Quadratic-form Identity Research Articles

The coupling integrable couplings of the generalized coupled Burgers equation hierarchy and its Hamiltonian structure

In this paper, we mainly give the Lie algebras E, F and H of three kinds and their commutator, respectively. Next, we establish three isospectral problems with the help of their corresponding loop algebras Ẽ, F̃, and H̃. Based on on the Tu scheme, coupling integrable couplings of three kinds for the generalized coupled Burgers equation hierarchy are obtained. Finally, we obtain the Hamiltonian structure of one of the coupling integrable couplings of the generalized coupled Burgers equation hierarchy by using the quadratic-form identity.

Hamiltonian forms of the two new integrable systems and two kinds of Darboux transformations

Based on the Lie algebra gl(2), taking a kind of corresponding loop algebra gl(2)∼, a new Lax integrable hierarchy can be obtained. Then, by means of the quadratic-form identity, the corresponding bi-Hamiltonian structure was worked out. Expanding Lie algebra gl(2), and making use of the new zero curvature equation Zhang (2008) [9], we obtain an integrable hierarchy and its Hamiltonian structure. At last, two kinds of Darboux transformations of the equation are generated.

The Semidirect Sum of Lie Algebras and Its Applications to C-KdV Hierarchy

By use of the loop algebraG-~, integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by Tu scheme and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.

A Hierarchy of Discrete Integrable Coupling System with Self-Consistent Sources

Integrable coupling system of a lattice soliton equation hierarchy is deduced. The Hamiltonian structure of the integrable coupling is constructed by using the discrete quadratic-form identity. The Liouville integrability of the integrable coupling is demonstrated. Finally, the discrete integrable coupling system with self-consistent sources is deduced.

The fractional quadratic-form identity and Hamiltonian structure of an integrable coupling of the fractional Ablowitz-Kaup-Newell-Segur hierarchy

Starting from a general isospectral problem of fractional order, we propose a fractional quadratic-form identity, from which the Hamiltonian structure of an integrable coupling of the fractional Ablowitz-Kaup-Newell-Segur hierarchy is derived. The method can be generalized to other fractional integrable couplings.

The Fractional Quadratic-Form Identity and Hamiltonian Structure of an Integrable Coupling of the Fractional Broer-Kaup Hierarchy

A fractional quadratic-form identity is derived from a general isospectral problem of fractional order, which is devoted to constructing the Hamiltonian structure of an integrable coupling of the fractional BK hierarchy. The method can be generalized to other fractional integrable couplings.

A Few Notes on Lax Integrability, Integrable Couplings and Computing Formula of the Constant γ

We point out in the paper that the current available definition on Lax integrability is necessary to be modified; the existed method for generating integrable couplings by the approach “original equations + symmetric equations” is wrong. Besides, a simple and efficient formula for calculating the constant γ appearing in the trace identity and the quadratic-form identity is proposed, which is universal for finding Hamiltonian structures of integrable dynamics.

New Integrable Couplings of Generalized Kaup—Newell Hierarchy and Its Hamiltonian Structures

A new isospectral problem is firstly presented, then we derive integrable system of soliton hierarchy. Also we obtain new integrable couplings of the generalized Kaup—Newell soliton equations hierarchy and its Hamiltonian structures by using Tu scheme and the quadratic-form identity. The method can be generalized to other soliton hierarchy.

A few Lie algebras and their applications for generating integrable hierarchies of evolution types

A Lie algebra consisting of 3 × 3 matrices is introduced, whose induced Lie algebra by using an inverted linear transformation is obtained as well. As for application examples, we obtain a unified integrable model of the integrable couplings of the AKNS hierarchy, the D-AKNS hierarchy and the TD hierarchy as well as their induced integrable hierarchies. These integrable couplings are different from those results obtained before. However, the Hamiltonian structures of the integrable couplings cannot be obtained by using the quadratic-form identity or the variational identity. For solving the problem, we construct a higher-dimensional subalgebra R and its reduced algebra Q of the Lie algebra A 2 by decomposing the induced Lie algebra and then again making some linear combinations. The subalgebras of the Lie algebras R and Q do not satisfy the relation ( G = G 1 ⊕ G 2 , [ G 1 , G 2 ] ⊂ G 2 ), but we can deduce integrable couplings, which indicates that the above condition is not necessary to generate integrable couplings. As for application example, an expanding integrable model of the AKNS hierarchy is obtained whose Hamiltonian structure is generated by the trace identity. Finally, we give another Lie algebras which can be decomposed into two simple Lie subalgebras for which a nonlinear integrable coupling of the classical Boussinesq–Burgers (CBB) hierarchy is obtained.

A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras G̃1 and G̃2 are constructed. Basing on G̃1 and G̃2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.

Coupling integrable couplings and bi-Hamiltonian structure associated with the Boiti–Pempinelli–Tu hierarchy

A kind of new Lie algebra is introduced whose corresponding loop algebra is followed to define. From which two isospectral problems are established whose compatibility condition gives rise to a coupling integrable coupling of the Boiti–Pempinelli–Tu (BPT) hierarchy. By making use of the quadratic-form identity, its bi-Hamiltonian structure is obtained. Finally, we reduce the coupling integrable coupling to obtain two integrable couplings of the BPT hierarchy. Furthermore, the well-known mKdV equation and a new coupled equation with two dependent variables are produced, respectively.

INTEGRABLE HAMILTONIAN HIERARCHIES ASSOCIATED WITH THE EQUATION OF HEAT CONDUCTION

Using a 4-dimensional Lie algebra g, an isospectral Lax pair is introduced, whose compatibility condition is equivalent to a soliton hierarchy of evolution equations with three components of potential functions. Its Hamiltonian structure is obtained by employing the quadratic-form identity proposed by Guo and Zhang. In order to obtain explicit Hamiltonian functions, a detailed computing formula for the constant appearing in the quadratic-form identity is obtained. One type of reduction equations of the hierarchy is also produced, which is further reduced to the standard equation of heat conduction. By introducing a loop algebra of the Lie algebra g, we obtain a soliton hierarchy with an arbitrary parameter which can be reduced to the previous equation hierarchy obtained, whose quasi-Hamiltonian structure is also worked out by the quadratic-form identity. Finally, we extend the Lie algebra g into a higher-dimensional Lie algebra so that a new integrable Hamiltonian hierarchy, which comprise integrable couplings, is produced; its reduced equations in particular contain two arbitrary parameters.

A New Liouville Integrable Hamiltonian System

With the help of a Lie algebra, an isospectral Lax pair is introduced for which a new Liouville integrable hierarchy of evolution equations is generated. Its Hamiltonian structure is also worked out by use of the quadratic-form identity.

Three kinds of coupling integrable couplings of the Korteweg–de Vries hierarchy of evolution equations

We introduce three kinds of column-vector Lie algebras Ls(s=1,2,3). By making invertible linear transformations we get the corresponding three induced Lie algebras. According to the defined loop algebras L̃s of the Lie algebras Ls(s=1,2,3), we establish three various isospectral problems. Then by applying Tu scheme, we obtain three different coupling integrable couplings of the Korteweg–de Vries (KdV) hierarchy and further reduce them to three kinds of explicit coupling integrable couplings of the KdV equation. One of the coupling integrable couplings of the KdV hierarchy of evolution equations possesses Hamiltonian structure obtained by using the quadratic-form identity and it is Liouville integrable.

Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure

By using a Lie algebra, an integrable couplings of the classical-Boussinesq hierarchy is obtained. Then, the Hamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-form identity.

TWO (2+1)-DIMENSIONAL INTEGRABLE COUPLINGS OF THE KN HIERARCHY AND CORRESPONDING HAMILTONIAN STRUCTURES

A (2+1) zero curvature equation is generated from one of the reduced equations of the self-dual Yang–Mills equations. As its applications, two (2+1)-dimensional integrable couplings of the famous KN hierarchy are obtained with the help of a subalgebra of the Lie subalgebra R9, which can be reduced to the Burgers equation. Furthermore, their Hamiltonian structures are worked out by taking use of the quadratic-form identity and the variational identity, respectively.

SPECTRAL RADIUS ANALYSIS OF MATRICES AND THE ASSOCIATED WITH INTEGRABLE SYSTEMS

An isospectral problem is introduced, a spectral radius of the corresponding spectral matrix is obtained, which enlightens us to set up an isospectral problem whose compatibility condition gives rise to a zero curvature equation in formalism, from which a Lax integrable soliton equation hierarchy with constraints of potential functions is generated along with 5 parameters, whose reduced cases present three integrable systems, i.e., AKNS hierarchy, Levi hierarchy and D-AKNS hierarchy. Enlarging the above Lie algebra into two bigger ones, the two integrable couplings of the hierarchy are derived, one of them has Hamiltonian structure by employing the quadratic-form identity or variational identity. The corresponding integrable couplings of the reduced systems are obtained, respectively. Finally, as comparing study for generating expanding integrable systems, a Lie algebra of antisymmetric matrices and its corresponding loop algebra are constructed, from which a great number of enlarging integrable systems could be generated, especially their Hamiltonian structure could be computed by the trace identity.

A NEW SOLITON HIERARCHY AND ITS TWO KINDS OF EXPANDING INTEGRABLE MODELS AS WELL AS HAMILTONIAN STRUCTURE

With the help of two different Lie algebras and the corresponding loop algebras, the first and second kind of expanding integrable models of a new soliton hierarchy of evolution equations are obtained, respectively. The Hamiltonian structure of the first one is worked out by the quadratic-form identity. The bi-Hamiltonian structure of the second one is also generated. From the paper, we conclude that various Lie algebras really produce different soliton hierarchies of evolution equations. The approach presented in the paper provides a way for generating different integrable soliton expanding systems of the known soliton hierarchy of equations.

Multi-component Harry–Dym hierarchy and its integrable couplings as well as their Hamiltonian structures

This paper obtains the multi-component Harry–Dym (H–D) hierarchy and its integrable couplings by using two kinds of vector loop algebras G̃3 and G̃6. The Hamiltonian structures of the above system are given by the quadratic-form identity. The method can be used to produce the Hamiltonian structures of the other integrable couplings or multi-component hierarchies.

A SOLITON HIERARCHY FROM THE LEVI SPECTRAL PROBLEM AND ITS TWO INTEGRABLE COUPLINGS, HAMILTONIAN STRUCTURE

A soliton-equation hierarchy from the D. Levi spectral problem is obtained under the framework of zero curvature equation. By employing two various multi-component Lie algebras and the loop algebras, we enlarge the Levi spectral problem and the corresponding time-part isospectral problems so that two different integrable couplings are produced. Using the quadratic-form identity yields the Hamiltonian structure of one of the two integrable couplings.