Liouville Integrable Hierarchy Research Articles

A combined Liouville integrable hierarchy associated with a fourth-order matrix spectral problem

This paper aims to propose a fourth-order matrix spectral problem involving four potentials and generate an associated Liouville integrable hierarchy via the zero curvature formulation. A bi-Hamiltonian formulation is furnished by applying the trace identity and a recursion operator is explicitly worked out, which exhibits the Liouville integrability of each model in the resulting hierarchy. Two specific examples, consisting of novel generalized combined nonlinear Schrödinger equations and modified Korteweg–de Vries equations, are given.

A Liouville integrable hierarchy with four potentials and its bi-Hamiltonian structure

"We aim to construct a Liouville integrable Hamiltonian hierarchy from a specific matrix spectral problem with four potentials through the zero curvature formulation. The Liouville integrability of the resulting hierarchy is exhibited by a bi-Hamiltonian structure explored by using the trace identity. Illustrative examples of novel four-component coupled Liouville integrable nonlinear Schr¨odinger equations and modified Korteweg-de Vries equations are presented."

A new 4-dimensional implicit vector-form loop algebra with arbitrary constants and the corresponding computing formula of constant γ in the Variation identity

A new 4-dimensional implicit vector-form loop algebra containing several arbitrary constants is presented, and the corresponding computing formula of constant γ in the Variation identity is obtained in this paper. As an application, a new Liouville integrable hierarchy is gotten which can be reduced to Dirac hierarchy, nonlinear Scho¨inger equation (NLS), M-KdV equation and so on.

New extended Lie algebra and the generalized integrable Liouville hierarchy

Abstract Starting from a new Lie algebra G 1 , the corresponding loop algebra G 1 ∼ is presented, from which a Liouville integrable hierarchy is given by using of variational identity. It follows that an expanding Lie algebra G 2 is obtained based on G 1 , furthermore, a related Lax integrable hierarchy is presented by making use of its related loop algebra G 2 ∼ . With the help of variational identity, it is not difficult to prove that the hierarchy has Hamiltonian structure and is Liouville integrable. It is also can be seen that the second hierarchy is the expanding model of the first one.

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.

A Lie algebra containing four parameters for the generalized Dirac hierarchy

A Lie algebra containing four parameters is obtained, whose commutation operation is concise, and the corresponding computing formula of constant γ in the variational identity is presented in this paper. As application, a new Liouville integrable hierarchy which can be reduced to Dirac hierarchy is derived by designing a special isospectral problem. We call it generalized Dirac hierarchy.

A Type of New Loop Algebra and a Generalized Tu Formula

A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing–Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula (GTF) is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 (1989) 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi-Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.

High-Order Binary Symmetry Constraints of a Liouville Integrable Hierarchy and Its Integrable Couplings

A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra Ã2. Furthermore, high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method. Finally, according to another new subalgebra of loop algebra Ã2, its integrable couplings are established.

A New Loop Algebra and the Corresponding Computing Formula of Constant γ in the Quadratic-Form Identity, as Well as the Generalized Burgers Hierarchy

A new loop algebra containing four arbitrary constants is presented, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper, which can be reduced to computing formula of constant γ in the trace identity. As application, a new Liouville integrable hierarchy which possess bi-Hamiltonian structure and generalized Burgers hierarchy are derived.

A New Loop Algebra and Corresponding ComputingFormula of Constant γ in Quadratic-Form Identity

A new loop algebra containing fourarbitrary constants is presented, whose commutation operation isconcise, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper, whichcan be reduced to computing formula of constant γ in thetrace identity. As application, a new Liouville integrable hierarchy,which can be reduced to AKNS hierarchy is derived.

A NEW LOOP ALGEBRA AND TWO NEW LIOUVILLE INTEGRABLE HIERARCHY

A new loop algebra containing four arbitrary constants is presented, and the corresponding computing formula of constant γ in the quadratic-form identity is obtained in this paper, which can be reduced to a computing formula of constant γ in the trace identity. As application, two new Liouville integrable hierarchy and Hamiltonian structures are derived.

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.

A type of loop algebra and the associated loop algebras

A higher-dimensional twisted loop algebra is constructed. As its application, a new Lax pair is presented, whose compatibility gives rise to a Liouville integrable hierarchy of evolution equations by making use of Tu scheme. One of the reduction cases of the hierarchy is an analogous of the well-known AKNS system. Next, the twisted loop algebra, furthermore, is extended to another higher dimensional loop algebra, from which a hierarchy of evolution equations with 11-potential component functions is obtained, whose reduction is just standard AKNS system. Especially, we prove that an arbitrary linear combination of the four Hamiltonian operators directly obtained from the recurrence relations is still a Hamiltonian operator. Therefore, the hierarchy with 11-potential functions possesses 4-Hamiltonian structures. Finally, an integrable coupling of the hierarchy is worked out.

An unified form of the AKNS and BPT hierarchies with two arbitrary parameters

Abstract We present a new basis of the loop algebra A ˜ 1 which is devote to setting up a new isospectral problem. By using Tu scheme, a new Liouville integrable hierarchy of soliton equations with two arbitrary parameters, which possesses the bi-Hamiltonian structure, is worked out. As reduction cases, it is shown that the hierarchy can be decomposed into the well-known AKNS hierarchy and the BPT hierarchy by taking different values of the parameters respectively. We call the hierarchy as an unified expression of the AKNS and the BPT hierarchies.

A Liouville integrable hierarchy, symmetry constraint, new finite-dimensional integrable systems, involutive solution and expanding integrable models

Abstract A new spectral problem and the associated integrable hierarchy of nonlinear evolution equations are presented in this paper. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. An explicit symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of the hierarchy. Moreover, the corresponding Lax pairs and adjoint Lax pairs are nonlinearized into a hierarchy of commutative, new finite-dimensional completely integrable Hamiltonian systems in the Liouville sense. Further, an involutive representation of solution of each equation in the hierarchy is given. Finally, expanding integrable models of the hierarchy are constructed by using a new Loop algebra.

A trick loop algebra and a corresponding Liouville integrable hierarchy of evolution equations

A subalgebra of loop algebra A 2 is first constructed, which has its own special feature. It follows that a new Liouville integrable hierarchy of evolution equations is obtained, possessing a tri-Hamiltonian structure, which is proved by us in this paper. Especially, three symplectic operators are constructed directly from recurrence relations. The conjugate operator of a recurrence operator is a hereditary symmetry. As reduction cases of the hierarchy presented in this paper, the celebrated MKdV equation and heat-conduction equation are engendered, respectively. Therefore, we call the hierarchy a generalized MKdV-H system. At last, a high-dimension loop algebra G is constructed by making use of a proper scalar transformation. As a result, a type expanding integrable model of the MKdV-H system is given.

A generalized Boite–Pempinelli–Tu (BPT) hierarchy and its bi-Hamiltonian structure

A subalgebra of loop algebra A ̃ 1 is presented. It follows that a new Liouville integrable hierarchy with 6 potential functions, which possesses a bi-Hamiltonian structure, is obtained. Since it can reduce to Boite–Pempinelli–Tu (BPT) hierarchy, we call it a generalized BPT system. Two symplectic operators of the system obtained are directly derived from recurrence relations, not from a recurrence operator. As a reduction case, the famous KdV-mKdV equation is given.