Constraints and Soliton Solutions for KdV Hierarchy and AKNS Hierarchy

It is well-known that the finite-gap solutions of the KdV equation can 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 a lower-dimensional or fewer variable integrable system is proposed.A direct result is that the n-soliton solutions of the KdV hierarchy can be completely depicted by 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.

Solving the AKNS Hierarchy by Its Bilinear Form: Generalized Double Wronskian Solutions

Through the Wronskian technique, a simple and direct proof is presented that the AKNS hierarchy in the bilinear form has generalized double Wronskian solutions. Moreover, by using a unified way, soliton solutions, rational solutions, Matveev solutions and complexitons in double Wronskian form for it are constructed.

THE CONFOCAL INVOLUTIVE SYSTEM AND THE INTEGRABILITY OF THE NONLINEARIZED LAX SYSTEMS FOR AKNS HIERARCHY

The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.

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 soIRon hierarchy, and Hamiltonian structures of the obtained tri-integrable couplings are constructed by the variational identity.

AN INTEGRABLE COUPLING OF THE GENERALIZED AKNS HIERARCHY

A direct method for establishing integrable couplings is proposed in this paper by constructing a new loop algebra G. As an illustration by example, an integrable coupling of the generalized AKNS hierarchy is given. Furthermore, as a reduction of the generalized AKNS hierarchy, an integrable coupling of the well-known G J hierarchy is presented. Again a simple example for the integrable coupling of the MKdV equation is also given. This method can be used generally.

Double Darboux Transformation for AKNS Hierarchy

In this paper we construct a double Darboux Transformation for AKNS hierarchy and give its decomposition theorem.A remarkable characteristic of the double Darboux Transforma- tion - incommutativity is proved.

TWO CLASSES OF INTEGRABLE COUPLING FOR AKNS HIERARCHY

A set of new matrix Lie algebra is constructed, which is devoted to obtaining a new loop algebra A 2M . Then we use the idea of enlarging spectral problems to make an enlarged spectral problems. It follows that the multi-component AKNS hierarchy is presented. Further, two classes of integrable coupling of the AKNS hierarchy are obtained by enlarging spectral problems.

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 solutionsare obtained through Hiorta's direct method.

Schroedinger Flows to Symmetric Spaces and the Second Matrix-AKNS Hierarchies

This article gives a unified geometric interpretation of the second Matrix-AKNS hierarchies via Schroedinger flows to symmetric spaces of Kaehler and paraKaehler types.

A Method for Obtaining Integrable Couplings

By making use of the vector product in R^3, a commuting operation is introduced so that R^3 becomes a Lie algebra. The resulting loop algebra R^-3 is presented, from which the well-known AKNS hierarchy is produced. Again via applying the superposition of the commuting operations of the Lie algebra, a commuting operation in R^6 is constructed so that R^6 becomes a Lie algebra. Thanks to the corresponding loop algebra R^3 of the Lie algebra R^3, the integrable coupling of the AKNS system is obtained. The method presented in this paper is rather simple and can be used to work out integrable coupling systems of the other known integrable hierarchies of soliton equations.

Conservation laws of some soliton equations with self-consistent sources

Infinitely many conservation laws for some （1＋1）-dimension soliton hierarchy with self-consistent sources are constructed from their corresponding Lax pairs directly. Three examples are given. Besides, infinitely many conservation laws for Kadomtsev-Petviashvili （KP） hierarchy with self-consistent sources are obtained from the pseudo-differential operator and the Lax pair.

Integrable Couplings of Generalized Glachette-Johnson (GJ) Hierarchy

A new loop algebra G is established to obtain integrable coupling of GJ hierarchy. In particular, iiitegrable coupling of the well-known AKNS hierarchy is presented. This method can be used generally.

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

We 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.