A type of multi-component integrable hierarchy

A new isospectral problem is established by constructing a simple interesting loop algebra. A commutation operation of the loop algebra is as straightforward as the loop algebra ?_1. It follows that a type of multi-component integrable hierarchy is obtained. This can be used as a general method.     

Multi-component SC Lax Integrable Hierarchy of Soliton Equations and Its Multi-component Integrable Coupling System with Two Arbitrary Functions

A new simple loop algebra ^-GM is constructed, which is devoted to establishing an isospectral problem.By making use of generalized Tu scheme, the multi-component SC hierarchy is obtained. Furthermore, an expanding loop algebra ^-FM of the loop algebra ^-GM is presented. Based on FM, the multi-component integrable coupling system of the multi-component SC hierarchy of soliton equations is worked out. How to design isospectral problem of mulitcomponent hierarchy of soliton equations is a technique and interesting topic. The method can be applied to other nonlinear evolution equations hierarchy.     

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M-dimensional loop algebra \tilde{X} is produced. By taking advantage of \tilde{X}, a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra \tilde{F}_M of the loop algebra \tilde{X} is presented. Based on the \tilde{F}_M, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.     

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.     

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.     

Multi-linear Variable Separation Approach to Solve a （1＋1）-Dimensional Coupled Integrable Dispersionless System

The multi-linear variable separation approach （MLVSA ） is very useful to solve （2＋1）-dimensional integrable systems. In this letter, we extend this method to solve a （1＋1）-dimensional coupled integrable dispersion-less system. Namely, by using a Backlund transformation and the MLVSA, we find a new general solution and define a new ＂universal formula＂. Then, some new （1＋1）-dimensional coherent structures of this universal formula can be found by selecting corresponding functions appropriately. Specially, in some conditions, bell soliton and kink soliton can transform each other, which are illustrated graphically.     

Multi-component C-KdV Hierarchy of Soliton Equations and Its Multi-component Integrable Coupling System

A new simple loop algebra G^-M is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component C-KdV hierarchy is obtained. Further, an expanding loop algebra F^-M of the loop algebra G^-M is presented. Based on F^-M , the multi-component integrable coupling system of the multi-component C-KdV hierarchy is worked out. The method can be used to other nonlinear evolution equations hierarchy.     

The multi-component Tu hierarchy of soliton equations and its multi-component integrable couplings system

A new simple loop algebra GM is constructed, which is devoted to the establishing of an isospectral problem. By making use of the Tu scheme, the multi-component Tu hierarchy is obtained.Furthermore, an expanding loop algebra FM of the loop algebra GM is presented. Based on the FM, the multi-component integrable coupling system of the multi-component Tu hierarchy has been worked out. The method can be applied to other nonlinear evolution equation hierarchies.     

（2＋2）-Dimensional Discrete Soliton Equations and Integrable Coupling System

In this paper, we extend a （2＋2）-dimensional continuous zero curvature equation to （2＋2）-dimensional discrete zero curvature equation, then a new （2＋2）-dimensional cubic Volterra lattice hierarchy is obtained. Fhrthermore, the integrable coupling systems of the （2＋2）-dimensional cubic Volterra lattice hierarchy and the generalized Toda lattice soliton equations are presented by using a Lie algebraic system sl（4）.     

An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system

A subalgebra of loop algebra ?_2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz－Kaup－Newell－Segur (AKNS) hierarchy and a generalized standard form of the Schr?dinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra ?_2 into loop algebra ?_1. Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.     

The extended trace identity and its application

The trace identity is extended to the general loop algebra. The Hamiltonian structures of the integrable systems concerning vector spectral problems and the multi-component integrable hierarchy can be worked out by using the extended trace identity. As its application, we have obtained the Hamiltonian structures of the Yang hierarchy, the Korteweg-de--Vries (KdV) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur (M-AKNS) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur Kaup--Newell (M-AKNS--KN) hierarchy and a new multi-component integrable hierarchy separately.     

Three New Integrable Hierarchies of Equations

A general Lie algebra Vs and the corresponding loop algebra Vx are constructed, from which the linear isospectral Lax pairs are established, whose compatibility presents the zero curvature equation. As its application, a new Lax integrable hierarchy containing two parameters is worked out. It is not Liouville-integrable, however, its two reduced systems are Liouville-integrable, whose Hamiltonian structures are derived by making use of the quadratic-form identity and the γ formula （i.e. the computational formula on the constant γ appeared in the trace identity and the quadratic-form identity）.     

Two new discrete integrable systems

In this paper,we focus on the construction of new(1+1)-dimensional discrete integrable systems according to a subalgebra of loop algebra A 1.By designing two new(1+1)-dimensional discrete spectral problems,two new discrete integrable systems are obtained,namely,a 2-field lattice hierarchy and a 3-field lattice hierarchy.When deriving the two new discrete integrable systems,we find the generalized relativistic Toda lattice hierarchy and the generalized modified Toda lattice hierarchy.Moreover,we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity.     

A generalized SHGI integrable hierarchy and its expanding integrable model

An anti-symmetric loop algebra \overline{A}_2 is constructed. It follows that an integrable system is obtained by use of Tu's scheme. The eminent feature of this integrable system is that it is reduced to a generalized Schr?dinger equation, the well-known heat-conduction equation and a Gerdjkov－Ivanov (GI) equation. Therefore, we call it a generalized SHGI hierarchy. Next, a new high-dimensional subalgebra \tilde{G} of the loop algebra ?_2 is constructed. As its application, a new expanding integrable system with six potential functions is engendered.     

Decomposition for a 2+1-Dimensional Discrete Integrable Model

A 2＋1-dimensional discrete is presented, which is decomposed into a new integrable symplectic map and a class of finite-dimensional integrable Hamiltonian systems, with aid of the nonlineaxization of Lax pairs. The system is completely integrable in the Liouville sense.     

High-Order Binary Symmetry Constraints of a Liouville Integrable Hierarchy and Its Integrable Couplings

A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra A^-2. Furthermore, high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method. Finally, according to another new subalgebra of loop algebra A^-2, its integrable couplings are established.     

Two types of loop algebras and their expanding Lax integrable models

Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation U_t-V_x+[U,V]=0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator \widetilde{J} is presented by constructing a subalgebra \widetilde{G } of the loop algebra \widetilde A₂. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their (2+1)-dimensional forms are derived from a (2+1)-dimensional zero-curvature equation.     

The Liouville integrable coupling system of the m-AKNS hierarchy and its Hamiltonian structure

In this paper a type of 9-dimensional vector loop algebra \tilde{F} is constructed, which is devoted to establish an isospectral problem. It follows that a Liouville integrable coupling system of the m-AKNS hierarchy is obtained by employing the Tu scheme, whose Hamiltonian structure is worked out by making use of constructed quadratic identity. The method given in the paper can be used to obtain many other integrable couplings and their Hamiltonian structures.     

A subalgebra of loop algebra A2^～ and its applications

A subalgebra of loop algebra A2^～ and its expanding loop algebra G^- are constructed. It follows that both resulting integrable Hamiltonian hierarchies are obtained. As a reduction case of the first hierarchy, a generalized nonlinear coupled Schroedinger equation, the standard heat-conduction and a formalism of the well known Ablowitz, Kaup, Newell and Segur hierarchy are given, respectively. As a reduction case of the second hierarchy, the nonlinear Schroedinger and modified Korteweg de Vries hierarchy and a new integrable system are presented. Especially, a coupled generalized Burgers equation is generated.     

A new multi-component integrable coupling system for AKNS equation hierarchy with sixteen-potential functions

It is shown in this paper that the upper triangular strip matrix of Lie algebra can be used to construct a new integrable coupling system of soliton equation hierarchy. A direct application to the Ablowitz--Kaup--Newell-- Segur(AKNS) spectral problem leads to a novel multi-component soliton equation hierarchy of an integrable coupling system with sixteen-potential functions. It is indicated that the study of integrable couplings when using the upper triangular strip matrix of Lie algebra is an efficient and straightforward method.