A NEW COMPLETELY INTEGRABLE LIOUVILLE' S SYSTEM, ITS LAX REPRESENTATION AND BI-HAMILTONIAN STRUCTURE

A new isospectral problem and the corresponding hierarchy of nonlinear evolution equations is presented. As a reduction, the well-known MKdV equation is obtained. It is shown that the hierarchy of equations is integrable in Liouville' s sense and possesses Bi-Hamiltonian structure. Under the constraint between the potentials and eigenfunctions, the eigenvalue problem can be nonlinearized as a finite dimensional completely integrable system.

HOW TO CONSTRUCT LAX REPRESENTATION FOR CONSTRAINED FLOWS OF THEBOUSSINESQ HIERARCHY VIA ADJOINT REPRESENTATIONS

By using the constraint relating potential and eigenfunctions, the decomposition of each equation in the Boussinesq hierarchy into two commuting finite-dimensional integrable Hamiltonian system (FDIHS) is presented. A method to construct the Lax representations for both x- and t(n)- constrained flows via reduction of the adjoint representations of the auxiliary linear problems is developed.

A FAMILY OF INTEGRABLE SYSTEMS OF LIOUVILLE AND LAX REPRESENTATION, DARBOUX TRANSFORMATIONS FOR ITS CONSTRAINED FLOWS

A family of integrable systems of Liouville are obtained by, Tu pattern. Using higher-order potential-eigenfunction constraints, the integrable systems are factorized to two x- and t(n)-integrable Hamiltonian systems whose Lax representation and three kinds of Darboux transformations are presented.

The Lax Representation and Darboux Transformation for Constrained Flows of the AKNS Hierarchy

By using a general scheme for decomposing a zero-curvature equation into two commut- ing x-and t_n-finite-dimensional integrable Hamiltonian systems (FDIHS),a systematic deduction of the Lax representation for all constrained flows of the AKNS hierarchy from the adjoint repre- sentation of the two auxiliary linear problems is presented.The Darboux transformation for these FDIHSs is derived.

REDUCTIONS OF ADJOINT REPRESENTATIONS TO LAX REPRESENTATIONS FOR CONSTRAINED FLOWS

Within framework of zero-curvature representation theory, the Lax representations for x- andtn-constrained flows of soliton hierarchy are obtained from reductions of adjoint representationsof the auxiliary linear problems. This method is applied to the third order spectral problem bytaking modified Boussinesq hierarchy as an illustrative example.

BCKLUND TRANSFORMATION AND LAX REPRESENTATION FOR A NONLINEAR DIFFERENTIAL EQUATION

In this paper, the Hirota bilinear method is applied to a nonlinear equation which is a deformation to a KdV equation with a source. Using the Hirota’s bilinear operator, we obtain its bilinear form and construct its bilinear Bcklund transformation. And then we obtain the Lax representation for the equation from the bilinear Bcklund transformation and testify the Lax representation by the compatibility condition.

NEW SYMPLECTIC MAPS: INTEGRABILITY AND LAX REPRESENTATION

New family of integrable symplectic maps are reduced from the Toda hierarchy via constraint for a higher flow of the hierarchy in terms of square eigenfunctions.Their integrability and Lax representation are deduced systematically from the discrete zero curvature representation of the Toda hierarchy. Also a discrete zero curvature representation for the Toda hierarchy with sources is presented.

LAX REPRESENTATIONS FOR DIRAC SOLITON HIERARCHY

This paper obtains Dirac soliton hierarchy and their Lax representations from a eigenvalue problem. The corresponding Lenard's recursive sequence can be solved explicitly.

Two New Multi-component BKP Hierarchies

We firstly propose two kinds of new multi-component BKP （mcBKP） hierarchy based on the eigenfunction symmetry reduction and nonstandard reduction, respectively. The first one contains two types of BKP equation with self-consistent sources whose Lax representations are presented. The two mcBKP hierarchies both admit reductions to the k-constrained BKP hierarchy and to integrable （1＋1）-dimensional hierarchy with self-consistent sources, which include two types of SK equation with self-consistent sources and of hi-directional SK equations with self-consistent

On Camassa-Holm Equation with Self-Consistent Sources and Its Solutions

Regarded as the integrable generalization of Camassa-Holm (CH) equation, the CH equation with selfconsistent sources (CHESCS) is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon solution, N-soliton, N-cuspon, N-positon, and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants.

r-Matrix Structure for a Restricted Flow with Bargmann Constraint

This paper deals with the integrability of a finite-dimensional Hamiltonian system linked with the generalized coupled KdV hierarchy. For this purpose the associated Lax representation is presented after an elementary calculation. It is shown that the Lax representation enjoys a dynamical r-matrix formula instead of a classical one in the Poisson bracket on R2N. Consequently the resulting system is proved to be completely integrable in view of its r-matrix structure.

Rosochatius Deformed Soliton Hierarchy with Self-Consistent Sources

Integrable Rosochatius deformations of finite-dimensional integrable systems are generalized to the solitonhierarchy with self-consistent sources.The integrable Rosochatius deformations of the Kaup-Newell hierarchy withself-consistent sources,of the TD hierarchy with self-consistent sources,and of the Jaulent Miodek hierarchy with self-consistentsources,together with their Lax representations are presented.

Recursion Operators of Two Supersymmetric Equations

From Lax representations,recursion operators for the supersymmetric KdV and the supersymmetric Kaup-Kupershimdt (SKK) equations are proposed explicitly.Under some special conditions,the recursion operator of the supersymmetric Sawada-Kotera equation can be recovered by the one of the SKK equation.

Infinitely Many Conservation Laws of a Differential-difference Equation and Backlund Transformation

In this paper, with the help of the Lax representation, we show the existence of infinitely many conservation laws for a differential-difference equation,which is one of the Ladic-Ablowitz hierarchy, and the conservation density and the associated flux are given formularlly. We also demonstrate the relation between a continuous partial differential equation and the differential-difference equation, and give Backlund transformation for the former.

Bilinear approach to N=2 supersymmetric KdV equations

The N=2 supersymmetric KdV equations are studied within the framework of Hirota bilinear method. For two such equations, namely N=2, a=4 and N=2, a=1 supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear Bcklund transformation is given for the N=2, a=1 supersymmetric KdV equation.

Finite Dimensional Integrable Systems Related to Generalized Schr dinger Equations

The binary nonlinearization method is applied to a 4×4 matrix eigenvalue problem. The typical system of the corresponding soliton hierarchy associated with this eigenvalue problem is the multi-component generalization of the nonlinear Schrodinger equation. With this method, Lax pairs and adjoint Lax pairs of the soliton hierarchy are reduced to two classes of finite dimensional Hamiltonian systems: a spatial finite dimensional Hamiltonian system and a hierarchy of temporal finite dimensional Hamiltonian systems. These finite dimensional Hamiltonian systems are commutative and Liouville integrable.

A New(γ_A,σ_B)-Matrix KP Hierarchy and Its Solutions

A new(γA,σB)-matrix KP hierarchy with two time series γA and σB,which consists of γA-flow,σB-flow and mixed γA and σB-evolution equations of eigenfunctions,is proposed.The reduction and constrained flows of(γA,σB)matrix KP hierarchy are studied.The dressing method is generalized to the(γA,σB)-matrix KP hierarchy and some solutions are presented.

A Hierarchy of Multidimensional Hénon-Heiles Systems

A hierarchy of multidimensional Hénon-Heiles(M-H-H)systems are constructed via the x-and t_n-higher-order-constrained flows of KdV hierarchy.The Lax representation for the M-H-H hierarchy is determined from the adjoint representation of the auxiliary linear problem for the KdV hierarchy.By using the Lax representation the classical Poisson structure and r-matrix for the hierarchy are found and the Jacobi inversion problem for the hierarchy is constructed.

RESTRICTED FLOWS OF A HIERARCHYOF INTEGRABLE DISCRETE SYSTEMS

The restricted flows of a hierarchy of discrete integrable systems are presented. It is shownthat integrals of motion and Lax representation for restricted flows as well as a discrete zerocurvature representation for the hierarchy with sources can be deduced from the zero-curvaturerepresentation for the hierarchy. Also it is studied how the hierarchy is related to the AKNShierarchy in the continuous limit.