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.     

Vector Loop Algebra and Its Applications to Tu Hierarchy

A vector loop algebra and its extended loop algebra are proposed, which are devoted to obtaining the Tu hierarchy. By making use of the extended trace identity, the Harniltonian structure of the Tu hierarchy is constructed. Furthermore, we apply the quadratic-form identity to the integrable coupling system of the Tu hierarchy.     

The trace identity and the planar Casimir effect

The familiar trace identity associated with the scale transformationx ^Μ → x′ ^Μ = e^-λ x ^Μ on the Lagrangian density for a noninteracting massive real scalar field in 2 + 1 dimensions is shown to be violated on a single plate on which the Dirichlet boundary condition Φ(t, x¹, x² = -a) = 0 is imposed. It is however respected in: (i) 1 + 1 dimensions in both free space and on a single plate on which the Dirichlet boundary condition Φ(t, x¹ = -a) = 0 holds and (ii) in 2 + 1 dimensions in free space, i.e. the unconstrained configuration. On the plate where Φ(t, x¹, x² = -a) = 0, the modified trace identity is shown to be anomalous with a numerical coefficient for the anomalous term equal to the canonical scale dimension, viz. 1/2. The technique of Bordaget al [Ann. Phys. (N.Y.),165, 162 (1985)] is used to incorporate the said boundary condition into the generating functional for the connected Green’s functions.     

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.     

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.     

The quadratic-form identity for constructing Hamiltonian structures of the Guo hierarchy

The trace identity is extended to the quadratic-form identity. The Hamiltonian structures of the multi-component Guo hierarchy, integrable coupling of Guo hierarchy and (2+1)-dimensional Guo hierarchy are obtained by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.     

Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure＊

By using a Lie algebra, an integrable couplings of the classicai-Boussinesq hierarchy is obtained. Then, the Hamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-form identity.     

A Type of New Loop Algebra and a Generalized Tu Formula

A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing-Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula （GTF） is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 （1989） 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi- Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.     

New DLW Hierarchy of an Integrable Coupling and Its Hamiltonian Structure

A type of higher-dimensional loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic identity.     

A New Liouville Integrable Hamiltonian System

With the help of a Lie algebra, an isospectral Lax pair is introduced for which a new Liouville integrable hierarchy of evolution equations is generated. Its Hamiltonian structure is also worked out by use of the quadratic-form identity.     

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 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.     

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.     

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.     

一类S-mKdV方程族及其扩展可积模型

由loop代数的一个子代数出发，构造了一个线性等谱问题，再利用屠格式计算出了一类Liouvelle意义下的可积系统及其双Hamilton结构，作为该可积系统的约化，得到了著名的Schrdinger方程和mKdV方程，因此称该系统为S-mKdV方程族.根据已构造的的子代数，又构造了维数为5的loop代数的一个新的子代数，由此出发设计了一个线性等谱形式，再利用屠格式求得了S-mKdV方程族的一类扩展可积模型.利用这种方法还可以求BPT方程族、TB方程族等谱系的扩展可积模型.因此本方法具有普遍应用价值.最后作为特例，求得了著名的Schrdinger方程和mKdV方程的可积耦合系统.     

The Application of Discrete Variational Identity on Semi-Direct Sums of Lie Algebras

With non-semisimple Lie algebras, the trace identity was generalized to discrete spectral problems. Then the corresponding discrete variational identity was used to a class of semi-direct sums of Lie algebras in a lattice hierarchy case and obtained Hamiltonian structures for the associated integrable couplings of the lattice hierarchy. It is a powerful tool for exploring Hamiltonian structures of discrete soliton equations.     

A New Loop Algebra and Corresponding Computing Formula of Constant γ in Quadratic-Form Identity

A new loop algebra containing four arbitrary constants is presented, whose commutation operation is concise, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper,which can be reduced to computing formula of constant γ in the trace identity. As application, a new Liouville integrable hierarchy, which can be reduced to AKNS hierarchy is derived.     

An integrable Hamiltonian hierarchy and associated integrable couplings system

This paper establishes a new isospectral problem. By making use of the Tu scheme, a new integrable system is obtained. It gives integrable couplings of the system obtained. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.     

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.     

Multi-component Harry--Dym hierarchy and its integrable couplings as well as their Hamiltonian structures

This paper obtains the multi-component Harry--Dym (H--D) hierarchy and its integrable couplings by using two kinds of vector loop algebras \widetilde{G}₃ and \widetilde{G}₆. The Hamiltonian structures of the above system are given by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.