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 53

Example 2.7 (Radial geometry) Consider the following ODE for the subspace:



β−1



, where β = 0, 1. (2.17)

This representation is related to the radial Laplace operator



N−1

for the (non-integer and, possibly, negative) dimension N = 2 − β.Herex > 0

denotes the radial space variable. The FSS of (2.17) is

(x ) = 1and f

(x ) = x

By (2.11), we readily ﬁnd that, up to a constant factor, the FSS of the conjugate

equation is

(x ) = x and g

(x ) = x

1−β

Then Theorem 2.1 implies that the linear space of nonlinear operators admitting

= L{1, x

} has the form

F[y] = A

, I

) + A

, I

with arbitrary functions A

and A

of the ﬁrst integrals

= xy



− βy and I

= x

1−β



The translation-invariantoperators satisfying (2.14) are spanned by

[y] =







β−1

, F

[y] = y, and F

[y] = 1.

The operator F

generates the following semilinear higher-order PDEs (m ≥ 2):

∂

∂x

+ λ(u

)

m−1

> 0), (2.18)

where λ is a constant. The operator on the right-hand side has the invariant subspace

= L{1, x

}, and hence, (2.18) admits solutions

u = C

(t) + C

(t)x

Substituting into the PDE yields C



= m! C

+λ(mC

)

m−1

, which leads

to the following dynamical system:





= m! C



= λ(mC

)

m−1

It is solved explicitly and gives the solutions of (2.18) of the form

u(x , t) =

(m−1)

m−1



λ(m−2)

−λt)

m−2

−λt)

m−1

+ a



for m = 2,

u(x , t) =



−

ln|a

− λt|+

−λt

+ a



for m = 2,

where a

and a

are arbitrary constants. Note that the change x =˜x

1/β

transforms

(2.17) into a simpler equation, y

˜x ˜x

= 0.

54 Exact Solutions and Invariant Subspaces

2.2 The optimal estimate on dimension of invariant subspaces

We consider the question of the maximal dimension of invariant subspaces for non-

linear operators of a ﬁxed differential order. This is an important, crucial aspect of

Problem I: F → {W

}. It is especially key for applications, since this establishes the

maximally possible order of dynamical systems as restrictions of the PDEs.

In all the previous examples, the dimensions of invariant subspaces were less than

or equal to three for the ﬁrst-order operators, and less than or equal to ﬁve for the

second-order ones. The following theorem establishes the maximal dimension of

invariant subspaces for arbitrary kth-order nonlinear ordinary differential operators.

Theorem 2.8 (“Theorem on maximal dimension”)

If a linear subspace

is in-

variant under a nonlinear ordinary differential operator

(2.3)

of the order

,then

n ≤ 2k + 1. (2.19)

Proof. Arguing by contradiction, assume that there exists an invariant subspace W

of (2.3), and (2.19) is not valid, i.e., n ≥ 2k + 2. Let us show that F is then a linear

operator. Let W

be deﬁned by the linear ODE

(n)

= a

(n−1)

+ ... + a

n−1



+ a

y. (2.20)

By the condition (2.4), the following identity must be true on solutions of (2.20):

F ≡ a

n−1

F + ... + a

n−1

DF + a

F, (2.21)

where D denotes the operator of the total derivative in x . For convenience, perform-

ing algebraic manipulations with the function F(·), the following notation is used:

= y

( j )

for all j ≥ 0 (y

= y), and F

∂ F

∂y

Then (2.20) reads y

= a

n−1

+... +a

. Differentiating F(x, y

, y

, ..., y

),and

keeping leading linear and quadratic terms yields

DF = y

k+1

+ ... , D

F = y

k+2

+ (y

k+1

)

+ ... ,

F = y

k+3

+ 3y

k+1

k+2

+ ... ,

F = y

k+4



k+1

k+3

+ 3(y

k+2

)



+ ... ,

etc. By induction, after any p ≥ 4steps,weﬁnd

F = y

k+p





[

]−1

i=1

k+p−i

k+i

+νC

[

]

k+p−[

]

k+[

]



+ ... ,

(2.22)

where





denotes the integer part, C

for i = 1, ...,





are binomial coefﬁcients,

and ν =

for p even, ν = 1 for odd. Here, we separate the ﬁrst linear term that

contains the highest-order derivative y

k+p

. The sum in square brackets in (2.22) is

composed of the quadratic (in higher-order derivatives) summands that exhibit the

maximal total order of both the derivatives: (k + p − i) + (k + i) = 2k + p. For

p = n, keeping in (2.22) only the quadratic terms, containing at least one derivative

of the order not less than n − 1, gives

F =





k+1

i=1

k+n−i

k+i



+ ... , (2.23)

2 Invariant Subspaces: Mathematics in 1D 55

where α

= C

for i = 1, ..., k, α

k+1

= C

k+1

if n > 2k + 2, and α

k+1

n = 2k +2. Here, we do not display the linear term. The total order of the summands

in the square brackets is equal to 2k + n. By (2.20), all the derivatives y

k+n−i

for

i = 1, 2, ..., k can be linearly expressed in terms of y

n−1

, ..., y

. Keeping in the

square brackets in (2.23) the terms containing y

n−1

,weﬁnd

F =





i=1

˜α

k+i

+ α

k+1

2k+1



n−1

+ ... , (2.24)

where ˜α

= α

and χ

are expressed via a

, ..., a

and their derivatives. There

exists a single quadratic term in (2.24) of the maximal total order 2k + n, namely,

k+1

2k+1

n−1

It is easy to see that such terms do not appear in the derivatives D

F with p < n.In

order for (2.21) to be valid, we should then set

= 0. (2.25)

Taking into account (2.25), we derive similar to (2.23) that

F =





k+1

i=1

k+n−i

k−1+i



k−1

+ ... ,

where β

are some positive coefﬁcients. In the summands, containing at least one

derivative of the order not less than n − 1, the total order of the derivatives is now

2k + n − 1. Excluding, as above, the derivatives y

k+n−i

for i = 1, 2, ..., k by using

(2.20) yields the unique quadratic term of the maximal total order

k+1

n−1

k−1

Such summands do not appear in any other derivative D

F for p < n. Equating this

coefﬁcient to zero implies

k−1

= 0.

Assume that the equalities

= F

k−1

= ... = F

k−( j −1)

= 0

have been proved,i.e., F

depends on the arguments x , y

, ..., y

k−j

only. Similarly

to the above calculus,

F =





k+1

i=1

k+n−i

k−j +i



k−j

+ ... , (2.26)

where γ

> 0 and the total order of the quadratic summands relative to higher deriva-

tives is 2k + n − j. Expressing y

k+n−j

again in terms of y

n−1

,...,y

yields that, in

(2.26), there exists the unique quadratic summand of the total order 2k + n − j ,

k+1

n−1

2k+1−j

k−j

No other derivatives D

F with p < k produce such a term. Hence,

k−j

= 0. (2.27)

By induction, it follows that (2.27) holds for all j = 0, 1, ..., k, i.e.,

F[y] = f

(x )y



F(x , y

, ... , y

k−1

56 Exact Solutions and Invariant Subspaces

Since the summand f

(x )y

generates the terms in (2.21) that are linear in y

, ...,

n−1

, the aboveconclusions based on the analysis of the quadratic terms can literally

be repeated for operator



F(x , y

, ... , y

k−1

), replacing k by k − 1 in all instances.

This yields



F[y] = f

k−1

(x )y

k−1



F(x , y

, ... , y

k−2

Acting in a similar fashion yields that the function F(x, y

, y

, ... , y

) is linear rel-

ative to all the arguments y

for i = k, ..., 0. Hence,

F[y] = f

(x )y

+ ... + f

(x )y

+ f (x),

completing the proof.

For arbitrary k, there exist nonlinear quadratic operators admitting invariant sub-

spaces of the maximal dimension 2k + 1 as the following example shows.

Example 2.9 (Subspace of maximal dimension)Thekth-order quadratic operator

F[y] =



(k)



(2.28)

admits the (2k+1)th-dimensionalsubspace

2k+1

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

In the next three remarks, we comment on some aspects of our analysis.

Remark 2.10 Let operator F be admitted by the linear ODE (2.2) that deﬁnes the

invariant subspace (2.1). Let

F[y] =



(i)

[y],

where the sum is ﬁnite or inﬁnite, and each F

[y] = F

(x , y, y



, ..., y

)

) is a homo-

geneous polynomial in y, y



, ..., and y

)

of degree i . In this case, every operator F

is also admitted by this ODE.

This directly follows from the invariance criterion (2.4) that is written down as



(i)



L[F

[y]]



L[y]=0



≡ 0.

Hence, in view of the homogenuity of all the summands, this splits into the system

L[F

[y]]



L[y]=0

≡ 0foralli.

Remark 2.11 Any change of variables

y = α(x)y, x = β(x), (2.29)

where α(x), β(x ) are given functions, transforms any homogeneous polynomial in

y, y



, ..., y

(k)

into a homogeneous polynomial of the same degree in y, y



, ...,y

(k)

Two operators F[y] = F(x , y, y



, ..., y

(k)

) and



F[y] =



F(x, y,y



, ..., y

(k)

) are

called equivalent, if there exists the change of variables (2.29) such that



F[y] =

α(x)

F[y]. (2.30)

If operator F preserves the given subspace

= L{f

(x ), ... , f

(x )},

2 Invariant Subspaces: Mathematics in 1D 57

then the equivalent operator



F admits the invariant subspace



= L{



(x), ... ,



(x)},

where



(x) = f

(x (x))/α(x (x)) for all i = 1, ... , n.

Remark 2.12 Subspace W

, invariant under the operator F[y], is also invariant un-

der



F[y] = F[y + f ], with any function f (x ) ∈ W

Example 2.13 (Exponential

2k+1

) The change of variables

x = e

˜x

, y = e

2k ˜x

˜y,

transforms (2.28) into the following quadratic operator

F[˜y] = e

−2k ˜x



−˜x

d˜x





2k ˜x

˜y



The invariant subspace is now of the exponential type rather than polynomial,

2k+1

= L



−2k ˜x

, ..., e

−˜x

, 1



Note that

F[˜y] is a translation-invariant operator.

2.3 First-order operators with subspaces of maximal dimension

Inviewof Theorem 2.8,fornonlinearﬁrst-orderoperators,the dimensionof invariant

subspaces cannot exceed three. We now describe all ﬁrst-order operators

F[y] = F(x, y, p), where p = y



, (2.31)

admitting 3D subspaces given by the linear ODEs



= a

(x )y



+ a

(x )y



+ a

(x )y, (2.32)

with sufﬁciently arbitrary coefﬁcients a

(x ) for k = 1, 2, 3. The invariance criterion

(2.4) then takes the form

F −



F + a

DF + a



≡ 0, (2.33)

where D is the operator of the full derivative via (2.32),

D =

∂

∂x

+ y



∂

∂y

+ y



∂

∂y





(x )y



+ a

(x )y



+ a

(x )y



∂

∂y



Lemma 2.14

Any operator

(2.31)

preserving 3D subspaces can be represented in

the following form:

F[y] = F

[y] + F

[y],

where

(2.34)

[y] = Lp

+ Myp + Ny

, F

[y] = Rp + Sy, F

[y] = f, (2.35)

and

L, M, N, R, S

,and

are arbitrary functions of

Proof. It is easy to check that the left-hand side of identity (2.33) is a third-degree

polynomial in the variable y



, while the coefﬁcient of (y



)

∂

∂p

. Equating this

coefﬁcient to zero and solving the resulting equation implies

F(x , y, p) = A

(x , y) p

+ A

(x , y) p + A

(x , y), (2.36)

58 Exact Solutions and Invariant Subspaces

where A

(x , y), A

(x , y),andA

(x , y) are some functions. Rewriting the coefﬁ-

cients of (y



)

for s = 2, 1, and 0 by using (2.36), we obtain polynomials in p with

coefﬁcients, depending only on x and y. Equating the coefﬁcients of the leading

powers of p in these polynomials to zero yields

∂ A

∂y

= 0,

∂

∂y

= 0,

∂

∂y

= 0.

In view of (2.36), this completes the proof.

From the representation (2.34) by using Remark 2.10, we see that this operator is

ODE. Note that the admissibility of F

implies that f (x) ∈ W

. In what follows, we

consider all operators up to such trivial summands, and up to multiplication by an

arbitrary non-zero number.

Let us consider the quadratic operator F

[y] in (2.35).

Lemma 2.15

Any operator

[y]

admitting a 3D subspace is equivalent to

F[y] = (y



)

+ Ny

where

N =−1, 0,

1. (2.37)

The 3D subspace invariant with respect to operator

(2.37)

is given by the ODE



+ Ny



= 0 (2.38)

and has the form

= L{1, e

−x

, e

}=L{1, cosh x , sinh x}

for

N =−1; (2.39)

= L{1, x , x

}

for

N = 0;

and

(2.40)

= L{1, cos x , sin x}

for

N = 1. (2.41)

Proof. Let us use the ﬁrst transformation in (2.29). By (2.30), operator F

reads



F[y] =

α(x)



x,αy,(αy)





= Lα(y



)

+(2Lα



+ Mα)y y





L(α



)

+ Mαα



+ Nα



y

(2.42)

Consider the following two cases: (i) L(x ) = 0, and (ii) L(x ) ≡ 0, M(x) = 0. In

the former case of (i), choosing α(x) from the condition



=−

M(x)

2L(x)

α, (2.43)

the operator (2.42) is reduced to



F[y] = αL(y



)

+ α



N −



y

, (2.44)

where the sign of α is picked so that α(x)L(x)>0.

Similarly, in the latter case of (ii), α(x) is chosen fromthe condition α



=−

N(x)

M(x)

α.

Then operator (2.42) takes the form



F[y] = αM ˜y ˜y



. (2.45)

Let us apply the second transformation in (2.29). Since

= β



(x )

dx

, by choos-

ing β from the condition



(x ) =



L(x)α(x)



−

(2.46)

admitted by the equation (2.32) iff each operator in (2.35) is also admitted by this

2 Invariant Subspaces: Mathematics in 1D 59

in the case of (2.44), and from β



= [M(x)α(x)]

−1

in the case of (2.45), we derive,

respectively, that



F[y(x)] = (y



x

)



N (x)y

, (2.47)

where



N(x) = α(x )



N(x) −

M(x)

4L(x)



,and



F[y(x )] = y y



x

. (2.48)

Assume now that, for the operator F

, there exists a 3D subspace. Then the equiv-

alent operators (2.47) and (2.48) must have the same property. A direct checking

via the invariance condition (2.33) shows that, in the case of (2.47), this is possible

only for



N = const. The subspace is then deﬁned by an equation of the form (2.38),

while, in the case of (2.48), 3D subspaces do not exist. It remains to note that, if



N = const = 0, (2.47) is reduced by extensions along x to the form (2.37) with

|N|=1.

The proof implies that the ﬁrst operator in (2.35) can preserve 3D subspaces iff

L(x) = 0 (2.49)

and



N(x) = α(x )



N(x) −

(x)

4L(x)



= const. In view of (2.43), after differentiating in

x, one can rewrite down the last condition in terms of the coefﬁcients of the operator

only. Namely,



N(x) −

(x)

4L(x)



≡

M(x)

2L(x)



N(x) −

(x)

4L(x)



. (2.50)

Conditions (2.49) and (2.50) make it possible to determine at once, by the coefﬁ-

cients of operator F

, whether it admits 3D subspaces or not.The functions α(x) and

β(x ) deﬁning the variable change (2.29) are found from equations (2.43) and (2.46),

respectively. Thisgivesthefollowingstraightforwardconsequencefor operatorswith

constant coefﬁcients.

Lemma 2.16

Up to a constant multiplier,there existexactly two translation-invariant

quadratic operators

(2.35)

that preserve 3D subspaces,

F[y] = (y



)

+ Ny

and

(2.51)

F[y] = (y



+ Cy)

, (2.52)

where

C = 0

and

are arbitrary constants. By the change of variables

y = e

−Cx

y

and

x = e

, (2.53)

operator

(2.52)

is transformed into

(2.51)

with

N = 0

Note that the 3D subspace of the operator

F[y] = (y



x

)

has the form

L{1,x,x

= e

−Cx



1, e

, e



= L



1, e

−

, e

−Cx



Let us now consider the linear summand F

[y] in (2.34). By Lemma 2.15, it suf-

ﬁces to describe all such operators for which equation (2.38) is invariant.The invari-

ance criterion implies the following result.

} (see (2.40)). By Remark 2.11, the 3D subspace of (2.52) is

60 Exact Solutions and Invariant Subspaces

Lemma 2.17

The linear operator

(2.35)

is admitted by equation

(2.38)

iff

R(x) ∈ W

and

S(x) =−R



(x ) + C,

where

is an arbitrary constant.

Combining Lemmas 2.14, 2.15, and 2.17 gives the complete description of opera-

tors (2.31) preserving 3D subspaces.

Theorem 2.18

Any nonlinear ﬁrst-order operator

(2.31)

admitting 3D subspaces is

reduced by transformations

(2.29)

(2.30)

to the form

F[y] = (y



)

+ Ny

+ R(x )y



+ [C − R



(x )]y, (2.54)

where

N =−1, 0

R(x)

satisﬁes

(2.38)

,and

is a constant.The corresponding

3D subspaces are deﬁned by equation

(2.38)

and are given by

(2.39)

–

(2.41)

Following Remark 2.12, let us replace y by y + f with a function f (x) ∈ W

obtain, up to trivial summands



f (x) ∈ W

, the operator



F[y] = (y



)

+ Ny

+ [R(x ) + 2 f



+ [S(x ) + 2Nf]y, (2.55)

where S(x ) =−R(x)



+C. Note that R(x ) ∈ W

implies that S(x) ∈ W

.IfN = 0,

setting f (x) =−

S(x )

, we see that the coefﬁcient of y vanishes, and the coefﬁcient

of y



becomes constant. Consequently, in this case, the operator (2.54) is reduced to

F[y] = (y



)

+ Ny

+ Ry



where N =±1andR = const. If N = 0, the coefﬁcients of operator (2.54) are

R(x) =−C

+ C

x + C

and S(x) =−R(x)



+ C =−2C

x + C

where C

, ... , C

are constants. Putting f (x ) =−



+ C



into (2.55)

implies that operator (2.54) is reduced to

F[y] = (y



)

+ R



+ (−2R

x + R

with constants R

and R

Since subspaces W

, which are invariant under operators (2.51) and (2.52), are

also invariant with respect to any linear operators with constant coefﬁcients, Lemma

2.16 implies the following result.

Theorem 2.19

The set of nonlinear translation-invariant ﬁrst-order operators that

admit 3D subspaces, up to a constant multiplier, is exhausted by

F[y] = (y



)

+ Ny

+ Ry



+ Sy

and

(2.56)

F[y] = (y



+ y)

+ Ry



+ Sy, (2.57)

where

N, R, S

,and

are arbitrary constants.

As follows from Lemma 2.16, the quadratic parts of operators (2.57) and (2.56)

for N = 0 are related by the change of variables (2.53). It is easy to check, however,

that this is not valid for full operators containing linear terms. Indeed, by the change

(2.53), the operator (2.57) for R = 0 is reduced to



F[y] =



y



˜x



x y



˜x

+ (S − RC)y,

2 Invariant Subspaces: Mathematics in 1D 61

depending explicitly on x.

2.4 Second-order operators with subspaces of maximal dimension

We now study subspaces of maximal dimension for nonlinear operators

F[y] = F(x , y, p, q), where p = y



and q = y



. (2.58)

By Theorem 2.8, the dimension cannot exceed ﬁve, so we will deal with 5D linear

subspaces given by the ODEs

(5)

= a

(x )y

(4)

+ a

(x )y



+ a

(x )y



+ a

(x )y



+ a

(x )y. (2.59)

In this case, the invariance criterion leads to a result similar to Lemma 2.14.

Lemma 2.20

Any operator

(2.58)

preserving a 5D subspace can be represented in

the form of

F[y] = F

[y] + F

[y],

where

[y] = b

2,2

+ b

1,2

pq +b

0,2

yq + b

1,1

+ b

0,1

yp + b

0,0

[y] = b

q + b

p + b

and

[y] = b,

where

i, j

and

for

i, j = 0

, 1, and 2, and

are functions of

Proof. This result is based on analyzing the condition of invariance of equation (2.59)

under the operator (2.58). In this case, the condition takes the form

F −



F + a

DF + a



≡ 0, where (2.60)

D =

∂

∂x

+ y

∂

∂y

+ y

∂

∂y

+ y

∂

∂y

+ y

∂

∂y



(x )y

+ ... + a

(x )y

+ a

(x )y



∂

∂y

As above, we have set

for k = 1, 2, ... .

The left-hand side of the identity (2.60) is a polynomial in the variables y

and y

with coefﬁcients that should be equalto zero. Consider the resulting conclusions that

have been obtained by using the package Reduce [282] for analytic manipulations.

Coefﬁcients of y

and y

lead to conditions

= F

= 0, (2.61)

where the lower indices of F denote the corresponding partial derivatives.

In view of (2.61), the coefﬁcient of y

implies the equation, which by differentiat-

ing in y

takes the form

+ 3F

= 0.

Simultaneously, the coefﬁcient of y

yields

+ 2F

= 0.

These equalities imply

= 0 and (2.62)

62 Exact Solutions and Invariant Subspaces

= 0. (2.63)

It follows from (2.61) and (2.62) that F

= C(x ), and hence, F has the form

F =

+ φ(x , y, y

+ ψ(x, y, y

). (2.64)

Later on, we supposethat C is a constant,since the general case C = C(x) is reduced

to the constant case by the change (2.29).

Using (2.64), the condition of vanishing the coefﬁcient of y

10(φ

+ a

C) = 3a

C, from whence come

≡ F

=−

C (2.65)

and F

= 0. (2.66)

Then the coefﬁcient of y

leads to

−60a



C + 20F

+ 30F

− 5a

C + 24a

C = 0,

which, on differentiation in y

and y, implies, respectively, that

= 0and2F

+ 3F

yyy

= 0. (2.67)

The coefﬁcient of y

, on differentiation in y

, yields 5F

+4F

yyy

= 0 that, by

the previous equality, means

= F

yyy

= 0. (2.68)

By differentiating in y, the same coefﬁcient implies that

yyy

= 0. (2.69)

Finally, consider the coefﬁcient of y

. Equating it to zero and differentiating in y

twice the resulting relation, we ﬁnd that

yyyy

+ F

xyyy

= 0.

Differentiating next in y

and using (2.69) yields

yyyy

= F

xyyy

= 0. (2.70)

We now apply the above results for deﬁning the coefﬁcients in (2.64). It follows

from (2.63), (2.66), and (2.68) that the function φ is linear in y and y

φ = φ

(x )y

+ φ

(x )y + φ

(x ).

Similarly, (2.67)–(2.70)imply that function ψ takes the form

ψ = C



ψ(x, y, y

where C

= constant, and



ψ is a second-degree polynomial in y and y

. Hence, the

right-hand side of (2.64) is a sum of homogeneous polynomials of the zero, ﬁrst,

and second degree in variables y, y

, y

, and of the third-degree polynomial C

Each of those polynomials determines a differential operator, and, clearly, equation

(2.59) is invariant with respect to any of them, and, in particular, with respect to



F = C

(see Remark 2.10). However,this is possible, providedthat C

= 0, since,