Explicit solution and Darboux transformation for a new discrete integrable soliton hierarchy with 4×4 Lax pairs

The Darboux transformation method with 4×4 spectral problem has more complexity than 2×2 and 3×3 spectral problems. In this paper, we start from a new discrete spectral problem with a 4×4 Lax pairs and construct a lattice hierarchy by properly choosing an auxiliary spectral problem, which can be reduced to a new discrete soliton hierarchy. For the obtained lattice integrable coupling equation, we establish a Darboux transformation and apply the gauge transformation to a specific equation and then the explicit solutions of the lattice integrable coupling equation are obtained. Copyright © 2017 John Wiley & Sons, Ltd.     

Noncommutative integrable coupling system of TC equation hierarchy

A noncommutative version of the TC soliton equation hierarchy is presented, which possesses the zero curvature representation. Then, we show that noncommutative (NC) TC equation can be derived from the noncommutative (anti-)self-dual Yang-Mills equation by reduction. Finally, an integrable coupling system of the NC TC equation hierarchy is constructed by using of the enlarged Lax pairs.     

RESTRICTED FLOWS OF A HIERARCHYOF INTEGRABLE DISCRETE SYSTEMS

1.IntroductionTherestrictedflowsofsolitonhierarchyhavebeenextensivelystudied(see,forexample,[1--7]).Theapproachforconstructingrestrictedflowsofsolitonhierarchycanalsobeappliedtoobtainrestrictedflows(discretemaps)ofahierarchyofdiscreteintegrablesystems(nonlineardifferential-differenceequations)IS,9].TheserestrictedflowshavetheformofLagrangeequationsandthereforecanmodelphysicallyinterestingprocesses.Wesupposethatthehierarchyofdiscreteintegrablesystems(DIS)isassociatedwithadiscreteisospectralp…     

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.     

Global properties of integrable Hamiltonian systems

This paper deals with Lagrangian bundles which are symplectic torus bundles that occur in integrable Hamiltonian systems. We review the theory of obstructions to triviality, in particular monodromy, as well as the ensuing classification problems which involve the Chern and Lagrange class. Our approach, which uses simple ideas from differential geometry and algebraic topology, reveals the fundamental role of the integer affine structure on the base space of these bundles. We provide a geometric proof of the classification of Lagrangian bundles with fixed integer affine structure by their Lagrange class.      

ON GENERATING EQUATIONS FOR THE KAUP-NEWELL HIERARCHY

It is shown that the Kanp-Newell hierarchy can be derived from the so-called gen- erating equations which are Lax integrable.Positive and negative flows in the hierarchy are derived simultaneously.The generating equations and mutual commutativity of these flows en- able us to construct new Lax integrable equations.     

Nonlinearization of the Lax pairs for discrete Ablowitz-Ladik hierarchy

The discrete Ablowitz-Ladik hierarchy with four potentials and the Hamiltonian structures are derived. Under a constraint between the potentials and eigenfunctions, the nonlinearization of the Lax pairs associated with the discrete Ablowitz-Ladik hierarchy leads to a new symplectic map and a class of finite-dimensional Hamiltonian systems. The generating function of the integrals of motion is presented, by which the symplectic map and these finite-dimensional Hamiltonian systems are further proved to be completely integrable in the Liouville sense. Each member in the discrete Ablowitz-Ladik hierarchy is decomposed into a Hamiltonian system of ordinary differential equations plus the discrete flow generated by the symplectic map.     

A HIERARCHY OF INTEGRABLE LATTICE SOLITON EQUMIONS AND ITS INTEGRABLE SYMPLECTIC MAP

1TheLatticeSolitonHierarchyDenoteEf(n)-f(n 1),nEZ,andwritef(n)=f,f(n k)=E*f.Considerthediscrete3x3matrixspectralproblem:whereajb,carethreepotentials,Aisaconstantspectralparameter.Theadjointrepresentationequationof(1)iswhereeachentryVj=Kj(B(A),C(A),D(A))0fthe3x3matrixVisaLaurentexpansionofA:Then(2)isequivaentt0'therecursi0nrelations:,Theaboverecursionequationscanbesolvedsuccessivelyt0deducethefollowingresults:Wedefine{Dj}bythefollowingrelation:From(3)we0btainKGj-1=JGj,Gj=(Aj,Bj,Cj…     

一族新的可积系及其双约束流的Hamilton正则表示

基于一个新的等谱问题，按屠格式导出了一族新的可积系，具有双Hamilton结构，通过建立双对称约束，得到了该方程族的两组约束流，并将其化为正则的Hamilton系统。     

A generalization of the coupled integrable dispersionless equations

The paper investigates an extension of the coupled integrable dispersionless equations, which describe the current‐fed string within an external magnetic field. By using the relation among the coupled integrable dispersionless equations, the sine‐Gordon equation and the two‐dimensional Toda lattice equation, we propose a generalized coupled integrable dispersionless system. N‐soliton solutions to the generalized system are presented in the Casorati determinant form with arbitrary parameters. By choosing real or complex parameters in the Casorati determinant, the properties of one‐soliton and two‐soliton solutions are investigated. It is shown that we can obtain solutions in soliton profile and breather profile. Copyright © 2016 John Wiley & Sons, Ltd.     

一个新的Liouville可积系统及其Lax表示，Bi—Hamilton结构

从一个特征值问题出发,首先推导一族非线性发展方程,其中包括著名MKdV方程做为特殊约化,进一步证明这族方程在Liounille意义下可积并具有Bi-Hamilton结构,而在位执函数和特征函数之间的一定约束下,特征值问题被非线性化为一完全可积的有限维Hamilton系统。     

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.     

Permutations of tori in integrable Hamiltonian systems and spectral series of pseudodifferential operators

In the present paper, we study integrable Hamiltonian systems with two degrees of freedom, whose regular level sets consist of several Liouville tori, and the bifurcation diagram has an isolated point. We study assumptions under which going around the singular point causes a permutation of the tori. We also consider a quantum analog of this situation and give model examples.     

Lax pairs for higher-dimensional evolution PDEs and a 3+1 dimensional integrable generalization of the Burgers equation

We construct Lax pairs for general dimensional evolution equations in the form , where depends on the field and its space derivatives. As an example we study a dimensional integrable generalization of the Burgers equation. We develop a procedure to generate some exact solutions of this equation, based on a class of discrete symmetries of the Darboux transformation type. In the one-dimensional limit, these symmetries reduce to the Cole-Hopf substitution for the Burgers equation. It is discussed how the technique can be used to construct exact solutions for higher-dimensional evolution PDEs in a broader context.

     

一个新的离散的演化方程族及其Hamilton结构

A new discrete isospectral problem is introduced, from which a hierarchy of Laxintegrable lattice equation is deduced. By using the trace identity, the correspondingHamiltonian structure is given and its Liouville integrability is proved.