Gauge Transformation for BCr-KP Hierarchy and Its Compatibility with Additional Symmetry

The BCr-KP hierarchy is an important sub-hierarchy of the KP hierarchy. In this paper, the BCr-KP hierarchy is investigated from three aspects. Firstly, we study the gauge transformation for the BCr-KP hierarchy.Different from the KP hierarchy, the gauge transformation must keep the constraint of the BCr-KP hierarchy. Secondly,we study the gauge transformation for the constrained BCr-KP hierarchy. In this case, the constraints of the Lax operator must be invariant under the gauge transformation. At last, the compatibility between the additional symmetry and the gauge transformation for the BCr-KP hierarchy is explored.     

Bi-Hamiltonian structure of the extended noncommutative Toda hierarchy

The noncommutative Toda hierarchy is studied with the help of Moyal deformation by a reduction on the non-commutative two dimensional Toda hierarchy. Further we generalize the noncommutative Toda hierarchy to the extended noncommutative Toda hierarchy. To survey on its integrability, we construct the bi-Hamiltonian structure and noncommutative conserved densities of the extended noncommutative Toda hierarchy by means of the R-matrix formalism. This extended noncommutative Toda hierarchy can be reduced to the extended multicomponent Toda hierarchy, extended Z_N?-Toda hierarchy, extended Toda hierarchy respectively by reductions on Lie algebras.     

A Lattice Hierarchy with a Free Function and Its Reductions to the Ablowitz-Ladik and Volterra Hierarchies

By embedding a free function into a compatible zero curvature equation, we propose a lattice hierarchy with the free function which still admits zero curvature representation. It is interesting that the hierarchy can reduce the Ablowitz-Ladik hierarchy, the Volterra hierarchy and a new hierarchy by properly choosing the embedded function. Moreover, the new hierarchy is integrable in Liouville’s sense and possess multi-Hamiltonian structure.     

New integrable lattice hierarchies

In this Letter we give a new integrable four-field lattice hierarchy, associated to a new discrete spectral problem. We obtain our hierarchy as the compatibility condition of this spectral problem and an associated equation, constructed herein, for the time-evolution of eigenfunctions. We consider reductions of our hierarchy, which also of course admit discrete zero curvature representations, in detail. We find that our hierarchy includes many well-known integrable hierarchies as special cases, including the Toda lattice hierarchy, the modified Toda lattice hierarchy, the relativistic Toda lattice hierarchy, and the Volterra lattice hierarchy. We also obtain here a new integrable two-field lattice hierarchy, to which we give the name of Suris lattice hierarchy, since the first equation of this hierarchy has previously been given by Suris. The Hamiltonian structure of the Suris lattice hierarchy is obtained by means of a trace identity formula.     

A Multi-component Matrix Loop Algebra and Its Application

A set of multi-component matrix Lie algebra is constructed. It follows that a type of new loop algebra AM-1 is presented. An isospectral problem is established. Integrable multi-component hierarchy is obtained by Tu pattern, which possesses tri-Hamiltonian structures. Furthermore, it can be reduced to the well-known AKNS hierarchy and BPT hierarchy. Therefore, the major result of this paper can be regarded as a unified expression integrable model of the AKNS hierarchy and the BPT hierarchy.     

A Multi-component Matrix Loop Algebra and Its Application

A set of multi-component matrix Lie algebra is constructed. It follows that a type of new loop algebra A^- M-1 is presented. An isospectral problem is established. Integrable multi-component hierarchy is obtained by Tu pattern, which possesses tri-Hamiltonian structures. Furthermore, it can be reduced to the well-known AKNS hierarchy and BPT hierarchy. Therefore, the major result of this paper can be regarded as a unified expression integrable model of the AKNS hierarchy and the BPT hierarchy.     

Recursion Operators for Dispersionless KP Hierarchy

Based on the corresponding theorem between dispersionless KP(dKP)hierarchy and -dependent KP ( KP) hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding KP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.     

The extended trace identity and its application

The trace identity is extended to the general loop algebra. The Hamiltonian structures of the integrable systems concerning vector spectral problems and the multi-component integrable hierarchy can be worked out by using the extended trace identity. As its application, we have obtained the Hamiltonian structures of the Yang hierarchy, the Korteweg-de--Vries (KdV) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur (M-AKNS) hierarchy, the multi-component Ablowitz--Kaup--Newell--Segur Kaup--Newell (M-AKNS--KN) hierarchy and a new multi-component integrable hierarchy separately.     

Recursion Operators for Dispersionless KP Hierarchy

Based on the corresponding theorem between dispersionless KP (dKP) hierarchy and h-dependent KP (hKP) hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding hKP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.     

A new six-component super soliton hierarchy and its self-consistent sources and conservation laws

A new six-component super soliton hierarchy is obtained based on matrix Lie super algebras. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy. After that, the selfconsistent sources of the new six-component super soliton hierarchy are presented. Furthermore, we establish the infinitely many conservation laws for the integrable super soliton hierarchy.     

On the Geometry of Darboux Transformations for the KP Hierarchy and its Connection with the Discrete KP Hierarchy

We tackle the problem of interpreting the Darboux transformation for the KP hierarchy and its relations with the modified KP hierarchy from a geometric point of view. This is achieved by introducing the concept of a Darboux covering. We construct a Darboux covering of the KP equations and obtain a new hierarchy of equations, which we call the Darboux-KP hierarchy (DKP). We employ the DKP equations to discuss the relationships among the KP equations, the modified KP equations, and the discrete KP equations. Our approach also handles the various reductions of the KP hierarchy. We show that the KP hierarchy is a projection of the DKP, the mKP hierarchy is a DKP restriction to a suitable invariant submanifold, and that the discrete KP equations are obtained as iterations of the DKP ones. Received: 23 July 1996 / Accepted: 6 January 1997     

The relation between the Toda hierarchy and the KdV hierarchy

Under three relations connecting the field variables of Toda flows and that of KdV flows, we present three new sequences of combinations of the equations in the Toda hierarchy which have the KdV hierarchy as a continuous limit. The relation between the Poisson structures of the KdV hierarchy and the Toda hierarchy in the continuous limit is also studied.     

Hierarchy of equations for reduced density matrices in the case of thermodynamic equilibrium

A hierarchy of equations for s-particle density matrices at thermodynamic equilibrium is obtained, with the equation for the nonequilibrium density matrix as the starting point. When deducing the hierarchy the hypothesis of maximum statistical independence for the density matrices is used. The hierarchy obtained is an analogue of the classical equilibrium BBGKY hierarchy and goes over into it when . It is shown that thermodynamic quantities can be expressed in terms of functions that enter only into the first hierarchy equations. The hierarchy is analysed in detail in the case of a uniform fluid. As an example in which the equations can be solved easily enough, a hard-sphere system wherein triplet correlations are neglected is considered. Different approximations that can be used when solving the equations derived are discussed. Comparisons are made with the results of other theoretical treatments.     

Canonical property of the Miura maps between the MKP and KP hierarchies,continuous and discrete

Apart from the case of the KP hierarchy, all known Miura maps between integrable Hamiltonian systems had been proven to be canonical. The remaining KP case is settled below. As a corollary, it is shown that the KP hierarchy is a factor — hierarchy of the mKP one, with the kernel consisting of a single scalar field. A discrete mKP hierarchy and the associated Miura map are constructed, and the latter is shown to be canonical as well. As in the continuous case, this implies that one can extend the discrete KP hierarchy by a single new field into an extended discrete KP hierarchy in such a way that the extended discrete Miura map mKPeKP is a canonical isomorphism.     

Darboux transformation for the non-isospectral AKNS hierarchy and its asymptotic property

In this Letter, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-Bäcklund transformation, because the integral constants of the hierarchy will be changed after the transformation. The transform rule of the integral constants will be also derived. By this means, the soliton solutions of the nonlinear equations derived by the non-isospectral AKNS hierarchy can be found.     

Special polynomials associated with some hierarchies

Special polynomials associated with rational solutions of a hierarchy of equations of Painlevé type are introduced. The hierarchy arises by similarity reduction from the Fordy-Gibbons hierarchy of partial differential equations. Some relations for these special polynomials are given. Differential-difference hierarchies for finding special polynomials are presented. These formulae allow us to obtain special polynomials associated with the hierarchy studied. It is shown that rational solutions of members of the Schwarz-Sawada-Kotera, the Schwarz-Kaup-Kupershmidt, the Fordy-Gibbons, the Sawada-Kotera and the Kaup-Kupershmidt hierarchies can be expressed through special polynomials of the hierarchy studied.     

Soliton solutions by Darboux transformation for a Hamiltonian lattice system

Starting from a new discrete iso-spectral problem, we derive a hierarchy of Hamiltonian lattice equations. A Darboux transformation is established for the lattice soliton hierarchy. As applications, the soliton solutions of resulted lattice hierarchy are given.     

On Recursion Operator of the q-KP Hierarchy

It is the aim of the present article to give a general expression of flow equations of the q-KP hierarchy. The distinct difference between the q-KP hierarchy and the KP hierarchy is due to q-binomial and the action of q-shift operator θ, which originates from the Leibnitz rule of the quantum calculus. We further show that the n-reduction leads to a recursive scheme for these flow equations. The recursion operator for the flow equations of the q-KP hierarchy under the n-reduction is also derived.