Abstract
We derive the first-order approximate symmetries for the Harry Dym equation by the method of approximate transformation groups proposed by Baikov et al. (1989, 1996). 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.
1. Introduction
The following nonlinear partial differential equation is known as the Harry Dym equation [1]. 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 [2] and rediscovered independently in [3, 4]. The Harry Dym equation shares many of the properties typical of the soliton equations. It is a completely integrable equation [5, 6], which can be solved by inverse scattering transformation [7–9]. 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 is a small parameter, with a method which was first introduced by Baikov et al. [12, 13]. 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 and Shtelen [14] and later followed by Euler et al. [15, 16]. For a comparison of these two methods, we refer the interested reader to [17, 18]. 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 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 invariants with respect to the generators of Lie algebra and optimal system. In Section 7, we summarize our results.
2. 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 [19]. If a function satisfies the condition it is written and is said to be of order less than . If the functions and are said to be approximately equal (with an error ) and written as or, briefly, when there is no ambiguity. The approximate equality defines an equivalence relation, and we join functions into equivalence classes by letting and be members of the same class if and only if . Given a function , let be the approximating polynomial of degree in obtained via the Taylor series expansion of in powers of about . Then any function (in particular, the function itself) has the form Consequently the expression (6) is called a canonical representative of the equivalence class of functions containing . Thus, the equivalence class of functions is determined by the ordered set of functions . In the theory of approximate transformation groups, one considers ordered sets of smooth vector-functions depending on ’s and a group parameter : with coordinates Let us define the one-parameter family of approximate transformations of points into points as the class of invertible transformations with vector-functions such that Here is a real parameter, and the following condition is imposed:
Definition 1. The set of transformations (10) is called a one-parameter approximate transformation group if for all transformations (11).
Definition 2. Let be a one-parameter approximate transformation group: An approximate equation is said to be approximately invariant with respect to or admits if whenever satisfies (16). If then (16) becomes an approximate differential equation of order , and is an approximate symmetry group of the differential equation.
Theorem 3. Equation (16) is approximately invariant under the approximate transformation group (15) with the generator if and only if or where is the prolongation of of order . The operator (18) satisfying (20) is called an infinitesimal approximate symmetry of or an approximate operator admitted by (16). Accordingly, (20) is termed the determining equation for approximate symmetries.
Theorem 4. If (16) admits an approximate transformation group with the generator , where , then the operator is an exact symmetry of the equation
Definition 5. Equations (22) and (16) are termed an unperturbed equation and a perturbed equation, respectively. Under the conditions of Theorem 4, the operator is called a stable symmetry of the unperturbed equation (22). The corresponding approximate symmetry generator for the perturbed equation (16) is called a deformation of the infinitesimal symmetry of (22) caused by the perturbation . In particular, if the most general symmetry Lie algebra of (22) is stable, we say that the perturbed equation (16) inherits the symmetries of the unperturbed equation.
3. 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 , , and for are unknown functions of , , and . Solving the determining equation for the exact symmetries of the unperturbed equation, we obtain where are arbitrary constants. Hence, Therefore, the unperturbed Harry Dym equation admits the five-dimensional Lie algebra with the basis
3.2. Approximate Symmetries
At first, we need to determine the auxiliary function by virtue of (19), (20), and (16), that is, by the equation Substituting the expression (27) of the generator into (29) we obtain the auxiliary function Now, calculate the operators by solving the inhomogeneous determining equation for deformations: So, the above determining equation for this equation is written as Solving the determining equation yields where are arbitrary constants.
Thus, we derive the following approximate symmetries of the perturbed Harry Dym equation: Table 1 of commutators, evaluated in the first order of precision, shows that the operators (34) span a ten-dimensional approximate Lie algebra and hence generate a ten-parameter approximate transformation group.
Remark 6. Equations (34) show that all symmetries (28) of (1) are stable. Hence, the perturbed equation (2) inherits the symmetries of the unperturbed equation (1).
4. 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 is nonsolvable, since The Lie algebra admits a Levi decomposition as the following semidirect product , where is the radical of (the largest solvable ideal contained in ) and is a semisimple subalgebra of .
The radical is solvable with the following chain of ideals: where The semisimple subalgebra of is isomorphic to the Lie algebra of the classification of three-dimensional Lie algebras in [20], by the following isomorphism:
5. Optimal System for Perturbed Harry Dym Equation
Definition 7. Let be a Lie group. An optimal system of -parameter subgroups is a list of conjugacy inequivalent -parameter subgroups with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of -parameter subalgebras forms an optimal system if every -parameter subalgebra of is equivalent to a unique member of the list under some element of the adjoint representation: , .
Proposition 8. Let and be connected, -dimensional Lie subgroups of the Lie group with corresponding Lie subalgebras and of the Lie algebra of . Then are conjugate subgroups if and only if 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 one-dimensional subalgebra is determined by a nonzero vector in . To compute the adjoint representation one uses the Lie series:
where , is the commutator for the Lie algebra and is a parameter. In this manner, one constructs Table 2 with the th entry indicating .
Theorem 9. An optimal system of one-dimensional approximate Lie algebras of the perturbed Harry Dym equation is provided by
Proof. Consider the approximate symmetry algebra of the unperturbed Harry Dym equation, whose adjoint representation was determined in Table 2. Given a nonzero vector
our task is to simplify as many of the coefficients as possible through judicious applications of adjoint maps to .
Suppose first that . Scaling if necessary, we can assume that . Referring to Table 2, if we act on such a by
we can make the coefficient of vanish. The remaining one-dimensional subalgebras are spanned by vectors of the above form with . If , we scale to make and then act on to cancel the coefficient of as follows:
We can further act on by the group generated by ; this has the net effect of scaling the coefficients of :
So, depending on the sign of , we can make the coefficient of either , , or 0. If and , we scale to make . So, the nonzero vector is equivalent to under adjoint maps:
If and , by scaling , we can assume that . Referring to Table 2, if we act on such a by the following adjoint map, we can arrange that the coefficients of vanish:
If and , we scale to make . Thus, is equivalent to under the adjoint representations:
If and , we scale to make . So, we can make the coefficients of , zero by using the following adjoint maps:
If and , by scaling , we can assume that . Therefore, we can arrange that the coefficients of vanish by simplifying the nonzero vector as follows:
We can further act on by the group generated by :
So, depending on the sign of , we can make the coefficient of either , , or . If and , by scaling , we can assume that . We can act on such a by the group generated by . So, depending on the sign of , we can make the coefficient of either , or . The case and , no further simplifications are possible. The last remaining case occurs when and , for which our earlier simplifications were unnecessary. Hence, the only remaining vectors are the multiples of , on which the adjoint representation acts trivially.
6. 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 . To determine the independent invariants , we need to solve the first-order partial differential equation: 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 differential 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 3.
7. Conclusions
In this paper, we investigate the approximate symmetry of the perturbed Harry Dym equation and discuss 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 invariants with respect to the generators of Lie algebra and optimal system.