Symmetry-Preserving Difference Models of Some High-Order Nonlinear Integrable Equations

In this paper, a procedure for constructing the symmetry-preserving difference models by means of the potential systems is employed to investigate some kinds of integrable equations. The invariant difference models for the Benjamin–Ono equation and the nonlinear dispersive Km,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\left( {m,n} \right)$$\end{document} equation are investigated. Four cases of Km,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\left( {m,n} \right)$$\end{document} equations which yield compactons are studied. The invariant difference models preserving all the symmetries are obtained. Furthermore, some linear combinations of the symmetries are used to construct the invariant difference models. The invariant difference model of the Hunter–Saxton equation is constructed. The idea of this paper can be further extended to discrete some other high-order nonlinear integrable equations.


Introduction
Lie symmetry analysis method is one of the most effective method to study the properties of partial differential equations [1][2][3][4][5][6][7]. One of the main application of Lie symmetry is to find the group invariant solutions [8][9][10]. The structure of the Lie algebra spanned by Lie symmetry operators provides important information about the set of the solutions. High dimensional transformation group (HDTG) may have more possible applications. Therefore, for a finite-difference model, it is an important property to preserve the symmetries of its original continuous model.

Preliminaries for Construction of Invariant Difference Models
In this section, we briefly introduce the bases of the finite-difference derivatives and a procedure for constructing the symmetry-preserving difference scheme. For the details, the readers are referred the references [19,23]. Let us consider the space of sequences x, u, u 1 , u 2 , … , where independent variable is x = x i ;i = 1, 2, … , n , dependent variable is u = u k ;k = 1, 2, … , m , u 1 = u k i represents the set of mn first derivatives, u 2 = u k ij is the set of second order partial derivatives, etc. In order to briefly introduce the finite-difference derivatives, we consider the simple case n = 2 , i.e.,

Proposition 2
The following condition is a necessary and sufficient condition to guarantee an orthogonal mesh w h to preserve its orthogonality in the plane (t, x) under any transformation of the symmetry group G 1 . where w t is a potential variable. The potential system (8) can be reduced to equation (7) by means of the compatibility condition w xt = w tx . Based on the theory of Bluman's nonlocally related systems [24][25][26], we know that each local symmetry of potential system (8) projects onto a local symmetry of the original equation (7). Then the symmetry-preserving difference models can be constructed with the aid of potential system (8) of Eq. (7) and a fact presented in the following theorem.

Theorem 1 If the potential system of the finite-difference equation (3)
where D , admits the symmetries then the difference model E(z) admits the symmetries of original equation (7). In the continuous limit, the difference model E(z) is changed to the evolution Eq. (7).

Invariant Difference Model of the Benjamin-Ono Equation
In this section, we consider the Benjamin-Ono equation which describes the long internal gravity waves in deep stratified fluids [27]. The potential system of Eq. (9) can be written as where w(x, t) and v(x, t) are the potential variables. With the aid of Lie symmetry analysis, we can conclude that the potential system (10) admits the five-dimensional algebra of infinitesimal operators Substituting all the operators into conditions (1) and (2), we conclude that the mesh w h remain uniform and orthogonality in the plane (t, x) under the action of the transformation group. It implies that the mesh equation has the form h + = h − , which satisfies the second condition in the Eq. (6). We consider the set of operators (11) The corresponding stencil is shown in Fig. 1. It is obvious that the discrete subspace is thirteen-dimensional, The symmetry operators (11) prolonged to the difference stencil variables has the form . To find the finite-difference invariants I j , one needs to solve the system of linear equations By solving above equations, we obtain eight invariants By means of above invariants, we obtain following explicit scheme for Eq. (9) which is is equivalent to Applying the difference operators D + , D +h and compatibility we obtain the invariant difference model for Eq. (9) Fig. 1 The difference stencil of the Benjamin-Ono equation  (14) where . Thus we prove that the difference model (13) admits the symmetry group with operators (11), which means the difference model preserves original continuous symmetries.

Invariant Difference Model of the Nonlinear Dispersive K 2, 3 Equation
In 1993, Rosenau and Hyman [28] derived a family of fully nonlinear KdV equations which is called K(m, n) equation The physical phenomenon described by this equation can explain the nonlinear dispersion can compactify solitary waves and generate compactons. When m = n = 2 or 3, Eq. (15) is reduced to the K(2, 2) and the K(3, 3) equation [29]. Firstly, we consider nonlinear dispersive K(m, n) equation as m = 2 and n = 3.
The potential system of Eq. (16) can be written as We can conclude that this potential system possesses following four symmetry operators through Lie symmetry analysis. All the operators of (17) satisfy conditions (1) and (2). Thus, one can directly utilize the orthogonal gride which is uniform in t and x directions. We consider the set of operators (17) in the space t, x, u, w, h, , u − , u + , w + ,ŵ . The corresponding stencil is shown in Fig. 2. In this case, the discrete subspace is ten-dimensional, M ∼ t, x, u, w, h, , u − , u + , w + ,ŵ . By solving the characteristic equations of the prolonged symmetry, one can obtain following six difference invariants  Fig. 2 The difference stencil of the nonlinear dispersive K(2,3) equation (19) preserves the symmetry operators (17).

Proof
The symmetry operators extended to the space t, x, u, w, , h, (17) where Ψ = u t +2u u After checking above identical equations, we can conclude that the difference equation (19) admits the symmetry group with operators (17), which means the difference model preserves original continuous symmetries.  [30,31]. Based on the technique of Painlevé analysis, we know all the four cases do not possess the Painlevé property [32]. For the K(2, 2) equation, its potential system can be written as The Lie point symmetries of this potential system consist of Since all the symmetries can not be extended the operators containing both difference stencil variables h and , these four symmetries can not be used to construct the symmetry-preserving difference models for K(2, 2) equation. K(3, 3) equation and K(2, 2) equation have similar symmetric structure, so it is impossible to construct symmetry-preserving difference model for the same reason. (19) (24) u xxt + 4u x u xx + 2uu xxx = 0. Remark 2 This section gives an example to illustrate the scheme presented in this paper can be used to construct the invariant difference model not only preserving all the symmetries, but also maintaining some linear combination of the symmetries. This idea inspires us to study some other types of integrable systems to investigate the invariant difference models to further study the numerical solutions. For the Camassa-Holm equation, it has the symmetry operators X 1 = t , X 2 = x and X 3 = −t t + u u . The symmetries of Camassa-Holm equation are consistent with K(2, 2) equation. Therefore, these symmetries can not be employed to construct invariant difference model.

Conclusions
In this paper, we focus on investigating the symmetry invariant difference models of high order continuous differential equations. The procedure of constructing symmetry-preserving difference meshes with the aid of potential systems of original equations is briefly introduced. Then the potential systems of the Benjamin-Ono equation and the nonlinear dispersive K(m, n) equation are employed to construct the difference models inherited the symmetries of the original equations. The symmetry invariance of the obtained difference models are also checked by considering the symmetries extended to the difference variables. For the K(m, n) equation, we consider four cases which yield compactons, i.e. m, n = 2, 3 . It is proved that one can construct the invariant difference models preserving all the symmetries when m ≠ n . In order to further expand the method of this paper, other types of equations are studied. The symmetries of these equations can not be used to construct difference invariants, but the linear combinations of partial symmetries can be used to construct difference invariants. The Hunter-Saxton equation is investigated as an example. The idea of this paper can be further utilized to investigate the symmetrypreserving difference models of some other types of integrable equations.