发展方程族Lax表示的广义结构*

本文给出非线性发展方程族的一个生成格式(该格式包含了保谱族与非保谱族作为其两个特殊情况),并提供该格式下发展方程族Lax表示的广义结构.最后,作为应用,我们讨论了Levi族发展方程.     

Decomposition of a hierarchy of nonlinear evolution equations

The generalized Hamiltonian structures for a hierarchy of nonlinear evolution equations are established with the aid of the trace identity. Using the nonlinearization approach, the hierarchy of nonlinear evolution equations is decomposed into a class of new finite-dimensional Hamiltonian systems. The generating function of integrals and their generator are presented, based on which the finite-dimensional Hamiltonian systems are proved to be completely integrable in the Liouville sense. As an application, solutions for the hierarchy of nonlinear evolution equations are reduced to solving the compatible Hamiltonian systems of ordinary differential equations.     

一类非线性演化方程族的可积耦合

Starting from a Tu Guizhang‘s isospectral‘problem, a Lax pair is obtained by means of Tu scheme ( we call it Tu Lax pair ). By applying a gauge transformation between matrices, the Tu Lax pair is changed to its equivalent Lax pair with the traces of spectral matrices being zero, whose compatibility gives rise to a type of Tu hierarchy of equations. By making use of a high order loop algebra constructed by us, an integrable coupling system of the Tu hierarchy of equations are presented. Especially, as reduction cases, the integrable couplings of the celebrated AKNS hierarchy, TD hierarchy and Levi hierarchy are given at the same time.     

A new extended dispersionless mKP hierarchy and hodograph solution

In this article, a new extended dispersionless mKP hierarchy (exdmKPH) is constructed to obtain two types of dispersionless mKP equations with self-consistent sources (dmKPSCS) and their associated conservation equations. Two reductions of this hierarchy are used to get two types of the corresponding dispersionless mKdV equations with self-consistent sources (dmKdVSCS). A hodograph solution for the first type of dmKdVSCS and Bäcklund transformation between the extended dispersionless KP hierarchy (exdKPH) and exdmKPH are also given.     

The Liouville integrability of integrable couplings of Volterra lattice equation

A hierarchy of integrable couplings of Volterra lattice equations with three potentials is proposed, which is derived from a new discrete six-by-six matrix spectral problem. Moreover, by means of the discrete variational identity on semi-direct sums of Lie algebra, the two Hamiltonian forms are deduced for each lattice equation in the resulting hierarchy. A strong symmetry operator of the resulting hierarchy is given. Finally, we prove that the hierarchy of the resulting Hamiltonian equations are all Liouville integrable discrete Hamiltonian systems.     

Eigenfunctions of Linearized Integrable Equations Expanded Around an Arbitrary Solution

Eigenfunctions of linearized integrable equations expanded around an arbitrary solution are obtained for the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy and the Korteweg–de Vries (KdV) hierarchy. It is shown that the linearization operators and the integrodifferential operator that generates the hierarchy are commutable. Consequently, eigenfunctions of the linearization operators are precisely squared eigenfunctions of the associated eigenvalue problem. Similar results are obtained for the adjoint linearization operators as well. These results make a simple connection between the direct soliton/multisoliton perturbation theory and the inverse-scattering based perturbation theory for these hierarchy equations.     

Solving the non-isospectral Ablowitz–Ladik hierarchy via the inverse scattering transform and reductions

The non-isospectral Ablowitz–Ladik hierarchy is integrated by the inverse scattering transform. In contrast with the isospectral Ablowitz–Ladik hierarchy, the eigenvalues of the non-isospectral Ablowitz–Ladik equations in the scattering data are time-dependent. The multi-soliton solution for the hierarchy is presented. The reductions to the non-isospectral discrete NLS hierarchy and the non-isospectral discrete mKdV hierarchy and their solutions are considered.     

Thermodynamics and Hydrodynamics (Statistical Foundations): 4. Kinetic Processes in Multicomponent Systems

We use the BBGKY hierarchy and the mass, momentum, and intrinsic energy conservation laws, which are consequences of this hierarchy, to obtain the hydrodynamic equations for multicomponent systems and the diffusion equations. We formulate several restrictions on the thermodynamic equations for irreversible processes.     

ON GENERATING EQUATIONS FOR THE KAUP-NEWELL HIERARCHY

It is shown that the Kanp-Newell hierarchy can be derived from the so-called gen- erating equations which are Lax integrable.Positive and negative flows in the hierarchy are derived simultaneously.The generating equations and mutual commutativity of these flows en- able us to construct new Lax integrable equations.     

Molecular extended thermodynamics: comparison between rarefied polyatomic and monatomic gas closures

Molecular extended thermodynamics is justified at the mesoscopic level by the moment equations associated with the Boltzmann equation. For polyatomic gases we have a binary hierarchy of moments in contrast with the usual single hierarchy for monatomic gases. In this paper, taking one-dimensional space variables for simplicity, we review the closure of the system of the moment equations for polyatomic gases with the use of the maximum entropy principle, which is equivalent to the entropy principle. Then we consider the singular limit where the degrees of freedom of a molecule approach 3, and we prove that, by imposing appropriate initial conditions, the solutions for polyatomic gases converge to the ones for monatomic gases. As examples of the singular limit, the asymptotic behaviors of linear waves and light scattering based on the linearized system of field equations are briefly presented.     

RIEMANN-HILBERT PROBLEMS OF A SIX-COMPONENT MKDV SYSTEM AND ITS SOLITON SOLUTIONS

Based on a 4 x 4 matrix spectral problem, an AKNS soliton hierarchy with six potentials is generated. Associated with this spectral problem, a kind of Riemann-Hilbert problems is formulated for a six-component system of mKdV equations in the resulting AKNS hierarchy. Soliton solutions to the considered system of coupled mKdV equations are computed, through a reduced Riemann-Hilbert problem where an identity jump matrix is taken.     

On the use of the Painlevé property for obtaining miura type transformation

We present a method for obtaining an associated hierarchy of evolution equations possessing the Painlevé property from a given hierarchy which possesses the Painlevé property. This method is applied to the classical Boussinesq hierarchy to obtain the Miura type transformation and the modified classical Boussinesq hierarchy. It is also used to construct a large hierarchy of evolution equations which possess the Painlevé property and include the classical Boussinesq the Jaulent Miodek, the dispersive long wave hierarchy as special cases. All these hierarchies have the same modified hierarchy.The projection supported by the National Natural Science Fundation of China.     

A hierarchy of differential-difference equations,conservation laws and new integrable coupling system

Starting from a discrete spectral problem with two arbitrary parameters, a hierarchy of nonlinear differential-difference equations is derived. The new hierarchy not only includes the original hierarchy, but also the well-known Toda equation and relativistic Toda equation. Moreover, infinitely many conservation laws for a representative discrete equation are given. Further, a new integrable coupling system of the resulting hierarchy is constructed.     

Truncating the hierarchy of moment equations based on point distribution—application to innovation diffusion

The phenomenon of innovation diffusion, modeled as a nonlinear birth process, leads to a hierarchy of moment equations. For gaining insight into the evolution of moments of a number of adopters in innovation diffusion, truncation procedures based on point distributions are proposed for truncating the hierarchy of moment equations. It is found that the results obtained by employing new truncation procedures are in close agreement with analytical results based on system size expansion for large population size.     

Bilinear identities for an extended B-type Kadomtsev–Petviashvili hierarchy

We construct bilinear identities for wave functions of an extended B-type Kadomtsev–Petviashvili (BKP) hierarchy containing two types of (2+1)-dimensional Sawada–Kotera equations with a self-consistent source. Introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the τ -function and bilinear identities for this extended BKP hierarchy. The bilinear identities generate all the Hirota bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, we obtain the Hirota bilinear equations for the two types of (2+1)-dimensional Sawada–Kotera equations in explicit form.     

AN ABLOWITZ-LADIK INTEGRABLE LATTICE HIERARCHY WITH MULTIPLE POTENTIALS

Within the zero curvature formulation, a hierarchy of integrable lattice equations is constructed from an arbitrary-order matrix discrete spectral problem of Ablowitz-Ladik type. The existence of infinitely many symmetries and conserved functionals is a consequence of the Lax operator algebra and the trace identity. When the involved two potential vectors are scalar, all the resulting integrable lattice equations are reduced to the standard AblowitzLadik hierarchy.     

新的耦合mKdV方程族及其Liouville可积的无限维Hamilton结构

根据第Ⅱ屠格式,从一个特征值问题出发,本文推得了一族新的耦合ｍＫｄＶ方程,然后用迹恒等式人出了其无限维Ｈａｍｉｌｔｏｎ结构。最后证明了该Ｈａｍｉｌｔｏｎ方程族是Ｌｉｏｕｖｉｌｌｅ可积的,并且有无穷多个彼此对合的公共守恒密度。     

Multi-component bi-Hamiltonian Dirac integrable equations

A specific matrix iso-spectral problem of arbitrary order is introduced and an associated hierarchy of multi-component Dirac integrable equations is constructed within the framework of zero curvature equations. The bi-Hamiltonian structure of the obtained Dirac hierarchy is presented be means of the variational trace identity. Two examples in the cases of lower order are computed.     

Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions

We show that a system of Hirota bilinear equations introduced by Jimbo and Miwa defines tau-functions of the modified KP (MKP) hierarchy of evolution equations introduced by Dickey. Some other equivalent definitions of the MKP hierarchy are established. All polynomial tau-functions of the KP and the MKP hierarchies are found. Similar results are obtained for the reduced KP and MKP hierarchies.     

A hierarchy of Liouville integrable lattice equations and its integrable coupling systems

A new discrete two-by-two matrix spectral problem with two potentials is introduced, followed by a hierarchy of integrable lattice equations obtained through discrete zero curvature equations. It is shown that the Hamiltonian structures of the resulting integrable lattice equations are established by virtue of the trace identity. Furthermore, based on a discrete four-by-four matrix spectral problem, the discrete integrable coupling systems of the resulting hierarchy are obtained. Then, with the variational identity, the Hamiltonian structures of the obtained integrable coupling systems are established. Finally, the resulting Hamiltonian systems are proved to be all Liouville integrable.