A discrete iso-spectral problem with positive and negative hierarchies and integrable coupling system

Two hierarchies of integrable positive and negative nonlinear lattice systems are derived from a discrete iso-spectral problem. When the Lax operators are expanded by virtue of the positive and negative power expansion with respect to the spectral parameter, we get the corresponding integrable hierarchies. Moreover, a direct matrix spectral method is used to get the associated integrable coupling system of the first resulting hierarchy.     

Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation

Two hierarchies of integrable positive and negative lattice equations in connection with a new discrete isospectral problem are derived. It is shown that they correspond to positive and negative power expansions respectively of Lax operators with respect to the spectral parameter, and each equation in the resulting hierarchies is Liouville integrable. Moreover, infinitely many conservation laws of corresponding positive lattice equations are obtained in a direct way. Finally, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations, by means of which the exact solutions are given.     

An integrable lattice hierarchy, associated integrable coupling, Darboux transformation and conservation laws

A new integrable lattice hierarchy is constructed from a discrete matrix spectral problem, some related properties of the new hierarchy are discussed. The Hamiltonian structures and Liouville integrability of the new hierarchy are established by using the discrete trace identity. A kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems. A Darboux transformation (DT) with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its Lax representation, the explicit solutions are obtained via the DT, the structures for those solutions are graphically investigated. All these properties might be helpful to understanding some physical phenomena.     

Two families of Liouville integrable lattice equations

By virtue of zero curvature representations, we are successful to generate the Lax representations of two hierarchies of discrete lattice equations respectively, which are derived from two new and interesting 3 × 3 matrix spectral problems. Moreover, by using the trace identity, the bi-Hamiltonian structures of the above systems are given, and it is shown that they are integrable in the Liouville sense. Finally, infinitely many conservation laws for the second hierarchy of lattice equations are given by a direct method.     

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.     

A hierarchy of Liouville integrable lattice equations and its integrable coupling systems

A new discrete two-by-two matrix spectral problem with two potentials is introduced, followed by a hierarchy of integrable lattice equations obtained through discrete zero curvature equations. It is shown that the Hamiltonian structures of the resulting integrable lattice equations are established by virtue of the trace identity. Furthermore, based on a discrete four-by-four matrix spectral problem, the discrete integrable coupling systems of the resulting hierarchy are obtained. Then, with the variational identity, the Hamiltonian structures of the obtained integrable coupling systems are established. Finally, the resulting Hamiltonian systems are proved to be all Liouville integrable.     

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.     

AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS

Within the zero curvature formulation, a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type. The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity. When the involved two potential vectors are scalar, all the resulting integrable lattice equations are reduced to the standard AblowitzLadik hierarchy.     

Hierarchies,integrable decompositions and conservation laws of Geng equation

In this paper, two hierarchies of the Geng equations are presented, including positive non-isospectral hierarchy and negative non-isospectral hierarchy. Moreover, integrable couplings of the corresponding Geng hierarchies are also constructed by enlarging the associated matrix spectral problem. Three new integrable decompositions and conservation laws of the isospectral Geng equation are also obtained. The Gauge transformations are used to obtain the associated binary symmetry constraints of the Geng equation at the first time.     

New integrable lattice hierarchies and associated properties

New positive hierarchy and negative hierarchy of Liouville integrable lattice equation and their Hamiltonian structure are generated by use of Tu model. Then, some properties of the obtained equation hierarchies are discussed.     

The Liouville integrability of integrable couplings of Volterra lattice equation

A hierarchy of integrable couplings of Volterra lattice equations with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebra, the two Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.     

离散晶格的正族、负族及其Hamilton结构

构造了一个新的等谱问题，利用相容性条件，推导出离散晶格方程的正族和负族。再利用迹恒等式，建立其Hamilton 结构。获得的离散方程族的达布变换、双线性化、对称、守恒率及其精确解也值得进一步研究。     

Outer automorphisms ofsl(2), integrable systems, and mappings

Outer automorphisms of infinite-dimensional representations of the Lie algebra sl(2) are used to construct Lax matrices for integrable Hamiltonian systems and discrete integrable mappings. The known results are reproduced, and new integrable systems are constructed. Classical r-matrices, corresponding to the Lax representation with the spectral parameter are dynamic. This scheme is advantageous because quantum systems naturally arise in the framework of the classical r-matrix Lax representation and the corresponding quantum mechanical problem admits a variable separation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 2, pp. 205–216, February, 1999.     

Invariant manifolds and Lax pairs for integrable nonlinear chains

We continue the previously started study of the development of a direct method for constructing the Lax pair for a given integrable equation. This approach does not require any addition assumptions about the properties of the equation. As one equation of the Lax pair, we take the linearization of the considered nonlinear equation, and the second equation of the pair is related to its generalized invariant manifold. The problem of seeking the second equation reduces to simple but rather cumbersome calculations and, as examples show, is effectively solvable. It is remarkable that the second equation of this pair allows easily finding a recursion operator describing the hierarchy of higher symmetries of the equation. At first glance, the Lax pairs thus obtained differ from usual ones in having a higher order or a higher matrix dimensionality. We show with examples that they reduce to the usual pairs by reducing their order. As an example, we consider an integrable double discrete system of exponential type and its higher symmetry for which we give the Lax pair and construct the conservation laws.     

A unified expressing model of the AKNS hierarchy and the KN hierarchy, as well as its integrable coupling system

A new subalgebra of loop algebra Ã₁ is first constructed. Then a new Lax pair is presented, whose compatibility gives rise to a new Liouville integrable system(called a major result), possessing bi-Hamiltonian structures. It is remarkable that two symplectic operators obtained in this paper are directly constructed in terms of the recurrence relations. As reduction cases of the new integrable system obtained, the famous AKNS hierarchy and the KN hierarchy are obtained, respectively. Second, we prove a conjugate operator of a recurrence operator is a hereditary symmetry. Finally, we construct a high dimension loop algebra to obtain an integrable coupling system of the major result by making use of Tu scheme. In addition, we find the major result obtained is a unified expressing integrable model of both the AKNS and KN hierarchies, of course, we may also regard the major result as an expanding integrable model of the AKNS and KN hierarchies. Thus, we succeed to find an example of expanding integrable models being Liouville integrable.     

The generalized Kupershmidt deformation for constructing new discrete integrable systems

It is known that the KdV6 equation can be represented as the Kupershmidt deformation of the KdV equation. We propose a generalized Kupershmidt deformation for constructing new discrete integrable systems starting from the bi-Hamiltonian structure of a discrete integrable system. We consider the Toda, Kac-van Moerbeke, and Ablowitz-Ladik hierarchies and obtain Lax representations for these new deformed systems. The generalized Kupershmidt deformation provides a new way to construct discrete integrable systems.     

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…     

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.     

The integrable coupling system of a 3 × 3 discrete matrix spectral problem

Integrable coupling with six potentials is first proposed by coupling a given 3 × 3 discrete matrix spectral problem. It is shown that coupled system of integrable equations can possess zero curvature representations and recursion operators, which yield infinitely many commuting symmetries. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebras, the Hamiltonian form is deduced for the lattice equations in the resulting hierarchy. Finally, we prove that the hierarchy of the resulting Hamiltonian equations is Liouville integrable discrete Hamiltonian system.