Bi-integrable Couplings Research Articles

Bi-Integrable Couplings Associated with so $$(3,{\mathbb R})$$ ( 3 , R )

By a class of zero curvature equations over a non-semisimple matrix loop algebra, we generate a new hierarchy of bi-integrable couplings for a soliton hierarchy associated with so(3, $$\mathbb {R}$$ ). The bi-Hamiltonian structures are found by the associated variational identity, which imply that all the presented coupling systems possess infinitely many commuting symmetries and conserved functionals and, thus, are Liouville integrable.

Bi-integrable and tri-integrable couplings of a soliton hierarchy associated with SO(4)

Abstract In our paper, the theory of bi-integrable and tri-integrable couplings is generalized to the discrete case. First, based on the six-dimensional real special orthogonal Lie algebra SO(4), we construct bi-integrable and tri-integrable couplings associated with SO(4) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by the variational identities.

Bi-Integrable and Tri-Integrable Couplings of a Soliton Hierarchy Associated with SO(3)

Based on the three-dimensional real special orthogonal Lie algebra SO(3), by zero curvature equation, we present bi-integrable and tri-integrable couplings associated with SO(3) for a hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Moreover, Hamiltonian structures of the obtained bi-integrable and tri-integrable couplings are constructed by applying the variational identities.

The Bi-Integrable Couplings of Two-Component Casimir-Qiao-Liu Type Hierarchy and Their Hamiltonian Structures

A new type of two-component Casimir-Qiao-Liu type hierarchy (2-CQLTH) is produced from a new spectral problem and their bi-Hamiltonian structures are constructed. Particularly, a new completely integrable two-component Casimir-Qiao-Liu type equation (2-CQLTE) is presented. Furthermore, based on the semidirect sums of matrix Lie algebras consisting of3×3block matrix Lie algebra, the bi-integrable couplings of the 2-CQLTH are constructed and their bi-Hamiltonian structures are furnished.

Bi-integrable couplings and tri-integrable couplings of the modified Ablowitz-Kaup-Newell-Segur hierarchy with self-consistent sources

We discuss the bi-integrable couplings and tri-integrable couplings of the modified Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy with self-consistent sources from the enlarged matrix spectral problem and the enlarged zero curvature equations. By using the variational identities over non-semisimple Lie algebras consisting of block matrices, we construct Hamiltonian formulations for integrable couplings, bi-integrable couplings, and tri-integrable couplings.

Bi-Integrable Couplings of a New Soliton Hierarchy Associated withSO(4)

Based on the six-dimensional real special orthogonal Lie algebraSO(4), a new Lax integrable hierarchy is obtained by constructing an isospectral problem. Furthermore, we construct bi-integrable couplings for this hierarchy from the enlarged matrix spectral problems and the enlarged zero curvature equations. Hamiltonian structures of the obtained bi-integrable couplings are constructed by the variational identity.

Bi-integrable couplings of a Kaup–Newell type soliton hierarchy and their bi-Hamiltonian structures

A new Kaup–Newell type soliton hierarchy is generated from an asymmetric matrix spectral problem associated with the three-dimensional special linear Lie algebra sl(2,R). Then based on semi-direct sums of matrix Lie algebras consisting of 3×3 block matrix Lie algebras, corresponding bi-integrable couplings of this hierarchy are constructed. Each equation in the resulting system has a bi-Hamiltonian structure furnished by the variational identity, which lead to Liouville integrability.

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.

Bi-integrable couplings of a new soliton hierarchy associated with a non-semisimple Lie algebra

We construct bi-integrable couplings for a new soliton hierarchy from zero curvature equations associated with a non-semisimple loop algebra. We demonstrate the Liouville integrability of the hierarchy by showing that the bi-integrable couplings generated are bi-Hamiltonian.

An so (3, ℝ) Counterpart of the Dirac Soliton Hierarchy and its Bi-Integrable Couplings

We derive a counterpart hierarchy of the Dirac soliton hierarchy from zero curvature equations associated with a matrix spectral problem from so (3, ℝ). Inspired by a special class of non-semisimple loop algebras, we construct a hierarchy of bi-integrable couplings for the counterpart soliton hierarchy. By applying the variational identities which cope with the enlarged Lax pairs, we generate the corresponding Hamiltonian structure for the hierarchy of the resulting bi-integrable couplings. To show Liouville integrability, infinitely many commuting symmetries and conserved densities are presented for the counterpart soliton hierarchy and its hierarchy of bi-integrable couplings.

Nonlinear bi-integrable couplings with Hamiltonian structures

Bi-integrable couplings of soliton equations are presented through introducing non-semisimple matrix Lie algebras on which there exist non-degenerate, symmetric and ad-invariant bilinear forms. The corresponding variational identity yields Hamiltonian structures of the resulting bi-integrable couplings. An application to the AKNS spectral problem gives bi-integrable couplings with Hamiltonian structures for the AKNS equations.

Bi-Integrable Couplings of a Nonsemisimple Lie Algebra by Toda Lattice Hierarchy

W. X. Ma established theory of bi-integrable couplings to construct Hamilton structure of continuous bi-integrable couplings. In our paper, the theory of bi-integrable couplings is generalized to the discrete case. First, based on semi-direct sums of Lie subalgebra , a class of higher-dimensional 6×6 matrix Lie algebras is constructed. Moreover, starting from a new 6-order matrix spectral problem with a parameter, the bi-integrable couplings of the Toda lattice hierarchy was obtained from the proposed nonsemisimple higher-dimensional Lie algebras. Finally, the obtained discrete bi-integrable coupling systems are all written into their bi-Hamiltonian forms by the discrete variational identity.

Non-Semisimple Lie Algebras of Block Matrices and Applications to Bi-Integrable Couplings

AbstractWe propose a class of non-semisimple matrix loop algebras consisting of 3 × 3 block matrices, and form zero curvature equations from the presented loop algebras to generate bi-integrable couplings. Applications are made for the AKNS soliton hierarchy and Hamiltonian structures of the resulting integrable couplings are constructed by using the associated variational identities.

A Block Matrix Loop Algebra and Bi-Integrable Couplings of the Dirac Equations

AbstractA non-semisimple matrix loop algebra is presented, and a class of zero curvature equations over this loop algebra is used to generate bi-integrable couplings. An illustrative example is made for the Dirac soliton hierarchy. Associated variational identities yield bi-Hamiltonian structures of the resulting bi-integrable couplings, such that the hierarchy of bi-integrable couplings possesses infinitely many commuting symmetries and conserved functionals.

An integrable system and associated integrable models as well as Hamiltonian structures

Starting from an existed Lie algebra introduces a new Lie algebra A1 = {e1, e2, e3} so that two isospectral Lax matrices are established. By employing the Tu scheme an integrable equation hierarchy denoted by IEH is obtained from which a few reduced evolution equations are presented. One of them is the mKdV equation. The elliptic variable solutions and three kinds of Darboux transformations for one coupled equation which is from the IEH are worked out, respectively. Finally, we take use of the Lie algebra A1 to generate eight higher-dimensional Lie algebras from which the linear integrable couplings, the nonlinear integrable couplings, and the bi-integrable couplings of the IEH are engendered, whose Hamiltonian structures are also obtained by the variational identity. Then further reduce one coupled integrable equation to get a nonlinear generalized mKdV equation.

Loop algebras and bi-integrable couplings

A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations. The variational identities under non-degenerate, symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings. A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.

Nonlinear Bi-Integrable Couplings of Multicomponent Guo Hierarchy with Self-Consistent Sources

Based on a well-known Lie algebra, the multicomponent Guo hierarchy with self-consistent sources is proposed. With the help of a set of non-semisimple Lie algebra, the nonlinear bi-integrable couplings of the multicomponent Guo hierarchy with self-consistent sources are obtained. It enriches the content of the integrable couplings of hierarchies with self-consistent sources. Finally, the Hamiltonian structures are worked out by employing the variational identity.