Galaktionov V.A., Svirshchevskii S.R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics

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2 Invariant Subspaces: Mathematics in 1D 73

2.6.1 Main Theorem for polynomial subspaces

From now on, we concentrate on subspaces of the polynomial type,

= L{1, x , ... , x

n−1

}, (2.96)

deﬁned by the simplest linear ODE

(n)

= 0. (2.97)

As explained in Section 2.1, we introduce the ﬁrst integrals via solutions g

of the

conjugate equation (−1)

(n)

= 0. Choosing g

(x ) = f

(x ) = x

i−1

for i = 1, ..., n

yields the complete set of ﬁrst integrals in the form of



j =1

(−1)

j −1

(i−1)!

(i−j )!

i−j

(n−j )

for i = 1, ..., n. (2.98)

Therefore, Theorem 2.1 reads as follows:

Theorem 2.23

The set

n−1

of nonlinear operators

preserving the polynomial

subspace

(2.96)

is generated by the operators

F[y] =



i=1

, ... , I

i−1

, (2.99)

where

, ... , I

)

i = 1, ..., n

, are any smooth functions of ﬁrst integrals

(2.98)

As was mentioned above, every operator (2.3) preserving subspace (2.1) deﬁnes

aLie-B¨acklund symmetry operator X = F[y]

∂

∂y

of the corresponding linear ODE

(2.2). The whole set F

n−1

of such operators forms an inﬁnite-dimensional Lie alge-

bra completelydescribed by Theorem 2.1. In its turn, Theorem 2.23 providesus with

all Lie-B¨acklund symmetries for equation (2.97) that were found in [12].

Along with the algebra F

n−1

, we consider its linearsubspaces F

with m ≤ n −2,

consisting of operators (2.3) of the order not greater than m. Operators of the order

not greater than n − 2 give a linear space F

n−2

. We call these the operators or

symmetries of submaximal order. One purpose of this section is to obtain a general

representation for the sets F

, with m ≤ n − 2, in the case of the subspace (2.96)

(equation (2.97)). In particular,we provethat the dimension of the linear space F

n−2

of the operators of submaximal order is expressed as

dimF

n−2

= C

2n−1

(2.100)

via the binomial coefﬁcient C

k!(n−k)!

For n = 3, this has been actually shown by Lie [393], who found that the maximal

dimension of the algebra of contact symmetries on the plane cannotexceedC

= 10,

and that this maximal value is attained on the equation y



= 0. A generalization of

this result for arbitrary ODEs of the order n = 4was obtained in [298].This, together

with Theorem 2.24, proved below, makes plausible the following conjecture (this is

OPEN PROBLEM).

Conjecture 2.1 For any nth-order ODEs (linear or not), the maximal dimension of

the linear space F

n−2

of symmetries of submaximal order is equal to C

2n−1

74 Exact Solutions and Invariant Subspaces

Bearing in mind applications to PDEs, we also consider the subspace



n−1

⊂

n−1

of translation-invariant quadratic operators and give its complete description.

Finally, we perform a description of all the translation-invariant operators preserving

(2.96).

symmetry operators,orsimplysymmetries of these ODEs.

2.6.2 Operators of submaximal and lower orders

A symmetry operator (2.3) for (2.2) is deﬁned by the invariance condition (2.4),

L[F[y]]



L[y]=0

≡ 0. (2.101)

We will assume that the function F in (2.3) is analytic and is represented by a

convergent power series in variables y, y



, ... , y

(n−1)

, with coefﬁcients depending

on x. Considering this series as an inﬁnite sum of homogeneous polynomials in

y, y



, ... , y

(n−1)

, and using the fact that the ODE (2.2) is linear, we obtain from

(2.101) that every such homogeneous polynomial deﬁnes a symmetry as well (see

Remark 2.10). Hence, we can suppose beforehand that operators (2.3) have the ho-

mogeneous form

F[y] =



+... +i

n−1

, ... ,i

n−1

(x ) y



)

·... ·(y

(n−1)

)

n−1

where k = 0, 1, 2, ... and 0 ≤ i

, ..., i

n−1

≤ k. Using (2.5), one can express y

and its derivatives as linear homogeneous functions of I

, ... , I

, thus obtaining the

representation

F[y] =



+... +i

, ... ,i

(x )(I

)

·... ·(I

)

, (2.102)

where, in accordance with Theorem 2.1, the coefﬁcients γ

, ... ,i

(x ) are linear com-

binations of the fundamental set of solutions {f

(x )}.

Let us return to the equation (2.97). Symmetries of the order n −2 are singled out

by the condition

∂ F

∂y

(n−1)

≡



i=1

i−1

∂ F

∂ I

= 0, (2.103)

where F is considered as a function of x , I

, ... , I

.Theﬁrst integrals of the corre-

sponding characteristic system are

x and J

= xI

− I

i+1

for i = 1, ..., n − 1,

and therefore, the general expression for F is given by F =



F(x , J

, ... , J

n−1

) with

an arbitrary function



F. Passing in (2.102) from I

, i = 1, ..., n, to the new variables

= I

, J

= xI

− I

i+1

(i = 1, ..., n − 1),

so that I

= J

and I

i+1

= x

−



j =1

i−j

, we obtain

F[y] =



+... +i

n−1

, ... ,i

n−1

(x )(J

)

·... ·(J

n−1

)

n−1

, (2.104)

In this section, we refer to operators (2.3) admitted by equations (2.2) or (2.97) as

2 Invariant Subspaces: Mathematics in 1D 75

where we have taken into account that F does not depend on J

Notice that δ

, ... ,i

n−1

(x ) and (J

)

·... ·(J

n−1

)

n−1

are some polynomials in x of

degrees s

, ... ,i

n−1

and k respectively, with coefﬁcients, depending on I

, with i =

1, ..., n. The leading term of x in (2.104) can be written as

s+k



+... +i

n−1

, ... ,i

n−1

)

·... ·(I

n−1

)

n−1

where C

, ... ,i

n−1

are some constants, and s = max s

, ... ,i

n−1

. It follows from (2.99)

that s +k ≤ n −1, providing us with estimates s

, ... ,i

n−1

≤ n −1−k for k ≤ n −1.

By Theorem 2.23, every monomial

)

·... ·(J

n−1

)

n−1

, (2.105)

with i

, ... , i

n−1

≥ 0andi

+ ... +i

n−1

≤ n − 1, deﬁnes a symmetry of the ODE

(2.97). Therefore, F

n−2

is a linear span of these monomials. Calculating the number

of operators (2.105) yields

Theorem 2.24

The linear space

n−2

of symmetry operators of submaximal order

for equation

(2.97)

is spanned by the operators

(2.105)

,and

(2.100)

holds.

Remark 2.25 The theorem remains valid without the analyticity assumption on F.

Indeed, applying the condition (2.103) to the general representation (2.99) yields



i=1





j =1

j −1



i−1

≡ 0, where A

∂ A

∂ I

Equating the coefﬁcients of the different powers of x to zero gives the system



(i+j =k)

= 0fork = 2, ..., 2n, (2.106)

that leads to the necessary result.

In the next examples, we apply Theorems 2.23 and 2.24 to the cases n = 2 and 3.

Example 2.26 (n = 2)Forn = 2, equation (2.97) is



= 0, (2.107)

and, for the spaces F

and F

, we obtain the following results:

(i) ThewholeLie-B¨acklund algebra F

of the equation (2.107) is generated by op-

erators

F[y] = A

, I

) + A

, I

)x , (2.108)

where A

and A

are arbitrary functions of the ﬁrst integrals I

= y



and I



− y.

(ii) The linear space F

is 3D spanned by operators x

)

, i

≥ 0, i

≤ 1),

or, explicitly,

1, x, J

, (2.109)

where J

= xI

− I

= y. The operators (2.109) deﬁne a 3D subalgebra of the

algebra of Lie point symmetries of equation (2.107).

Illustrating Remark 2.25 above, note that, in this case, system (2.106) reads

= 0, A

+ A

= 0, A

= 0.

76 Exact Solutions and Invariant Subspaces

Its solutions are A

= C

− C

and A

= C

+ C

, that after substitution into

(2.108) yield F[y] = C

+ C

x + C

(xI

− I

). This coincides with (2.109).

Example 2.27 (n = 3)Forn = 3, equation (2.97) is



= 0, (2.110)

and, for the spaces F

and F

, Theorems 2.23 and 2.24 yield the following.

(i) ThewholeLie-B¨acklund algebra F

of the equation (2.110) is generated by

F[y] = A

, I

) + A

, I

)x + A

, I

where A

, A

,andA

are arbitrary functions of the ﬁrst integrals

= y



, I

= xy



− y



, and I

= x



− 2xy



+ 2y. (2.111)

(ii) The subspace F

(coinciding with the subalgebra of contact symmetries) is 10D

spanned by the operators x

)

, i

≥ 0, i

≤ 2), or,

1, x , x

, J

, xJ

, J

, xJ

,(J

)

, J

,(J

)

, where (2.112)

= xI

− I

= y



and J

= xI

− I

= xy



− 2y. (2.113)

Plugging the last expressions into (2.112) yields a basis of the well-known 10D al-

gebra of contact symmetries of (2.110) described by Lie in [393].

Using a generalrepresentationof the operators F ∈ F

n−2

given by Theorem 2.24,

and successively applying conditions ∂ F/∂y

(p)

= 0 with p = n − 2, n − 3, ... , 1,

one can obtain a complete description of the linear spaces F

n−1−k

for arbitrary k =

1, ..., n − 1. The result is expressed in terms of differences deﬁned recursively by

= xJ

k−1

− J

k−1

i+1

for i = 1, ..., n − k, (2.114)

with J

= I

for i = 1, ..., n,andJ

= J

for i = 1, ..., n − 1. Taking into account

that J

for i = 1, ..., n − k are polynomials in x of the degree k with coefﬁcients,

linearly depending on I

,...,I

, we arrive at the following theorem which includes

Theorem 2.24 as a special case.

Theorem 2.28

The linear space

n−1−k

1 ≤ k ≤ n − 1

, of symmetry operators of

the order not greater than

n − 1 −k

for equation

(2.97)

is spanned by the operators

)

·... ·(J

n−k

)

n−k

where

, i

, ... , i

n−k

are nonnegative integers satisfying the condition

+ k(i

... + i

n−k

) ≤ n − 1.

Example 2.29 (n = 4)Letn = 4, so equation (2.97) is

(4)

= 0. (2.115)

Theorems 2.24 and 2.28 yield the following conclusions:

(i) (k = 1) The subspace F

of the operators of submaximal order is 35D spanned

by the operators

)

for i

, i

≥ 0, i

≤ 3,

2 Invariant Subspaces: Mathematics in 1D 77

where J

= xI

− I

i+1

for i = 1, 2, and 3, and the ﬁrst integrals are written as

= y



, I

= xy



− y



, I

= x



− 2xy



+ 2y



and I

= x



− 3x



+ 6xy



− 6y.

(2.116)

Substituting (2.116) into the expressions for J

, one obtains

= y



, J

= xy



− 2y



, and J

= x



− 4xy



+ 6y. (2.117)

(ii) (k = 2) Space F

, corresponding to the algebra of point symmetries, is spanned

by x

)

, i

≥ 0, i

+ 2(i

) ≤ 3, or, explicitly,

1, x, x

, x

, K

, xK

, K

, xK

where K

= xJ

−J

= 2y



and K

= xJ

−J

= 2xy



−6y. (Here we set J

= K

and J

= K

(iii) (k = 3) Space F

is spanned by x

)

, i

≥ 0, i

+ 3i

≤ 3, or

1, x, x

, x

, L

where L

= xK

− K

= 6y. (We denote J

= L

Example 2.30 (n ≥ 4) The subspace F

= F

n−1−(n−2)

(k = n − 2) for equation

(2.97) with n ≥ 4, coinciding with the algebra of point symmetries, is spanned by

operators

n−2

)

n−2

)

, i

≥ 0, i

+ (n − 2)(i

) ≤ n − 1, or

1, x , ... , x

n−1

, J

n−2

, xJ

n−2

, J

n−2

, xJ

n−2

. (2.118)

Then, for the expressions J

, k = 0, ..., n − 1, the following representation holds:

= k! x

i+k



−1−k

(n−i−k)



(i−1)

for i = 1, ..., n − k, (2.119)

and, therefore, one obtains

n−2

= (n − 2)! x

n−1

1−n



) = (n − 2)! y



and

n−2

= (n − 2)! x

1−n



= (n − 2)![xy



+ (1 − n)y].

Plugging into (2.118) yields the well-known (n+4)-dimensional algebra of Lie point

symmetries of (2.97) with n ≥ 4, [374].

2.6.3 Translation-invariant operators

In the context of applications to evolution PDEs, it makes sense to obtain a descrip-

tion of operators admitted by the equation (2.97) which do not depend explicitly on

x. We denote by



n−1

the linear subspace of such translation-invariant operators

of orders not greater than n − 1. In this case,



n−1

is a subalgebra of the symme-

try algebra F

n−1

. Using the representation (2.99) for the operators F ∈ F

n−1

,the

translation-invariant operators are singled out by the condition (2.14), i.e.,

∂ F

∂x

≡ 0.

78 Exact Solutions and Invariant Subspaces

Example 2.31 (n = 2) Let us start with the case n = 2, i.e., with the ODE y



= 0.

Applying condition (2.14) to operator (2.108) yields the identity

+ A

) + xI

= 0,

where A

≡ ∂ A

/∂ I

,sothatI

=−A

and A

= 0. This system has the general

solution A

= I

A(I

), A

=−I

A(I

) + B(I

) with arbitrary functions A(I

) and

B(I

). Plugging these expressions into (2.108) yields

F[y] = (xI

− I

)A(I

) + B(I

) = yA(y



) + B(y



Example 2.32 (n = 3) Acting in a similar way in the case n = 3 with the ODE



= 0, one obtains the following general description of operators F belonging to



F[y] = (y



)

) + y



) + C(

where A, B,andC are arbitrary functions of the expressions

= I

and

−(I

)

viathe ﬁrst integrals(2.111)which imply

= y



and

∼ 2yy



−(y



)

Generalizing these examples yields the following result.

Theorem 2.33

The set



n−1

of translation-invariantoperators admitted by the equa-

tion

(2.97)

is generated by

F[y] =



i=1

(

, ... ,

n−1

)(y

(n−2)

)

i−1

where

are arbitrary functions of the homogeneouspolynomials

= I

− (I

)

= (I

)

k−1

k+1

− (I

)

−



k−2

i=1

k−i

)

, 3 ≤ k ≤ n − 1,

(2.120)

of the ﬁrst integrals

(2.98)

Proof. We apply condition (2.14) to operator (2.99). Using that ∂ I

/∂x = 0and

∂ I

/∂x = (k − 1)I

k−1

for k = 2, ..., n yields the identity

n−1



i=1





k=2

(k − 1)I

k−1

+ iA

i+1



i−1





k=2

(k − 1)I

k−1



n−1

≡ 0,

which is reduced to the following system of linear equations:

=−iA

i+1

(i = 1, ..., n − 1), XA

= 0 (2.121)

with X ≡



k=2

(k − 1)I

k−1

∂/∂ I

. One can verify that, for functions (2.120), the

relations X

= 0(i = 1, ..., n − 1) hold, while X (I

) = 1, and, hence, in

variables

(i = 1, ..., n − 1) and t

the operator X is transformed into ∂/∂t

. The system (2.121) is then rewritten as

∂ A

/∂t

=−iA

i+1

(i = 1, ..., n − 1), ∂ A

/∂t

= 0, or, in the equivalent form,

(−1)

k−1

(k−1)!

∂

k−1

∂t

k−1

(k = 2, ..., n),

∂

∂t

= 0.

2 Invariant Subspaces: Mathematics in 1D 79

This provides us with general expressions for A

= (−1)

k−1



i=k

k−1

i−1

)

i−k

α

, ... , t

n−1

)(k = 1, ..., n),

with arbitrary functionsα

(i = 1, ..., n). Substituting these into (2.99) yields

F[y] =



k=1



i=k

k−1

i−1

(−x)

k−1

)

i−k

α



i=1





k=1

k−1

i−1

(−x)

k−1

)

i−k



α



i=1

− x )

i−1

α



i=1

(xI

− I

)

i−1

where α

=α

/(−I

)

i−1

.SincexI

− I

= y

(n−2)

, this completes the proof.

Applying Theorem 2.33 to the case n = 4 leads to

Example 2.34 (n = 4) For equation (2.115), y

(4)

= 0, the set



consists of opera-

tors

F[y] =



i=1



)

i−1

(

where α

(·) are arbitrary functions of the homogeneous polynomials

= I

− (I

)

and

= (I

)

− (I

)

− 3

Substituting the expressions (2.116) for I

, I

,andI

yields

= y



= 2y





− (y



)

, and

=−6y(y



)

+ 6y







− 2(y



)

2.6.4 Translation-invariant quadratic operators

We return to the study of quadratic operators and present a more detailed description

of operators admitting polynomial subspaces. Namely, we study the special class

n−1

⊂

n−1

of translation-invariant quadratic operators preserving the subspace

(2.96) (the ODE (2.97)). For exponential subspaces, a related problem was solved

earlier in Section 1.5.2 (Example 1.49). We will again use the notation y

= y

(i)

Let Q be the linear span of all the monomials y

for i, j = 0, ..., n −1. In what

follows, we represent Q as a direct sum of the form

Q =⊕

n−1

k=0

n−1−k

, (2.122)

where Q

n−1−k

denotes a linear space of quadratic operators of the order n − 1 − k,

with the basis that is constructed below. The linear space

n−1

⊂ Q will then be

expressed in terms of Q

n−1−k

’s.

Theorem2.28 and representation(2.102)implythat the setF

n−1−k

ofall quadratic

operators of the order not greater than n − 1 − k admitted by the equation (2.97) is

generated by

F =



n−1−2k





n−k

(i, j =1,i≤j )

,i, j



, (2.123)

where α

,i, j

are arbitrary constants and k satisﬁes the inequality

0 ≤ k ≤

n−1

. (2.124)

The subset

n−1−k

⊂ F

n−1−k

of translation-invariant operators is singled out from

(2.123) by the condition (2.14).

80 Exact Solutions and Invariant Subspaces

The construction of the basis of Q

n−1−k

is fulﬁlled in a few steps.

Step 1. We start with a particular case of operators of the form

F =



n−k

(i, j =1,i≤j )

i, j

. (2.125)

Here, instead of (2.124),we assume that 0 ≤ k ≤ n −1. In this case, the invariance of

the subspace (2.96) can be violated, since the deﬁnition of J

at the right-hand side of (2.125) is a polynomial of x of the degree 2k, whose

coefﬁcients depend on ﬁrst integrals of the equation (2.97). Therefore, the operator

(2.125) maps W

given by (2.96) onto W

2k+1

= L{1, ..., x

}, so that there is no

invariance for k >

n−1

. Nevertheless, at the ﬁrst step, we construct all the operators

of the form (2.125) that are translation-invariant. The linear space of such operators

is denoted by Q

n−i−k

From (2.119), the following identities are obtained:

∂ J

∂x

= 0and

∂ J

∂x

= (i − 1)J

i−1

for i = 2, ..., n − k. (2.126)

Then (2.114) and (2.126) imply that it sufﬁces to consider operators (2.125) with

i + j = s,2≤ s ≤ 2(n − k) that can be written as

F =





[

]

i=1

s−i

for s = 2, ..., n − k + 1,



[

]

i=s−n+k

s−i

for s = n − k + 2, ..., 2(n − k).

Condition (2.114) leads to the system of linear homogeneous equations for the co-

efﬁcients {α

}. It can be shown that a nontrivial solution exists only for even s that

belong to the ﬁrst interval, i.e., in the case of the operators



i=1

s−i

for even s ∈{2, ..., n − k + 1}. (2.127)

Applying (2.119) and (2.126) to (2.127) yields (for s ≥ 4) the identity

0 =

∂

∂x



i=1



(i − 1)J

i−1

s−i

+ (s −i −1)J

s−i−1





i=1



iα

+ (s − i −1)α



s−i−1



− 1



−1

This gives the following system for the coefﬁcients:



i+1

=−

s−i−1

for i = 1, ...,

− 2 (s ≥ 6),

=−

2(s−2)

−1

Setting α

= 1 yields

= (−1)

i−1

s−2

for i = 1, ...,

− 1,

= (−1)

−1

−2

s−3

Substituting these expressions into (2.127) and taking into account the obvious case

s = 2, we obtain a basis of Q

n−1−k

= (J

)



−1

i=1

(−1)

i−1

s−2

s−i

+ (−1)

−1

−2

s−3





(2.128)

(see (2.114)) implies

2 Invariant Subspaces: Mathematics in 1D 81

where s ∈{4, ..., n − k + 1} is even.

Step 2. Operators (2.128) provide us with a part of the basis of the linear space

n−1−k

. The rest of the basis can be constructedas follows. Note that, for k = n −1,

the only operator that remains in (2.128) is F

2(n−1)

= (J

n−1

)

.Letk ≤ n − 2.

Replacing k by k +1 in (2.128) leads to the (n −2 −k)th-order operators F

2k+2

for

even s ∈{2, ..., n − k}. Using operator D of the total derivative in x , we introduce

the following (n − 1 − k)th-order operators:

2k+1

= DF

2k+2

for even s ∈{2, ..., n − k}. (2.129)

From (2.119), one can derive that

k+1

= (k + 1)J

for i = 1, ..., n − k − 1.

Applying this allows us to rewrite (2.129) in terms of {J

} and {J

k+1

} as

2k+1

= DF

2k+2

= 2(k + 1)J

k+1

2k+1

= DF

2k+2

= (k + 1)





−1

i=1

(−1)

i−1

s−2



k+1

s−i

+ J

k+1

s−i



+ 2(−1)

−1

−2

s−3

k+1



(2.130)

for evens ∈{2, ..., n −k}. Operators (2.130) map W

onto W

2k+2

= L{1, ..., x

2k+1

Let Q

2k+1

n−1−k

denote the linear envelope of these operators.

Notice that the set of operators (2.128)and (2.130)is linearly independentand that

every operator F ∈

n−1−k

is represented as a linear combination of these operators

and operators of less orders. (This is readily seen from the explicit representation

of the operators given below.) The subspace Q

n−1−k

is then deﬁned as the linear

span of the operators (2.128) and (2.130). Obviously, the representation Q

n−1−k

⊕ Q

2k+1

n−1−k

holds.

Step 3. In order to obtain the explicit representation of the operators (2.128), we

use the formulae (2.119). It follows that

= (−1)

i−1

(i + k − 1)! y

n−i−k

+ ... for i = 1, ..., n − k,

whereby “...” we denote the x-dependent summands that are mutually cancelled by

plugging into (2.128). Substituting these expressions into (2.128) yields

∼ (y

n−1−k

)

, F

∼



−1

i=1

(−1)

s−i−1

i+k−1

×C

s−i+k−1

n−i−k

n−s+i−k

(−1)

−1



+k−1





n−

−k



(2.131)

for even s ∈{4, ..., n − k + 1}. Using (2.129) (or (2.130)) yields

2k+1

∼ y

n−1−k

n−2−k

2k+1

∼ C

k+1

s−1+k

n−1−k

n−s−k



−1

i=1

(−1)

s−i−1



k+1

i+k

k+1

s−i+k

− C

k+1

i+1+k

k+1

s−i−1+k



n−i−k−1

n−s+i−k

(2.132)

for even s ∈{2, ..., n −k}. Leaving in (2.131) and (2.132)only the terms containing

82 Exact Solutions and Invariant Subspaces

the leading order derivative y

n−1−k

yields

∼ y

n−1−k

n−s+1−k

+ ... for even s ∈{2, ..., n − k + 1}, and

2k+1

∼ y

n−1−k

n−s−k

+ ... for even s ∈{2, ..., n − k}.

Here, all the monomials of the form y

n−1−k

n−i−k

for i = 1, ..., n − k are avail-

able and moreover each of these appear in a single operator only. Hence, opera-

tors (2.131), and (2.132) are linearly independent, and their envelope Q

n−1−k

con-

tains all the operators of the order n − 1 − k up to adding lower order opera-

tors. The dimensions of the subspaces are given by the following integer parts:

dim Q

n−1−k



n−1−k



and dim Q

2k+1

n−1−k



n−k



. Hence,

dim Q

n−1−k

= dim Q

n−1−k

+ dim Q

2k+1

n−1−k

= n − k.

For Q



=⊕

n−1

k=0

n−1−k

,wehave

dim Q





n−1

k=0

(n − k) =

n(n+1)

= dim Q.

Therefore, Q



= Q, completing the construction of the representation (2.122).

Noting that subspace (2.96) is invariant with respect to F ∈ Q

n−1−k

(r = 2k or

r = 2k +1) iff r ≤ n − 1 and that all such operators form the set

n−1

, we arrive at

the following result.

Theorem 2.35

The linear space

n−1

of translation-invariantquadratic operators of

the order not greater than

n − 1

, which preserve subspace

(2.96)

, is represented as:

(i)

n−1

= Q

n−1

⊕



⊕

n−3

k=0

n−1−k



, dim

n−1

+4n+9

for odd

and

(ii)

n−1

=⊕

n−2

k=0

n−1−k

, dim

n−1

n(3n+2)

for even

It is seen that the set of translation-invariant quadratic operators of the order k,

whichpreservethe polynomialsubspace (2.96) of the maximaldimensionn = 2k+1,

coincides with the subspace Q

. Setting n = 2k + 1 in (2.131) yields

Theorem 2.36

The basis of the linear subspace

of translation-invariant

th-

order operators preserving the polynomial subspace of the maximal dimension

n =

2k + 1

is given by

∼ (y

)

, F

∼



−1

i=1

(−1)

s−i−1

i+k−1

s−i+k−1

k+1−i

k+1−s+i

(−1)

−1



+k−1





k+1−



(2.133)

for even

s ∈{4, ..., k + 2}

. The dimension is

dim Q



k+2



Example 2.37 (n = 5)Forn = 5, the subspace and its equation are

= L{1, ..., x

} and y

= 0. (2.134)

Applying(2.128), (2.131), and (2.130), we obtain Table 2.1, where the boldface sub-

spaces Q

are composed of operators admitted by the ODE in (2.134). Second-order