A New Nonlinear Integrable Couplings of Yang Equations Hierarchy and Its Hamiltonian Structure

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.     

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.     

GENERALIZED FRACTIONAL TRACE VARIATIONAL IDENTITY AND A NEW FRACTIONAL INTEGRABLE COUPLINGS OF SOLITON HIERARCHY

Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.     

Generations of integrable hierarchies and exact solutions of related evolution equations with variable coefficients

We first propose a way for generating Lie algebras from which we get a few kinds of reduced 6 6 Lie algebras, denoted by R6, R8 and R1,R6/2, respectively. As for applications of some of them, a Lax pair is introduced by using the Lie algebra R6 whose compatibility gives rise to an integrable hierarchy with 4- potential functions and two arbitrary parameters whose corresponding Hamiltonian structure is obtained by the variational identity. Then we make use of the Lie algebra R6 to deduce a nonlinear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is also obtained. Again,via using the Lie algebra R62, we introduce a Lax pair and work out a linear integrable coupling hierarchy of the mKdV equation whose Hamiltonian structure is obtained. Finally, we get some reduced linear and nonlinear equations with variable coefficients and work out the elliptic coordinate solutions, exact traveling wave solutions, respectively.     

The algebraic structure of discrete zero curvature equations associated with integrable couplings and application to enlarged Volterra systems

An algebraic structure of discrete zero curvature equations is established for integrable coupling systems associated with semi-direct sums of Lie algebras. As an application example of this algebraic structure, a τ-symmetry algebra for the Volterra lattice integrable couplings is engendered from this theory.     

THE INTEGRABLE COUPLINGS OF THE GENERALIZED COUPLED mKdV HIERARCHY AND ITS HAMILTONIAN STRUCTURES

We construct a loop algebra 3 , then a new 4×4 isospectral problem is presented. By Tu scheme, the generalized coupled mKdV equation hierarchy is derived. Based on an expanding loop algebra F3 of the loop algebra 3 , the integrable couplings of the generalized coupled mKdV hierarchy is solved. Finally, the Hamiltonian structures of the integrable couplings of the generalized coupled mKdV hierarchy is obtained by the quadratic-form identity.     

Super Jaulent-Miodek hierarchy and its super Hamiltonian structure,conservation laws and its self-consistent sources

A super Jaulent-Miodek hierarchy and its super Hamiltonian structures are constructed by means of a kind of Lie super algebras and super trace identity. Moreover, the self-consistent sources of the super Jaulent-Miodek hierarchy is presented based on the theory of self-consistent sources. Further- more, the infinite conservation laws of the super Jaulent-Miodek hierarchy are also obtained. It is worth noting that as even variables are boson variables, odd variables are fermi variables in the spectral problem, the commutator is different from the ordinary one.     

扩展的可积模型及其约化与达布变换

In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras?H and?T, respectively. By using the loop algebra?H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra?T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra?H to obtain an integrable hierarchy with variable coefficients.     

Binary Nonlinearization of the Nonlinear Schr?dinger Equation Under an Implicit Symmetry Constraint

By modifying the procedure of binary nonlinearization for the AKNS spectral problem and its adjoint spectral problem under an implicit symmetry constraint,we obtain a finite dimensional system from the Lax pair of the nonlinear Schr¨odinger equation.We show that this system is a completely integrable Hamiltonian system.     

THE CONFOCAL INVOLUTIVE SYSTEM AND THE INTEGRABILITY OF THE NONLINEARIZED LAX SYSTEMS FOR AKNS HIERARCHY

The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.     

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.     

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.     

Nonlinear continuous integrable Hamiltonian couplings

Based on a kind of special non-semisimple Lie algebras, a scheme is presented for constructing nonlinear continuous integrable couplings. Variational identities over the corresponding loop algebras are used to furnish Hamiltonian structures for the resulting continuous integrable couplings. The application of the scheme is illustrated by an example of nonlinear continuous integrable Hamiltonian couplings of the AKNS hierarchy of soliton equations.     

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.     

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.     

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.     

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.     

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.     

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.