Discrete Integrable Couplings Research Articles

Some novel soliton solution, breather solution and Darboux transformation for a generalized coupled Toda soliton hierarchy

A few of discrete integrable coupling systems(DICSs) of previous papers are linear discrete integrable couplings(LDICS). We take a special matrix Lie algebra system(non-semisimple) to construct the Lax pairs, and establish a method for deriving the nonlinear discrete integrable coupling systems(NDICS). From the Lax pairs of the generalized Toda(G-Toda) spectral problem, we can derive a novel NDICS, which is a real NDICS. For the obtained lattice integrable coupling equation, we establish a Darboux transformation (DT) with 4 × 4 Lax pairs, and apply the gauge transformation to a specific equation, then the explicit solutions of the lattice integrable coupling equation are given, which contains discrete soliton solution, breather solution and rogue wave solution. Furthermore, we can derive the discrete explicit solutions with free parameters to depict their dynamic behaviors.

Construction and separability of nonlinear soliton integrable couplings

The paper is motivated by recent works of several authors, initiated by articles of Ma and Zhu [W. X. Ma, Z. N. Zhu, Constructing nonlinear discrete integrable Hamiltonian couplings, Comput. Math. Appl. 60 (2010) 2601] and Ma [W. X. Ma, Nonlinear continuous integrable Hamiltonian couplings, Appl. Math. Comput. 217 (2011) 7238], where new class of soliton systems, being nonlinear integrable couplings, was introduced. Here, we present a general construction of such class of systems and we develop the decoupling procedure, separating them into copies of underlying original equations.

TWO HIERARCHIES OF NONLINEAR SOLITON EQUATIONS, NEW INTEGRABLE SYMPLECTIC MAP AND DISCRETE INTEGRABLE COUPLINGS

Two hierarchies of nonlinear soliton equations are derived from a discrete spectral problem. It is shown that the hierarchies are completely integrable Hamiltonian systems. Moreover, a new integrable symplectic map is obtained using the binary nonlinearization method. With the help of semi-direct sum of Lie algebra, discrete integrable couplings are constructed.

Constructing nonlinear discrete integrable Hamiltonian couplings

Beginning with Lax pairs from special non-semisimple matrix Lie algebras, we establish a scheme for constructing nonlinear discrete integrable couplings. Discrete variational identities over the associated loop algebras are used to build Hamiltonian structures for the resulting integrable couplings. We illustrate the application of the scheme by means of an enlarged Volterra spectral problem and present an example of nonlinear discrete integrable Hamiltonian couplings for the Volterra lattice equations.

INTEGRABLE COUPLING SYSTEMS OF HAMILTONIAN LATTICE EQUATIONS BY SEMI-DIRECT SUMS OF LIE ALGEBRAS

By considering a discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations are derived. The relation to the Toda type lattice is achieved by variable transformation. With the help of Tu scheme, the Hamiltonian structure of the resulting lattice hierarchy is constructed. The Liouville integrability is then demonstrated. Semi-direct sum of Lie algebras is proposed to construct discrete integrable couplings. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

A NEW INTEGRABLE LATTICE HIERARCHY AND ITS TWO DISCRETE INTEGRABLE COUPLINGS

Starting from a discrete isospectral problem, a hierarchy of nonlinear Liouville integrable lattice soliton equations are derived. Through enlarging associated spectral problems, two kinds of discrete integrable couplings are constructed for the resulting lattice hierarchy.

A Bi-Hamiltonian Lattice System of Rational Type and Its Discrete Integrable Couplings

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

Constructing New Discrete Integrable Coupling System for Soliton Equation by Kronecker Product

It is shown that the Kronecker product can be applied to constructing new discrete integrable coupling system of soliton equation hierarchy in this paper. A direct application to the fractional cubic Volterra lattice spectral problem leads to a novel integrable coupling system of soliton equation hierarchy. It is also indicated that the study of discrete integrable couplings by using the Kronecker product is an efficient and straightforward method. This method can be used generally.

The quadratic-form identity for constructing the Hamiltonian structures of the discrete integrable systems

The quadratic-form identity is extended to the discrete version which can be used to construct the Hamiltonian structures of the discrete integrable systems associated with the Lie algebra possessing degenerate Killing forms. Especially, it can be used to work out the Hamiltonian structures of some kinds of discrete integrable couplings. Then a kind of integrable coupling of the Toda hierarchy is obtained and its Hamiltonian structure is worked out by using the discrete quadratic-form identity. Moreover, the Liouville integrability of the integrable coupling is demonstrated.

A New Integrable Lattice Hierarchy and Its Two Discrete Integrable Couplings

A new and efficient way is presented for discrete integrable couplings with the help of two semi-direct sum Lie algebras. As its applications, two discrete integrable couplings associated with the lattice equation are worked out. The approach can be used to study other discrete integrable couplings of the discrete hierarchies of solition equations.

Constructing integrable coupling systems of Hamiltonian lattice equations using semi-direct sums of Lie algebras

In this article, by considering a discrete isospectral problem, a hierarchy of Hamiltonian lattice equations are derived. Two types of semi-direct sums of Lie algebras are proposed, using which a practicable way to construct discrete integrable couplings is introduced. As an application, two kinds of discrete integrable couplings of the resulting system are worked out.

Coupling commutator pairs and integrable systems

According to classification of the matrix Lie algebras, a type of explicit Lie algebras are constructed which can be decomposed into a few Lie subalgebras. These subalgebras constitute several coupling commutator pairs from which some continuous multi-integrable couplings could be generated if the proper isospectral Lax pairs could be set up. Then the above Lie algebras are again decomposed into a kind of Lie algebras which are also closed under the matrix multiplication. From such the Lie algebras, some discrete multi-integrable couplings could be worked out. Finally, a few examples are given. However, the Hamiltonian structures of the (continuous and discrete) integrable couplings obtained by the above Lie algebras cannot be computed by using the trace identity or the quadratic-form identity, which is a strange and interesting problem. The phenomenon indicates that the importance of the Lie-algebra classification. The problem also needs us to try to find an efficient scheme to deal with.

Discrete integrable couplings associated with modified Korteweg–de Vries lattice and two hierarchies of discrete soliton equations

A direct way to construct integrable couplings for discrete systems is presented by use of two semi-direct sum Lie algebras. As their applications, the discrete integrable couplings associated with modified Korteweg–de Vries (m-KdV) lattice and two hierarchies of discrete soliton equations are developed. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards the complete classification of integrable couplings.

Three semi-direct sum Lie algebras and three discrete integrable couplings associated with the modified K d V lattice equation

Three semi-direct sum Lie algebras are constructed, which is an efficient and new way to obtain discrete integrable couplings. As its applications, three discrete integrable couplings associated with the modified K d V lattice equation are worked out. The approach can be used to produce other discrete integrable couplings of the discrete hierarchies of soliton equations.

Discrete Integrable Couplings of the Volterra Lattice

A practicable way to construct discrete integrable couplings is proposedby making use of two types of semi-direct sum Lie algebras. As itsapplication, two kinds of discrete integrable couplings of the Volterralattice are worked out.

Discrete integrable couplings associated with Toda-type lattice and two hierarchies of discrete soliton equations

With the help of two semi-direct sum Lie algebras, an efficient way to construct discrete integrable couplings is proposed. As its applications, the discrete integrable couplings of the Toda-type lattice equations are obtained. The approach can be devoted to establishing other discrete integrable couplings of the discrete lattice integrable hierarchies of evolution equations.

Semidirect sums of Lie algebras and discrete integrable couplings

A relation between semidirect sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrable couplings is further proposed. An application of the resulting general theory to the generalized Toda spectral problem yields two classes of integrable couplings for the generalized Toda hierarchy of lattice equations. The construction of integrable couplings using semidirect sums of Lie algebras provides a good source of information on complete classification of integrable lattice equations.