Approximate symmetries of the Harry Dym equation

In this paper, we derive the first order approximate symmetries for the Harry Dym equation by the method of approximate transformation groups proposed by Baikov, Gaszizov and Ibragimov. Moreover, we investigate the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. We compute the one-dimensional optimal system of subalgebras as well as point out some approximately differential invariants with respect to the generators of Lie algebra and optimal system.


Introduction
The following nonlinear partial differential equation is known as the Harry Dym equation [7]. This equation was obtained by Harry Dym and Martin Kruskal as an evolution equation solvable by a spectral problem based on the string equation instead of Schrödinger equation. This result was reported in [9] and rediscovered independently in [15], [16]. The Harry Dym equation shares many of the properties typical of the soliton equations. It is a completely integrable equation which can be solved by inverse scattering transformation [3], [17], [18]. It has a bi-Hamiltonian structure and an infinite number of conservation laws and infinitely many symmetries [10], [11].
In this paper, we analyze the perturbed Harry Dym equation where ε is a small parameter, with a method which was first introduced by Baikov, Gazizov and Ibragimov [1], [2]. This method which is known as "approximate symmetry" is a combination of Lie group theory and perturbations. There is a second method which is also known as "approximate symmetry" due to Fushchich anf Shtelen [6] and later followed by Euler et al [4], [5]. For a comparison of these two methods, we refer the interested reader to the papers [12], [14]. our paper is organized as follows: In section 2, we present some definitions and theorems in the theory of approximate symmetry. In section 3, we obtain the approximate symmetry of the perturbed Harry Dym equation. In section 4, we discuss on the structure of its Lie algebra. In section 5, we construct the one-dimensional optimal system of subalgebras. In section 6, we compute some approximately differential invarints with respect to the generators of Lie algebra and optimal system. In section 7, we summarize our results.

Notations and Definitions
In this section, we will provide the background definitions and results in approximate symmetry that will be used along this paper. Much of it is stated as in [8]. If a functionf (x, ε) satisfies the condition it is written f (x, ε) = o(ε p ) and f is said to be oforder less than ε p . If the functions f and gare said to be approximately equal (with an error o(ε p )) and written as or or, briefly f ≈ g when there is no ambiguity. The approximate equality defines an equivalence relation, and we join functions into equivalence classes by letting f (x, ε) and g(x, ε) to be members of the same class if and only if f ≈ g. Given a function f (x, ε), let be the approximating polynomial of degree p in ε obtained via the Taylor series expansion of f (x, ε) in powers of ε about ε = 0. Then any function g ≈ f (in particular, the function f itself) has the form Consequently the expression (6) is called a canonical representative of the equivalence class of functions containing f Thus, the equivalence class of functions g(x, ε)f (x, ε) is determined by the ordered set of p+1 functions f 0 (x), f l (x), · · · , f p (x). In the theory of approximate transformation groups, one considers ordered sets of smooth vector-functions depending on x's and a group parameter a: with coordinates Let us define the one-parameter family G of approximate transformations of points x = (x 1 , · · · , x n ) ∈ R n into pointsx = (x 1 , · · · ,x n ) ∈ R n as the class of invertible transformationsx with vector-functions f = (f 1 , · · · , f n ) such that Here a is a real parameter, and the following condition is imposed: Definition. The set of transformations (10) is called a one-parameter approximate transformation group if for all transformations (11). Definition. Let G be a one-parameter approximate transformation group: An approximate equation is said to be approximately invariant with respect to G, or admits G if whenever z = (z l , · · · , z N ) satisfies Eq. (16). If z = (x, u, u (1) , · · · , u (k) ) then (16) becomes an approximate differential equation of order k, and G is an approximate symmetry group of the differential equation. Theorem. Eq. (16) is approximately invariant under the approximate transformation group (15) with the generator if and only if where X (k) is the prolongation of X of order k. The operator (18) satisfying Eq. (20) is called an infinitesimal approximate symmetry of, or an approximate operator admitted by Eq. (16). Accordingly, Eq. (20) is termed the determining equation for approximate symmetries. Theorem. If Eq. (16) admits an approximate tramformation group with the generator X = X 0 + εX 1 , where X 0 = 0, then the operator is an exact symmetry ofthe equation Definition. Eqs. (22) and (16) are termed an unperturbed equation and a perturbed equation, respectively. Under the conditions of Theorem 2.3, the operator X 0 is called a stable symmetry of the unperturbed equation (22). The corresponding approximate symmetry generator X = X 0 + εX 1 for the perturbed equation (16) is called a deformation of the infinitesimal symmetry X 0 of Eq. (22) caused by the perturbation εF 1 (z). In particular, if the most general symmetry Lie algebra of Eq. (22) is stable, we say that the perturbed equation (16) inherits the symmetries of the unperturbed equation.

Approximate symmetries of the perturbed Harry Dym equation
Consider the perturbed Harry Dym equation By applying the method of approximate transformation groups, we provide the infinitesimal approximate symmetries (18) for the perturbed Harry Dym equation (2).
3.1. Exact symmetries. Let us consider the approximate group generators in the form where rξ i , τ i and φ i for i = 0, 1 are unknown functions ofx, t and u. Solving the determining equation for the exact symmetries X 0 of the unperturbed equation, we obtain where A 1 , · · · , A 5 are arbitrary constants. Hence, Therefore, the unperturbed Harry Dym equation, admits the five-dimensional Lie algebra with the basis  20) and (16), i.e., by the equation Substituting the expression (26) of the generator X 0 into Eq. (28) we obtain the auxiliary function Now, calculate the operators X 1 by solving the inhomogeneous determining equation for deformations: So, the above determinig equation for this equation is written as solving the determining equation yields, where C 1 , · · · , C 5 are arbitrary constants. Thus, we derive the following approximate symmetries of the perturvbed Harry Dym equation: The following table of commutators, evaluated in the first-order of precision, shows that the operators (33) span an ten-dimensional approximate Lie algebra , and hence generate an ten-parameter approximate transformations group.

The structure of the Lie algebra of symmetries
In this section, we determine the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. The Lie algebra g is non-solvable, since (2) , g (2) ] = g (2) The Lie algebra g admits a Levi decomposition as the following semi-direct product g = r ∝ s, where is the radical of g (the largest solvable ideal contained in g) and is a semi-simple subalgebra of g.
The radical r is solvable with the following chain of ideals where The semi-simple subalgebra s of g is isomorphic to the Lie algebra A 3,8 of the classification of three dimensional Lie algebras in [13], by the following isomorphism.

Optimal system for perturbed Harry Dym equation
Definition. Let G be a Lie group. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subgroups with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras forms an optimal system if every s-parameter subalgebra of g is equivalent to a unique member of the list under some element of the adjoint representation:h = Adg(h)), g ∈ G.
Proposition. Let H andH be connected, s-dimensional Lie subgroups of the Lie group G with corresponding Lie subalgebras h andh of the Lie algebra g of G. ThenH = gHg −1 are conjugate subgroups if and only ifh = Adg(h)are conjugate subalgebras.(Proposition 3.7 of [11]) Actually, the proposition says that the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. For one-dimensional subalgebras, this classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation, since each onedimensional subalgebra is determined by a nonzero vector in g. To compute the adjoint representation we use the Lie series where [v i , v j ], i, j = 1, · · · , 10 is the commutator for the Lie algebra and µ is a parameter. In this manner, we construct the table with the (i, j)-th entry indicating Ad(exp(µv i ))v j . Table 2. Adjoint representation of approximate symmetry of the perturbed Harry Dym equation Theorem. An optimal system of one-dimensional approximate Lie algebras of the perturbed Harry Dym equation is provided by Proof. Consider the approximate symmetry algebra g of the unperturbed Harry Dym equation, whose adjoint representation was determined in the table 2. Given our task is to simplify as many of the coefficients ai as possible through judicious applications of adjoint maps to v.
Suppose first that a 10 = 0. Scaling v if necessary, we can assume that a 10 = 1. Referring to table (3.24), if we act on such a v by 6 v 6 + a 7 v 7 + a 9 v 9 + v 10 , Table 3. One-dimensional optimal system of the perturbed Harry Dym equation we can make the coefficient of a 8 vanish. The remaining one-dimensional subalgebras are spanned by vectors of the above form with a 10 = 0. If a 9 = 0, we scale to make a 9 = 1, and then act on v to cancel the coefficient of a 7 as follows: We can further act on v ′ by the group generated by v 4 ; this has the net effect of scaling the coefficients of v 2 : So, depending on the sign of a 2 , we can make the coefficient of v 2 either +1, −1 or 0. If a 10 = a 9 = 0 and a 4 = 0, we scale to make a 4 = 1. So, the non-zero vector v is equivalent to v ′ under adjoint maps: If a 10 = a 9 = a 4 = 0 and a 3 = 0, by scaling v, we can assume that a 3 = 1.
Referring to the table, if we act on such a v by the following adjoint map, we can arrange that the coefficients of a 6 vanish.
If a 10 = a 9 = a 4 = a 3 = 0 and a 5 = 0, we scale to make a 5 = 1. Thus, v is equivalent to v ′ under the adjoint representations: If a 10 = a 9 = a 4 = a 3 = a 5 = 0 and a 1 = 0, we scale to make a 1 = 1. So, we can make the coefficients of a 6 , a 8 zero by using the following adjoint maps: If a 10 = a 9 = a 4 = a 3 = a 5 = a 1 = 0 and a 2 = 0, by scaling v, we can assume that a 2 = 1. Therefore, we can arrange that the coefficients of a 7 vanish by simplifying the non-zero vector v as follows: We can further act on v ′ by the group generated by v 3 ; So, depending on the sign of a 6 , we can make the coefficient of v 6 either +1, −1 or 0. If a 10 = a 9 = a 4 = a 3 = a 5 = a 1 = a 2 = 0 and a 6 = 0, by scaling v, we can assume that a 6 = 1. We can act on such a v by the group generated by v 4 ; So, depending on the sign of a 7 , we can make the coefficient of v 7 either +1, −1 or 0. The remaining cases a 10 = a 9 = a 4 = a 3 = a 5 = a 2 = a 1 = a 3 = 0 and a 7 = 0, no further simplifications are possible. The last remaining cace occurs when a 10 = a 9 = a 4 = a 3 = a 5 = a 1 = a 2 = a 4 = a 6 = a 7 = 0 and a 8 = 0, for which our earlier simplifications were unnecessary. Since, the only remaining vectors are the multiples of v 8 , on which the adjoint representation acts trivially.

Approximately differential invariants for the perturbed Harry Dym equation
In this section, we compute some approximately differential invariants of the perturbed Harry Dym equation with respect to the optimal system. Consider the operator v 2 . To determine the independent invariants I, we need to solve the first order partial differential equation (ε ∂ ∂t + aεx ∂ ∂x + aεu ∂ ∂u )(I(x, t, u)) = 0, that is which is a first order homogeneous PDE. The solution can be found by integrating the corresponding characteristic system of ordinary differential equation, which is Hence, the independent approximately differntial invariants are as follows: In this manner, we investigate some independent approximately differential invariants with respect to the optimal system which are listed in Table 4.  8 − ln(ax + 1) aε + t, u ax + 1 v 1 + av 2 + bv 7 −bεx − aε + t, u av 1 + bv 2 + v 5 + cv 6 + dv 7 −dε − b √ cε + a arctan x √ cε + a + t, u x 2 + cε + a

Conclusions
In this paper, we investigate the approximate symmetry of the perturbed Harry Dym equation and discuss on the structure of its Lie algebra. Moreover, we compute optimal system of one-dimensional approximate Lie algebras of the perturbed Harry Dym equation and derive some approximately differential invarints with respect to the generators of Lie algebra and optimal system.