Yang族一个新的非线性可积耦合及其哈密尔顿结构(英文)

Based on a kind of non-semisimple Lie algebras, we establish a way to construct nonlinear continuous integrable couplings. Variational identities over the associated loop algebras are used to furnish Hamiltonian structures of the resulting continuous couplings.As an illustrative example of the scheme is given nonlinear continuous integrable couplings of the Yang hierarchy.     

Guo族非线性可积耦合的哈密顿结构及其守恒

基于可积耦合的基本理论,我们给出了构造孤子族非线性可积耦合的一般方法,并用相应圈代数上的变分恒等式来求可积耦合的哈密顿结构.作为应用,我们给出了Guo族的非线性可积耦合及其哈密顿结构.最后,给出了Guo族非线性可积耦合的守恒律.     

利用李代数构造非线性可积及双可积哈密顿耦合

With the help of a Lie algebra,two kinds of Lie algebras with the forms of blocks are introduced for generating nonlinear integrable and bi-integrable couplings.For illustrating the application of the Lie algebras,an integrable Hamiltonian system is obtained,from which some reduced evolution equations are presented.Finally,Hamiltonian structures of nonlinear integrable and bi-integrable couplings of the integrable Hamiltonian system are furnished by applying the variational identity.The approach presented in the paper can also provide nonlinear integrable and bi-integrable couplings of other integrable system.     

Expanding integrable models of the nonlinear Schrödinger equation

By using a few Lie algebras and the corresponding loop algebras, we establish some isospectral problems whose compatibility conditions give rise to a few various expanding integrable models (including integrable couplings) of the well-known nonlinear Schrödinger equation. The Hamiltonian forms of two of them are generated by making use of the variational identity. Finally, we propose an efficient method for generating a nonlinear integrable coupling of the nonlinear Schrödinger equation.     

Coupling commutator pairs and integrable systems

According to classification of the matrix Lie algebras, a type of explicit Lie algebras are constructed which can be decomposed into a few Lie subalgebras. These subalgebras constitute several coupling commutator pairs from which some continuous multi-integrable couplings could be generated if the proper isospectral Lax pairs could be set up. Then the above Lie algebras are again decomposed into a kind of Lie algebras which are also closed under the matrix multiplication. From such the Lie algebras, some discrete multi-integrable couplings could be worked out. Finally, a few examples are given. However, the Hamiltonian structures of the (continuous and discrete) integrable couplings obtained by the above Lie algebras cannot be computed by using the trace identity or the quadratic-form identity, which is a strange and interesting problem. The phenomenon indicates that the importance of the Lie-algebra classification. The problem also needs us to try to find an efficient scheme to deal with.     

Kaup-Newell族的分数阶非线性双可积耦合及其Hamilton结构

本文研究了Kaup-Newell族的分数阶非线性双可积耦合.利用分数阶等谱问题和非半单矩阵Lie代数上的非退化、对称双线性形式,得到了Kaup-Newell族的分数阶非线性双可积耦合,并求出了Kaup-Newell族双可积耦合的分数阶Hamilton结构.本文的方法还可以应用于其它孤子族分数阶可积耦合.     

Integrable couplings,bi‐integrable couplings and their Hamiltonian structures of the Giachetti–Johnson soliton hierarchy

On the basis of zero curvature equations from semi‐direct sums of Lie algebras, we construct integrable couplings of the Giachetti–Johnson hierarchy of soliton equations. We also establish Hamiltonian structures of the resulting integrable couplings by the variational identity. Moreover, we obtain bi‐integrable couplings of the Giachetti–Johnson hierarchy and their Hamiltonian structures by applying a class of non‐semisimple matrix loop algebras consisting of triangular block matrices. Copyright © 2014 John Wiley & Sons, Ltd.     

Three nonlinear integrable couplings of the nonlinear Schrödinger equations

Based on two types of expanding Lie algebras of a Lie algebra G, three isospectral problems are designed. Under the framework of zero curvature equation, three nonlinear integrable couplings of the nonlinear Schröding equations are generated. With the help of variational identity, we get the Hamiltonian structure of one of them. Furthermore, we get the result that the hierarchy is also integrable in sense of Liouville.     

Applications of a few Lie algebras

Two isomorphic groups R ² andM are firstly constructed. Then we extend them into the differential manifold R ²ⁿ and n products of the group M for which four kinds of Lie algebras are obtained. By using these Lie algebras and the Tu scheme, integrable hierarchies of evolution equations along with multi-component potential functions can be generated, whose Hamiltonian structures can be worked out by the variational identity. As application illustrations, two integrable Hamiltonian hierarchies with 4 component potential functions are obtained, respectively, some new reduced equations are followed to present. Specially remark that the integrable hierarchies obtained by taking use of the approach presented in the paper are not integrable couplings. Finally, we generalize an equation obtained in the paper to introduce a general nonlinear integrable equation with variable coefficients whose bilinear form, B¨acklund transformation, Lax pair and infinite conserved laws are worked out, respectively, by taking use of the Bell polynomials.     

Constructing integrable coupling systems of Hamiltonian lattice equations using semi-direct sums of Lie algebras

In this article, by considering a discrete isospectral problem, a hierarchy of Hamiltonian lattice equations are derived. Two types of semi-direct sums of Lie algebras are proposed, using which a practicable way to construct discrete integrable couplings is introduced. As an application, two kinds of discrete integrable couplings of the resulting system are worked out.     

Lie algebras and Lie super algebra for the integrable couplings of NLS–MKdV hierarchy

Lie algebras and Lie super algebra are constructed and integrable couplings of NLS–MKdV hierarchy are obtained. Furthermore, its Hamiltonian and Super-Hamiltonian are presented by using of quadric-form identity and super-trace identity. The method can be used to produce the Hamiltonian structures of the other integrable and super-integrable systems.     

The coupling integrable couplings of the modified Korteweg–de Vries (mKdV) hierarchy

Two kinds of coupling integrable couplings of the mKdV hierarchy are obtained, respectively, by making use of two higher-dimensional Lie algebras in the vector forms. The Hamiltonian structure of one reduced coupling integrable coupling of them is worked out by employing the variational identity.     

AKNS-KN孤子方程族的可积耦合与Hamilton结构

首先通过引入高维圈代数,在零曲率方程框架下得到了AKNS-KN孤子族(记为AKNS-KN-SH)的一个新的可积耦合系统;再由二次型恒等式得到了该系统的双-Hamilton结构形式.最后引进了一个新的Lie代数A_4,可通过建立其不同的圈代数与等价的列向量Lie代数,研究AKNS-KN-SH的多分量可积耦合系统及其Hamilton结构.     

Loop Algebras and Bi-integrable Couplings

A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations.The variational identities under non-degenerate,symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings.A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.     

A few new higher-dimensional Lie algebras and two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy

Four higher-dimensional Lie algebras are introduced. With the help of their different loop algebras and the block matrices of Lax pairs for the zero curvature representations of two given integrable couplings, the two types of coupling integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively, which fill up the consequences obtained by Ma and Gao (2009) [9]. The coupling integrable couplings of the AKNS hierarchy obtained in the paper again reduce to the coupling integrable couplings of the nonlinear Schrödinger equation and the modified KdV (mKdV) equation, which are different from the resulting results given by Ma and Gao.     

一个超可积族的非线性超可积耦合及其超哈密顿结构（英文）

Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction,special cases of this nonlinear super integrable couplings were obtained.     

Broer-Kaup-Kupershmidt族的非线性双可积耦合及其自相容源

基于新的非半单矩阵李代数,介绍了构造孤子族非线性双可积耦合的方法,由相应的变分恒等式给出了孤子族非线性双可积耦合的Hamilton结构.作为应用,给出了Broer-Kaup-Kupershmidt族的非线性双可积耦合及其Hamilton结构.最后指出了文献中的一些错误,利用源生成理论建立了新的公式,并导出了带自相容源Broer-Kaup-Kupershmidt族的非线性双可积耦合方程.     

Invertible linear transformations and the Lie algebras

With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.     

多分量的KN族与可积耦合及其它们的哈密顿结构

Firstly,a vector loop algebra (~G)3 is constructed,by use of it multi-component KN hierarchy is obtained.Further,by taking advantage of the extending vector loop algebras (~G)6 and (~G)9 of (~G)3 the double integrable couplings of the multi-component KN hierarchy are worked out respectively.Finally,Hamiltonian structures of obtained system are given by quadratic-form identity.     

An integrable coupling hierarchy of the Mkdv_integrable systems, its Hamiltonian structure and corresponding nonisospectral integrable hierarchy

A four-by-four matrix spectral problem is introduced, locality of solution of the related stationary zero curvature equation is proved. An integrable coupling hierarchy of the Mkdv_integrable systems is presented. The Hamiltonian structure of the resulting integrable coupling hierarchy is established by means of the variational identity. It is shown that the resulting integrable couplings are all Liouville integrable Hamiltonian systems. Ultimately, through the nonisospectral zero curvature representation, a nonisospectral integrable hierarchy associated with the resulting integrable couplings is constructed.