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The method of approximate transformation groups, which was proposed by Baikov et al. (1988 and 1996), is extended on Hamiltonian and bi-Hamiltonian systems of evolution equations. Indeed, as a main consequence, this extended procedure is applied in order to compute the approximate conservation laws and approximate recursion operators corresponding to these types of equations. In particular, as an application, a comprehensive analysis of the problem of approximate conservation laws and approximate recursion operators associated to the Gardner equation with the small parameters is presented.

The investigation of the exact solutions of nonlinear evolution equations has a fundamental role in the nonlinear physical phenomena. One of the significant and systematic methods for obtaining special solutions of systems of nonlinear differential equations is the classical symmetries method, also called Lie group analysis. This well-known approach originated at the end of nineteenth century from the pioneering work of Lie [

As it is well known, Hamiltonian systems of differential equations are one of the famous and significant concepts in physics. These important systems appear in the various fields of physics such as motion of rigid bodies, celestial mechanics, quantization theory, fluid mechanics, plasma physics, and so forth. Due to the significance of Hamiltonian structures, in this paper, by applying the linear behavior of the Euler operator, characteristics, prolongation, and Fréchet derivative of vector fields, we have extended ASM on the Hamiltonian and bi-Hamiltonian systems of evolution equations, in order to investigate the interplay between approximate symmetry groups, approximate conservation laws, and approximate recursion operators.

The structure of the present paper is as follows. In Section

In this section, we will mention some necessary preliminaries regarding Hamiltonian structures. In order to be familiar with the general concepts of the ASM, refer to [

Let

A

A generalized vector field

Among all the generalized vector fields, those in which the coefficients

A manifold

Let

Therefore, in the given coordinate chart, Hamilton’s equations take the form of

If

The principal innovations needed to convert a Hamiltonian system of ordinary differential equations (

replacing the Hamiltonian function

replacing the vector gradient operation

replacing the skew-symmetric matrix

Let

Consider a system of perturbed evolution equations:

Substituting according to (

Considering the above assumptions, some useful relevant theorems and definitions could be rewritten as follows.

An approximate evolutionary vector field

Any approximate conservation law of a system of perturbed evolution equations takes the form of

Let

In other words, the Hamiltonian system corresponding to a distinguished functional is completely trivial:

Now, according to [

Let

The Gardner equation

In [

For the first Hamiltonian operator

For the above seven characteristics, we have

Keeping on this procedure recursively, further approximate conservation laws could be generated. But, this procedure will be done in the next section by applying approximate recursion operators.

Let

For nonlinear perturbed systems, there is an analogous criterion for a differential operator to be an approximate recursion operator, but to state it we need to introduce the notion of the (formal) Fréchet derivative of a differential function.

Let

If

Suppose that

Suppose that

From (

Consider the potential Burgers’ equation

The corresponding characteristics of the first twelve approximate symmetries are

Inspection of

There is thus an infinite hierarchy of approximate symmetries, with characteristics

Consider the Gardner equation, which was shown to have two Hamiltonian structures with

Note that the Fréchet derivative for the right-hand side of Gardner’s equation is

Let

To prove this theorem, we apply the similar method applied in Theorem 5.31 of [

We proceed by induction on

To prove this fact, note that the formal adjoint of the approximate recursion operator

A pair of skew-adjoint

If

Let

Consider the approximate Hamiltonian operators

A differential operator

Now, according to [

Let

for each

the corresponding approximate evolutionary vector fields

the approximate Hamiltonian functionals

We have seen that given an approximate bi-Hamiltonian system, the operator

Let

Let

Judging from

The approximate recursion operators of the Gardner equation are

But if we set

Now, for

Sometimes, differential equations appearing in mathematical modelings are written with terms involving a small parameter which is known as the perturbed term. Taking into account the instability of the Lie point symmetries with respect to perturbation of coefficients of differential equations, the approximate (perturbed) symmetries for such equations are obtained. Different methods for computing the approximate symmetries of a system of differential equations are available in the literature [

The approximate symmetry method proposed by Fushchich and Shtelen [

Since prolongation and Fréchet derivative of vector fields are linear, both of the approximate symmetry methods can be extended on the Hamiltonian structures. But due to the significance of vector fields in Hamiltonian and bi-Hamiltonian systems, the approximate symmetry method proposed by Baikov et al. [

It is a pleasure to thank the anonymous referees for their constructive suggestions and helpful comments which have improved the presentation of the paper. The authors wish to express their sincere gratitude to Fatemeh Ahangari for her useful advice and suggestions.