The hierarchy of an integrable coupling and its Hamiltonian structure

A type of higher dimensional loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic 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.     

An integrable hierarchy including the AKNS hierarchy and its strict version

We present an integrable hierarchy that includes both the AKNS hierarchy and its strict version. We split the loop space g of gl₂ into Lie subalgebras g≥0 and g<0 of all loops with respectively only positive and only strictly negative powers of the loop parameter. We choose a commutative Lie subalgebra C in the whole loop space s of sl₂ and represent it as C = C≥0⊕C<0. We deform the Lie subalgebras C≥0 and C<0 by the respective groups corresponding to g<0 and g≥0. Further, we require that the evolution equations of the deformed generators of C≥0 and C<0 have a Lax form determined by the original splitting. We prove that this system of Lax equations is compatible and that the equations are equivalent to a set of zero-curvature relations for the projections of certain products of generators. We also define suitable loop modules and a set of equations in these modules, called the linearization of the system, from which the Lax equations of the hierarchy can be obtained. We give a useful characterization of special elements occurring in the linearization, the so-called wave matrices. We propose a way to construct a rather wide class of solutions of the combined AKNS hierarchy.     

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.     

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.     

Integrable coupling of the Ablowitz–Ladik hierarchy and its Hamiltonian structure

The trace identity is generalized to work for the discrete zero-curvature equation associated with the Lie algebra possessing degenerate Killing forms. Then a kind of integrable coupling of the Ablowitz–Ladik (AL) hierarchy is obtained and its Hamiltonian structure is worked out. Moreover, Liouville integrability of the integrable coupling is demonstrated.     

The multi-component KdV hierarchy and its multi-component integrable coupling system

Construction of a type of simple loop algebra is devoted to establishing an isospectral problem. It follows that the multi-component KdV hierarchy of soliton equations is obtained. Further, an expanded loop algebra of the above algebra is presented, which is used to work out the multi-component integrable coupling system of the multi-component KdV 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.     

Non-symmetry constraints of the AKNS system yielding integrable Hamiltonian systems

This paper aims to show that there exist non-symmetry constraints which yield integrable Hamiltonian systems through nonlinearization of spectral problems of soliton systems, like symmetry constraints. Taking the AKNS spectral problem as an illustrative example, a class of such non-symmetry constraints is introduced for the AKNS system, along with two-dimensional integrable Hamiltonian systems generated from the AKNS spectral problem.     

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.     

A new multi-component matrix loop algebra and a multi-component integrable couplings of the NLS-MKdV hierarchy and its Hamiltonian structure

A new multi-component matrix loop algebra is constructed, which is devoted to establishing an isospectral problem. By making use of Tu scheme, the multi-component integrable couplings of the NLS-MKdV hierarchy is obtained, then the bi-Hamiltonian structure of the above system is given.     

Constructing super D-Kaup-Newell hierarchy and its nonlinear integrable coupling with self-consistent sources

How to construct new super integrable equation hierarchy is an important problem. In this paper, a new Lax pair is proposed and the super D-Kaup-Newell hierarchy is generated, then a nonlinear integrable coupling of the super D-Kaup-Newell hierarchy is constructed. The super Hamiltonian structures of coupling equation hierarchy is derived with the aid of the super variational identity. Finally, the self-consistent sources of super integrable coupling hierarchy is established. It is indicated that this method is a straight- forward and efficient way to construct the super integrable equation hierarchy.     

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.     

A generalized AKNS hierarchy and its bi-Hamiltonian structures

First we construct a new isospectral problem with 8 potentials in the present paper. And then a new Lax pair is presented. By making use of Tu scheme, a class of new soliton hierarchy of equations is derived, which is integrable in the sense of Liouville and possesses bi-Hamiltonian structures. After making some reductions, the well-known AKNS hierarchy and other hierarchies of evolution equations are obtained. Finally, in order to illustrate that soliton hierarchy obtained in the paper possesses bi-Hamiltonian structures exactly, we prove that the linear combination of two-Hamiltonian operators admitted are also a Hamiltonian operator constantly. We point out that two Hamiltonian operators obtained of the system are directly derived from a recurrence relations, not from a recurrence operator.     

A New 4 × 4 AKNS Spectral Problem and Its Associated Integrable Decomposition of the AKNS Equation

A new approach to construct a new 4×4 matrix spectral problem from a normal 2×2 matrix spectral problem is presented.AKNS spectral problem is discussed as an example.The isospectral evolution equation of the new 4×4 matrix spectral problem is nothing but the famous AKNS equation hierarchy.With the aid of the binary nonlino earization method,the authors get new integrable decompositions of the AKNS equation. In this process,the r-matrix is used to get the result.