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.     

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.     

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.     

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.     

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.     

A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras G1 and G2 are constructed. Basing on G1 and G2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.     

A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras \tilde{G₁} and \tilde{G₂} are constructed. Basing on \tilde{G₁} and \tilde{G₂}, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.     

Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebras are obtained. Two expanding integrable systems are produced with the help of the generalized zero curvature equation. One of them has complex Hamiltion structure with the help of generalized Tu formula （GTM）.     

Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebras are obtained. Two expanding integrable systems are produced with the help of the generalized zero curvature equation. One of them has complex Hamiltion structure with the help of generalized Tu formula (GTM).     

A New Integrable Lattice Hierarchy and Its Two Discrete Integrable Couplings

A new and efficient way is presented for discrete integrable couplings with the help of two semi-direct sum Lie algebras. As its applications, two discrete integrable couplings associated with the lattice equation are worked out. The approach can be used to study other discrete integrable couplings of the discrete hierarchies of solition equations.     

A Method for Obtaining Integrable Couplings

By making use of the vector product in R³, a commuting operation is introduced so that R³ becomes a Lie algebra. The resulting loop algebra \tilde R³ is presented, from which the well-known AKNS hierarchy is produced. Again via applying the superposition of the commuting operations of the Lie algebra, a commuting operation in R⁶ is constructed so that R⁶ becomes a Lie algebra. Thanks to the corresponding loop algebra \tilde R³ of the Lie algebra R³, the integrable coupling of the AKNS system is obtained. The method presented in this paper is rather simple and can be used to work out integrable coupling systems of the other known integrable hierarchies of soliton equations.     

New Matrix Lie Algebra and Its Application

A set of new matrix Lie algebra and its corresponding loop algebra are constructed. By making use of Tu scheme, a Liouville integrable multi-component hierarchy of soliton equation is generated. As its reduction cases, the multi-component Tu hierarchy is given. Finally, the multi-component integrable coupling system of Tu hierarchy is presented through enlarging matrix spectral problem.     

The Double Integrable Couplings of Tu Hierarchy

Two types of Lie algebras are presented, from which two integrable couplings associated with the Tu isospectral problem are obtained, respectively. One of them possesses the Hamiltonian structure generated by a linear isomorphism and the quadratic-form identity. An approach for working out the double integrable couplings of the same integrable system is presented in the paper.     

The Double Integrable Couplings of Tu Hierarchy

Two types of Lie algebras are presented, from which two integrable couplings associated with the Tu isospectral problem are obtained, respectively. One of them possesses the Hamiltonian structure generated by a linear isomorphism and the quadratic-form identity. An approach for working out the double integrable couplings of the same integrable system is presented in the paper.     

Upper Triangular Matrix of Lie Algebra and a New Discrete Integrable Coupling System

The upper triangular matrix of Lie algebra is used to construct integrable couplings of discrete solition equations. Correspondingly, a feasible way to construct integrable couplings is presented. A nonlinear lattice soliton equation spectral problem is obtained and leads to a novel hierarchy of the nonlinear lattice equation hierarchy. It indicates that the study of integrable couplings using upper triangular matrix of Lie algebra is an important step towards constructing integrable systems.     

Two Integrable Couplings of Modified Tu Hierarchy

Two different integrable couplings of the modified Tu hierarchy are obtained under the zero curvature equation by using two higher dimension Lie algebras. Furthermore, a complex Hamiltonian structures of the second integrable couplings is presented by taking use of the variational identity.     

A New Three-Dimensional Lie Algebra and a Modified AKNS Hierarchy of Soliton Equations

A new three-dimensional Lie algebra and its corresponding loop algebra are constructed, from which a modified AKNS soliton-equation hierarchy is obtained.