Integrable Coupling System Research Articles

An integrable Hamiltonian hierarchy and associated integrable couplings system

This paper establishes a new isospectral problem. By making use of the Tu scheme, a new integrable system is obtained. It gives integrable couplings of the system obtained. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.

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.

Group-invariant solutions of a integrable coupled system

In this paper, a new idea is put forward to modify the Clarkson–Kruskal's (CK's) direct method. By using the classical Lie group approach and the modified the CK's direct method, symmetry reductions and exact solutions are discussed for a integrable coupled KdV system. The group explanation for all the results obtained by the modified direct method is also given.

Hamiltonian systems and its integrable coupling associated with a new discrete spectral problem

Starting from a new discrete spectral problem, the corresponding positive and negative hierarchies of nonlinear lattice equations are proposed. It is shown that the both lattice soliton hierarchies possess the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, we construct two integrable coupling systems for the positive hierarchy through enlarging Lax pair method.

A Hierarchy of Differential-Difference Equations and TheirIntegrable Couplings

Starting from a discrete spectral problem, the correspondinghierarchy of nonlinear differential-difference equation is proposed.It is shown that the hierarchy possesses the bi-Hamiltionianstructures. Further, two integrable coupling systems for the hierarchyare constructed through enlarged Lax pair method.

The applications of a higher-dimensional Lie algebra and its decomposed subalgebras

With the help of invertible linear transformations and the known Lie algebras, a higher-dimensional 6 × 6 matrix Lie algebra sμ(6) is constructed. It follows a type of new loop algebra is presented. By using a (2 + 1)-dimensional partial-differential equation hierarchy we obtain the integrable coupling of the (2 + 1)-dimensional KN integrable hierarchy, then its corresponding Hamiltonian structure is worked out by employing the quadratic-form identity. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra sμ(6), then a discrete lattice integrable coupling system is produced. A remarkable feature of the Lie algebras sμ(6) and E is used to directly construct integrable couplings.

A new multi-component discrete integrable hierarchy and multi-component cubic Volterra integrable hierarchy as well as their coupling systems

By using a new algebraic system G M , a new multi-component discrete integrable hierarchy and multi-component cubic Volterra integrable hierarchy are obtained. Moreover, an new extended algebraic system G ∼ M of G M is presented,from which the integrable couplings system of the multi-component lattice hierarchy are obtained.

A new fractional order soliton equation hierarchy and its integrable coupling system

A fractional order soliton equation hierarchy is presented, which is a new Liouville integrable hierarchy and possesses bi-Hamiltonian structure. Finally, we obtain a integrable coupling system of this fractional order soliton equation hierarchy by applying a upper triangular matrix of Lie algebras.

(2+1)-DIMENSIONAL TD HIERARCHY AND ITS INTEGRABLE COUPLINGS

Under the frame of the (2 + 1)-dimensional zero curvature equation and Tu model, the (2 + 1)-dimensional TD hierarchy is obtained. Again, by using the expanding loop algebra, the integrable coupling system of the above hierarchy is given.

The Liouville integrable coupling system of the m-AKNS hierarchy and its Hamiltonian structure

In this paper a type of 9-dimensional vector loop algebra is constructed, which is devoted to establish an isospectral problem. It follows that a Liouville integrable coupling system of the m-AKNS hierarchy is obtained by employing the Tu scheme, whose Hamiltonian structure is worked out by making use of constructed quadratic identity. The method given in the paper can be used to obtain many other integrable couplings and their Hamiltonian structures.

Multi-component Dirac equation hierarchy and its multi-component integrable couplings system

A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M-dimensional loop algebra is produced. By taking advantage of , a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra M of the loop algebra is presented. Based on the M, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.

A new multi-component hierarchy and its integrable expanding model

A set of multi-component matrix Lie algebra is constructed, it follows that a type of new loop algebra is presented and multi-component integrable hierarchy is obtained. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system is worked out. As reduction of the hierarchy, some well-known hierarchy such as DNLS, KN, CLL hierarchy are established.

On the integrable discrete versions of the Leznov lattice: Determinant solutions and pfaffianization

In this paper, we present the Casorati form of the N-soliton solution for an integrable fully-discrete version and two integrable semi-discrete versions of the Leznov lattice, which arise from the integrable discretization of the two-dimensional Leznov lattice. By using the pfaffianization procedure of Hirota and Ohta, a new integrable coupled system is generated from the semi-discrete version of the Leznov lattice in the y-direction.

(2+1)-Dimensional Dirac hierarchy and its integrable couplings as well as multi-component integrable system

Abstract Under the frame of the (2 + 1)-dimensional zero curvature equation and Tu model, (2 + 1)-dimensional Dirac hierarchy is obtained. Again by use of the expanding loop algebra the integrable coupling system of the above hierarchy is given.

Constructing a multi-function equation hierarchy and its integrable coupling system with MAPLE

In this article by considering a multi-dimensional Lie algebra system, a new multi-function integrable equation hierarchy is presented with the help of MAPLE. Then the integrable coupling system of the new multi-function integrable equation hierarchy is obtained through enlarging matrix spectral problem.

On the [formula omitted]-dimensional Gardner equation: Determinant solutions and pfaffianization

In this paper, we first present the Casorati and Grammian determinant solutions to the ( 2 + 1 ) -dimensional Gardner equation. Then, by using the pfaffianization procedure of Hirota and Ohta, a new integrable coupled system is generated from the Gardner equation. Moreover, Gramm-type pfaffian solutions to the pfaffianized system are proposed.

Two Kinds of New Lie Algebras for Producing IntegrableCouplings

Two types of Lie algebras are constructed, which are directly usedto deduce the two resulting integrable coupling systems withmulti-component potential functions. Many other integrablecouplings of the known integrable systems may be obtained by the approach.

A higher-dimensional Lie algebra and its decomposed subalgebras

A higher-dimensional 4×4 matrix Lie algebra G is presented, which is devoted to setting up an isospectral Lax pair whose compatibility generates an integrable coupling system. By employing the quadratic-form identity, its Hamiltonian structure is obtained. Furthermore, a higher-dimensional Lie algebra denoted by E, is given by decomposing the Lie algebra G. If E is decomposed into two subalgebras E1 and E2 and we require the closure between E1 and E2 under the matrix multiplication, that is, E1E2, E2E1⊂E2, then a discrete lattice integrable coupling system is worked out. A remarkable feature of the Lie algebras G and E is used to directly construct integrable couplings.

Reduced equations of the self-dual Yang–Mills equations and applications

A few reduced equations from the self-dual Yang–Mills equations are presented, which are seemly extended zero curvature equations. As their applications, a good many nonlinear evolution equations could be obtained. In this paper, a (2 + 1)-dimensional AKNS hierarchy of soliton equations is generated from one of reduced equations of the self-dual Yang–Mills equations. With the help of a proper loop algebra, the Hamiltonian structure of its expanding integrable model (actually, its integrable couplings) is worked out by using the quadratic-form identity, which is Liouville integrable. The way presented in the paper has extensive applications. That is to say, making use of the approach specially a few higher-dimensional zero curvature equations in the paper could produce a lots of higher-dimensional integrable coupling systems and their corresponding Hamiltonian structures.

A Method for Obtaining Integrable Couplings

By making use of the vector product in R3, a commutingoperation is introduced so that R3 becomes a Lie algebra. Theresulting loop algebra R̃3 is presented, from which thewell-known AKNS hierarchy is produced. Again via applying thesuperposition of the commuting operations of the Lie algebra, acommuting operation in R6 is constructed so that R6 becomes aLie algebra. Thanks to the corresponding loop algebra R̃3of the Lie algebra R3, the integrable coupling of the AKNSsystem is obtained. The method presented in this paper is rathersimple and can be used to work out integrable coupling systems ofthe other known integrable hierarchies of soliton equations.