Approximate Hamiltonian Symmetry Groups and Recursion Operators for Perturbed Evolution Equations

In this paper, the method of approximate transformation groups which was proposed by Baikov, Gazizov and Ibragimov, 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.


Introduction
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 [7]. The fact that symmetry reductions for many PDEs can not be determined via the classical symmetry method, motivated the creation of several generalizations of the classical Lie group approach for symmetry reductions. Consequently, several alternative reduction methods have been proposed, going beyond Lie's classical procedure and providing further solutions. One of these techniques which is extremely applied particularly for nonlinear problems is perturbation analysis. It is worth mentioning that sometimes differential equations which appear in mathematical modelings are presented with terms involving a parameter called the perturbed term. Because of the instability of the Lie point symmetries with respect to perturbation of coefficients of differential equations, a new class of symmetries has been created for such equations, which are known as approximate (perturbed) symmetries. In the last century, in order to have the utmost result from the methods, combination of Lie symmetry method and perturbations are investigated and two different so called approximate symmetry methods(ASM) have been developed. The first method is due to Baikov, Gazizov and Ibragimov [1,2]. The second procedure was proposed by Fushchich and Shtelen [5] and later followed by Euler et al [3,4]. This method is generally based on the perturbation of dependent variables. In [10,11], a comprehensive comparison of these two methods is presented. 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, etc. 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 2, some necessary preliminaries regarding to the Hamiltonian structures are presented. In section 3, a comprehensive investigation of the approximate Hamiltonian symmetries and approximate conservation laws associated to the perturbed evolution equations is proposed. Also, as an application of this procedure, approximate Hamiltonian symmetry groups, approximate bi-Hamiltonian Structures and approximate conservation laws of the Gardner equation are computed. In section 4, the approximate recursion operators are studied and the proposed technique is implemented for the Gardner equation as an application. Finally, some concluding remarks are mentioned at the end of the paper.

Preliminaries
In this section, we will mention some necessary preliminaries regarding to Hamiltonian structures. In order to be familiar with the general concepts of the ASM, refer to [6]. It is also worth mentioning that most of this paper's definitions, theorems and techniques regarding to Hamiltonian and bi-Hamiltonian structures are inspired from [9]. Let M Ă XˆU denote a fixed connected open subset of the space of independent and dependent variables x " px 1 ,¨¨¨, x p q and u " pu 1 ,¨¨¨, u q q. The algebra of differential functions P px, u pnq q " prus over M is denoted by A . We further define A l to be the vector space of l-tuples of differential functions, P rus " pP 1 rus, . . . , P l rusq, where each P j P A . A generalized vector field will be a (formal) expression of the form v " in which ξ i and φ α are smooth differential functions. The Prolonged generalized vector field can be defined as follows: , whose coefficients are determined by the formula with the same notation as before. Given a generalized vector field v, its infinite prolongation (or briefly prolongation) is the formally infinite sum where each φ J α is given by (2.2), and the sum in (2.3) now extends over all multi-indices J " pj 1 , . . . , j k q for k ě 0, 1 ď j k ď p.
Among all the generalized vector fields, those in which the coefficients ξ i rus of the B Bx i are zero play a distinguished role. Let Qrus " pQ 1 rus, . . . , Q q rusq P A q be a q-tuple of differential functions. The generalized vector field v Q " q ÿ α"1 Q α rus B Bu α is called an evolutionary vector field, and Q is called its characteristic.
A manifold M with a Poisson bracket is called a Poisson manifold, the bracket defining a Poisson structure on M . Let M be a Poisson manifold and H : M Ñ R be a smooth function. The Hamiltonian vector field associated with H is the unique smooth vector fieldv H on M satisfying the following identitŷ v H " tF, Hu "´tH, F u In other words, in order to compute the Poisson bracket of any pair of functions in some given set of local coordinates, it suffices to know the Poisson brackets between the coordinate functions themselves. These basic brackets, Alternatively, using (2.6), we could write this in the "bracket form" as follows: dx dt " tx, Hu, the i-th component of the right-hand side being tx i , Hu. A system of first order ordinary differential equations is said to be a Hamiltonian system if there is a Hamiltonian function Hpxq and a matrix of functions Jpxq determining a Poisson bracket (2.8) whereby the system takes the form (2.9). If D " ÿ J P J rusD J , P J P A is a differential operator, its (formal)adjoint is the differential operator D˚which satisfies for every pair of differential functions P, Q P A which vanish when u " 0. Also, for every domain Ω Ă R p and every function u " f pxq of compact support in Ω. An operator D is self-adjoint if D˚" D; it is skew-adjoint if D˚"´D.
The principal innovations needed to convert a Hamiltonian system of ordinary differential equations (2.9) to a Hamiltonian system of evolution equations are as follows (refer to [9] for more details):

Approximate Hamiltonian Symmetries and Approximate Conservation Laws
Consider a system of perturbed evolution equations in which P ru, εs " P px, u pnq , εq P A q , x P R p , u P R q and ε is a parameter. Substituting according to (3.1) and its derivatives, we see that any evolutionary symmetry must be equivalent to one whose characteristic Qru, εs " Qpx, t, u pmq , εq depends only on x, t, u, ε and the xderivatives of u. On the other hand, (3.1) itself can be considered as the equations corresponding to the flow expptv p q of the evolutionary vector field with characteristic P . The symmetry criterion (2.4), which in this case is can be readily seen to be equivalent to the following Lie bracket condition on the two approximate generalized vector fields. Indeed, this point generalizes the correspondence between symmetries of systems of first order perturbed ordinary differential equations and the Lie bracket of the corresponding vector fields. Considering above assumptions, some useful relevant theorems and definitions could be rewritten as follows: Proposition 3.1. An approximate evolutionary vector field v Q is a symmetry of the system of perturbed evolution equations u t " P ru, εs if and only if holds identically in px, t, u pmq , εq. (Here Bv Q {Bt denotes the evolutionary vector field with characteristic BQ{Bt.) Any approximate conservation law of a system of perturbed evolution equations takes the form D t T`Div X " opε p q, (3.4) in which Div denotes spatial divergence. Without loss of generality, the conserved density T px, t, u pnq , εq can be assumed to depend only on x-derivatives of u. Equivalently, for Ω Ă X, the functional is a constant, independent of t, for all solutions u such that T px, t, u pnq , εq Ñ 0 as x Ñ BΩ. Note that if T px, t, u pnq , εq is any such differential function, and u is a solution of the perturbed evolutionary system u t " P ru, εs, then where B t " B{Bt denotes the partial t-derivative. Thus T is the density for a conservation law of the system if and only if its associated functional T satisfies the following identity In the case that our system is of Hamiltonian form, the bracket relation (2.11) immediately leads to the Noether relation between approximate Hamiltonian symmetries and approximate conservation laws.
Definition 3.2. Let D be a qˆq approximate Hamiltonian differential operator. An approximate distinguished functional for D is a functional G P F satisfying DδG " opε p q for all x, u.
In other words, the Hamiltonian system corresponding to a distinguished functional is completely trivial: u t " 0. Now, according to [9], the perturbed Hamiltonian version of Noether's theorem can be presented as follows: Theorem 3.3. Let u t " DδH be a Hamiltonian system of perturbed evolution equations. An approximate Hamiltonian vector fieldv P with characteristic DδP, P P F , determines an approximate generalized symmetry group of the system if and only if there is an equivalent functionalP « P´G differing only from by a time-dependent approximate distinguished functional G rt; u, εs, such thatP determines an approximate conservation law.
Example 3.4. The Gardner equation can in fact be written in Hamiltonian form in two distinct ways. Firstly, we see where D " D x and is an approximate conservation law. Note that D is certainly skew-adjoint, and Hamiltonian. The Poisson bracket is tP, L u " ż δP¨D x pδL qdx The second Hamiltonian form is in which H 0 ru, εs " ż 1 2 u 2 dx E is skew-adjoint and satisfies the Jacobi identity. Therefore it is Hamiltonian.
In [8], we have comprehensively analyzed the problem of approximate symmetries for the Gardner equation. We have shown that the approximate symmetries of the Gardner equation are given by the following generators: with corresponding characteristics (up to sign). For the first Hamiltonian operator D " D x , there is one independent nontrivial approximate distinguished functional, the mass P 0 " M " ş u dx which is consequently approximately conserved. For the above seven characteristics, we have with the following approximately conserved functionals: For the second Hamiltonian operator E " 4uD x`2 u x`3 εpuu x`u 2 D x q´D 3 x , the following approximately conserved functionals are the corresponding approximate conservation laws: In this case, nothing new is obtained. Note that the other approximate conservation law P 5 did not arise from one of the geometrical symmetries. According to Theorem 3.3, however, there is an approximate Hamiltonian symmetry which gives rises to it, namelyv P5 . The characteristic of this approximate generalized symmetry isQ ote thatQ 5 happens to satisfy the Hamiltonian condition (3.6) for D with the following functional Consequently, another approximate conservation law is provided for the Gardner equation.
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.

Approximate Recursion Operators
Definition 4.1. Let ∆ be a system of perturbed differential equations. An approximate recursion operator for ∆ is a linear operator R : A q Ñ A q in the space of q-tuples of differential functions with the property that whenever v Q is an approximate evolutionary symmetry of ∆, so vQ is withQ « RQ.
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.
Definition 4.2. Let P ru, εs " P px, u pnq , εq P A r be an r-tuple of differential functions. The Fréchet derivative of P is the perturbed differential operator D P : A q Ñ A r defined so that D P pQq " d dǫˇˇˇǫ"0 P ru`ǫ Qru, εss (4.1) for any Q P A q . Proposition 4.3. If P P A r and Q P A q then Theorem 4.4. Suppose that ∆ru, εs " 0 be a system of perturbed differential equations. If R : A q Ñ A q is a linear operator such that for all solutions u of ∆, whereR : A q Ñ A q is a linear differential operator, then R is an approximate recursion operator for the system.
Suppose that ∆ru, εs " u t´K ru, εs is a perturbed evolution equation. Then D ∆ " D t´DK . If R is an approximate recursion operator, then it is not hard to observe that the operatorR in (4.3) must be the same as R. Therefore, the condition (4.3) in this case reduces to the commutator condition R t « rD K , Rs for an approximate recursion operator of a perturbed evolution equation. From (4.4), we can conclude that if R is an approximate recursion operator, then for all l ě 1 in which ε l R ‰ 0, ε l R is an approximate recursion operator In order to illustrate the significance of the above theorem, we discuss a couple of examples, including the potential Burgers' equation and the Gardner equation. In the first example, we apply some technical methods, used in Examples 5.8 and 5.30 of [9].
Example 4.5. Consider the potential Burgers' equation As mentioned in [10], approximate symmetries of the potential Burgers' equation are given by the following twelve vector fields " εpxB x`2 tB t q " εv 3 , v 10 " εp2tB x´x uB u q " εv 4 , v 11 " εuB u " εv 5 , v 12 " εp4xtB x`4 t 2 B t´p x 2`2 tquB u q " εv 6 , plus the infinite family of vector fields v f,g "´f px, tqp1´εuq`εgpx, tq¯B u , where f, g are arbitrary solutions of the heat equation u t " u xx . The corresponding characteristics of the first twelve approximate symmetries are (up to sign). Inspection of Q 1 , Q 2 , Q 7 , Q 8 leads us to the conjecture that R 1 " D x`ε u x is an approximate recursion operator, since Q 3 " R 1 Q 1 , Q 8 " R 1 Q 7 , etc. To prove this, we note that the Fréchet derivative for the right-hand side of potential Burgers' equation is We must verify (4.4). The time derivative of the first approximate recursion operator R 1 on the solutions of the potential Burgers' equation is the multiplication operator On the other hand, the commutator is computed using Leibniz' rule for differential operators: Comparing these two verifies (4.4) and proves that R 1 is an approximate recursion operator for the potential Burgers' equation.
There is thus an infinite hierarchy of approximate symmetries, with characteristics R k 1 Q 1 , k " 0, 1, 2, . . . For example, the next characteristic after Q 12 in the sequence is To obtain the characteristics depending on x and t, we require a second approximate recursion operator, which by inspection, we guess to be Using the fact that R 1 satisfies (4.4), we readily find proving that R 2 is also an approximate recursion operator. There is thus a doubly infinite hierarchy of approximate generalized symmetries of potential Burgers' equation, with characteristics R l 2 R k 1 Q 1 , k, l ě 0. For instance, Q 2 " R 1 Q 1 , Q 3 " 2R 2 R 1 Q 1 and so on.
x . Hence, the operator connecting our hierarchy of approximate Hamiltonian symmetries is x . Therefore, our results on approximate bi-Hamiltonian systems will provide ready-made proofs of the existence of infinitely many approximate conservation laws and approximate symmetries for the Gardner equation.
Note that the Fréchet derivative for the right-hand side of Gardner's equation is Theorem 4.7. LetQ 0 " εu x . For each k ě 0, the differential polynomialQ k " R kQ 0 is a total xderivative,Q k " D x R k , and hence we can recursively defineQ k`1 " RQ k . EachQ k is the characteristic of an approximate symmetry of the Gardner equation.
Proof. To prove this theorem, we apply the similar method applied in theorem 5.31 of [9].
We proceed by induction on k, so suppose thatQ k " R kQ 0 for some R k P A . From the form of the approximate recursion operator, f we can prove that for some differential polynomial S k P A , uQ k " D x S k , we will indeed have proved that Q k`1 " D x R k`1 , where R k`1 is the above expression in brackets. Consequently, the induction step will be completed.
To prove this fact, note that the formal adjoint of the approximate recursion operator εR is εR˚" εp4u´2D´1 x¨ux´D 2 x q " D´1 x εR D x . We apply this in order to integrate the expression uQ k , by parts, sō Q k " uR k rεu x s " u x¨p εR˚q k rus`D x A k for some differential function A k P A . On the other hand, using a further integration by parts, for some B k P A the following identity holds: Substituting into the previous identity, we conclude which proves our claim.
in the case of the second approximate Hamiltonian operator for the Gardner equation, we have trivially, by the properties of the wedge product, it is deduced that: Thus D and E form an approximately Hamiltonian pair.
Definition 4.12. A differential operator D : A r Ñ A s is approximately degenerate if there is a nonzero differential operatorD : A s Ñ A such thatD¨D " opε p q Now, according to [9], we are in a situation to state the main theorem on approximate bi-Hamiltonian systems.
Theorem 4.13. Let u t " K 1 ru, εs « DδH 1 « E δH 0 be an approximate bi-Hamiltonian system of perturbed evolution equations. Assume that the operator D of the approximately Hamiltonian pair is approximate nondegenerate. Let R " E¨D´1 be the corresponding approximate recursion operator, and let K 0 « DδH 0 . Assume that for each n " 1, 2, . . . we can recursively define K n « RK n´1 , n ě 1, meaning that for each n, K n´1 lies in the image of D. Then there exists a sequence of functionals H 0 , H 1 , H 2 , . . . such that (i) for each n ě 1 the perturbed evolution equation is an approximate bi-Hamiltonian system; (ii) the corresponding approximate evolutionary vector fields v n " v Kn all mutually commute: rv n , v m s " opε p q, n, m ě 0; (iii) the approximate Hamiltonian functionals H n are all in involution with respect to either Poisson bracket: tH n , H n u D " opε p q " tH n , H n u E , n, m ě 0, (4.8) and hence provide an infinite collection of approximate conservation laws for each of the approximate bi-Hamiltonian systems (4.5).
We have seen that given an approximate bi-Hamiltonian system, the operator R " E¨D´1, when applied successively to the initial equation K 0 " DδH 0 , produces an infinite sequence of approximate generalized symmetries of the original system (subject to the technical assumptions contained in Theorem 4.13). It is still not clear that R is a true approximate recursion operator for the system, in the sense that whenever v Q is an approximate generalized symmetry, so is v RQ . So far, we only know it for approximate symmetries with Q " K n for some n. In order to establish this more general result, we need a formula for the infinitesimal change of the approximate Hamiltonian operator itself under a Hamiltonian flow.
Lemma 4.14. Let u t " K « DδH be an approximate Hamiltonian system of perturbed evolution equations with corresponding vector field v K "v H . Then prv H pDq « D K¨D`D¨DK Theorem 4.15. Let u t " K « DδH 1 « E δH 0 be an approximate bi-Hamiltonian system of perturbed evolution equations. Then the operators R l " ε l E¨D´1, 0 ď l ď p, are approximate recursion operators for the system.
Judging from R p l " opε p q, when l ‰ 0, this type of approximate recursion operators have less significance than R 0 .
and we can apply R 0 to the right-hand side of the Gardner equation to obtain the approximate symmetries. The first step in this recursion is the flow which is not approximately total derivative, so we can not re-apply the approximate recursion operator to get a meaningful approximate generalized symmetry. But if we set K 1 ru, εs " Q 5 " εK 1 ru, εs " εp6uu x´uxxx q,H 0 "P 5 " εH 0 ,H 1 " P 5 " εH 1 , then we can apply R 0 successively toK 1 in order to obtain the approximate symmetries. The first phase become u t « E δH 1 « DδH 2 « R 0K1 « εpu xxxxx´1 0 u u xxx´2 0 u x u xx`3 0 u 2 u x q in whichH 2 "P 5 " ε 2 ż pu 2 xx´5 u 2 u xx`5 u 4 q dx is another approximate conservation law. Now, forK 2 " R 0K1 we have u t « E δH 2 « DδH 3 « R 0K2 « εp´u xxxxxxx`1 4 uu xxxxx`4 2u x u xxxx q 70 εpu xx u xxx´u 2 u xxx`2 u 3 u x´4 uu x u xx´u 3 x q whereH 3 " 7ε ż p u 2 xxx 14`u u 2 xx`5 u 2 u 2 x`u 5 q dx is a further approximate conservation law.

Concluding Remarks
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 [1,2,5].
The approximate symmetry method proposed by Fushchich and Shtelen [5] is based on a perturbation of dependent variables. This method has so many advantages such as producing more approximate groupinvariant solutions, consistence with the perturbation theory, solving singular perturbation problems [10,11] and close relationship with approximate homotopy symmetry method [12]. But despite above mentioned benefits, this procedure converts a perturbed evolution equation to an equivalent perturbed evolutionary system. In his case, obtaining the corresponding Hamiltonian formulation will be hard. Due to increase of the dimensions of Hamiltonian operators D, E , computation of the approximate recursion operator R " E¨D´1 is difficult. 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, Gazizov and Ibragimov [1,2] seems to be more consistent.

Acknowledgements
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 advise and suggestions.