Nonlocal symmetries of systems of evolution equations

We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. Based on this fact is our method of group classification of potential symmetries of systems of evolution equations having non-trivial Lie symmetry. Next, we modify the above method to generate more general nonlocal symmetries, which yields a purely algebraic approach to classifying nonlocal symmetries of evolution type systems. Several examples are considered.


Introduction
Lie symmetries and their various generalizations have become an inseparable part of the modern physical description of wide range of phenomena of nature from quantum physics to hydrodynamics. Such success of a purely mathematical theory of continuous groups developed by Sophus Lie in 19th century [1] is explained by the remarkable fact that the overwhelming majority of mathematical models of physical, chemical and biological processes possess nontrivial Lie symmetry.
One can even argue that this very property, invariance under Lie symmetries, distinguish the popular models of mathematical and theoretical physics from a continuum of possible models in the form of differential or integral equations (see, e.g., [2,3]). Based on this observation is the symmetry selection principle stating that if an equation describing some physical process contains arbitrary elements, then the latter should be so chosen that the resulting model possesses the highest possible symmetry. In this sense Lie theory effectively predicts which equation is the best candidate to serve as a mathematical model of a specific physical, chemical or biological process.
The process of choosing from a prescribed class of differential equations those enjoying the highest Lie symmetry is called group classification. In the case when non-Lie symmetries are involved, the more general term, symmetry classification, is used.
There is a lot papers devoted to group classification of different subclasses of the class of partial differential equations (PDEs) of the form (1) (see, e.g. [4]- [7] and the references therein). The major tool utilized in these studies is the infinitesimal Lie approach enabling to reduce the problem of exhaustive description of Lie transformation groups admitted by (1) to integrating some linear system of PDEs (for further details, see [8]- [10]).
However, with all importance and power of the traditional Lie approach, it does not provide all the answers to the mounting challenges of the modern nonlinear physics. By this very reason there were numerous attempts of generalize Lie symmetries so that the generalized symmetries retain the most important features of Lie symmetries and allow for broader scope of applicability. The natural move in this direction would be to allow for the coefficients of infinitesimal generators of Lie symmetries to contain not only independent and dependent variables and their derivatives but integrals of dependent variables, as well. In this way, the so called nonlocal symmetries were introduced into mathematical physics.
The concept of nonlocal symmetry of linear PDEs is well understood by now (see, e.g., [11]). However, this is not the case for nonlinear equations. The problem of developing regular methods for constructing nonlocal symmetries of nonlinear PDEs is still waiting for its Sophus Lie. On the other hand, there is a number of results on nonlocal symmetries for specific equations. One of the possible approaches to construction of nonlocal symmetries has been suggested by Bluman [12]- [14]. He put forward the concept of po-tential symmetry, which is a special case of nonlocal symmetry. The basic idea of the method for constructing potential symmetries of PDEs can be formulated in the following way. Consider evolution equation Suppose that it can be rewritten in the form of a conservation law By force of (3) we can introduce new dependent variable v = v(t, x) and rewrite equation (1) as follows Now if system of two equations (4) admits Lie symmetry such that at least one of the coefficients of its infinitesimal operators depends on v = ∂ −1 x G(t, x, u), then this symmetry is the nonlocal symmetry for the initial evolution equation (2).
This nonlocal symmetry is also called potential symmetry of (2).
Pucci and Saccomandi proved that potential symmetries can be derived using non-classical symmetries of the (2). Recently, we established much stronger assertion by associating potential symmetries with classical (contact) symmetries [17]. More precisely, we proved that any potential symmetry of evolution equation (2) can be reduced to contact symmetry by a suitable nonlocal transformation of dependent and independent variables. As a consequence, one can obtain exhaustive description of potential symmetries of (2) through classification of contact symmetries of PDEs of the form (2).
Some applications of potential symmetries to specific subclasses of (2) can be found in [18]- [23].
In the present paper we generalize results of [17] for system of evolution equations (1) and prove that any potential symmetry of the system in question reduces to classical Lie symmetry under a suitable nonlocal transformation of dependent and independent variables (Sections 1, 2). Next, we suggest in Section 3 a more general approach to constructing nonlocal symmetries that goes far beyond of the concept of potential symmetries. This approach enables generating systems of evolution equations associated with a given system of the system (1), provided the latter admits non-trivial Lie symmetry. We give several examples of application of the approach in Section 3.

Conservation law representation and classical symmetries
Definition 1 We say that system (1) admits complete conservation law representation (CLR) if it can be written in the form Here u, F, G are m-component vectors.
Definition 2 We say that system (1) admits partial CLR if it can be written in the form Here u, F, G and w, H are r-component and m − r-component vectors, respectively.
Below we present theorems that provide exhaustive characterization of conservation law representability in terms of classical Lie symmetries. We give the detailed proof of the assertion regarding complete CLR, the case of partial CLR is handled in a similar way.
and besides,

Proof. Suppose system (1) admits CLR (5). Introducing new m-component
and eliminating u from (5) Integrating the obtained system of PDEs with respect to x yields Note that the integration constant w(t) is absorbed into the function v.
What is more, the coefficients of the basis elements of the algebra L m satisfy condition (8).
Let us prove now the that the inverse assertion is also true. Suppose that (1) admits Lie algebra L m = e 1 , . . . , e m , whose basis elements have the form (7) and satisfy (8). Then there is a change of variables (see, e.g. [8]) reducing basis elements of L m to the form e i = ∂ūi, i = 1, . . . , m. In what follows we drop the bars. Now (1) necessarily takes the form Differentiating (12) with respect to x and making the (nonlocal) change of dependent variables v x = u, we finally get which completes the proof. Note 1. The fact that symmetry operators, e 1 , . . ., e m are of specific form (7) is crucial for the whole procedure of reducing a system of evolution equations to a 'conserved' form (5). If a symmetry group generated by some operator e i does not preserve the temporal variable, t (which means that the coefficient of ∂ t in e i is non-zero for some i), then this operator cannot be reduced to the canonical form ∂ v i and the reduction routine does not work.
and besides, 3 Potential symmetries Potential symmetries of system of evolution equations (1) appear in the same way as they do for a single evolution equation. For simplicity, we consider the case of complete CLR. By force of (5) we can introduce the new dependent variable v, so that Note that v is nonlocal variable since v = ∂ −1 x G(t, x, u). Suppose now that system (14) admits Lie symmetry such that one of the derivatives does not vanish identically. Rewriting group (15) in terms of variables t, x, u and taking into account that v = ∂ −1 x u yield the nonlocal symmetry of the initial system of evolution equations (1). This means, in particular, that symmetry in question cannot be obtained within the Lie infinitesimal approach.
What we are going to prove, is that this symmetry can be derived by regular Lie approach if the later is combined with the nonlocal transformation of the dependent variables. Indeed, let system (1) admit complete CLR (5). In addition, we suppose that (1) possesses potential symmetry. Making the nonlocal change of we rewrite (5) in the form (10). As initial system (1) admits a potential symmetry, system (14) is invariant under the Lie transformation group of the form (15).
Integrating (10) with respect to x yields system of evolution equations Next we rewrite Lie symmetry (15) by eliminating u according to (16) which yields By construction, Lie transformation group (18) maps the set of solutions of (17) into itself. Consequently, (18) is the Lie group of contact symmetries of system of evolution equations (17). It is a common knowledge that any contact symmetry of a system of PDEs boils down to the first prolongation of a classical symmetry [24]. Consequently, the derivatives of T, X, V with respect to the third argument vanish identically and we get This group is nothing else than the standard Lie symmetry group of system (17). The same assertion holds true for the case of partial CLR. Theorem 3 Let system of evolution equations (1) admit complete or partial CLR and be invariant under a potential symmetry. Then there exist a (nonlocal) change of variables mapping (1) into another system of the form (1) so that the potential symmetry of (1) becomes the standard Lie symmetry of the transformed system. This assertion is in fact a no-go theorem for potential symmetries of systems of evolution equations. It states that the concept of potential symmetry does not produce essentially new symmetries. The system admitting potential symmetry is equivalent to the one admitting standard Lie symmetry, which is the image of the potential symmetry in question.
However, there is more to it. Theorem 3 imply the regular algorithm for group classification system of nonlinear evolution equations admitting nonlocal symmetries. Again, for the sake of simplicity, we consider the case of complete CLR. Indeed, let system of evolution equations (1) be invariant under (m + 1)-dimensional Lie algebra L m+1 = e 1 , . . . , e m+1 . Here e 1 , . . ., e m are commuting operators of the form (7) and their coefficients satisfy constraint (8). Basis operator e m+1 is of the generic form Making an appropriate change of variables we can reduce the operators e 1 , . . ., e m to the canonical forms, namely, e i = ∂ u i , i = 1, . . . , m. Then system (1) necessarily takes the form (17).
Let (19) be Lie transformation group generated by the symmetry operator e m+1 . Calculating the first prolongation of formulas (19) we get the transformation rule for the first derivatives of v Now we differentiate (10) with respect to x and make the following change of dependent variables equations, which yield Formulas (19), (20) provide the image of the transformation group (19) under the mapping (20), so that Here Consequently, if one of the derivatives, ∂T /∂v i , ∂X/∂v i , ∂W/∂v i , does not vanish identically, then (23) is the nonlocal symmetry group of system of evolution equations (22).
The same line of reasonings applies to the case when system (1) admits partial CLR.
We summarize the above speculations in the form of the multi-step algorithm for group classification of nonlocal symmetries of systems of evolutions equations associated with a given system of the form (1).
Let system of evolution equations (1) be invariant under N-dimensional Lie symmetry algebra L N . For simplicity we formulate the algorithm for the case of complete CLR. 3. For each commutative subalgebra M m perform change of variables reducing its basis elements to the canonical forms ∂ v 1 , . . ., ∂ v m and transform correspondingly initial system (1).

Perform nonlocal transformation (21).
5. Eliminate 'old' dependent variables v from (19) in order to derive symmetry group (23) of the transformed system of evolution equations (22). 6. Verify that there is, at least, one derivative from the list ∂T /∂v i , ∂X/∂v i , ∂W/∂v i that does not vanish identically. If this is the case, then (23) is the nonlocal (potential) symmetry of (22).
The steps needed to implement the above algorithm for the case of system of evolution equations admitting partial CLR are the same, the only difference is that intermediate formulas (19)-(23) are more cumbersome, since we need to distinguish between two sets of dependent variables u and w (see, (6)).
Note that by force of Theorems 1,2 any potential symmetry of equations of the form (1) can be obtained in the above described manner.

Some generalizations
Denote the class of partial differential equations of the form (1) as E n . Then any system of the form (i) belongs to E n , and, (ii) its image under nonlocal transformation u = v x also belongs to E n . Existence of such nonlocal transformation is in the core of our approach to classifying nonlocal symmetries of systems of evolution equations. It is not but natural to ask whether there are other types of nonlocal transformations of the class E n that can be utilized to generate nonlocal symmetries. Remarkably, such nonlocal transformations do exist. Sokolov [25] put forward the idea of group approach to generating such transformations for a single evolution equation. It is straightforward to modify his approach to handle systems of evolution equations, as well. As an illustration, we consider system (24). It is invariant under the m-dimensional Lie algebra L m = ∂ u 1 , . . . , ∂ u m . The simplest set of (m + 2) functionally-independent invariants of the algebra L m can be chosen as follows t, x, u 1 x , . . ., u m x . Now we define the transformation t = T (t, x, u, u x , u xx , . . .),x = X(t, x, u, u x , u xx , . . .), u = U(t, x, u, u x , u xx , . . .) (25) so that T, X, U are invariants of the symmetry group of the system under study. In the case under consideration, we have T = t, X = x, U = u x . As we established in Section 1, applying this transformation to any equation of the form (24) yields system of evolution equations that belongs to E n . What is more, Lie symmetry group of (24) is mapped into symmetry group of the transformed system and some of the basis operators of the latter become nonlocal ones. Consider, as the next example system of evolution equations This system is invariant under the 2m-dimensional Lie algebra L 2m = ∂ u 1 , . . ., ∂ u m , x∂ u 1 , . . ., x∂ u m . The simplest set of m+ 2 functionally independent first integrals reads as t, x, u 1 xx , . . ., u m xx . Consequently, change of variables (25) takes the form Note that we dropped the bars and replacedū with v. Transforming (26) according to (27) we get or, equivalently, Note that integration constants w 1 (t)x + w 2 (t) are absorbed by the function v.
So that nonlocal transformation (27) maps a subset of equations from E n into E n . Consequently, it can be used to generate nonlocal symmetries of the initial system (26).
Let system (26) be invariant under the Lie transformation group Computing the second prolongation of the above formulas we get the transformation law for the functions Combining (29) and (30) yields the symmetry group of system of evolution equations (28), It is important to emphasize that the symmetry algebra L m is not obliged to be commuting. The necessary condition is that the corresponding transformation group has to preserve the temporal variable, t, i.e., basis elements of L m have to be of the form As an illustration, consider the following system of second-order evolution equations: Taking into account that the operators ∂ ∂t and ∂ −1 x commute, differentiating (34) with respect to x, and replacing v with w x we finally get The above system is obtained from the initial one through the change of dependent variables u i = ∂ −1 x exp(w i ), i = 1, . . . , m. Consequently, if system (33) admits symmetry (29), then system (35) admits the following transformation group: does not vanish identically, then (36) is the nonlocal invariance group of system of evolution equations (34). The algorithm for group classification of nonlocal symmetries of system (1) suggested in the previous section yields those nonlocal symmetries which are potential, since the nonlocal transformation was chosen a priori. Allowing for a nonlocal transformation to be determined by symmetry group of the system under study, yields a more general algorithm for constructing nonlocal symmetries.
Let system of evolution equations (1) be invariant under N-dimensional Lie symmetry algebra L N . Then the following multi-step algorithm can used to construct nonlocal symmetries of (1).
Algorithm 2. Classification of nonlocal symmetries of (1) 1. Calculate inequivalent subalgebras M of the algebra L N .
4. Eliminate 'old' dependent variables u from (37) in order to derive symmetry group G of the transformed system of evolution equations.

5.
Verify that there is, at least, one function from the list {ω t , ω x , ω 1 , . . ., ω r } that depends on u i for some 1 ≤ i ≤ r. If this is the case, then G is the nonlocal symmetry of (22).

Conclusion.
One of the principal results of the paper is Theorem 3 from Section 2 stating that any potential symmetry of system of evolution equations (1) reduces to a Lie symmetry by an appropriate nonlocal transformation of dependent and independent variables. The nonlocal transformation in question is a superposition of the local change of variables t = t,x = X(t, x, u),ū = U(t, x, u) and of the nonlocal change of dependent variables v =ū x .
The explicit form of transformations (38) is defined by the Lie symmetry admitted by the corresponding system (1). we obtain as an by-product exhaustive characterization of systems (1), that can be represented in the form of conservation law(s), in terms of Lie symmetries preserving the temporal variable, t, t ′ = t, x ′ = X(t, x, u, θ), u ′ = U(t, x, u, θ) (see, Theorems 1,2).
In Section 3, we generalize the above reasonings in order to obtain nonlocal symmetries which are not potential. The basic idea is replacing (39) with a more general nonlocal transformation. This transformations is determined by invariants of Lie symmetry algebra of the system under study.
We intend to devote one of our future publications to systematic study of nonlocal symmetries of systems of nonlinear evolution equations (1) within the framework of the approach developed in Section 3.