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We hereafter assume that [MATH] and [MATH] are non-vacuously transversal , i.e. [MATH] is not empty and at every point [MATH] one has that [MATH] . This turns [MATH] into a submanifold of [MATH] |
. As illustrated by examples in this work and other previous ones , this assumption seems to be reasonable. In particular, if [MATH] , then [MATH] and [MATH] with [MATH] for the cases of interest in the Clairin theory of conditional symmetries. |
The following notion of a Lie algebra of conditional symmetries represents an intrinsic formulation of the definition given in 23 , Definition 2.2] |
Definition 7.1 Lie algebra of conditional symmetries of an [MATH] -th order system of PDEs [MATH] is a Lie algebra [MATH] of vector fields on [MATH] such that [MATH] is tangent to [MATH] for every [MATH] |
As we did not assume that [MATH] projects onto [MATH] or its rank is [MATH] , the previous definition does not need to give rise to a section [MATH] . As a consequence, this definition is more general than that of the Clairin approach and it covers more general Lie algebras of conditional symmetries |
. Relevantly, our definition is purely geometrical and it does not rely on coordinates expressions of characteristics or the basis of [MATH] |
Let us comment on the definition of a Lie algebra of conditional symmetries and other related notions. We say that a vector field [MATH] is a conditional Lie symmetry of the system of PDEs given by [MATH] if [MATH] is a Lie algebra of conditional symmetries of [MATH] . This amounts to saying that [MATH] is a nonclassic... |
Note that if [MATH] on [MATH] , then [EQUATION] and Corollary 5.4 ensures that [MATH] is also tangent to [MATH] . Consequently, [MATH] is tangent to [MATH] and [MATH] becomes a conditional Lie symmetry of [MATH] . Therefore, Lie point symmetries of [MATH] induce a conditional Lie symmetry. Nevertheless, a conditional L... |
in references therein). Many Lie algebras of conditional symmetries in the literature admit a basis of the form ( 2.6 ). Nevertheless, we want to study the theory of Lie algebras of conditional symmetries given by rectifiable PDE Lie algebras of vector fields. To start with, the following theorem shows that a rectifiab... |
Theorem 7.2 A rectifiable PDE Lie algebra of vector fields [MATH] is a Lie algebra of conditional symmetries of [MATH] if and only if an associated rectified PDE Lie algebra of vector fields [MATH] consists of conditional symmetries. |
Proof. Let us prove the direct part of the theorem. By definition, if [MATH] is a rectifiable PDE Lie algebra of vector fields, then there exists a rectified PDE Lie algebra [MATH] such that [MATH] . Let [MATH] be a basis of [MATH] and let [MATH] be a basis of [MATH] . Hence, to prove that [MATH] is a Lie algebra of co... |
[EQUATION] By virtue of Proposition 5.1 , since [MATH] is a rectified PDE Lie algebra associated with [MATH] , both spaces of vector fields span the same distribution, namely [MATH] and [MATH] for [MATH] and some functions [MATH] with [MATH] . Lemma 4.1 shows that |
[MATH] on [MATH] . Hence, the vector fields [MATH] vanish on [MATH] over [MATH] if the [MATH] do. The converse is analogous. Consequently, the obtention of a rectifiable PDE Lie algebra of conditional symmetries reduces to the study of rectified PDE Lie algebras of conditional symmetries. Conversely, the knowledge of a... |
and it will be demonstrated in Section , where we apply our techniques to physically motivated examples. In fact, this is a generalisation of the cases studied in the above-mentioned works. |
Since previous comments relate rectifiable PDE Lie algebras of conditional symmetries to the standard Clairin theory of conditional symmetries dealing with rectified PDE Lie algebras of conditional symmetries |
, the elements of such rectifiable PDE Lie algebras can be called Clairin conditional symmetries Let us now study the determination of rectified PDE Lie algebras of conditional symmetries. In this case, the system of PDEs associated with [MATH] is given, in an appropriate coordinate system where a basis of [MATH] takes... |
[EQUATION] and all the total derivatives of this system with respect to the [MATH] with [MATH] can be determined. Recall that in view of Proposition 2.1 the compatibility condition for ( 7.1 ), which in coordinates takes the form ( 2.7 ), amounts to the fact that [MATH] is Abelian. From a practical point of view, this ... |
. The solutions of the system [MATH] are therefore given by the following system of PDEs: [EQUATION] with [MATH] If [MATH] is a Lie algebra of conditional symmetries of [MATH] , then one has the additional condition |
[EQUATION] The descriptions of these conditions in coordinates are frequently called the invariant surface conditions (see ). Consequently, the obtention of a rectified PDE Lie algebra of conditional symmetries requires the solving of a nonlinear system of PDEs, which is in general much more complicated than the standa... |
. Nevertheless, recall that this method is applied when standard Lie point symmetries do not provide enough information about the system of PDEs under study. Moreover, the conditional symmetry approach may offer Lie symmetries of particular families of solutions of the original system of PDEs, which may not be related ... |
, and the DCs can be used to obtain particular solutions of the initial system of PDEs, which is a topic that was studied in Section 2.1 |
The following theorem allows us to simplify the system of PDEs necessary to obtain Lie algebras of conditional symmetries of [MATH] in the Clairin approach. |
Theorem 7.3 Let [MATH] be a rectifiable PDE Lie algebra of vector fields and let [MATH] be an associated rectified PDE Lie algebra with a basis |
[MATH] where [MATH] . Consider the [MATH] -th order system of PDEs, [MATH] , described by the zeroes of the functions [MATH] , with [MATH] . If [MATH] , with [MATH] and [MATH] , then [MATH] is a Lie algebra of conditional Lie symmetries of [MATH] |
Proof. Theorem 7.2 shows that [MATH] is a Lie algebra of conditional symmetries of [MATH] if and only if [MATH] is a rectified PDE Lie algebra of conditional symmetries of [MATH] |
Let us prove that [MATH] is a Lie algebra of conditional symmetries if it satisfies the conditions of our theorem. Take a point [MATH] . Since [MATH] is a rectified PDE Lie algebra, Corollary 6.6 states that [MATH] is a locally solvable [MATH] -th order system of PDEs, every [MATH] , with [MATH] , is tangent to [MATH] ... |
[EQUATION] By the assumptions we have made, [MATH] for [MATH] . Thus, [EQUATION] Consequently, [MATH] is tangent to the intersection [MATH] and [MATH] becomes a Lie algebra of conditional symmetries. |
8. Conditional symmetries and PDE Lie systems The study of conditional symmetries requires the solving of a nonlinear system of PDEs ( 7.2 ), which is more complicated than the linear one appearing in the determination of classical Lie point symmetries |
. As mentioned previously, the solution of ( 7.2 ) is still justified by the fact that we assume that Lie point symmetries are not sufficient to study the properties of the system of PDEs, [MATH] , under study. |
In this work, we improve the approach initiated in and suggest appropriate assumptions allowing for the description of conditional Lie symmetries through the so-called PDE Lie systems , which provides many techniques to obtain the solutions of such nonlinear PDEs, such as the reduction and integration methods (cf. |
). Consider a family of functions on the coordinates of [MATH] that is closed relative to linear combinations, products and derivatives in terms of the coordinates of [MATH] . For instance, consider the families |
[MATH] [MATH] [MATH] Let us consider the family [MATH] . Our methods can be accomplished similarly for [MATH] or other sets of families satisfying the described properties. Assume now that the functions [MATH] , which depend on the dependent and independent coordinates, admit a polynomial expansion in the dependent coo... |
[EQUATION] where [MATH] with [MATH] . We define [MATH] , where the [MATH] are certain [MATH] -dependent functions, and the sum in ( 8.1 ) is over arbitrary multi-indices [MATH] . In this case, the system determining conditional symmetries ( 7.1 ) reads |
[EQUATION] Instead of assuming that [MATH] is a second-order polynomial as in or making assumptions on the reductions of this system of differential equations as in |
, we propose to consider that equation ( 8.2 ) gives rise to a general polynomial PDE Lie system , which can be written in the form of an integrable system (in the sense of satisfying the compatibility condition) given by |
[EQUATION] for certain [MATH] -dependent functions [MATH] and vector fields [MATH] spanning a finite-dimensional Lie algebra of vector fields. The Lie algebra spanned by [MATH] is called a Vessiot–Guldberg Lie algebra of the PDE Lie system |
. The Vessiot–Guldberg Lie algebra shows whether the PDE Lie system can be straightforwardly integrated or allows for the construction of superposition rules , which enables ones to obtain the general solution of the PDE Lie system from a finite set of particular solutions and a set of constants |
If we restrict the expansion of [MATH] to the case where its multi-indices satisfy [MATH] , then the right-hand side of ( 8.2 ) becomes a second-order polynomial, which suggests that we impose conditions on the [MATH] to match the form of the so-called partial differential matrix Riccati equations |
, which is an integrable first-order system of PDEs (in the sense of satisfying the compatibility condition) of the form [EQUATION] |
where [MATH] is a vertical vector [MATH] , the [MATH] belong to [MATH] , and [MATH] are [MATH] matrices for every [MATH] . In this case, the Vessiot–Guldberg Lie algebra, [MATH] , is isomorphic to [MATH] (cf. |
). Partial differential matrix Riccati equations appear, for instance, in the study of the Wess-Zumino-Novikov-Witten (WZNW) equations |
. They also appear in the study of Bäcklund transformations of relevant systems of PDEs The previous conditions on the coefficients [MATH] are quite general. If [MATH] all PDE Lie systems are such that a change of variables may map the Lie algebra spanned by the [MATH] onto [MATH] (see |
). When [MATH] , not all PDE Lie systems can be mapped by means of a change of variables on [MATH] onto a partial differential matrix Riccati equation (cf. 21 , Table 4] ). For instance, no change of variables on [MATH] can map a generic PDE Lie system with a Vessiot–Guldberg Lie algebra of vector fields [MATH] into a ... |
. In particular, one can assume that the [MATH] are polynomials of any order (cf. or 21 , Table 4] ). For instance, one may consider the PDE Lie system |
[EQUATION] with [MATH] and where the [MATH] -dependent functions are chosen so that the system satisfies the compatibility conditions. This PDE Lie system is related to a Vessiot–Guldberg Lie algebra of type [MATH] (cf. |
). Consequently, there exists no diffeomorphism on [MATH] on the plane mapping this PDE Lie system into a subcase of ( 8.4 ) because, for instance, the dimension of [MATH] is larger than the dimension of [MATH] . Obviously, studying PDE Lie systems more general than partial differential matrix Riccati equations offers ... |
Once the form of the system ( 8.2 ) has been established, it is convenient to look for methods to determine Lie algebras of conditional symmetries. In particular, we first focus on the case when the Lie algebra of conditional symmetries is spanned by |
[EQUATION] As already mentioned, this amounts to the fact that the Lie algebra must be Abelian. Recall also that this latter fact ensures that the differential constraints ( 7.1 ) give rise to an integrable first-order system of PDEs that therefore admits solutions for every initial condition [MATH] . In coordinates, t... |
[EQUATION] More specifically, the substitution of ( 8.1 ) into the compatibility condition ( 2.7 ) of the system ( 8.2 ) gives [EQUATION] |
where [MATH] is an arbitrary multi-index for the polynomials on [MATH] To ensure that the above expressions are zero for every [MATH] , we must impose that the coefficients of the expansion in different polynomials in the variables [MATH] of ( 8.1 ) vanish. This in turn can be considered as a first-order system of PDEs... |
[EQUATION] for a certain set of [MATH] functions [MATH] Additionally, one has to recall that the [MATH] must still satisfy [EQUATION] |
At this point, we assume that the functions [MATH] are polynomials with [MATH] -dependent coefficients in the functions of [MATH] and their total derivatives of arbitrary order. In view of our assumptions for [MATH] , this implies that [MATH] can be written as a polynomial of the form |
[EQUATION] By assuming that the coefficients [MATH] are solutions of the family of differential equations [MATH] , for all the multi-indices [MATH] , and ( 8.5 ), one obtains that all solutions of the PDE Lie system ( 8.3 ) become solutions of the higher-order system of PDEs under analysis, namely [MATH] . Hence, all s... |
9. Applications This section is aimed at illustrating the results given in previous parts of the work through three different physical models: a nonlinear wave equation, a Gauss-Codazzi equation for minimal surfaces, and a generalised Liouville equation. |
9.1. Nonlinear wave equation Let us study the conditional symmetries of a nonlinear wave equation [EQUATION] where [MATH] is, so far, an underdetermined function depending on [MATH] Our study will specify several theoretical and practical details not explained in |
The nonlinear wave-equation ( 9.1 ) is a PDE on the second-order jet bundle [MATH] relative to [MATH] and [MATH] Consider a rectified PDE Lie algebra [MATH] of conditional symmetries |
[EQUATION] where [MATH] and [MATH] are, for the time being, some functions on [MATH] to be determined. In any case, [MATH] is a projectable regular distribution of rank two. Therefore, the characteristic system in [MATH] is given by the conditions |
[EQUATION] and their total derivatives [EQUATION] give rise to the characteristic system [MATH] . In view of Proposition 2.1 , the condition [MATH] amounts to the compatibility condition for ( 9.2 ), which in turn ensures that [MATH] , which yields that [MATH] is a section of [MATH] by virtue of Theorem 4.5 |
In the Clairin approach to Lie algebras of conditional symmetries, it is assumed that [MATH] and [MATH] are such that the first-order system of PDEs ( 9.2 ) is integrable. In view of Proposition 2.1 and Theorem 4.5 , system [MATH] is locally solvable. In view of Corollary 5.7 , the solutions to the system ( 9.1 ), ( 9.... |
). In order to use PDE Lie systems to study the conditional symmetries of ( 9.1 ), consider the expansion of [MATH] and [MATH] up to second order in the dependent variable [MATH] . Then, |
[EQUATION] for functions [MATH] whose particular form must be determined to ensure that [MATH] is a Lie algebra of conditional symmetries |
In view of ( 9.1 ) and ( 9.2 ), we get that [MATH] . Therefore, one can restrict oneself to the particular case [MATH] for certain constants [MATH] (other more general cases could also be considered). Note that therefore the system of PDEs ( 9.1 ) satisfies the conditions of the formalism given in Section for [MATH] an... |
By substituting the expansion ( 9.4 ) and the differential constraints ( 9.2 ) into the wave-equation ( 9.1 ), one obtains [EQUATION] |
To ensure that all solutions of ( 9.2 ) are solutions of ( 9.1 ), we assume that the coefficients in [MATH] are all zero. Then, the compatibility condition for ( 9.2 ) amounts to the fact that |
[EQUATION] In view of previous equalities, we obtain [EQUATION] If the coefficients of the expansion ( 9.4 ) satisfy the above conditions, then Theorem 7.3 ensures that [MATH] and [MATH] are conditional Lie symmetries of ( 9.1 ). It is therefore not necessary to consider the extension of the vector fields [MATH] to [MA... |
Let us provide a simple case of conditional symmetries for ( 9.1 ). A particular solution to the equations ( 9.5 ), e.g. [MATH] , gives rise to |
[EQUATION] which span a Lie algebra of conditional symmetries of [EQUATION] Moreover, all solutions of [EQUATION] namely [MATH] with [MATH] , are particular solutions of ( 9.6 ). |
We can obtain a new Lie algebra of conditional Lie symmetries by obtaining a rectifiable family of vector fields [MATH] whose [MATH] will match [MATH] , for instance, |
[EQUATION] In fact, one gets that [MATH] . It is worth noting that the new vector fields can be interesting if they leave invariant a geometric structure on [MATH] whereas the vector fields [MATH] do not (see |
for examples of this). Note that we could have repeated the whole above procedure by using the set of functions [MATH] . In that case, we would have added differential constraints given by a PDE Lie system with a Vessiot–Guldberg Lie algebra given by |
[EQUATION] In turn, this gives rise to the study of the sine-Gordon equation [MATH] , which admits pseudo-spherical surfaces and gives rise to Bäcklund transformations |
. Another set of functions [MATH] would have given rise to the Vessiot–Guldberg Lie algebras [EQUATION] and new results and Lie algebras of conditional symmetries could be derived for the obtained types of Sinh-Gordon equations |
9.2. Gauss-Codazzi equations Let us now apply our formalism to the study of a class of Gauss–Codazzi equations , which take the form |
[EQUATION] where [MATH] [MATH] and [MATH] are real functions, and [MATH] is a complex valued function. We will focus on the first part of the Gauss–Codazzi equations. Although it is a complex differential equation due to the appearance of [MATH] and [MATH] , it is simple to see that it can be considered as a real diffe... |
In order to apply the conditional symmetry theory, we consider the differential constraints [EQUATION] whose compatibility condition amounts to |
[EQUATION] The system of PDEs ( 9.8 ) can easily be rewritten as a PDE Lie system of the form [EQUATION] related to a Vessiot–Guldberg Lie algebra [MATH] . This allows us to study two-dimensional Lie algebras of conditional symmetries spanned by the vector fields |
[EQUATION] To verify whether the vector fields related to our differential constraints give rise to conditional Lie symmetries, we have to substitute ( 9.8 ) in ( 9.7 ). Then, we obtain |
[EQUATION] Let us assume that all coefficients accompanying the different exponentials of the variable [MATH] are zero. This ensures that all solutions of ( 9.8 ) give rise to solutions of the Gauss–Codazzi equations. In view of Theorem 7.3 , this also ensures that the vector fields ( 9.10 ) span a Lie algebra of condi... |
[EQUATION] Since [MATH] is a real-valued function, equation ( 9.15 ) implies that [MATH] and the Gauss–Codazzi equations ( 9.7 ) show that [MATH] . It is immediate to verify that the latter fact, along with ( 9.12 )–( 9.14 ), allows us to ensure that the compatibility conditions ( 9.9 ) are satisfied. |
Let us now obtain the conditional symmetries of this system. Since [MATH] , one has from ( 9.12 ) that [MATH] . The equations ( 9.12 ) can easily be solved to obtain that |
[EQUATION] for an arbitrary function [MATH] Substituting this in ( 9.16 ), we get [EQUATION] which is acceptable since [MATH] is a holomorphic function. Consequently, the initial Gauss–Codazzi equation reduces under previous assumptions to |
[EQUATION] whereas [EQUATION] where we recall that [MATH] The latter is a PDE Lie system related to a solvable Vessiot–Guldberg Lie algebra [MATH] . Consequently, it is solvable (cf. |
). In fact, a simple method to solve it goes as follows. Let us rewrite ( 9.17 ) in terms of the variable [MATH] as [EQUATION] The homogeneous part of the system reads |
[EQUATION] and a particular solution reads [MATH] . Substituting [MATH] into ( 9.17 ), we obtain that [EQUATION] and the final solution for ( 9.17 ) follows immediately. It is remarkable that [MATH] implies that the obtained surfaces are minimal |
9.3. A generalised Liouville equation Finally, let us study the generalised Liouville equation in [MATH] introduced by Santini . Let [MATH] be a general point in [MATH] and let [MATH] be the gradient operator in [MATH] relative to the Euclidean metric [MATH] on [MATH] . Then, the generalised Liouville equation is given... |
[EQUATION] where [MATH] is a function on [MATH] We make use of the set of functions [MATH] and we assume that the conditional symmetry is given by |
[EQUATION] The reason for this ansatz is that the vector fields [EQUATION] span a Lie algebra of vector fields isomorphic to [MATH] . Thus, if we choose [MATH] , with [MATH] , so that ( 9.21 ) is integrable, then we get that ( 9.21 ) is a PDE Lie system. The [MATH] , with [MATH] , give rise to three [MATH] -dependent v... |
[EQUATION] Therefore, the particular conditions [EQUATION] ensure that every solution of the DCs ( 9.21 ) gives rise to a particular solution of ( 9.20 ). Using that [MATH] and assuming that [MATH] , the compatibility condition for ( 9.21 ) reads |
[EQUATION] The first condition implies that there exists a function [MATH] such that [EQUATION] In particular, [MATH] and substituting this in the first equality of ( 9.22 ), one obtains that |
[EQUATION] and [MATH] becomes a particular solution of ( 9.20 ). Using ( 9.24 ) in the second compatibility condition in ( 9.23 ), we obtain that |
[EQUATION] Therefore, [EQUATION] In particular, [MATH] . Using [MATH] in the second expression of ( 9.23 ), we obtain [EQUATION] |
Under previous assumptions, the DCs ( 9.21 ) reduce to [EQUATION] As this is a PDE Lie system related to the solvable Lie algebra of vector fields [MATH] , its general solution can be obtained |
. In fact, it follows from ( 9.26 ) that [EQUATION] and [EQUATION] is the general solution of the system of equations given by ( 9.20 ) and ( 9.26 ). This can be viewed as a generalised type of auto-Bäcklund transformation mapping solutions of the generalised Liouville equation satisfying the DC ( 9.21 ) into new solut... |
Let us present several particular solutions. If [MATH] and [MATH] for any [MATH] -dependent function [MATH] , then equation ( 9.25 ) reduces to [MATH] and [MATH] becomes any harmonic function on [MATH] , e.g. the real and imaginary parts [MATH] and [MATH] , respectively, of a [MATH] -dependent holomorphic function [MAT... |
[EQUATION] The above procedure gives a new approach to , explains many details not given in there, and avoids several of its ad-hoc constructions, e.g. our approach does not need any ad-hoc transformation. Let us finally explain how to obtain a particular solution proposed in |
without a detailed explanation. Every function [MATH] , for arbitrary functions [MATH] and [MATH] , is a particular solution of ( 9.20 ). In particular, if we assume [MATH] , then |
[EQUATION] is a particular solution of ( 9.20 ) and [MATH] is a solution of ( 9.25 ). According to ( 9.27 ), we obtain a solution with the freedom of [MATH] arbitrary functions of one variable |
[EQUATION] of the system of PDEs given by ( 9.20 ) and ( 9.21 ). This reproduces the multi-mode particular solution given in without a detailed explanation. |
10. Conclusions This work has provided a geometric approach to the Clairin theory of conditional symmetries for higher-order systems of PDEs. Many of the hypotheses employed in this theory have been analysed and their geometrical meaning has been explained. In certain cases, it was stated that standard assumptions can ... |
After the study of the Clairin theory of conditional symmetries, we believe that we should continue the analysis of other conditional symmetries and to consider the case of conditional symmetries given by the characteristics of non-Lie point symmetries. It seems to us that the geometric formalism developed in this work... |
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