Approximate Symmetries of the Harry Dym Equation

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 Schr？dinger 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