Soliton Solutions for a Negative Order AKNS Equation Hierarchy

In this paper, bilinear form of a negative order AKNS equation hierarchy is given. The soliton solutions are obtained through Hiorta's direct method.     

Soliton Solutions for a Negative Order AKNS Equation Hierarchy

In this paper, bilinear form of a negative order AKNS equation hierarchy is given. The soliton solutions are obtained through Hiorta＇s direct method.     

A Discrete Spectral Problem and Related Hierarchy of Discrete Hamiltonian Lattice Equations

Staring from a discrete matrix spectral problem, a hierarchy of lattice soliton equations is presented though discrete zero curvature representation. The resulting lattice soliton equations possess non-local Lax pairs. The Hamiltonian structures are established for the resulting hierarchy by the discrete trace identity. Liouville integrability of resulting hierarchy is demonstrated.     

A Discrete Spectral Problem and Related Hierarchy of Discrete Hamiltonian Lattice Equations

Staring from a discrete matrix spectral problem, a hierarchy of lattice soliton equations is presented though discrete zero curvature representation. The resulting lattice soliton equations possess non-local Lax pairs. The Hamiltonian structures are established for the resulting hierarchy by the discrete trace identity. Liouville integrability of resulting hierarchy is demonstrated.     

Constraints and Soliton Solutions for KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equationcan be generated by its recursion operator.We generalize the result to a special form of Lax pair,from which a method to constrain the integrable system to alower-dimensional or fewer variable integrable system is proposed.A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depictedby a series of ordinary differential equations (ODEs), which may be gotten by a simple but unfamiliar Lax pair. Furthermore the AKNS hierarchy is constrained to a series of univariate integrable hierarchies. The key is a special form of Lax pair for the AKNS hierarchy. It is proved that under the constraints all equations of the AKNS hierarchy are linearizable.     

Integrable Coupling of KN Hierarchy and Its Hamiltonian Structure

The Hamiltonian structure of the integrable couplings obtained by our method has not been solved. In this paper, the Hamiltonian structure of the KN hierarchy is obtained by making use of the quadratic-form identity.     

A Difference Hamiltonian Operator and a Hierarchy of Generalized Toda Lattice Equations

A difference Ha-miltonian operator with three arbitrary constants is presented. When the arbitrary constants -in the Hamiltonian operator are suitably chosen, a pair of Hamiltonian operators are given. The resulting Hamiltonian pair yields a difference hereditary operator. Using Magri scheme of bi-Hamiltonian formulation, a hierarchy of the generalized Toda lattice equations is constructed. Finally, the discrete zero curvature representation is given for the resulting hierarchy.     

A Difference Hamiltonian Operator and a Hierarchy of Generalized Toda Lattice Equations

A difference Hamiltonian operator with three arbitrary constants is presented. When the arbitrary constants in the Hamiltonian operator are suitably chosen, a pair of Hamiltonian operators are given. The resulting Hamiltonian pair yields a difference hereditary operator. Using Magri scheme of bi-Hamiltonian formulations a hierarchy of the generalized Toda lattice equations is constructed. Finally, the discrete zero curvature representation is given for the resulting hierarchy.     

The Second mKdV Equation and Its Hereditary Symmetry and Hamiltonian Structure

The isospectral problem of the second mKdV equation is found out firstly. It follows that the strong hereditary symmetry and the Hamiltonian structure of the second mKdV equation are presented.     

New Lattice Equation Hierarchies and Darboux Transformation

A discrete integrable system and its Hamiltonian structure are generated by use of Tu model. Then, its Darboux transformation is obtained, which can get the expression of the new solutions.     

Hamiltonian Tri-Integrable Couplings of the AKNS Hierarchy

We propose a systematic approach for generating Hamiltonian tri-integrable couplings of soliton hierarchies. The resulting approach is based on semi-direct sums of matrix Lie algebras consisting of 4 × 4 block matrix Lie algebras. We apply the approach to the AKNS soliton hierarchy, and Hamiltonian structures of the obtained tri-integrable couplings are constructed by the variational identity.     

Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure＊

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

Integrable Coupling of （2＋1）-Dimensional Multi-component DLW Integrable Hierarchy and Its Hamiltonian Structure

A new multi-component Lie algebra is constructed, and a type of new loop algebra is presented. A （2＋1）-dimensional multi-component DLW integrable hierarchy is obtained by using a （2＋1）-dimensional zero curvature equation. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system and its corresponding Hamiltonian structure are worked out.     

A New Three-Dimensional Lie Algebra and a Modified AKNS Hierarchy of Soliton Equations

A new three-dimensional Lie algebra and its corresponding loop algebra are constructed, from which a modified AKNS soliton-equation hierarchy is obtained.     

A generalized Tu formula and Hamiltonian structures of fractional AKNS hierarchy

In this Letter, a generalized Tu formula is firstly presented to construct Hamiltonian structures of fractional soliton equations. The obtained results can be reduced to the classical Hamiltonian hierarchy of AKNS in ordinary calculus.     

Coupling Integrable Couplings of an Equation Hierarchy

Based on a kind of Lie algebra G proposed by Zhang, one isospectral problem is designed. Under the framework of zero curvature equation, a new kind of integrable coupling of an equation hierarchy is generated using the methods proposed by Ma and Gao. With the help of variational identity, we get the Hamiltonian structure of the hierarchy.     

A Hierarchy of Integrable Lattice Soliton Equations and New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy of integrable lattice soliton equations is derived. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses discrete bi-Hamiltonian structure. A new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a Bäcklund transformation for the resulting integrable lattice equations. At last, conservation laws of the hierarchy are presented.     

New DLW Hierarchy of an Integrable Coupling and Its Hamiltonian Structure

A type of higher-dimensional loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic identity.     

A Hierarchy of Nonlinear Lattice Soliton Equations and Its Darboux Transformation

A hierarchy of nonlinear lattice soliton equations is derived from a new discrete spectral problem. The Hamiltonian structure of the resulting hierarchy is constructed by using a trace identity formula. Moreover, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations. The exact solutions are given by applying the Darboux transformation.