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. As shown in this section, the literature on conditional symmetries lacks a purely geometric definition of characteristic systems and the study of characteristics systems for higher-order PDEs is scarcely considered |
. This section aims to fill this gap. Moreover, our definition is not limited to characteristic systems for Lie algebras of vector fields |
, which will allow us to dispense, in the next sections, with certain unnecessary technical conditions on characteristic systems given in previous works |
Definition 3.1 Let [MATH] be a finite-dimensional linear space of vector fields on [MATH] . The characteristic system of [MATH] in [MATH] is the subset of [MATH] given by |
[EQUATION] where [MATH] is the space of contact forms on [MATH] Let us describe Definition 3.1 in coordinates in order to compare it with previous definitions |
and to justify the term ‘characteristic system’. If [MATH] admits a basis (as a linear space) of the form [MATH] , with [MATH] , then the basis [MATH] of [MATH] , with [MATH] and [MATH] , allows us, by using ( 2.3 ), to write that |
[EQUATION] for [MATH] and [MATH] Hence, the functions [MATH] , where [MATH] , are the characteristics [MATH] of the vector field [MATH] and the [MATH] are all the compositions of up to [MATH] total derivatives of the characteristics [MATH] of [MATH] . Then, [MATH] if and only if [MATH] for [MATH] and the basis of basic... |
In the theory of conditional symmetries, a characteristic system in [MATH] is mostly defined to be the subset of zeros of the characteristics (and their total derivatives up to order [MATH] ) of a basis of a Lie algebra of vector fields |
. Hence, Definition 3.1 reproduces the standard definition when [MATH] is a Lie algebra. Our definition is purely geometrical as it does not rely on the use of characteristics of a basis of [MATH] and it does not need to give their coordinate expressions as in previous works. |
Some definitions of characteristic systems demand additional technical conditions on the elements of [MATH] (see ). The reasons to assume these conditions will be explained in the following sections. Meanwhile, Olver and Roseneau proposed a very general definition of differential constraints, the so-called side conditi... |
. This generalisation covers Definition 3.1 as a particular case. Nevertheless, Olver and Roseneau’s definition is not linked to characteristics of vector fields due to its generality 32 , p. 15] , which makes it inappropriate to the study of conditional symmetries, which are strongly related to characteristics. |
Note that since [MATH] can be understood as the [MATH] -th order system of PDEs determined by the characteristics and their successive total differentials up to order [MATH] of the elements of a linear space of vector fields [MATH] , it makes sense to call [MATH] a characteristic system. |
As the Cartan distribution [MATH] is the intersection of all the kernels of contact forms on [MATH] , the definition of [MATH] can be rewritten in the following dual equivalent manner. |
Definition 3.2 Let [MATH] be a finite-dimensional linear space of vector fields on [MATH] . The characteristic system of [MATH] in [MATH] is the subset [MATH] where the prolongations to [MATH] of the vector fields of [MATH] take values in the Cartan distribution [MATH] |
4. On characteristic systems and sections of jet bundles In the Clairin approach to the theory of conditional symmetries , characteristic systems are employed to study systems of PDEs in normal form. This section studies this relation in geometric terms. |
Note that [MATH] does not need to be a submanifold of [MATH] . For instance, if [MATH] and [MATH] , then [MATH] is not a submanifold of [MATH] . Nevertheless, in the next sections, [MATH] will be proven to be a submanifold due to the nature of the Clairin theory of conditional symmetries |
. The following results will explain the geometrical meaning of such assumptions and specify which of them can be dismissed. We begin with the following lemma. |
Lemma 4.1 Let [MATH] be a basis of a linear space [MATH] of vector fields on [MATH] . If [MATH] , then one has the following equality at points of [MATH] |
[EQUATION] where the functions [MATH] on the right-hand side are considered to be functions on [MATH] in the natural way, namely as functions on [MATH] depending only on [MATH] and [MATH] (see |
for further details). Proof. In coordinates we write [EQUATION] for [MATH] . To simplify the notation, we write [MATH] . In view of ( 2.3 ), one has that |
[MATH] Since all total derivatives [MATH] of order [MATH] of the characteristics of [MATH] vanish on [MATH] , we obtain that [MATH] for [MATH] and [MATH] In view of ( 2.3 ), one has that [MATH] , for [MATH] [MATH] , and formula ( 4.1 ) follows easily from the expression ( 2.3 ) for the prolongations to [MATH] of [MATH] |
We have already explained that [MATH] amounts to an [MATH] -th order system of PDEs. We are now concerned with establishing when such a system is in normal form. We prove the following rather simple fact to stress that [MATH] is a section of [MATH] if and only if its associated [MATH] -th order system of PDEs can be wr... |
Proposition 4.2 A characteristic system [MATH] amounts to an [MATH] -th order system of PDEs in normal form if and only if [MATH] is a section of [MATH] |
Proof. If [MATH] is a section of [MATH] , then every coordinate [MATH] of points in [MATH] can be written as a function [MATH] . Therefore, [MATH] can be written as the subset of [MATH] where the series of conditions [MATH] , with [MATH] and [MATH] are obeyed. Such conditions amount to an [MATH] -th order system of PDE... |
Let us characterise, via the space [MATH] , when [MATH] amounts to an [MATH] -th order system of PDEs in normal form. Theorem 4.3 |
Let [MATH] be a linear space of vector fields on [MATH] . Then, [MATH] is a section of the bundle [MATH] if and only if the vector fields of [MATH] span a regular distribution [MATH] of rank [MATH] and [MATH] projects onto [MATH] under [MATH] |
Proof. The linear space [MATH] admits a basis [MATH] of the form ( 4.2 ). Let us prove first the converse part of our theorem. If [MATH] is regular of rank [MATH] and its projection onto [MATH] is [MATH] (relative to [MATH] ), then [MATH] span [MATH] and |
[EQUATION] span [MATH] for every [MATH] . Consequently, we can assume without loss of generality that the first vector fields [MATH] from [MATH] are such that [MATH] are linearly independent and span [MATH] at a point [MATH] . If [MATH] is the [MATH] matrix with coefficients [MATH] of [MATH] , then [MATH] on an open su... |
[MATH] , with [MATH] . Since [MATH] generate the distribution [MATH] , in view of Lemma 4.1 , the [MATH] is locally given by the zeroes of the functions [MATH] , with [MATH] This shows that |
[EQUATION] Thus, the value of [MATH] is determined for every [MATH] univocally in terms of the [MATH] and [MATH] . Hence, [MATH] is locally a section of [MATH] . The smoothness of the vector fields [MATH] ensures that [MATH] is locally a smooth section of [MATH] . As the same procedure can be applied to any point [MATH... |
Let us prove the direct part of this theorem. If [MATH] is a smooth section of [MATH] , then the points [MATH] of [MATH] are such that for every [MATH] , the possible values of [MATH] are univocally defined by the conditions |
[EQUATION] The values of [MATH] for every [MATH] are univocally determined if and only if the extended and the standard matrix of coefficients of the previous system have rank [MATH] . The matrix of coefficients has rank [MATH] if and only if the projections of the vector fields of [MATH] onto [MATH] span a distributio... |
Example Theorem 4.3 shows that [MATH] may be a section of [MATH] even if [MATH] is not a Lie algebra and/or its elements do not span an integrable distribution, these conditions being the standard used in other works |
. For instance, consider [MATH] [MATH] and [MATH] , which is not a Lie algebra. Then [MATH] is given by the zeroes of the conditions [MATH] , which amount to a section on [MATH] |
Example Theorem 4.3 states that [MATH] may be a section of [MATH] when [MATH] is a Lie algebra of dimension different from [MATH] . This case does not appear in the Clairin approach, where [MATH] , (see |
). To illustrate such a new possibility, consider the case where [MATH] and [MATH] . The Lie algebra [MATH] is a three-dimensional Lie algebra giving rise to a section on [MATH] given by the equations [MATH] |
Example Let us provide an example of when a Lie algebra [MATH] on [MATH] does not give rise to a section of [MATH] . Consider [MATH] [MATH] , and the Lie algebra [MATH] . The Lie algebra [MATH] gives rise to a regular distribution of rank one, but the distribution does not project onto [MATH] under [MATH] (it has a pro... |
Let us now extend Proposition 4.3 to [MATH] . This will lead to the requirement that the distribution spanned by the elements of [MATH] be involutive in cases relevant to us, namely where we are studying systems of PDEs and therefore [MATH] . Before proving this result, we need the following lemma. |
Lemma 4.4 If [MATH] gives rise to a section [MATH] of [MATH] , then the first-order system of PDEs ( 2.5 ) associated with [MATH] is such that the vector fields ( 2.6 span the same distribution as [MATH] |
Proof. Let [MATH] be a basis of [MATH] given by ( 4.2 ). In view of Theorem 4.3 , the distribution [MATH] has rank [MATH] and there exist [MATH] vector fields among [MATH] , let us say without loss of generality that these are [MATH] , whose coefficients allow us to determine a first-order system of PDEs given by ( 4.3... |
[EQUATION] span the same distribution as the vector fields of [MATH] Theorem 4.5 Let [MATH] be a linear space of vector fields on [MATH] . Then, [MATH] , with [MATH] , is a smooth section of the bundle [MATH] if and only if the vector fields of [MATH] span an involutive distribution [MATH] of rank [MATH] projecting ont... |
Proof. Let us prove the converse part of the theorem by induction relative to [MATH] . Assume that [MATH] given by ( 4.2 ) is a basis of the linear space [MATH] . In view of Theorem 4.3 and the considered assumptions, [MATH] is a section of [MATH] and then [MATH] |
for certain functions [MATH] on [MATH] , with [MATH] and [MATH] Let us prove that if the [MATH] , where [MATH] is any multi-index with [MATH] for a natural number [MATH] , can be written as functions depending on [MATH] only on the subset [MATH] , then the [MATH] for [MATH] can also. Recall that one can guarantee on [M... |
that [EQUATION] By our induction hypothesis, the coordinates [MATH] of points of [MATH] for [MATH] can be written as functions of [MATH] only, namely [MATH] . Hence, rewriting the right-hand side of ( 4.4 ), we obtain that |
[EQUATION] for certain functions [MATH] with [MATH] [MATH] , and [MATH] that gather all the terms of [MATH] with derivatives of the [MATH] up to order [MATH] Since [MATH] is a section of [MATH] , Theorem 4.3 ensures that there exist [MATH] elements of the basis of [MATH] , which are assumed without loss of generality t... |
Let us prove that the system ( 4.5 ) is compatible. If [MATH] , then this is obvious as the expressions ( 4.5 ) determine each derivative of [MATH] in terms of the unique independent variable uniquely. If [MATH] , then [MATH] for [MATH] but the expressions for [MATH] and [MATH] obtained from ( 4.5 ) may give different ... |
Let us prove the direct part of the theorem. If [MATH] is a section of [MATH] , then [MATH] is a section of [MATH] and, using Theorem 4.3 , we obtain that [MATH] has rank [MATH] and projects onto [MATH] under [MATH] . It is left to prove that [MATH] is involutive. If [MATH] , the result is immediate. If [MATH] , we can... |
[EQUATION] for [MATH] and [MATH] . Since [MATH] is a section, one has that the equations [MATH] and [MATH] , with [MATH] , must lead to a unique solution for [MATH] at every [MATH] . In terms of the Lemma 4.1 , one has that [MATH] , which implies that the vector fields |
[EQUATION] commute among themselves. In view of Lemma 4.4 , the vector fields of [MATH] are linear combinations with functions in [MATH] of the vector fields ( 4.6 ). Then, they span an involutive distribution. |
Remark 4.6 The proof of Theorem 4.5 shows that equations ( 4.5 ) determine the value of each coordinate [MATH] , with [MATH] , of points of a section [MATH] of [MATH] with [MATH] , in terms of the [MATH] with [MATH] Thus, two integrable sections [MATH] and [MATH] , with [MATH] , sharing the same projection onto [MATH] ... |
5. On the geometry of characteristic systems It is frequently assumed in the Clairin theory of conditional symmetries that [MATH] is an Abelian Lie algebra of dimension [MATH] admitting a basis of a particular type |
. This ensures that [MATH] is a section of [MATH] amounting to an integrable system of PDEs and the vector fields of [MATH] are tangent to [MATH] . This section provides necessary and sufficient conditions on [MATH] to ensure previous results. More particularly, we will find that the conditions appearing in |
can be significantly relaxed. Proposition 5.1 Let [MATH] and [MATH] be linear spaces of vector fields on [MATH] whose characteristic systems in [MATH] are sections of [MATH] . Then, [MATH] if and only if the vector fields of [MATH] and [MATH] span the same distribution. |
Proof. Let us prove the direct part. Since [MATH] is a section of [MATH] , the projection of [MATH] to [MATH] via [MATH] gives rise to a section of [MATH] . Hence, [MATH] and [MATH] determine the same first-order system of PDEs in normal form, let us say ( 2.5 ). By virtue of Lemma 4.4 , the distributions [MATH] and [M... |
Conversely, if [MATH] and [MATH] span the same distribution, and since [MATH] and [MATH] are sections of [MATH] , then the ranks of [MATH] [MATH] are, by virtue of Theorems 4.3 and 4.5 , equal to [MATH] and there exists a family of functions [MATH] on [MATH] giving rise to an invertible [MATH] matrix [MATH] mapping [MA... |
Lemma 5.2 If the vector fields [MATH] given by [MATH] span an involutive distribution on [MATH] , then [MATH] is an Abelian Lie algebra. |
Proof. By the involutivity assumption on the distribution spanned by the elements of [MATH] , we have that [EQUATION] for certain functions [MATH] with [MATH] . Since the left-hand side projects onto zero relative to [MATH] , the right-hand side does also. Hence, |
[EQUATION] for every [MATH] . Therefore, [MATH] for all possible indices [MATH] and the vector fields [MATH] are in involution. Theorem 5.3 |
Let [MATH] be a section of [MATH] . The prolongations to [MATH] of the elements of [MATH] are tangent to [MATH] if and only if the distribution [MATH] spanned by the elements of [MATH] is involutive. |
Proof. Assume first that [MATH] is involutive. The prolongations to [MATH] of the elements of [MATH] are tangent to [MATH] if and only if the functions [MATH] , where [MATH] is any element of [MATH] and [MATH] is any contact form on [MATH] , are first-integrals of any [MATH] with [MATH] . Now, |
[EQUATION] Let us analyse both right-hand terms on [MATH] . First, as [MATH] is a prolongation to [MATH] of [MATH] , one has that [MATH] is a contact form and, by the definition of [MATH] , one has |
[EQUATION] Second, we make the assumption that [MATH] is a section of [MATH] . Then Theorem 4.5 states that [MATH] has order [MATH] . Since [MATH] is assumed to be involutive, we have [MATH] for the functions [MATH] and some elements [MATH] , with [MATH] , chosen from a basis of [MATH] . In view of Lemma 4.1 , one gets... |
[EQUATION] Using ( 5.2 ) and ( 5.3 ) to simplify ( 5.1 ), we get that [MATH] . Consequently, the prolongations to [MATH] of the elements of [MATH] are tangent to [MATH] |
We now prove the converse by contradiction. Assume that [MATH] is not involutive and that the prolongations to [MATH] of the elements of [MATH] are tangent to [MATH] . Then, there exist [MATH] such that [MATH] does not take values in [MATH] . Thus, the elements of the linear space [MATH] span a distribution [MATH] of r... |
The theory of conditional symmetries focuses on the case where [MATH] is a Lie algebra. Then, [MATH] is involutive and, therefore, an immediate consequence of Theorem 5.3 is the following trivial corollary. As we do not assume that [MATH] is a section of [MATH] , the corollary can be applied to general conditional symm... |
Corollary 5.4 If [MATH] is a Lie algebra of vector fields on [MATH] , then the prolongations to [MATH] of the elements of [MATH] are tangent to [MATH] |
Any section of [MATH] , let us say [EQUATION] where [MATH] for [MATH] , leads to an [MATH] -th order system of PDEs in normal form |
[EQUATION] for [MATH] [MATH] [MATH] and vice versa. This fact is frequently employed in the theory of conditional symmetries, where [MATH] is constructed in such a way that it is a section of |
[MATH] and, consequently, amounts to a system of PDEs in normal form (cf. ). From now on we assume that all characteristic systems [MATH] are sections of [MATH] . Theorem 4.5 then ensures that the distribution [MATH] spanned by the elements of [MATH] has rank [MATH] and its projection to [MATH] (via [MATH] ) is [MATH] ... |
Proposition 5.5 Let [MATH] be a linear space of vector fields whose [MATH] is a section of [MATH] . Then, [MATH] is locally solvable if and only if the vector fields of [MATH] span an involutive distribution [MATH] |
Proof. Assume first that [MATH] is involutive. Theorem 5.3 ensures that [MATH] is tangent to [MATH] . Given any point [MATH] , there exists a unique integral submanifold [MATH] of [MATH] passing through this point. Since [MATH] projects onto [MATH] , this can be considered as a section of [MATH] . As the elements of [M... |
The projection of [MATH] to [MATH] is given by the points of [MATH] satisfying the condition that the characteristics of elements of [MATH] vanish. This is the condition characterising [MATH] . Hence, the projection of [MATH] to [MATH] is [MATH] . If [MATH] is locally solvable, then its projection to [MATH] via [MATH] ... |
The above proposition justifies the following definition. Definition 5.6 A section [MATH] of [MATH] is integrable when its associated system of PDEs is locally solvable. |
Finally, let us give a rather immediate consequence of Proposition 5.5 that will allow us to simplify the application of the Clairin theory of conditional symmetries. |
Corollary 5.7 If [MATH] is integrable, then every initial condition admits a unique solution. Moreover, the solutions of [MATH] and [MATH] are the same. |
6. Generalising a standard assumption in the theory of conditional symmetries The Clairin theory of conditional symmetries mainly focuses on the case where [MATH] is a [MATH] -dimensional Lie algebra of conditional symmetries admitting a basis of a particular type |
. As far as we know, no work deals with more general types of [MATH] . This section is aimed at characterising this type of Lie algebra. We will also define a larger family of Lie algebras whose properties will make them more useful to study Lie algebras of conditional symmetries, as will be seen in the following secti... |
Proposition 6.1 A Lie algebra [MATH] of vector fields on [MATH] admits a basis of the form ( 2.6 ) if and only if [MATH] is a [MATH] -dimensional Abelian Lie algebra and the distribution [MATH] |
[MATH] projects onto [MATH] via [MATH] Proof. Let us prove the direct part of the proposition. If [MATH] admits a basis [MATH] given by ( 2.6 ), then [MATH] , and therefore all the elements of [MATH] , are projectable onto [MATH] . The projections of [MATH] are [MATH] , with [MATH] , and therefore [MATH] projects onto ... |
Let us now prove the converse. If [MATH] is a Lie algebra of vector fields projectable onto [MATH] , then the elements of a basis [MATH] of [MATH] are also projectable. Since [MATH] is Abelian, the projections span an Abelian Lie algebra. Since [MATH] projects onto [MATH] under [MATH] , the projections [MATH] , with [M... |
[EQUATION] Adding to the coordinates [MATH] a new set of coordinates to form a coordinate system on [MATH] , we obtain that the vector fields [MATH] can be written in the form |
[EQUATION] for certain functions [MATH] , with [MATH] and [MATH] . This finishes the converse part of our proposition. Due to their appearance in the literature (cf. |
) and in this work, the Lie algebras studied in the above proposition deserve a special name. Definition 6.2 rectified PDE Lie algebra is a [MATH] -dimensional Abelian Lie algebra of vector fields on [MATH] whose projections onto [MATH] span [MATH] |
We aim to show in the following sections that one can significantly enlarge the Clairin theory of conditional symmetries by considering Lie algebras [MATH] of conditional symmetries giving a section [MATH] of [MATH] such that there exists a rectified PDE Lie algebra [MATH] satisfying [MATH] . The following lemma is key... |
Lemma 6.3 Let [MATH] be a linear space of vector fields on [MATH] whose [MATH] is an integrable section of [MATH] and such that the distribution [MATH] has rank [MATH] and projects onto [MATH] under [MATH] . Then, there exists a rectified PDE Lie algebra [MATH] such that [MATH] |
Proof. Under the considered assumptions on [MATH] , one has that [MATH] admits a family [MATH] of elements spanning [MATH] such that [MATH] . Assume that the vector fields [MATH] take the form ( 4.2 ). Since [MATH] projects onto [MATH] relative to [MATH] , one has that the functions [MATH] for [MATH] are the coefficien... |
[EQUATION] Since [MATH] coincides with the distribution spanned by the elements [MATH] , Lemma 5.2 shows that [MATH] is an involutive Lie algebra and [MATH] becomes a rectified PDE Lie algebra. Since [MATH] , Proposition 5.1 ensures that [MATH] |
Lemma 6.3 and other results obtained in the following sections will allow us to extend findings concerning rectified PDE Lie algebras [MATH] of conditional symmetries to more general Lie algebras [MATH] . This justifies the introduction of the following definition. |
Definition 6.4 rectifiable linear space of vector fields [MATH] is a linear space of vector fields on [MATH] whose characteristic system is equal to the characteristic system of a rectified PDE Lie algebra [MATH] . Then, [MATH] is called an associated rectified PDE Lie algebra of [MATH] |
Example Assume that [MATH] and [MATH] . Define [MATH] . A short calculation shows that [MATH] . But then, [MATH] . Thus [MATH] for [MATH] , which is a rectified PDE Lie algebra associated with [MATH] . Then, [MATH] is a rectifiable linear space of vector fields. |
The following proposition determines straightforwardly, in terms of the distribution [MATH] , when [MATH] is a rectifiable linear space of vector fields. |
Proposition 6.5 A linear space of vector fields [MATH] is rectifiable if and only if the distribution [MATH] is involutive of rank [MATH] and its projection, via [MATH] , is [MATH] |
Proof. Assume that [MATH] is rectifiable. By Definition 6.4 and Proposition 5.1 , one has that [MATH] and the rectified PDE Lie algebra [MATH] span the same distribution [MATH] . In view of Proposition 6.1 , the distribution [MATH] is [MATH] -dimensional and projects onto [MATH] via [MATH] |
Let us prove the inverse. If [MATH] projects onto a distribution [MATH] , then there must exist [MATH] vectors on [MATH] , for arbitrary [MATH] , projecting onto [MATH] for [MATH] , respectively. Therefore, [MATH] admits [MATH] vector fields of the form |
[EQUATION] Since [MATH] has rank [MATH] , the elements of [MATH] generate the distribution [MATH] and, since [MATH] is involutive, one has in view of Lemma 5.2 that [MATH] is [MATH] -dimensional and Abelian. Consequently, [MATH] is rectifiable. |
The following corollary is an immediate consequence of previous results that will be useful in the remaining sections of our work. |
Corollary 6.6 If [MATH] is a rectifiable family of vector fields, then [MATH] is an integral section of [MATH] , the prolongations of the elements of [MATH] to [MATH] span the tangent space to [MATH] , and [MATH] is a locally solvable differential equation. |
7. Conditional symmetries This section introduces the theory of conditional symmetries with special emphasis on the Clairin approach. Next, we use the properties of the submanifolds [MATH] obtained in previous sections to investigate the Lie algebras of conditional symmetries of a system of PDEs [MATH] . Recall that ou... |
In general, [MATH] does not need to be a manifold and it can even be empty, e.g. the problem on [MATH] [MATH] given by [EQUATION] |
gives rise to an empty set [MATH] . Hence, the corresponding system of PDEs has no solutions. This problem is not commented in applications, which are indeed focused on those cases where [MATH] is a manifold and admits particular solutions of physical relevance |
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