Two Kinds of New Lie Algebras for Producing Integrable Couplings

Two types of Lie algebras are constructed, which are directly used to deduce the two resulting integrable coupling systems with multi-component potential functions. Many other integrable couplings of the known integrable systems may be obtained by the approach.     

Expansion of Lie Algebra and Its Application

Firstly we expand a finite-dimensional Lie algebra into a higher-dimenslonal one. By making use of the later and its corresponding loop algebra, the expanding integrable model of the multi-component NLS-mKdV hierarchy is worked out.     

Symmetries and Algebras of a （2＋1）-Dimensional MKdV-Type System

In this paper, a （2＋1）-dimensional MKdV-type system is considered. By applying the formal series symmetry approach, a set of infinitely many generalized symmetries is obtained. These symmetries constitute a closed infinite-dimensional Lie algebra which is a generalization of w∞ type algebra. Thus the complete integrability of this system is confirmed.     

Two New Expanding Lie Algebras and Their Integrable Models

A new Lax integrable hierarchy is obtained by constructing an isospectraJ problem with constrained conditions. Two kinds of integrable couplings are obtained by constructing two new expanding Lie algebras of the Lie algebra Be, respectively.     

The multicomponent （2＋1）-dimensional Glachette-Johnson （GJ） equation hierarchy and its super-integrable coupling system

This paper presents a set of multicomponent matrix Lie algebra, which is used to construct a new loop algebra A^-M. By using the Tu scheme, a Liouville integrable multicomponent equation hierarchy is generated, which possesses the Hamiltonian structure. As its reduction cases, the multicomponent （2＋1）-dimensional Glachette-Johnson （G J） hierarchy is given. Finally, the super-integrable coupling system of multicomponent （2＋1）-dimensional GJ hierarchy is established through enlarging the spectral problem.     

Matrix Lie Algebras and Integrable Couplings

Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively.     

Exact Solutions of Two Coupled Harmonic Oscillators Related to the Sp（4,R） Lie Algebra

Exact solutions of the eigenvalue problem of two coupled harmonic oscillators related to the Sp(4,R) Lie algebra are derived by using an algebraic method. It is found that the energy spectrum of the system is determined by one-boson excitation energies built on a vector coherent state of Sp(4,R)\supset U(2).     

Liouville Integrable System and Associated Integrable Coupling

Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.     

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.     

Integrable Couplings of the Coupled Burgers Hierarchy

In this letter, a new loop algebra G is constructed, from which a new isospectral problem is established. It follows that integrable couplings of the well-known coupled Burgers hierarchy are obtained.     

Lie Algebras Associated with Group U(n)

Starting from the subgroups of the group U(n), the corresponding Lie algebras of the Lie algebra A1 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.     

Matrix Lie Algebras and Integrable Couplings

Three kinds of higher-dimensional Lie algebras are given which can be used to directly construct integrable couplings of the soliton integrable systems. The relations between the Lie algebras are discussed. Finally, the integrable couplings and the Hamiltonian structure of Giachetti-Johnson hierarchy and a new integrable system are obtained, respectively.     

Characteristic Numbers of Matrix Lie Algebras

A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.     

Lie Algebras for Constructing Nonlinear Integrable Couplings

Two new explicit Lie algebras are introduced for which the nonlinear integrable couplings of the Giachetti-Johnson (GJ) hierarchy and the Yang hierarchy are obtained, respectively. By employing the variational identity their Hamiltonian structures are also generated. The approach presented in the paper can also
provide nonlinear integrable couplings of other soliton hierarchies of evolution equations.     

Two Kinds of New Lie Algebras for Producing Integrable Couplings

Two types of Lie algebras are constructed, which are directly used to deduce the two resulting integrable coupling systems with multi-component potential functions. Many other integrable couplings of the known integrable systems may be obtained by the approach.     

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C（（t^1/p））.     

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z₊-graded vertex algebra is construsted on the vacuum representation V_k(\hat{g}[ θ]) of \hat{g}[θ], which is a one-dimentional central extension of θ-invariant subspace on the loop algebra Lg=g\otimes C((t^1/p)).