A precise definition of reduction of partial differential equations

Journal of Mathematical Analysis and Applications 238, 101᎐123 Ž1999. Article ID jmaa.1999.6511, available online at http:rrwww.idealibrary.com on

A Precise Definition of Reduction of Partial Differential Equations R. Z. Zhdanov,U I. M. Tsyfra,† and R. O. Popovych‡ Institute of Mathematics of NAS of Ukraine, 3 Tereshchenki¨ ska Street, 252004 Kyi¨ , Ukraine Submitted by M. C. Nucci Received November 17, 1998

We give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry Žclassical and non-classical . reductions of partial differential equations. Using the introduced definition of reduction of differential equations we establish equivalence of the non-classical Žconditional symmetry. and direct ŽAnsatz. approaches to reduction of partial differential equations. As an illustration we give an example of non-classical reduction of the nonlinear wave equation in 1 q 3 dimensions. The conditional symmetry approach when applied to the equation in question yields a number of non-Lie reductions which are far-reaching generalizations of the well-known symmetry reductions of the nonlinear 䊚 1999 Academic Press wave equations.

1. INTRODUCTION The notion of a non-classical symmetry was introduced by Bluman and Cole as early as in 1969 w1x. However, non-trivial examples of non-classical symmetries for nonlinear partial differential equations ŽPDEs. appeared much later in the papers by Olver and Rosenau w2, 3x and by Fushchych and Tsyfra w4x. These papers, together with the ones by Fushchych and Zhdanov w5x, Clarkson and Kruskal w6x, and Levi, and Winternitz w7x, gave a start to an intensive search for non-classical symmetries of a wide range of nonlinear differential equations. Following the suggestion by Fushchych w8᎐10x we call these kinds of non-Lie symmetries conditional symmetries ŽCSs.. * E-mail: [email protected]. † E-mail: [email protected]. ‡ E-mail: [email protected]. 101 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.

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ZHDANOV, TSYFRA, AND POPOVYCH

The vast majority of the papers devoted to constructing CSs of nonlinear PDEs consider equations having two independent variables only. This is explained by the fact that the determining PDEs for CSs are nonlinear and have the dimension which is equal to the sum of the number of dependent and independent variables of the PDE under study. That is why there is no systematic general procedure for obtaining CSs of multi-dimensional nonlinear PDEs. Constructing CS for a specific multi-dimensional differential equation requires preliminary guesswork enabling one to reduce the dimension of the system of determining PDEs. In the papers w5, 11᎐18x devoted to studying CSs of multi-dimensional nonlinear equations of quantum field theory Žwave, Dirac, Levi-Leblond and SUŽ2. Yang᎐Mills ` equations. we developed an efficient approach based on fixing a special Ansatz for a conditionally invariant solution to be found. The underlying idea for choosing such an Ansatz was a proper use of Lie symmetry properties of the equation under consideration Ža complete account of the results obtained in this way can also be found in w19x.. There exist a number of different approaches to utilizing CSs of PDEs in order to reduce these to equations with fewer independent variables. However, with all the differences between these methods they can be classified into two major groups. The first one is composed of the methods that are close to the traditional Lie approach and can be regarded as the ‘‘infinitesimal methods for finding CS’’ w1᎐5, 7, 20᎐25x. The central role is played by infinitesimal CSs within the class of first-order differential operators. Given an operator of conditional symmetry, we can construct an Ansatz reducing the dimension of PDE under study. The second group of methods is the ‘‘direct’’ ones w6, 11᎐18, 26᎐28x Žsee also w29x and the references therein . that goes up, probably, to the papers by Fourier and Euler devoted to finding particular solutions of the two-dimensional heat equation with the help of a substitution of a special Žseparated. form. Namely, the methods in question are based on fixing a special Ansatz for a solution to be found. As a rule, these Ansatze ¨ contain arbitrary functions which are to be so chosen that some reduction requirements must be met. One of the principal motivations for writing the present article is studying interrelations between these approaches. A necessary ingredient of such study is a precise mathematical definition of reduction of PDEs. We attempt to give this definition which is the core result of the paper. Based on this definition is our proof of equivalence of the above two approaches to reduction of PDEs provided some reasonable restrictions are met Žsee also w30x.. The present paper is a natural continuation of our earlier papers w20, 31x, where some ideas presented below were indicated. We present these ideas in a rigorous mathematical form which, as we believe, should give new insights into the theory of conditional symmetries of PDEs.

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A PRECISE DEFINITION OF REDUCTION

¨ 2. ANSATZE AND INVOLUTIVE SETS OF OPERATORS Consider a family of first-order differential operators in the variables x s Ž x 1 , . . . , x n ., u

⭸

n

Qq s

Ý ␰ ai Ž x, u . ⭸ x

is1

q ␩a Ž x, u . i

⭸ ⭸u

,

a s 1, . . . , m,

Ž 1.

where ␰ ai , ␩a are some continuously differentiable functions in an open domain in ⺢ nq 1, m - n. The variable u is regarded as dependent; i.e., it corresponds to the function u s uŽ x .. In a sequel, we suppose that the conditions of the theorem about implicit function are fulfilled, wherever applicable. DEFINITION 1. A family of first-order differential operators Ž1. is called involutive if there exist smooth functions ␮ cab Ž x, u., a, b, c s 1, . . . , m, such that m

w Q a , Qb x s Ý ␮ cab Qc ,

a, b s 1, . . . , m.

Ž 2.

cs1

The simplest example of an involutive family of operators is given by first-order differential operators forming a Lie algebra. In this case ␮ cab s const, a, b, c s 1, . . . , m are called structure constants of the Lie algebra. This implication has far-reaching consequences in the modern theory of non-Lie reductions of PDEs. This is explained by the fact that involutive families of operators of the form Ž1. play the same role in the theory of non-classical symmetry reductions of PDEs having non-trivial conditional symmetries as that played by finite-dimensional Lie algebras in the theory of symmetry reductions of invariant PDEs. In a sequel we consider involutive families of operators Ž 1 . satisfying an additional constraint rank ␰ ai Ž x, u .

m n as1 is1

s rank ␰ ai Ž x, u . , ␩ Ž x, u .

m n as1 is1

s m. Ž 3 .

By direct computation we check that, if operators Ž1. form an involutive family, then the family of differential operators QXa s

m

Ý ␭ ab Ž x, u . Qb ,

det ␭ ab Ž x, u .

m a, bs1

/0

Ž 4.

bs1

is also involutive Žsee also w20x.. Furthermore, the involutive family Ž4. is easily seen to obey the condition Ž3..

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Provided the relation of the form Ž4. holds, two involutive families of operators  Q a4 and  QXa4 are called equivalent. This equivalence relation splits the set of involutive families of m operators into equivalence classes forming the quotient set. We denote this set as I . It is a common knowledge that conditions Ž2. are sufficient for the system of PDEs

⭸u

n

Ya Ž x, u, u . s 1

Ý ␰ ai Ž x, u . ⭸ x

is1

y ␩a Ž x, u . s 0,

a s 1, . . . , m Ž 5 .

i

to be compatible Žthe Frobenius theorem w33x.. Its general solution can be locally represented in the form F Ž W1 , . . . , Wnq1ym . s 0,

Ž 6.

where F is an arbitrary smooth function of the variables Wj , Wj s Wj Ž x, u., j s 1, . . . , n q 1 y m are functionally independent first integrals of system of PDEs Ž5.. Due to constraint Ž3. there exists a first integral Wk Ž x, u. such that the condition ⭸ Wk r ⭸ u / 0 holds locally, since otherwise integrals W1 , W2 , . . . , Wnq1ym would be functionally dependent. Changing, if necessary, enumeration, we can put k s 1. Solving Ž6. with respect to W1 and introducing the notations

␻ Ž x, u . s W1 Ž x, u . ,

␻ j Ž x, u . s Wjq1 Ž x, u . ,

j s 1, . . . , n y m

we get the following expression

␻ Ž x, u . s ␸ Ž ␻ 1 Ž x, u . , . . . , ␻ nym Ž x, u . . ,

Ž 7.

where ␸ is an arbitrary smooth function of the variables ␻ 1 , . . . , ␻ nym . DEFINITION 2. We call an expression of the form Ž7., where ␸ is an arbitrary smooth function, ␻ Ž x, u., ␻ 1Ž x, u., . . . , ␻ nymŽ x, u. are functionally independent and ⭸␻r⭸ u / 0, an Ansatz for the field u s uŽ x .. LEMMA 1. There is one-to-one correspondence between the set of Ansatze ¨ for the field u s uŽ x . and the elements of the space I . Proof. While constructing the general solution of system Ž5. we have shown that each involutive family obeying Ž3. gives rise to the Ansatz of the form Ž7.. Furthermore, by construction equivalent involutive families of operators have the same set of functionally independent first integrals. Hence we conclude that each element of I corresponds to one and only one Ansatz Ž7..

A PRECISE DEFINITION OF REDUCTION

105

Let us prove the inverse, namely, that each Ansatz Ž7. correspond to one and only one element of the space I . Choose the functions ␪aŽ x, u., a s 1, . . . , m so that the expressions

␪a Ž x, u . ,

␻ Ž x, u . ,

␻ j Ž x, u . ,

a s 1, . . . , m,

j s 1, . . . , n y m

Ž 8.

are functionally independent. Then the functions Ž8. form the new coordinate system in the space of variables x, u ya s ␪a Ž x, u . ,

z j s ␻ j Ž x, u . ,

a s 1, . . . , m,

¨ s ␻ Ž x, u . ,

j s 1, . . . , n y m.

Ž 9.

Rewriting Ž7. in the new variables y, z, ¨ we arrive at the following expression: ¨ s ␸ Ž z1 , . . . , z nym . .

Ž 10 .

Evidently, the formula Ž10. give the general solution of the system of PDEs ⭸ ¨ r⭸ ya s 0, a s 1, . . . , m. The operators Q a s ⭸r⭸ ya , a s 1, . . . , m form an involutive family Žsince they commute. and fulfill the condition Ž3.. These properties are preserved after rewriting the operators Q a in the initial variables x, u. Thus we have constructed an involutive family of operators which corresponds to a given Ansatz for the field u s uŽ x .. However, this correspondence is not one-to-one, since the choice of the functions ␪aŽ x, u. is ambiguous. Let us show that choosing another set of functions ␹ 1Ž x, u., . . . , ␹mŽ x, u. will lead to an involutive family which is equivalent to the above obtained involutive family. Indeed, consider the transformation of variables yXa s ␹ a Ž x, u . ,

zXj s ␻ j Ž x, u . ,

a s 1, . . . , m,

X

¨ s ␻ Ž x, u . ,

j s 1, . . . , n y m

Ž 11 .

which reduces the initial Ansatz to become Ž10.. Comparing Ž9. and Ž11. we conclude that the map relating the coordinate systems y, z, ¨ and yX , zX ,¨ X is of the form yXa s Fa Ž y, z, ¨ . ,

zXj s z j ,

X

¨ s¨

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with a s 1, . . . , m, j s 1, . . . , n y m. Consequently, the operators ⭸r⭸ ya , after being rewritten in the new variables yX , zX , ¨ X read

⭸ ⭸ ya

m

s

Ý bs1

⭸ yXb ⭸ ⭸ ya ⭸

m

yXb

s

Ý bs1

⭸ Fb ⭸ ⭸ ya ⭸

yXb

m

s

Ý bs1

⭸ Fb ⭸ ya

QXb ,

a s 1, . . . , m.

Since det 5 ⭸ Fbr⭸ ya 5 m a, bs1 / 0, hence it follows that the involutive families Q a s ⭸r⭸ ya and QXa s ⭸r⭸ yXa are equivalent. Thus each involutive family that corresponds to a fixed Ansatz for the field u s uŽ x . belongs to the same equivalence class. This is the same as what was to be proved.

3. CONDITIONAL SYMMETRY OF PDES Consider a PDE of the form L Ž x, u, u, . . . , ur . s 0,

Ž 12 .

1

where x s Ž x 1 , x 2 , . . . , x n ., u s uŽ x . is a sufficiently smooth function and the symbol us stands for the set of partial derivatives of the function uŽ x . of the order s, i.e.,

½

def

us s u i1 . . . i s s

⭸ su ⭸ x i1 ⭈⭈⭈ ⭸ x i s

5

, 1 F i1 F n, . . . , 1 F i s F n .

Within the local approach Žused throughout the paper. PDE Ž12. is treated as an algebraic equation in the jet space J Ž r . of the order r. Then L is a smooth function from D into ⺢, where D is an open domain in J Ž r .. Denote the manifold defined by the equation L s 0 in J Ž r . by L , the set of all differential consequences of the system of PDEs Ž5. of the order not higher than r y 1 Žremember that r is the order of the initial equation Ž12.. by the symbol M and the corresponding manifold in J Ž r . by M . The most widely used definition of conditional invariance is the following one. DEFINITION 3. PDE Ž12. is conditionally invariant with respect to involutive family of operators Ž1. if the relation s0

Qa L Žr.

Ž 13 .

Ll M

holds ᭙a s 1, . . . , m. Here the symbol Q stands for the r th prolongation Žr. of the operator Q a .

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A PRECISE DEFINITION OF REDUCTION

This definition is very useful when computing CSs for a specific PDE. However, for theoretical considerations it is preferable to utilize the alternative definition of conditional invariance given below. DEFINITION 4. PDE Ž12. is conditionally invariant with respect to involutive family of operators Ž1. if the relation s 0, where ⌳ s L < M ,

Qa ⌳ Žr.

Ž 14 .

Ll M

holds ᭙a s 1, . . . , m. It will be shown below that Definitions 3 and 4 are equivalent. Note that there are some other ways to define CS w1, 2, 7, 34x; however, Definition 4 is the most convenient for the purposes of this paper. Note. Definitions 3 and 4 make sense provided L l M / ⭋. If this is not the case, namely, if there exists an involutive family such that L l M s ⭋, then we suppose by definition that PDE Ž12. is conditionally invariant with respect to this family. LEMMA 2. Let the system of PDEs Ž12. be conditionally in¨ ariant under in¨ oluti¨ e family of differential operators Ž1.. Then, it is conditionally in¨ ariant under in¨ oluti¨ e family Ž4. with arbitrary smooth functions ␭ ab . Proof. To prove the lemma we use the special representation for the coefficients of the sth prolongation of a first-order operator Q given in w32x Q sQq

s

n

Ý

Ý

ks1 i 1 , . . . , i k s1

Ž s.

␩i1 ⭈ ⭈ ⭈ i k

⭸ ⭸ u i1 . . . i k ⭸

n

if Q s

Ý ␰ i Ž x, u . ⭸ x

is1

q ␩ Ž x, u . i

⭸ ⭸u

where n

␩i1 ⭈ ⭈ ⭈ i k s Di1 ⭈⭈⭈ Di k ␩ y

ž

n

Ý ␰ i ui is1

i1 , . . . , i k s 1, . . . , n,

/

q

Ý ␰ i ui

1

⭈ ⭈ ⭈ ik i ,

is1

k s 1, . . . , s

and Di s

⭸ ⭸ xi

q ui

⭸ ⭸u

q

⬁

n

Ý

Ý

ps1 i 1 , . . . , i ps1

u i1 ⭈ ⭈ ⭈ i p i

⭸ ⭸ u i1 ⭈ ⭈ ⭈ i k

,

i s 1, . . . , n

is a total differentiation operator with respect to the variable x i .

,

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ZHDANOV, TSYFRA, AND POPOVYCH

Using the above identity yields the chain of equations that corresponds to condition Ž13. for the operators Ž4.: m

QXa L

s

žÝ

␭ ab Ž x, u . Q b L

bs1

Ll M

Žr.

Žr.

/

Ll M

m

Ý ␭ ab Ž x, u .

s

bs1

ž

Qb L Ll M

Žr.

/

s 0.

Ž 15 .

Evidently, the same arguments apply if we use Definition 4. The chain of equations analogous to the above equations Ž15. is obtained, where one should replace L by ⌳. The lemma is proved. One of the important consequences of the above lemma is that while studying conditional symmetry of PDEs we can restrict our considerations to elements of the quotient space I . This enables one to choose the most simple representative of each equivalence class in the way described below. Let Ž1. be an involutive family of differential operators satisfying condition Ž3.. Then it is possible to choose the functions ␭ ab Ž x, u. and, if necessary, to change enumeration of the variables x 1 , . . . , x n in such a way that operators Ž4. take the form QXa s

m

n

⭸

Ý ␭ ab Ý ␰ b i ⭸ x

bs1

ž

is1

q ␩b i

⭸ ⭸u

/

s

⭸ ⭸ xa

n

q

Ý jsmq1

␰ aX j

⭸ ⭸ xj

q ␩aX

⭸ ⭸u

,

a s 1, . . . , m. Since the family of operators QXa , a s 1, . . . , m is also involutive, there exist functions ␮ ˜ cab Ž x, u. such that m

w QXa , QXb x s Ý ␮ ˜ cab QXc ,

a, b s 1, . . . , m.

cs1

Computing commutators on the left-hand sides of the above equalities and equating coefficients of the linearly independent differential operators ⭸r⭸ x 1 , . . . , ⭸r⭸ x n , we have ␮ ˜ cab s 0, a, b, c s 1, . . . , m. Consequently, opX erators Q a form a commutative Lie algebra. Hence, we conclude that there is a local coordinate system Ž9. such that the operators QXa after being rewritten in the variables y, z, ¨ read QXa s

⭸ ⭸ ya

,

a s 1, . . . , m.

Ž 16 .

Consequently, without loss of generality we may consider commuting families of operators. This fact simplifies calculations, since the latter can always be represented in the form Ž16..

A PRECISE DEFINITION OF REDUCTION

LEMMA 3.

109

Relation Ž14. holds true if and only if relation Ž13. holds true.

Proof. It suffices to consider the case L l M / ⭋. Let us fix an arbitrary point j 0 s Ž x 0 , u 0 , u 0 , . . . , ur 0 . g J Ž r . : j 0 g L l M . 1 As established above we can suppose without loss of generality that the operators Q a commute. Choosing an appropriate coordinate transformation Ž9. in a neighborhood of Ž x 0 , u 0 . we reduce them to become Q a s ⭸r⭸ ya . Now the manifold M is determined by the following set of N algebraic equations in the space of variables y, z, ¨1 , . . . , ¨r

½

M s Ž y, z, ¨1 , . . . , ¨r .

᭙s s 1, . . . , r , ᭙ i1 , . . . , i s s 1, . . . , n

Ž ᭚k s 1, . . . , s: i k F m .

5

: ¨ i1 ⭈ ⭈ ⭈ i s s 0 ,

where the variable ¨ i1 ⭈ ⭈ ⭈ i s of the jet space J Ž r . correspond to the derivative ⭸ s ¨ rŽ ⭸ t i1 ⭈⭈⭈ ⭸ t i s ., t a s ya ; a s 1, . . . , m and t j s z jym , j s m q 1, . . . , n. Taking into account the fact that the relation L Ž j1 . s ⌳ Ž j1 . holds for any point j1 g L l M and using the definition of the partial derivative yield the equality

⭸L ⭸ ya

Žj0 . s

⭸⌳ ⭸ ya

Žj0 . .

Since j 0 is an arbitrary point of L l M , the equation

⭸L ⭸ ya

s Ll M

⭸⌳ ⭸ ya

Ž 17 . Ll M

holds. Now taking into-account the fact that an arbitrary order prolongation of the operator ⭸r⭸ ya is equal to ⭸r⭸ ya we see that the left-hand side of Ž17. coincides with the left-hand side of Ž13. and the right-hand side of Ž17. coincides with the left-hand side of Ž14.. Hence the validity of the assertion of the lemma follows.

4. REDUCTION OF PDES We say that Ansatz Ž7. reduces PDE Ž12. if the substitution of formulae Ž7. into Ž12. gives rise to an equation which is equi¨ alent to PDE containing ‘‘new’’ independent ␻ 1 , . . . , ␻ nym and dependent ␸ variables only. To

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ZHDANOV, TSYFRA, AND POPOVYCH

give a formal definition let us insert Ansatz Ž7. into the initial equation Ž12.. As a result, we get some pth Ž p F r . order PDE W Ž x, u, ␸ , ␸ , . . . , ␸ . s 0, 1

p

where the symbol ␸ stands for the set of kth order derivatives of the k function ␸ with respect to the variables ␻ 1 , . . . , ␻ nym . Eliminating the variables x, u with the help of formulae Ž8. yields W X ␪ 1 , . . . , ␪m , ␻ 1 , . . . , ␻ nym , ␸ , ␸ , . . . , ␸ s 0.

ž

p

1

/

DEFINITION 4. Ansatz Ž7. reduces PDE Ž12. if the relation W X s H ␪ 1 , . . . , ␪m , ␻ 1 , . . . , ␻ nym , ␸ , ␸ , . . . , ␸

ž

p

1

/

˜ Ž ␻ 1 , . . . , ␻ nym , ␸ , ␸ , . . . , ␸ . =L 1

Ž 18 .

p

holds with some function H that does not vanish in D l M . The equation ˜ s 0 is called the reduced differential equation. L Remark. The reduced differential equation is determined up to a non-vanishing multiplier depending on ␻ 1 , . . . , ␻ nym , ␸ , ␸ , . . . , ␸ . 1

p

As mentioned in Section 1 there exist two different approaches to reduction of PDEs that are based on their conditional symmetry. The first one is solving the determining equations Ž13. in order to obtain an involutive family of operators Q a such that the equation under study is conditionally invariant with respect to these operators. According to w20x an Ansatz corresponding to the thus obtained involutive family reduces the PDE under study in the sense of Definition 4. Alternatively, one can try to construct an Ansatz Ž7. reducing the PDE under study without solving an intermediate problem of finding involutive families of operators obeying Ž13.. The first approach is usually addressed to as the non-classical or conditional symmetry reduction method. The second one is called the Ansatz or direct reduction method. Note that within the framework of the direct reduction method one always supposes an explicit dependence of an Ansatz on u, thus restricting the choice of Ansatze to the following ¨ particular form: u s f Ž x, ␸ Ž ␻ 1 Ž x . , . . . , ␻ nym Ž x . . . .

Ž 19 .

This assumption simplifies substantially the calculation involved but, on the other hand, it may result in losing some classes of Ansatze ¨ which have implicit dependence on u. This is indeed the case for the relativistic

A PRECISE DEFINITION OF REDUCTION

111

eikonal equation where some invariant Ansatze ¨ cannot be represented in the form Ž19. w29x. Now we are going to prove that the conditional symmetry reduction and Ansatz approaches are equivalent. THEOREM 1. Let system of PDEs Ž12. be conditionally in¨ ariant under the in¨ oluti¨ e family of differential operators Ž1. satisfying condition Ž3. and let the function ⌳ s L < M ha¨ e the maximal rank on L l M or be identically equal to 0. Then, the Ansatz Ž7. corresponding to Ž1. reduces system of PDEs Ž12.. In¨ ersely, let Ansatz Ž7. reduce PDE Ž12.. Then there is an in¨ oluti¨ e family of operators Ž1. obeying Ž3. and corresponding to Ansatz Ž7. such that PDE Ž12. is conditionally in¨ ariant with respect to this in¨ oluti¨ e family. Proof. Conditional symmetry « reduction: Let PDE Ž12. be conditionally invariant with respect to an involutive family of differential operators Ž1. obeying the relation Ž3. and let the function ⌳ have the maximal rank ˜ s 1 in Ž18., on L l M . If L l M s ⭋, then we can choose H s W X , L which means that PDE Ž12. is reduced to the incompatible equation 1 s 0. Suppose now that L l M / 0. Using the arguments analogous to those applied to prove Lemma 3, we rewrite Ž14. as follows:

⭸⌳ ⭸ ya

s 0. Ll M

Making use of the Hadamard lemma, we represent the above relation in the equivalent form: ⌳ y a s Fa ⌳ ,

a s 1, . . . , m.

Ž 20 .

Consequently, the function ⌳ is a solution of the over-determined system of PDEs Ž20., where Fa are smooth functions of the variables y, z, ¨ , ¨ Ž z ., . . . , ¨r Ž z .. Here the symbol ¨s Ž z . corresponds to the set of the 1 s-order derivatives of the function ¨ with respect to the variables z only. The necessary and sufficient compatibility conditions of Ž20. read

ž

⭸ Fa ⭸ yb

y

⭸ Fb ⭸ ya

/

⌳ s 0,

a, b s 1, . . . , m.

Ž 21 .

⌳s the function ⌳ has the maximal rank on L l M s  M s 0, ⌳ s 04 , in an arbitrary neighborhood of any point j g L l M in M there exists a point jX such that ⌳ŽjX . / 0. In view of this we get from Ž21. the following

112

ZHDANOV, TSYFRA, AND POPOVYCH

system of PDEs:

⭸ Fa ⭸ yb

s

⭸ Fb ⭸ ya

,

a, b s 1, . . . , m.

Consequently, there is a function F of the variables y, z, ¨ , ¨1 Ž z ., . . . , ¨r Ž z . such that Fa s ⭸ Fr⭸ ya , ᭙a. Hence we get the general solution of Ž20.

˜ ⌳ s exp  F 4 L,

Ž 22 .

˜ is an arbitrary function of z, ¨ , ¨ , . . . , ¨r . where L 1 Inserting the Ansatz ¨ s ␸ Ž z1 , . . . , z nym . invariant under the family of operators ⭸r⭸ y 1 , . . . , ⭸r⭸ ym into L yields ˜ < ¨ s␸ Ž z1 , . . . , z ny m . . L < ¨ s␸ Ž z 1 , . . . , z ny m . s ⌳ < ¨ s␸ Ž z 1 , . . . , z ny m . s exp  F 4 L For the case ⌳ ' 0 the proof is obvious Žfor example, we can choose H ' 1 and L ' 0.. Reduction « conditional symmetry: Let the Ansatz Ž7. reduce PDE Ž12.. Let us make the change of variables Ž9. in order to represent Ž7. in the form Ž10.. Then a corresponding involutive family of differential operators can be chosen as follows: Q a s ⭸r⭸ ya . Since the function ␸ is arbitrary, to insert the Ansatz Ž10. into PDE Ž12. written in the new variables y, z, ¨ Ž y, z . is the same as considering the intersection of the manifold L with the manifold M with a subsequent identifying ␸ with ¨ . By assumption of the theorem a relation of the form

˜ Ž z1 , . . . , z nym , ␸ , ␸ , . . . , ␸ . L < ¨ s␸ Ž z 1 , . . . , z ny m . s H Ž y, z, ␸ , ␸ , . . . , ␸ . L p

1

1

p

holds with some non-vanishing H. Consequently,

˜ z1 , . . . , z nym , ¨ , ¨ Ž z . , . . . , ¨p Ž z . . ⌳ s L < M s H y, z, ¨ , ¨1 Ž z . , . . . , ¨p Ž z . L 1

ž

/ ž

/

As the r th prolongation of the operator Q a s ⭸r⭸ ya is equal to ⭸r⭸ ya , ᭙a, we have Qa ⌳ s Žr.

s

⭸ ⭸ ya

⌳

⭸ H y, z, ¨ , ¨1 Ž z . , . . . , ¨p Ž z .

ž

⭸ ya

/ L˜

žz ,..., z 1

nym , ¨ , ¨ 1

Ž z.

, . . . , ¨p Ž z . .

/

A PRECISE DEFINITION OF REDUCTION

113

Next, as the function H does not vanish in D l M , the set of solutions of the equation ⌳ s 0 coincides with the set of solutions of the equation ˜ s 0. Consequently, the relation L

ž

Qa ⌳ Žr.

⌳s 0

/

s Qa ⌳ M

s0 Ll M

Žr.

holds ᭙a s 1, . . . , m. Hence, we conclude that the initial PDE Ž12. written in the variables y, z, ¨ Ž y, z . is conditionally invariant with respect to the involutive family ⭸r⭸ ya . Rewriting Ž12. and the involutive family ⭸r⭸ ya in the initial variables x, u completes the proof of the theorem.

5. APPLICATION: THE NONLINEAR WAVE EQUATION Now we are going to consider a specific example enlightening the peculiarities of the Ansatz Ždirect. and conditional Žnon-classical . symmetry approaches to the problem of dimensional reduction of PDEs. As a basic model we take the nonlinear Ž1 q 3.-dimensional wave equation I u s F Ž u. .

Ž 23 .

Here I s ⭸ 2r⭸ x 02 y ⌬ is the d’Alembertian, u s uŽ x . is a real-valued function of four real variables x 0 , x 1 , x 2 , x 3 , and F is an arbitrary continuous function. First, we apply the Ansatz approach to reduction of PDE Ž23.. To this end we utilize an idea suggested in w5x and make use of the Lie symmetry properties of the equation under study for the sake of elucidating of a possible structure of the Ansatz for the uŽ x .. As is well known the maximal in Lie’s sense symmetry group admitted by Eq. Ž23. with an arbitrary F is the ten-parameter Poincare ´ group P Ž1, 3. having the generators P␮ s

⭸ ⭸x␮

,

J␮␯ s x␮ P␯ y x␯ P␮ ,

␮ , ␯ s 0, 1, 2, 3, ␮ - ␯ . Ž 24 .

Hereafter raising and lowering the indices is performed with the help of the metric tensor of the Minkowski space g␮␯ s diagŽ1, y1, y1, y1. and the summation convention is used. For example, x ␮ s g␮␯ x␯ s

½

x0 , yx a ,

␮ s 0, ␮ s a s 1, 2, 3.

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ZHDANOV, TSYFRA, AND POPOVYCH

Symmetry reduction of Eq. Ž23. by subgroups of the Poincare ´ group has been performed in w27, 45x. An analysis of thus obtained invariant Ansatze ¨ for the scalar field uŽ x . shows that they have the same structure uŽ x . s ␸ Ž ␻ Ž x . . .

Ž 25 .

The form of a real-valued function ␻ Ž x . is determined by the choice of a specific subgroup of the group P Ž1, 3.. Thus as a first step of our approach we fix the Ansatz for solutions of Eq. Ž23. to be of the form Ž25.. However, we do not impose a priori restrictions on the choice of unknown function ␻ Ž x .. The only requirement to be met is that inserting the expression Ž25. into Eq. Ž23. should yield an ordinary differential equation ŽODE. for the function ␸ Ž ␻ .. This requirement gives rise to a compatible over-determined system of nonlinear partial differential equations for the functions ␻ Ž x .. Any solution of the latter after being inserted into formula Ž25. yields an Ansatz for the scalar field uŽ x . reducing Eq. Ž23. to ODE. Inserting Ž25. into the nonlinear wave equation Ž23. gives

ž

⭸␻ ⭸␻ ␮

⭸ x␮ ⭸ x

d 2␸

/

d␻

2

q I␻

d␸ d␻

s FŽ ␸. .

Ž 26 .

As the above equation has to be equivalent to ODE for the function ␸ Ž ␻ . under arbitrary F, the coefficients of d 2␸rd ␻ 2 , d ␸rd ␻ have to be some functions of ␻ . This requirement yields that there exist real-valued functions f 1Ž ␻ ., f 2 Ž ␻ . such that

⭸␻ ⭸␻ ⭸ x␮ ⭸ x ␮

s f 1Ž ␻ . ,

I ␻ s f2 Ž ␻ . .

Ž 27 .

System of nonlinear PDEs Ž27. is the necessary and sufficient condition for the Ansatz Ž25. to reduce the nonlinear wave equation Ž23. to an ordinary differential equation. And, what is more, the equation for the function ␸ Ž ␻ . reads f 1Ž ␻ .

d 2␸ d␻

2

q f2 Ž ␻ .

d␸ d␻

s FŽ ␸. .

Ž 28 .

Summing up we conclude that any solution of over-determined system of nonlinear PDEs Ž27. gives rise to an Ansatz for the field uŽ x . reducing Eq. Ž23. to an ODE of the form Ž28.. In particular, any Ansatz corresponding to the Lie symmetry of the nonlinear wave equation Ž23. can be obtained in this way. However, the Lie Ansatze ¨ do not exhaust the set of all possible

A PRECISE DEFINITION OF REDUCTION

115

substitutions of the form Ž25. reducing Eq. Ž23. to ODEs. This is explained by an existence of wide classes of Ansatze ¨ Ž25. that correspond to conditional symmetry of the nonlinear wave equation and cannot be, in principle, obtained within the framework of the Lie symmetry approach. Now we utilize the conditional symmetry approach for obtaining Ansatz Ž25. that reduce a PDE in four dimensions Ž23. to an ODE. Consider conditional symmetry of the nonlinear wave equation within the class of first-order differential operators Q s ␰␮ Ž x .

⭸ ⭸ x␮

.

Ž 29 .

As we are looking for conditional symmetries that enable reduction of Ž23. to an ODE, it is necessary to consider an involutive family of three differential operators of the form Ž29., namely, Q a s ␰ a ␮Ž x .Ž ⭸r⭸ x␮ ., a s 1, 2, 3. And, what is more, we require that the restriction Ž3. is respected, which means that 3

rank ␰ a ␮ Ž x .

3 as1 ␮ s0

s 3.

Taking into account the above relation, Lemma 2, and also making use of the Poincare ´ invariance of the equation under study we can always transform the operators Q1 , Q2 , Q3 to become Qa s

⭸ ⭸ xa

y faŽ x .

⭸ ⭸ x0

,

a s 1, 2, 3.

Ž 30 .

It is straightforward to check that the family of operators Ž30. is involutive if and only if Q1 , Q2 , Q3 commute each with another. Hence we get the system of three PDEs for the functions f 1Ž x ., f 2 Ž x ., f 3 Ž x .

⭸ fa ⭸ xb

y fb

⭸ fa ⭸ x0

s

⭸ fb ⭸ xa

y fa

⭸ fb ⭸ x0

,

where a, b s 1, 2, 3, a - b. Its general solution can be represented in the form Žsee, e.g., w25x. fa s

⭸␻ ⭸ xa

⭸␻

ž / ⭸ x0

y1

,

a s 1, 2, 3,

Ž 31 .

where ␻ s ␻ Ž x . is an arbitrary twice continuously differentiable function, ⭸␻r⭸ x 0 / 0.

116

ZHDANOV, TSYFRA, AND POPOVYCH

The condition Ž13. of invariance of the nonlinear wave equation Ž23. with respect to operators Ž30. after some involved straightforward algebraic manipulations reduces to the over-determined system of six PDEs for the functions f 1 , f 2 , f 3 I fa y 2

⭸ fa ⭸ fb ⭸ xb ⭸ x0

⭸ fa

s 0,

⭸ x0

y fb

⭸ fa ⭸ xb

s 0,

a s 1, 2, 3. Ž 32 .

Inserting the expressions for f a Ž31. into Ž32. and rearranging the obtained PDEs for the function ␻ Ž x . yields

ž

⭸␻

⭸

⭸ x0 ⭸ x a

ž

y

⭸␻

⭸␻

⭸

⭸ x a ⭸ x0

⭸

⭸ x0 ⭸ x a

y

⭸␻

/

⭸␻ ⭸␻ ⭸ x␮ ⭸ x ␮ ⭸

/

⭸ x a ⭸ x0

s 0,

I␻ s 0

with a s 1, 2, 3. Hence we conclude that there are smooth functions f 1Ž ␻ ., f 2 Ž ␻ . such that the relations Ž ⭸␻r⭸ x␮ .Ž ⭸␻r⭸ x ␮ . s f 1Ž ␻ ., I ␻ s f 2 Ž ␻ . hold and we arrive at system of PDEs Ž27.. Consequently, the involutive family Ž30. necessarily takes the form Qa s

⭸ ⭸ xa

y

⭸␻ ⭸ xa

⭸␻

ž / ⭸ x0

y1

⭸ ⭸ x0

,

a s 1, 2, 3,

Ž 33 .

where the function ␻ s ␻ Ž x . is a solution of system Ž27.. As the function ␻ s ␻ Ž x . is the first integral of the system of PDEs Q a f Ž x . s 0, a s 1, 2, 3, the Ansatz for the field uŽ x . corresponding to the family Q1 , Q2 , Q3 is given by Ž25.. Thus both Ansatz Ždirect. and conditional symmetry Žnon-classical . approaches to reduction of the nonlinear wave equation Ž23. to ODEs lead to the same reduction conditions, namely, to the system of differential equations Ž23. consisting of the nonlinear wave and relativistic Hamilton᎐Jacobi equations. Following w12x we call this system the d’Alembert᎐Hamilton system. The d’Alembert᎐Hamilton system in three dimensions was studied by Jacobi w35x, Smirnov and Sobolev w36, 37x, and later on by Collins w38x. Collins constructed the general solution of system of nonlinear PDEs Ž26. for a complex-valued function of three complex variables. Some exact solutions of the d’Alembert᎐Hamilton system in four dimensions have been constructed by Cartan w39x, Bateman w40x, and Erugin w41x. Recently, we have constructed the general solution of system Ž26. for the complexvalued function of four complex variables w42, 43x.

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A PRECISE DEFINITION OF REDUCTION

As established in w19x, system of PDEs Ž26. for the real-valued function ␻ Ž x . is compatible if and only if it is locally equivalent to the system I ␻ s ⑀ N␻y1 ,

Ž ⭸␮ ␻ . Ž ⭸ ␮␻ . s ⑀ ,

⑀ s "1, 0,

Ž 34 .

where N s 0, 1, 2, 3. The real form of the general solution of the system of PDEs Ž34. is given by one of the formulae below w15, 19x I. ⑀ s y1 Ž1. N s 0

␻ s A␮ Ž ␶ . x ␮ q R1 Ž ␶ . ,

Ž 35 .

where ␶ s ␶ Ž x . is determined in implicit way B␮ Ž ␶ . x ␮ q R 2 Ž ␶ . s 0 and A␮Ž␶ ., B␮Ž␶ ., R1Ž␶ ., R 2 Ž␶ . are arbitrary smooth real-valued functions satisfying the conditions A␮ Ž ␶ . A ␮ Ž ␶ . s y1,

A␮ Ž ␶ . B ␮ Ž ␶ . s 0,

A˙␮ Ž ␶ . B ␮ Ž ␶ . s 0,

B␮ Ž ␶ . B ␮ Ž ␶ . s 0; Ž2.

Ns1 2

2

␻ 2 s Ž d␮ x ␮ q g 2 . y Ž a␮ x ␮ q g 1 . , 2

Ž 36 .

2

␻ 2 s Ž b␮ x ␮ q C1 . q Ž c␮ x ␮ q C2 . ,

Ž 37 .

where g i s g i Ž a␮ x ␮ q d␮ x ␮ . g C 2 ŽR1 , R1 . are arbitrary functions; Ž3. N s 2 Ža. 2

␻ 2 s y Ž x␮ q A␮ Ž ␶ . . Ž x ␮ q A ␮ Ž ␶ . . y  B␮ Ž ␶ . Ž x ␮ q A ␮ Ž ␶ . . 4 , Ž 38 . where ␶ s ␶ Ž x . is determined in implicit way

Ž x␮ q A␮ Ž ␶ . . B˙␮ Ž ␶ . s 0,

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ZHDANOV, TSYFRA, AND POPOVYCH

A␮Ž␶ ., B␮Ž␶ . are arbitrary smooth real-valued functions satisfying the conditions B␮ Ž ␶ . B ␮ Ž ␶ . s y1,

B˙␮ Ž ␶ . B˙␮ Ž ␶ . s 0,

A˙␮ Ž ␶ . s R Ž ␶ . B˙␮ Ž ␶ .

with an arbitrary RŽ␶ . g C 1 ŽR1, R1 .; Žb. 2

␻ 2 s y Ž x␮ q A␮ Ž ␶ . . Ž x ␮ q A ␮ Ž ␶ . . q  b␮ Ž x ␮ q A ␮ Ž ␶ . . 4 , Ž 39 . where ␶ s ␶ Ž x . is determined in implicit way

Ž x␮ q A␮ Ž ␶ . . Ž A˙␮ Ž ␶ . q b ␮ b␯ A˙␯ Ž ␶ . . s 0, A␮Ž␶ . are arbitrary smooth real-valued functions satisfying the condition A˙␮ Ž ␶ . A˙␮ Ž ␶ . q b␮ A˙␮ Ž ␶ .

ž

2

/

s 0;

Žc. 2

2

2

␻ 2 s Ž b␮ x ␮ q C1 . q Ž c␮ x ␮ q C2 . q Ž d␮ x ␮ q C3 . ; Ž4.

Ž 40 .

Ns3

␻ 2 s y Ž x␮ q A␮ Ž ␶ . . Ž x ␮ q A ␮ Ž ␶ . . ,

Ž 41 .

where ␶ s ␶ Ž x . is determined in implicit way

Ž x␮ q A␮ Ž ␶ . . B ␮ Ž ␶ . s 0, A␮Ž␶ ., B␮Ž␶ . are arbitrary smooth real-valued functions satisfying the conditions A˙␮ Ž ␶ . B ␮ Ž ␶ . s 0,

B␮ Ž ␶ . B ␮ Ž ␶ . s 0.

Ž 42 .

II. ⑀ s 1 Ž1. N s 0

␻ s a␮ x ␮ q C1 ; Ž2.

Ž 43 .

Ns1 2

2

␻ 2 s Ž a␮ x ␮ q C1 . y Ž d␮ x ␮ q C2 . ;

Ž 44 .

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A PRECISE DEFINITION OF REDUCTION

Ns2

Ž3.

2

2

2

␻ 2 s Ž a␮ x ␮ q C1 . y Ž c␮ x ␮ q C2 . y Ž d␮ x ␮ q C3 . ; Ns3

Ž4.

␻ 2 s Ž x␮ q C␮ . Ž x ␮ q C ␮ . . III.

Ž 45 .

Ž 46 .

⑀ s 0, N s 0 A␮ Ž ␻ . x ␮ q B Ž ␻ . s 0,

where A␮ , B are arbitrary smooth real-valued functions such that A␮ A ␮ s 0. In the above formulae C0 , . . . , C3 are arbitrary real constants and a␮ , b␮ , c␮ , d␮ are arbitrary real constants satisfying the conditions a␮ a ␮ s yb␮ b ␮ s yc␮ c ␮ s yd␮ d ␮ s 1, a␮ b ␮ s a␮ c ␮ s a␮ d ␮ s b␮ c ␮ s b␮ d ␮ s c␮ d ␮ s 0, and dot over the symbol means differentiation with respect to ␶ . Let us emphasize that all the functions ␻ Ž x . defined by formulae Ž35., Ž36., Ž38., Ž39., and Ž41. give rise to conditionally invariant Ansatze ¨ for the field uŽ x . of the form Ž25.. Using these one can construct broad families of new Žnon-Lie. exact solutions even for such a well studied model as the nonlinear wave equation Žsee also w46x.. Consider, for example, the conformally invariant nonlinear wave equation I u s ␭u3 .

Ž 47 .

The Ansatz u s ␸ y Ž x␮ q A␮ Ž ␶ . . Ž x ␮ q A ␮ Ž ␶ . .

ž

1r2

/,

where ␶ s ␶ Ž x . is defined in Ž41. and A␮Ž␶ ., B␮Ž␶ . are arbitrary smooth functions satisfying Ž42., reduces Ž47. to an ODE for ␸ s ␸ Ž ␻ . d 2␸ d␻

2

q 3 ␻y1

d␸ d␻

s y␭␸ 3 .

Two particular solutions of the latter ␸ s ␭y1 r2␻y1 and ␸ s aŽ ␻ 2 q ␭ ar8.y1 , a s const give rise to two families of new exact solutions of the

120

ZHDANOV, TSYFRA, AND POPOVYCH

cubic wave equation Ž47. u Ž x . s ␭y1r2 y Ž x␮ q A␮ Ž ␶ . . Ž x ␮ q A ␮ Ž ␶ . . uŽ x . s a

␭ a2 8

y1 r2

,

y1

␮

␮

y Ž x␮ q A␮ Ž ␶ . . Ž x q A Ž ␶ . .

.

Choosing arbitrary functions A␮Ž␶ . to be constant yields the well-known exact solutions of Ž47. obtained in w45x within the symmetry reduction routine. However, if A␮Ž␶ . are not constants, the constructed solutions are new and cannot be found using the symmetry reduction procedure.

6. CONCLUDING REMARKS Thus introducing a rigorous definition of reduction of PDEs enables a systematic treatment of the problem of studying interrelations between the Ansatz Ždirect. and non-classical Žconditional symmetry. approaches to dimensional reductions of multi-dimensional PDEs. We have proved that the direct approach, taken in a full generality, is equivalent to the nonclassical approach provided some natural restrictions are met Žsee Theorem 1.. When we say ‘‘in a full generality’’ we mean that the most general form of the similarity Ansatz should be taken. For example, the Ansatz Ž25. is a particular case of the general similarity Ansatz for PDE Ž23. U Ž x, u . s ␸ Ž ␻ Ž x, u . . .

Ž 48 .

Imposing the restrictions UŽ x, u. s u, ␻ Ž x, u. s ␻ Ž x . results in losing some reductions. On the other hand, with this choice of the form of the Ansatz we were able to get a full solution of the problem of constructing the corresponding conditionally invariant Ansatze ¨ of the form Ž25. Žsee formulae Ž35. ᎐ Ž46.., as the system of nonlinear determining equations for ␻ Ž x . proves to be integrable. Integrating it yields broad classes of principally new reductions and exact solutions for nonlinear wave equations containing several arbitrary functions of one argument. So both direct and nonclassical approaches can be used on equal footing and the choice of one of them is, in fact, a matter of taste. Nevertheless, the direct approach has an evident benefit of being comparatively simple, since only some basics of the standard university course on partial differential equations are required for understanding and implementing it. Another merit of the direct approach is its flexibility. A similarity Ansatz can be easily modified in order to yield, for example, ‘‘nonlinear separation of variables’’ in the spirit of w47x Žsee also w21, 48x.. However, if we wish to

A PRECISE DEFINITION OF REDUCTION

121

take into consideration the case of implicit Ansatze ¨ Žsay, of the form Ž48.., then the nonclassical approach is preferable. These points are illustrated by the considerations of Section 5, where both approaches are applied to the nonlinear wave equation and the direct method provides a shorter way to obtain conditional symmetries of the equation under study. The fact that we restrict our considerations to scalar PDEs, namely, to PDEs with one dependent variable, is explained by the major difficulties arising when handling systems of PDEs. The first problem is the fact that different equations of the system may have different orders. Next, if the number of equations is greater than the number of dependent variables, there arises a natural question of compatibility of this system. However, the implication conditional in¨ ariance « reduction can be proved in almost the same way as it is done for the case of a single PDE in Theorem 1 w20x. The problem is how to modify the proof in order to establish a validity of an assertion reduction « conditional in¨ ariance. We postpone the investigation of this problem to our future publications.

ACKNOWLEDGMENTS This work is partially supported by the Ukrainian DFFD Foundation under the project 1.4r356.

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