Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

Some New Lie Symmetry Groups of Differential-Difference Equations Obtained from a Simple Direct Method

In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A: Math. Gen. 38 (2005) L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and (2+1)-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.     

Some New Lie Symmetry Groups of Differential-Difference Equations Obtained from a Simple Direct Method

In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A： Math. Gen. 38 （2005） L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and （2＋1）-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.     

Symmetry Analysis and Similarity Solutions of a Resonant Davey-Stewartson System

Lie point symmetry analysis is performed to a recently proposed （2＋1）-dimensional nonlinear system, the resonant Davey-Stewartson equation. Some similarity solutions of the RDS equation are thus obtained.     

Conditional Similarity Reductions of Jimbo-Miwa Equation via the Classical Lie Group Approach

Recently, the Clarkson and Kruskal direct method has been modified to find new similarity reductions (conditional similarity reductions) of nonlinear systems and the results obtained by the modified direct method cannot be obtained by the current classical and/or non-classical Lie group approach. In this paper, we show that the conditional similarity reductions of the Jimbo-Miwa equation can be reobtained by adding an additional constraint equation to the original model to form a conditional equation system first and then solving the model system by means of the classical Lie group approach.     

Mei Symmetry and Lie Symmetry of Relativistic Hamiltonian System

The Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are studied. The definition and criterion of the Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are given. The relationship between them is found. The conserved quantities which the Mei symmetry and the Lie symmetry lead to are obtained.An example is given to illustrate the application of the result.     

Mei Symmetry and Lie Symmetry of Relativistic Hamiltonian System

The Mei symmetry and the Lie symmetry of the relativistic Hamiltonian system are studied. The definition and criterion of the Mei symmetry and the Lie symmetry of the reoativistic Hamiltonian system are given. The relationship between them is found. The conserved quantities which the Mei symmetry and the Lie symmetry lead to are obtained. An example is given to illustrate the application of the result.     

Generating Lie Point Symmetry Groups of (2+1)-Dimensional Broer-Kaup Equation via a Simple Direct Method

Using the (2+1)-dimensional Broer-Kaup equation as an simple example, a new direct method is developed to find symmetry groups and symmetry algebras and then exact solutions of nonlinear mathematical physical equations.     

Mei Symmetry and Lie Symmetry of the Rotational Relativistic Variable Mass System

The Mei symmetry and the Lie symmetry of a rotational relativistic variable masssystem are studied. Thedefinitions and criteria of the Mei symmetry and the Lie symmetry of the rotational relativistic variable mass system aregiven. The relation between the Mei symmetry and the Lie symmetry is found. The conserved quantities which the Meisymmetry and the Lie symmetry lead to are obtained. An example is given to illustrate the application of the result.     

Mei Symmetry and Lie Symmetry of the Rotational Relativistic Variable Mass System

The Mei symmetry and the Lie symmetry of a rotational relativistic variable mass system are studied. The definitions and criteria of the Mei symmetry and the Lie symmetry of the rotational relativistic variable mass system are given. The relation between the Mei symmetry and the Lie symmetry is found. The conserved quantities which the Mei symmetry and the Lie symmetry lead to are obtained. An example is given to illustrate the application of the result.     

Thermo-Hydrodynamic Lattice BGK Schemes with Lie Symmetry Preservation

A new class of lattice Bhatnagar-Gross-Krook （BGK） models is proposed, based on the Lie symmetry preservation ansatz for the local equilibria. This class extends the range of stability of previous models, especially for thermohydrodynamic lattice BGK schemes.     

Lie Symmetry Analysis,Conservation Laws and Exact Power Series Solutions for Time-Fractional Fordy-Gibbons Equation

In this paper, the time fractional Fordy-Gibbons equation is investigated with Riemann-Liouville derivative. The equation can be reduced to the Caudrey-Dodd-Gibbon equation, Savada-Kotera equation and the Kaup-Kupershmidt equation, etc. By means of the Lie group analysis method, the invariance properties and symmetry reductions of the equation are derived. Furthermore, by means of the power series theory, its exact power series solutions of the equation are also constructed. Finally, two kinds of conservation laws of the equation are well obtained with aid of the self-adjoint method.     

Symmetry and Similarity Solutions of a （2＋1）-Dimensional Generalized Broer-Kaup System

We study the symmetries of a （2＋1）-dimensional generalized Broer-Kaup system by means of the classical Lie group theory. The corresponding group algebra is constructed. Based on the symmetries, severaJ types of similarity solutions are obtained.     

New Formal Solutions of Davey-Stewartson Equation via Combined tanh Function Method with Symmetry Method

We combine the tanh function method with the symmetry group method to construct new type of solutions of Davey-Stewartson equation and implemente it in a computer algebraic system. As a result, some new types of solutions are obtained. This method is also applied to other differential equations if the nonlinear evolution equations admit nontrivial one-parameter group of transformation.     

Similarity Reductions for the Davey-Stewartson Equations Using a Direct Method

By the direct method that involves no group theory, the Davey-Stewartson equations (DSE's)in (2+1) dimensions have been reduced to some (1+1)-or 2-dimensional partial differentialequations which have included all the types of symmetry reductions using the standard Liegroup theory. As a special case we obtained the 1-dimensional reducing equations fromDSE's in (2+1) dimensions directly.     

Primary Branch Solutions of First Order Autonomous Scalar Partial Differential Equations via Lie Symmetry Approach

A primary branch solution (PBS) is defined as a solution with m independent n ? 1 dimensional arbitrary functions for an m order n dimensional partial differential equation (PDE). PBSs of arbitrary first order scalar PDEs can be determined by using Lie symmetry group approach companying with the introduction of auxiliary fields. Because of the intrusion of the arbitrary function, the PBSs have abundant and complicated structure. Usually, PBSs are implicit solutions. In some special cases, explicit solutions such as the instanton (rogue wave like) solutions may be obtained by suitably fixing the arbitrary function of the PBS.     

Similarity Reductions and Nonlocal Symmetry of the KdV Equation

Using some local and nonlocal symmetries of the KdV equation we get two types of nontrivial new similarity reductions. The first type of reduction equation can be solved by means of the Weierstrass elliptic function and the Riemann's zeta function while the solutions of the other type of reduction can be changed to the Painlevd Ⅱ equation.     

一道经典竞赛题的简捷解法

有这样一道竞赛题：如图1所示，将一支正六棱柱形铅笔放在斜面上，斜面倾角α=40°，铅笔棱与水平方向成θ角，铅笔静止，试问：     

Conditional and Lie Symmetry of Nonlinear Wave Equation

Abstract

Group classification of the nonlinear wave equation is carried out and the conditional invariance of the wave equation with the nonlinearity F (u) = u is found.