Classical Lie Symmetry Research Articles

A Note on Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

Applying the classical Lie symmetry approach to the barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a β-plane channel, we find a new symmetry, which is not presented in a previous work [F. Huang, Commun. Theor. Phys. (Beijing, China) 42 (2004) 903]. A general finite transformation group is obtained based on the full Lie point symmetry. Furthermore, two new types of similarity reduction solutions are obtained.

Lie symmetries and form-preserving transformations of reaction–diffusion–convection equations

New Lie symmetry classification of the known class of reaction–diffusion–convection equations is presented. The classification method is based on combining the standard group classification method and the form-preserving transformation approach.

A note on a class of solitary-like solutions of the Tzitzéica equation generated by a similarity reduction

In this paper, new class of solutions of the Tzitzéica equation are derived by using the classical Lie symmetry analysis. The important aspect of this paper however is the fact that the analysis results in a new class of solitary-like solutions, the so-called cusp-solitary solutions. This special type of solutions are not found in the current literature and represents a necessary contribution for the whole solution manifold. The studied equation was originally found in the field of geometry, otherwise the Tzitzéica equation takes place in many branches of non-linear sciences. Therefore, explicit class of solutions connected by a physical meaning are of great importance. The analysis is restricted to the case of traveling waves represented by a similarity variable describing any wave propagation. A complete characterization of the group properties is given. The classical Lie point symmetries are derived and algebraic properties are determined. Similarity solutions and transformations are given in a most general form and have been derived for the first time in terms of Jacobian elliptic functions. It is worth to mention that the application of known powerful algebraic methods (e.g. special function transform methods) are not appropriate to study the solution manifold. Hence, the present paper is therefore suitable to create a deeper insight into the solution manifold with respect to the traveling wave picture.

On the classification of similarity solutions of a two-dimensional diffusion–advection equation

We consider the two-dimensional diffusion-advection equation ut=Duyy-dkduux. We present the classification of Lie symmetries of this equation. For the case k(u)=u^2 we give the complete list of similarity solutions. Exact and numerical solutions are obtained.

Lie symmetries and conservation laws of non-linear multidimensional reaction–diffusion systems with variable diffusivities

This work is devoted to the classical Lie symmetry analysis of a class of systems of two quasilinear multidimensional reaction-diffusion (RD) equations having variable diffusivities. Moreover, conservation laws for RD systems containing such diffusivities are constructed. Some generalizations of the results to the case of multicomponent RD systems with greater than two components are also presented.

New Exact Solutions of Broer–Kaup Equations**Theproject supported by the Natural Science Foundation of Zhejing Province of China under Grant No. 102037

By a known transformation, (2+1)-dimensional Brioer–Kaupequations are turned to a single equation. The classical Liesymmetry analysis and similarity reductions are performed for thissingle equation. From some of reduction equations, new exactsolutions are obtained, which contain previous results, and moreexact solutions can be created directly by abundant knownsolutions of the Burgers equations and the heat equations.

Non-linear reaction–diffusion systems with variable diffusivities: Lie symmetries, ansätze and exact solutions

This work first considers the classical Lie symmetry analysis of a class of systems of two quasilinear reaction–diffusion equations having variable diffusivities. Subsequently, non-Lie reductions to systems of first order ordinary differential equations are obtained for a subclass of these systems. In particular, families of exact solutions of a diffusive Lotka–Volterra type system are constructed.

Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation**The project supported by National Natural Science Foundation of China under Grant No. 40305009

The (2+1)-dimensional nonlinear barotropic and quasi-geostrophic potential vorticity equation without forcing and dissipation on a beta-plane channel is investigated by using the classical Lie symmetry approach. Some types of group-invariant wave solutions are expressed by means of the lower-dimensional similarity reduction equations. In addition to the known periodic Rossby wave solutions, some new types of exact solutions such as the ring solitary waves and the breaking soliton type of vorticity solutions with nonlinear and nonconstant shears are also obtained.

Symmetry reductions and new exact invariant solutions of the generalized Burgers equation arising in nonlinear acoustics

We perform a complete Lie symmetry classification of the generalized Burgers equation arising in nonlinear acoustics. We obtain seven functional forms of the ray tube area that allow symmetry reductions. We use symmetries to obtain reduced equations and exact solutions when possible. We also investigate the existence of potential symmetries for the generalized Burgers equation. It is found that only the classical Burgers equation admits true potential symmetries. We further obtain all conditional symmetries of the second kind and indicate a possible route for obtaining conditional symmetries of the first kind. The conditional symmetries of the second kind leads to symmetry reductions and exact solutions not obtainable from Lie point symmetries.

Symmetries and form-preserving transformations of generalised inhomogeneous nonlinear diffusion equations

We consider the variable coefficient inhomogeneous nonlinear diffusion equations of the form f( x) u t =[ g( x) u n u x ] x . We present a complete classification of Lie symmetries and form-preserving point transformations in the case where f( x)=1 which is equivalent to the original equation. We also introduce certain nonlocal transformations. When f( x)= x p and g( x)= x q we have the most known form of this class of equations. If certain conditions are satisfied, then this latter equation can be transformed into a constant coefficient equation. It is also proved that the only equations from this class of partial differential equations that admit Lie–Bäcklund symmetries is the well-known nonlinear equation u t =[ u −2 u x ] x and an equivalent equation. Finally, two examples of new exact solutions are given.

All Solutions of Standard Symmetric Linear Partial Differential Equations Have Classical Lie Symmetry

It is proven that every solution of any linear partial differential equation with an independent-variable-deforming classical Lie point symmetry is invariant under some classicalLie point symmetry. This is true for any number of independent variables and for equations of any order higher than one. Although this result makes use of the infinite-dimensional component of the Lie symmetry algebra due to linear superposition, it is shown that new similarity solutions, previously thought not to be classical, can be recovered prospectively by allowing symmetries to include superposition of similarity solutions already known from the finite part of the symmetry algebra. This result applies to all constant-coefficient equations and to many variable-coefficient equations such as Fokker–Planck equations.

The Nonclassical Group Analysis of the Heat Equation

The nonclassical method of reduction was devised originally by Bluman and Cole in 1969, to find new exact solutions of the heat equation. Much success has been had by many authors using the method to find new exact solutions of nonlinear equations of mathematical and physical significance. However, the defining equations for the nonclassical reductions of the heat equation itself have remained unsolved, although particular solutions have been given. Recently, Arrigo, Goard, and Broadbridge showed that there are no nonclassical reduction solutions of constant coefficient linear equations that are not already classical Lie symmetry reduction solutions. Their arguments leave open the problem of what is the general nonclassical group action, and its effect on the relevant solution of the heat equation. In this article, both these problems are solved. In the final section we use the methods developed to solve the remaining outstanding case of nonclassical reductions of Burgers' equation.

Symmetry classification and exact solutions of the Kramers equation

The complete classification of Lie symmetries for the Kramers equation is described. Group invariance under infinitesimal transformations is used to generate a wide class of exact solutions. Some of the known solutions are obtained as particular cases. The general results can also be used to solve some types of moving-boundary problems. We researched some Q-conditional symmetry properties of this equation.

Symmetry analysis of the Kramers equation

The complete classification of Lie symmetries for the Kramers equation is described. Also some Q-conditional symmetry properties of this equation have been investigated. The Q-conditional symmetry has been used to construct the well-known Boltzmann solution.

Direct Similarity Solution Method and Comparison with the Classical Lie Symmetry Solutions

Abstract We study the general applicability of the Clarkson–Kruskal’s direct method, which is known to be related to symmetry reduction methods, for the similarity solutions of nonlinear evolution equations (NEEs). We give a theorem that will, when satisfied, immediately simplify the reduction procedure or ansatz before performing any explicit reduction expansions. We shall apply the method to both scalar and vector NEEs in either 1+1 or 2+1 dimensions, including in particular, a variable coefficient KdV equation and the 2+1 dimensional Khokhlov–Zabolotskaya equation. Explicit solutions that are beyond the classical Lie symmetry method are obtained, with comparison discussed in this connection.

Symmetry Classification for a Coupled Nonlinear Schrödinger Equations

We do a Lie symmetry classification for a system of two nonlinear coupled Schrodinger equations. Our system under consideration is a generalization of the equations which follow from the analysis of optical fibres. Reductions of some special equations are given.