Prolongation structure of the variable coefficient KdV equation

The prolongation structure methodologies of Wahlquist--Estabrook [Wahlquist H D and Estabrook F B 1975 J. Math. Phys. 16 1] for nonlinear differential equations are applied to a variable-coefficient KdV equation. Based on the obtained prolongation structure, a Lie algebra with five parameters is constructed. Under certain conditions, a Lie algebra representation and three kinds of Lax pairs for the variable- coefficient KdV equation are derived.     

ONEOptimal:A Maple Package for Generating One-Dimensional Optimal System of Finite Dimensional Lie Algebra

We present a Maple computer algebra package, ONEOptimal, which can calculate one-dimensional optimal system of finite dimensional Lie algebra for nonlinear equations automatically based on Olver＇s theory. The core of this theory is viewing the Killing form of the Lie algebra as an invariant for the adjoint representation. Some examples are given to demonstrate the validity and efficiency of the program.     

Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra

By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 （2006） 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.     

Symmetry analysis and explicit solutions of the (3+1)-dimensional baroclinic potential vorticity equation

<正>This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-(3+l)- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method.Its symmetry algebra, symmetry group and group-invariant solutions are analysed.Otherwise,some exact explicit solutions are obtained from the corresponding(2+1)-dimensional equation,the inviscid barotropic nondivergent vorticy equation.To show the properties and characters of these solutions,some plots as well as their possible physical meanings of the atmospheric circulation are given.     

Block （or Hamiltonian） Lie Symmetry of Dispersionless D-Type Drinfeld-Sokolov Hierarchy

In this paper, the dispersionless D-type Drinfeld-Sokolov hierarchy, i.e. a reduction of the dispersionless two-component BKP hierarchy, is studied. The additional symmetry flows of this hierarchy are presented. These flows form an infinite-dimensional Lie algebra of Block type as well as a Lie algebra of Hamiltonian type.     

Statistical Properties of the nonlinear SU（1,1） Coherent States

Using non-Hermitian realizations of SU(1,1) Lie algebra in terms of an f-oscillator,we generalize the notion of nonlinear coherent states to the single-mode and two-mode nonlinear SU(1,1) coherent states.Taking the nonlinearity function f(k)=Lk^1(η^2)[(k 1)Lk^0(η^2)]^-1,their statistical properties are studied.     

New Matrix Loop Algebra and Its Application

A new matrix Lie algebra and its corresponding Loop algebra are constructed firstly, as its application, the multi-component TC equation hierarchy is obtained, then by use of trace identity the Hamiltonian structure of the above system is presented. Finally, the integrable couplings of the obtained system is worked out by the expanding matrix Loop algebra.     

A Maple Package to Compute Lie Symmetry Groups and Symmetry Reductions of （1＋1）-Dimensional Nonlinear Systems

Armed with the computer algebra system Maple, using a direct algebraic substitution method, we obtain Lie point symmetries, Lie symmetry groups and the corresponding symmetry reductions of one component nonlinear integrable and nonintegrable equations only by clicking the ‘Enter＇ key. Abundant （1＋1）-dimensional nonlinear mathematical physical systems are analysed effectively by using a Maple package LieSYMGRP proposed by us.     

Lie Algebras Associated with Group U（n）

Starting from the subgroups of the group U（n）, the corresponding Lie algebras of the Lie algebra Al are presented, from which two well-known simple equivalent matrix Lie algebras are given. It follows that a few expanding Lie algebras are obtained by enlarging matrices. Some of them can be devoted to producing double integrable couplings of the soliton hierarchies of nonlinear evolution equations. Others can be used to generate integrable couplings involving more potential functions. The above Lie algebras are classified into two types. Only one type can generate the integrable couplings, whose Hamiltonian structure could be obtained by use of the quadratic-form identity. In addition, one condition on searching for integrable couplings is improved such that more useful Lie algebras are enlightened to engender. Then two explicit examples are shown to illustrate the applications of the Lie algebras. Finally, with the help of closed cycling operation relations, another way of producing higher-dimensional Lie algebras is given.     

A Complex Higher-Dimensional Lie Algebra with Real and Imaginary Structure Constants as Well as Its Decomposition

A new Lie algebra G of the Lie algebra sl（2） is constructed with complex entries whose structure constants are real and imaginary numbers. A loop algebra G corresponding to the Lie algebra G is constructed, for which it is devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvature equation which is derived from the compatibility of the isospectral problems expressed by Hirota operators. Finally, we decompose the Lie algebra G to obtain the subalgebras G1 and G2. Using the G2 and its one type of loop algebra G2, a Liouville integrable soliton hierarchy is obtained, furthermore, we obtain its bi-Hamiltonian structure by employing the quadratic-form identity.     

Kac-Moody-Virasoro Symmetry Algebra of （2＋1）-Dimensional Dispersive Long-Wave Equation with Arbitrary Order Invariant

By Lie symmetry method, the Lie point symmetries and its Kac Moody-Virasoro （KMV） symmetry algebra of （2＋1）-dimensional dispersive long-wave equation （DLWE） are obtained, and the finite transformation of DLWE is given by symmetry group direct method, which can recover Lie point symmetries. Then KMV symmetry algebra of DLWE with arbitrary order invariant is also obtained. On basis of this algebra the group invariant solutions and similarity reductions are also derived.     

Explicit solutions from residual symmetry of the Boussinesq equation

The Bcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained: a new type of Backlund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.     

Lie Algebra and Lie Super Algebra for Integrable Couplings of C-KdV Hierarchy 1

Based on the constructed Lie algebra and Lie super algebra, the integrable couplings and super-integrable couplings of the C-KdV hierarchy are obtained respectively. Furthermore, its super-Hamiltonian structures are presented by using super-trace identity.     

Two expanding forms of a Lie algebra and their application

With the help of a known Lie algebra,two new high order Lie algebras are constructed.It is remarkable that they have different constructing approaches.The first Lie algebra is constructed by the definition of integrable couplings.the second one by an extension of Lie algebra,Then by making use of Tu scheme,a generalized AKNS hierarchy and another new hierarchy are obtained.As a reduction case of the first hierarchy,a kind of coupled KdV equation is presented.As a reduction case of the second one,a new coupled Schroedinger equation is given.     

Lie symmetries and conserved quantities for a two-dimensional nonlinear diffusion equation of concentration

The Lie symmetries and conserved quantities of a two-dimensional nonlinear diffusion equation of concentration are considered. Based on the invariance of the two-dimensional nonlinear diffusion equation of concentration under the infinitesimal transformation with respect to the generalized coordinates and time, the determining equations of Lie symmetries are presented. The Lie groups of transformation and infinitesimal generators of this equation are obtained. The conserved quantities associated with the nonlinear diffusion equation of concentration are derived by integrating the characteristic equations. Also, the solutions of the two-dimensional nonlinear diffusion equation of concentration can be obtained.     

Lie symmetries and conserved quantities for a two-dimentional nonlinear diffusion equation of concentration

The Lie symmetries and conserved quantities of a two-dimensional nonlinear diffusion equation of concentration are considered. Based on the invariance of the two-dimensional nonlinear diffusion equation of concentration under the infinitesimal transformation with respect to the generalized coordinates and time, the determining equations of Lie symmetries are presented. The Lie groups of transformation and infinitesimal generators of this equation are obtained. The conserved quantities associated with the nonlinear diffusion equation of concentration are derived by integrating the characteristic equations. Also, the solutions of the two-dimensional nonlinear diffusion equation of concentration can be obtained.     

A new eight-dimensional Lie superalgebra and two corresponding hierarchies of evolution equations

A new eight-dimensional Lie superalgebra is constructed and two isospectral problems with six potentials are designed. Corresponding hierarchies of nonlinear evolution equations, as well as super-AKNS and super-Levi, are derived. Their super-Hamiltonian structures are established by making use of the supertrace identity, and they are integrable in the sense of Liouville.     

N = 4 Supersymmetric Morse Oscillator and Its Spectrum-Generating Algebra

In this paper N = 4 supersymmetry of generalized Morse oscillators in one dimension is studied. Both bound states and scattering states of its four superpartner Hamiltonians are analyzed by using unitary irreducible representations of the noncompact Lie algebra su（1,1）. The spectrum-generating algebra governing the Hamiltonian of the N = 4 supersymmetric Morse oscillator is shown to be connected with the realization of Lie superalgebra osp（1,2） or B（0,1） in terms of the variables of a supersymmetric two-dimensional harmonic oscillator.     

Lie symmetry algebra of one-dimensional nonconservative dynamical systems

Lie symmetry algebra of linear nonconservative dynamical systems is studied in this paper. By using 1--1 mapping, the Lie point and Lie contact symmetry algebras are obtained from two independent solutions of the one-dimensional linear equations of motion.     

Similarity Reductions of Barotropic and Quasi-geostrophic Potential Vorticity Equation

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.