⳵x ,t⳵t⫹ ⳵x ,u⳵u
3
x
⳵t ,⳵x ,t⳵t⫹ ⳵x ,⳵u
3
x
2
⳵t ,⳵x ,t⳵t⫹ ⳵x⫺ u⳵u , n⫽0
3
n
x
2
⳵t ,⳵x ,t⳵t⫹ ⳵x⫺ u⳵u ,t⳵x⫹⳵u
3
3
10
1
e u ⫹c
⳵t ,⳵x ,t⳵t⫹ 3 共⳵x⫹2ct兲⳵x⫺ 3 ⳵u
11
1
1
⳵t ,⳵x ,t⳵t⫹ 3 共x⫹2t兲⳵x ,u⳵u ,g共x,t兲⳵u
g t ⫹g xxx ⫹g x ⫽0
1
2
1
u t ⫽u xxx ⫹u 2 u x .
A 23,7
A 14,3
A 13,7
A 13,7 丣 A 1
A 13,7 丣 A 1
A 23,7
A 14,6
A 23,7
Linear equation
(N⫽6 in Table I兲
共7.8兲
Since the maximal symmetry algebra of the mKdV equation is three-dimensional, it is not isomorphic to the KdV algebra. This implies that there is no point transformation, transforming the
mKdV equation into KdV equation. In this respect let us mention that there is the well-known
nonlocal transformation 共Miura transformation兲,
ũ⫽u 2 ⫾ 冑6iu x ,
taking the mKdV 共7.8兲 into the KdV equation ũ t ⫽ũ xxx ⫹ũũ x . Another integrable equation which
can be obtained from our classification is23
u t ⫽u xxx ⫹3 共 u xx u 2 ⫹3uu 2x 兲 ⫹3u 4 u x .
2
. Note that this equation can be linearized by a change
Its symmetry algebra is isomorphic to A 3,7
of dependent variable.
Let us mention that a classification based on higher order symmetries of third order integrable
nonlinear equations of the form
u t ⫽u xxx ⫹F 共 u,u x ,u xx 兲
共7.9兲
is given in Ref. 24. We should also note that the question of finding PDEs admitting Lie point
symmetries is different than finding integrable PDEs. In the latter case one requires the existence
of a generalized one as opposed to Lie point symmetries. For the classification of integrable PDEs
we refer the reader to Refs. 25–27.
Finally, let us point out that in a very recent work28 a class of integrable 共in the sense of
existence of an infinite number of generalized symmetries兲 third order evolution equations of the
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J. Math. Phys., Vol. 45, No. 6, June 2004
KdV-type nonlinear evolution equations
2313
form 共7.9兲 for specific F admitting recursion operators have been analyzed. Among others, the
special cases corresponding to F̃⫽3 ␻ and F̃⫽3/2⫺3 ␻ of 共6.1兲 produce the following equations
1
丣 A1 :
with 4-dimensional symmetry algebra A 3,7
u t ⫽u xxx ⫹3u ⫺1 u x u xx ,
u t ⫽u xxx ⫺3u ⫺1 u x u xx ⫹ 23 u ⫺2 u 3x ,
both of which were shown to admit recursion operators. This fact indicates that many equations
with relatively large symmetry groups in our classification are among the most probable candidates for being integrable.
We note that the maximal symmetry algebra of the first equation of the above list is infinitedimensional with basis elements:
X 1⫽ ⳵ t ,
X 2⫽ ⳵ x ,
x
u
X 3 ⫽t ⳵ t ⫹ ⳵ x ⫹ ⳵ u ,
3
2
X 共 ␳ 兲 ⫽ ␳ 共 x,t 兲 u ⫺1 ⳵ u ,
␳ t ⫽ ␳ xxx .
The existence of an infinite-dimensional symmetry algebra suggests linearizability of the equation
by point transformations and, indeed, it is linearized by the change of dependent variable
v (x,t)⫽u 2 (x,t)/2.
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