Symmetry-preserving Difference Research Articles

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

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 Kleft( {m,n} right) equation are investigated. Four cases of Kleft( {m,n} right) 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.

The symmetry-preserving difference schemes and exact solutions of some high-dimensional differential equations

A direct and effective method is employed to construct the discrete models of high-dimensional generalized Zakharov–Kuznetsov and diffusion–convection equations. Through the compatibility condition, we construct their potential systems, whose local symmetries can be projected into the local and nonlocal symmetries of the original equations. Furthermore, based on the resulting Lie symmetries, the invariant difference models and symmetry-preserving difference models of the original two equations can be derived by using the orthogonal meshes which are uniform in space. Finally, some exact solutions of these two equations are also obtained with their graphic analysis.

On symmetry-preserving difference scheme to a generalized Benjamin equation and third-order Burgers equation

In this paper, an exposition of a method is presented for discretizing a generalized Benjamin equation and third-order Burgers equation while preserving their Lie point symmetries. By using local conservation laws, the potential systems of original equation are obtained, which can be used to construct the invariant difference models and symmetry-preserving difference models of original equation, respectively. Furthermore, this method is very effective and can be applied to discrete high-order nonlinear evolution equations.

A symmetry-preserving difference scheme for high dimensional nonlinear evolution equations

In this paper, a procedure for constructing discrete models of the high dimensional nonlinear evolution equations is presented. In order to construct the difference model, with the aid of the potential system of the original equation and compatibility condition, the difference equations which preserve all Lie point symmetries can be obtained. As an example, invariant difference models of the (2+1)-dimensional Burgers equation are presented.

The Using of Conservation Laws in Symmetry-Preserving Difference Scheme

In this paper, by means of the potential systems of the given nonlinear evolution equations, a procedure of symmetry preserving discretization of differential equations is presented. The specific process will be given detailed in section 2. This extended method is effective for discreting the high-order (high-dimensional) nonlinear evolution equations. As examples, the invariant difference models of the mKdV equation and the Boussinesq equation are constructed.

The Lie point symmetry-preserving difference scheme of holonomic constrained mechanical systems

In this paper a Lie point symmetry- preserving difference schemes approximating holonomic constrained mechanical systems is presented. The procedure is as follows: Firstly we find the Lie point symmetry groups of the original equation, and prolong them to three points of the lattice. Secondly the discrete invariants are obtained by solving discrete characteristic equations, then the invariant difference scheme is constructed by using these discrete invariants; and this invariants difference scheme will give the original equation under the continuous limit. Finally an example is presented to illustrate the applications of the result.

Symmetry-adapted moving mesh schemes for the nonlinear Schrödinger equation

In this paper we consider symmetry-preserving difference schemes for the nonlinear Schrodinger equation where n is the number of space dimensions. This equation describes one-dimensional waves in n space dimensions in many physical situations, including phenomena in plasma physics and nonlinear optics. We will consider the nonintegrable case n≥2 for which the equation admits solutions that blow up in a finite time, and construct discretizations based upon moving mesh schemes that have the same Lie group properties and Lagrangian structures as the continuous counterpart.

Symmetry-preserving difference schemes for some heat transfer equations

Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modelling helps to retain qualitative properties of the differential equations in their difference counterparts.