We prove that any potential symmetry of a system of evolution equations reduces to a Lie symmetry through a nonlocal transformation of variables. This fact is in the core of our approach to computation of potential and more general nonlocal symmetries of systems of evolution equations having nontrivial Lie symmetry. Several examples are considered.

The 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 Lie and Engel in 19th century [

One can even argue that this very property, invariance under Lie symmetries, distinguishes 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., [

The procedure of selecting partial differential equations (PDEs) enjoying the highest Lie symmetry from a prescribed class of PDEs is called group classification. In the case when non-Lie symmetries are involved, the more general term, symmetry classification, is used.

In this paper we study symmetries of systems of evolution equations in one spatial variable

The problem of the Lie group classification of PDEs of the form (

However, with all its importance and power, the traditional Lie approach does not provide all the answers to mounting challenges of the modern nonlinear physics. By this very reason there were numerous attempts of generalization of Lie symmetries so that the generalized symmetries retain the most important features of Lie symmetries and allow for a broader scope of applicability. A natural move in this direction would be letting the coefficients of infinitesimal generators of the Lie symmetries to contain not only independent and dependent variables and their derivatives but also integrals of dependent variables, as well. In this way, the so-called nonlocal symmetries have been introduced into mathematical physics.

The concept of nonlocal symmetry of linear PDEs is relatively well understood (see, e.g., [

One of the possible approaches to construction of nonlocal symmetries has been suggested by Bluman et al. [

Pucci and Saccomandi [

Some applications of potential symmetries to specific subclasses of the class of PDEs (

In the present paper, we generalize the results of [

One says that system (

One says that system (

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.

System (

Suppose that system (

Let us prove now that the inverse assertion is also true. Suppose that (

Now, (

The fact that symmetry operators

System (

Potential symmetries of system of evolution equations (

Suppose now that system (

Indeed, let system (

Integrating (

It is a common knowledge that any contact symmetry of a system of PDEs boils down to the first prolongation of a classical symmetry [

The same assertion holds true for the case of partial CLR.

Let system of evolution equations (

This assertion is, in fact, the 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 the standard Lie symmetry, which is the image of the potential symmetry in question.

However, there is more to it. Theorem

Indeed, let system of evolution equations (

Making an appropriate change of variables, we can reduce the operators

Let (

Now, we differentiate (

Consequently, if one of the derivatives,

The same line of reasoning applies to the case when system (

We summarize the above speculations in the form of the procedure for computation of nonlocal symmetries of systems of evolution equations associated with a given system of the form (

Let system of evolution equations (

Calculate inequivalent subalgebras

Select those subalgebras

For each commutative subalgebra

Perform nonlocal transformation (

Eliminate “old” dependent variables

Verify that there is, at least, one derivative from the list

The steps needed to implement the above procedure for the case of system of evolution equations admitting partial CLR are the same, the only difference is that intermediate formulas (

Note that by force of Theorems

As an example, we consider the Galilei-invariant nonlinear Schrödinger equation introduced in [

Operators

Denote the class of partial differential equations of the form (

It is not but natural to ask whether there are other types of nonlocal transformations of the class

Consider as the next example system of evolution equations

Transforming (

So that nonlocal transformation (

Let system (

It is important to emphasize that the symmetry algebra

As an illustration, consider the following system of second-order evolution equations:

The procedure for calculation of nonlocal symmetries of system (

Let system of evolution equations (

Calculate inequivalent subalgebras

Select those subalgebras

For each

Eliminate “old” dependent variables

Verify that there is, at least, one function from the list

One of the principal results of the paper is Theorem

We obtain as a by-product exhaustive characterization of systems (

In Section

We intend to devote one of our future publications to systematic study of nonlocal symmetries of systems of nonlinear evolution equations (