A new integrable generalized Neumann system related to soliton hierarchy

In the present paper, the nonlinearization approach is applied to the soliton hierarchy associated with 3 × 3 matrix spectral problems. A new finite-dimensional integrable generalized C. Neumann system is obtained. The involutive system of conserved integrals is constructed by a direct method. Moreover the involutive solution of the soliton hierarchy is also given.     

The integrability and parametric representation of a hierarchy of soliton equations

In this paper after having obtained the Lax pair of a hierarchy of soliton equations,we discuss the parametric representation for finite-band solutions of the stationary solitonequation, and prove it can be represented as a Hamiltonian system which is integrable inLiouville sense. The nonconfocal involutive integral representations {Fm} are obtained also.In the condition of finite-band solutions of the soliton equation, the time and space can bedevided inio two Hamiltonian systems, so the fi…     

一族Liouville可积的有限维Hamilton系统

本文生成了一族Liouville可积的Hamilton相流彼此可交换的有限维Hamilton系统,并且给出了一串对合的显式公共运动积分及其一组对合的显式生成元.     

一个Bargmann系统与耦合Burgers族解的对合表示

本文给出耦合Burgers族的换位表示，并通过对耦合Burgers族Lax系统的非线性化得到一个Bargmann系统，证明该系统为Liouville完全可积的，还给出耦合Burgers族解的对合表示.     

Involutive Solutions and Commutator Representations of the Dirac Hierarchy

ConsidertheDiracspectralproblemwherep,qaretwopotentials,Aisaspectralparameter.L*isaninjectivehomomorphism.ThefunctionalgradientVA=(2RR,ri-of)TofeigenvalueAwithrespecttop,qsatisfiesarecalledtheLenard'soperatorpairof(1).Theorem1LetG(1)(x),G(z)(x)betwoarbitarysmoothfunctions,G=(G(1),G(2))".ThenthefollowingoperatorequationwithrespecttoV=V(G),possessestheoperatorsolutionwhereL.'-jisthecommutator;L=L(p,q),K,Jaredefinedby(1)l(4)respectively.ProofSubstitute(6)into(5),directlycalculate.Defin…     

On the complete integrability of nonlinearized lax systems for the classical Boussinesq hierarchy

TheN involutive integrals of motion with linearly independent gradients for the nonlinearized eigenvalue problem corresponding to the classical Boussinesq (CB) hierarchy are given. It is shown that whenn=1, 2, 3, the nonlinearized time parts of Lax systems for the CB hierarchy are transformed into three finite-dimensional integrable Hamiltonian systems under the constraint of the nonlinearized spatial part.     

Two types of hierarchies of evolution equations associated with the extended Kaup–Newell spectral problem with an arbitrary smooth function

In this paper we consider an extended Kaup–Newell (EKN) isospectral problem with an arbitrary smooth function and the corresponding two kinds of Lax integrable hierarchies by introducing two types of auxiliary spectral problems. The Hamiltonian structure of the second hierarchy is established. It is shown that the Hamiltonian system are integrable in Liouville’s sense and the set of Hamiltonian functions is the conserved densities of the second hierarchy, as well as they are in involutive in pairs under the Poisson bracket.     

Relation among nonlinear evolution equations and their reductions

The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.     

孤子族的生成及换位表示的一般结构

本文通过对谱问题ψｘ＝Ｕ（ｕ，λ）ψ的直接研究，利用谱梯度提供一条获得孤子方程族的途径，进一步，我们给出了孤子方程换位表示的一般结构，同时我们还将看到同一个谱问题可产生两族不同的孤子发展方程。     

Solving AKNS equations via Jacobi inversion problem

1.IntroductionRecelltlymuchworkhasbeencarriedoutinthestudyoftheseparationofvariablesofacompletelyintegrableHalniltoniansystemll--6].Forclassicalilltegrablesystemssubjecttoinversescatteringtransformationthestandardconstructionoftheaction-anglevariablesusingthepolesoftheBaker-Anheizerfullctionisequivalenttotheseparationofvariablesl31.Theabategapsolutionsofthesolitonequationsareconstructedduetotheseparationofvariablesofthestationarysolitonequationsll].Forsomekindoffinite-dimensionalintegrableHt…     

Integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of Lie algebras

Staring from a new spectral problem, a hierarchy of the soliton equations is derived. It is shown that the associated hierarchies are infinite-dimensional integrable Hamiltonian systems. By the procedure of nonlinearization of the Lax pairs, the integrable decomposition of the whole soliton hierarchy is given. Further, we construct two integrable coupling systems for the hierarchy by the conception of semidirect sums of Lie algebras.     

Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint

Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given.     

与薛丁谔型谱问题相联系的孤子族的分解

The soliton hierarchy associated with a Schrodinger type spectral problem with four potentials is decomposed into a class of new finite-dimensional Hamiltonian systems by using the nonlinearized approach.It is worth to point that the solutions for the soliton hierarchy are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.     

Nonlinear integrable couplings of a generalized super Ablowitz‐Kaup‐Newell‐Segur hierarchy and its super bi‐Hamiltonian structures

In this paper, a new generalized 5×5 matrix spectral problem of Ablowitz‐Kaup‐Newell‐Segur type associated with the enlarged matrix Lie superalgebra is proposed, and its corresponding super soliton hierarchy is established. The super variational identities are used to furnish super Hamiltonian structures for the resulting super soliton hierarchy.     

THE NEW INTEGRABLE SYSTEM OF THE COMPLEX FORM AND THECOUPLED MKDV HIERARCHY

In this paper, the new coupled MKdV hierarchy and their Lsx pairs are oinained,Through introducing a suitable complex form of symplectic structure [5,8], a new integraHe sys-tem of the complex form in the Liouville sense is generated. Moreover, the representations of the solution for the coupled MKdV hierarchy are given by the invohifive solutions of the commutable .     

Generalized Fractional Trace Variational Identity and A New Fractional Integrable Couplings of Soliton Hierarchy

Based on fractional isospectral problems and general bilinear forms, the generalized 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.     

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.     

A tri-Hamiltonian formulation of a new soliton hierarchy associated with so(3,R)

A third Hamiltonian operator is presented for a new hierarchy of bi-Hamiltonian soliton equations, thereby showing that this hierarchy is tri-Hamiltonian. Additionally, an inverse hierarchy of common commuting symmetries is also presented.     

一族Liouville可积系及其3-Hamilton结构

构造了loop代数A↑～1的一个高阶子代数,设计了一个新的Lax对,利用屠格式获得了含8个位势的孤立子方程族;利用Gauteax导数直接验证了所得3个辛算子的线性组合仍为辛算子．因此该孤立族具有3-Hamilton结构,具有无穷多个对合的公共守恒密度,故Liouville可积．作为约化情形,得到了2个可积系,其中之一是著名的AKNS方程族．