New Positive and Negative Hierarchies of Integrable Differential-Difference Equations and Conservation Laws

Two hierarchies of nonlinear integrable positive and negative lattice equations are derived from a discrete spectrak problem. The two lattice hierarchies are proved to have discrete zero curvature representations associated with a discrete spectral problem, which also shows that the positive and negative hierarchies correspond to positive and negative power expansions of Lax operators with respect to the spectral parameter, respectively. Moreover, the integrable lattice models in the positive hierarchy are of polynomial type, and the integrable lattice models in the negative hierarchy are of rational type. Further, we construct infinite conservation laws about the positive hierarchy.     

A novel hierarchy of differential integral equations and their generalized bi-Hamiltonian structures

With the aid of the zero-curvature equation, a novel integrable hierarchy of nonlinear evolution equations associated with a 3 x 3 matrix spectral problem is proposed. By using the trace identity, the bi-Hamiltonian structures of the hierarchy are established with two skew-symmetric operators. Based on two linear spectral problems, we obtain the infinite many conservation laws of the first member in the hierarchy.     

Painlevé analysis and integrability aspects of some nonlinear partial differential equations

A brief review of the Painlevé singularity structure analysis of some autonomous and nonautonomous nonlinear partial differential equations is discussed. We point out how the Painlevé analysis of solutions of these equations systematically provides the integrability properties of the equation. The Lax pair, Bäcklund transformation and bilinear forms are constructed from the analysis.     

Hamiltonian System and Infinite Conservation Laws Associated with a New Discrete Spectral Problem

Starting from a new discrete spectral problem, the corresponding hierarchy of nonlinear lattice equations is proposed. It is shown that the lattice soliton hierarchy possesses the bi-Hamiltonian structures and infinitely many common commuting conserved functions. Further, infinite conservation laws of the hierarchy are presented.     

Integrability,multi-soliton and rational solutions,and dynamical analysis for a relativistic Toda lattice system with one perturbation parameter

Under investigation in this paper is a relativistic Toda lattice system with one perturbation parameter α abbreviated as RTL_(α) system by Suris, which may describe the motions of particles in lattices interacting through an exponential interaction force. First of all, an integrable lattice hierarchy associated with an RTL_(α) system is constructed, from which some relevant integrable properties such as Hamiltonian structures, Liouville integrability and conservation laws are investigated. Secondly, the discrete generalized(m, 2 N-m)-fold Darboux transformation is constructed to derive multi-soliton solutions, higher-order rational and semirational solutions, and their mixed solutions of an RTL_(α) system. The soliton elastic interactions and details of rational solutions are analyzed via the graphics and asymptotic analysis. Finally, soliton dynamical evolutions are investigated via numerical simulations,showing that a small noise has very little effect on the soliton propagation. These results may provide new insight into nonlinear lattice dynamics described by RTL_(α) system.     

A Hierarchy of Lax Integrable Lattice Equations,Liouville Integrability and a New Integrable Symplectic Map

A discrete matrix spectral problem and the associated hierarchy of Lax integrable lattice equations are presented, and it is shown that the resulting Lax integrable lattice equations are all Liouville integrable discrete Hamiltonian systems. A new integrable symplectic map is given by binary Bargmann constraint of the resulting hierarchy. Finally, an infinite set of conservation laws is given for the resulting hierarchy.     

Painlevé test for integrability for a combination of Yang’s self-dual equations for <Emphasis Type="Italic">SU</Emphasis> (2) gauge fields and Charap’s equations for chiral invariant model of pion dynamics and a comparative discussion among the three

Painlevé test for integrability for the combined equations generated from Yang’s self-dual equations for SU (2) gauge fields and Charap’s equations for chiral invariant model of pion dynamics faces some peculiar situations that allow none of the stages (leading order analysis, resonance calculation and checking of the existence of the requisite number of arbitrary functions) to be conclusive. It is also revealed from a comparative study with the previous results that the existence of abnormal behaviour at any of the stated stages may have a correlation with the existence of chaotic property or some other properties that do not correspond to solitonic behaviour.      

Perturbation,symmetry analysis,Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain B?cklund transformation associated with truncated Painlevéexpansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal B?cklund transformations of the derived conservation laws for the first time.     

Cosmological evolution of quintessence and phantom with a new type of interaction in dark sector

In the present work, motivated by the work of Cai and Su [Phys. Rev. D 81 (2010) 103514], we propose a new type of interaction in dark sector, which can change its sign when our universe changes from deceleration to acceleration. We consider the cosmological evolution of quintessence and phantom with this type of interaction, and find that there are some scaling attractors which can help to alleviate the cosmological coincidence problem. Our results also show that this new type of interaction can bring new features to cosmology.     

Dynamical instability and adiabatic evolution of the atom--homonuclear--trimer dark state in a condensate system

This paper investigates the dynamical instability and adiabatic evolution of the atom--homonuclear--trimer dark state of a condensate system in a stimulated Raman adiabatic passage aided by Feshbach resonance. It obtains analytically the regions for the appearance of dynamical instability caused by the interparticle interactions. Moreover, the adiabatic property of the dark state is also studied in terms of a newly defined adiabatic fidelity. It shows that the nonlinear collisions have a negative effect on the adiabaticity of the dark state and hence reduce the conversion efficiency.     

On a revisit to the Painlevé test for integrability and exact solutions for Yang’s self-dual equations forSU (2) gauge fields

Painlevé test (Jimboet al [1]) for integrability for the Yang’s self-dual equations forSU(2) gauge fields has been revisited. Jimboet al analysed the complex form of the equations with a rather restricted form of singularity manifold. They did not discuss exact solutions in that context. Here the analysis has been done starting from the real form of the same equations and keeping the singularity manifold completely general in nature. It has been found that the equations, in real form, pass the Painlevé test for integrability. The truncation procedure of the same analysis leads to non-trivial exact solutions obtained previously and auto-Backlund transformation between two pairs of those solutions     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

An algorithm and its application for obtaining some kind of infinite-dimensional Hamiltonian canonical formulation

Using factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical evolution equation, infinite-dimensional Hamiltonian canonical system, factorization of differential operator, commutator,evolution equation, infinite-dimensional Hamiltonian canonical system, factorization of differential operator, commutatorProject supported by the National Natural Science Foundation of China (Grant No~10562002) and the Natural Science Foundation of Nei Mongol, China (Grant No~200508010103).2007-05-09{\partial}/{\partial x}Corresponding author．E-mail：alatanca@imu．edu.cn/qk/85823A/200711/25754042.html0200, 03405/9/2007 12:00:00 AMUsing factorization viewpoint of differential operator, this paper discusses how to transform a nonlinear evolution equation to infinite-dimensional Hamiltonian linear canonical formulation. It proves a sufficient condition of canonical factorization of operator, and provides a kind of mechanical algebraic method to achieve canonical $`{\partial}/{\partial x}'$-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.http://cpb.iphy.ac.cn/CN/10.1088/1009-1963/16/11/002https://cpb.iphy.ac.cn/CN/article/downloadArticleFile.do?attachType=PDF&id=1088382007-11-20'-type expression, correspondingly. Then three examples are given, which show the application of the obtained algorithm. Thus a novel idea for inverse problem can be derived feasibly.     

奇异系统Hamilton正则方程的Mei对称性、Noether对称性和Lie对称性

研究奇异系统Hamilton正则方程的形式不变性即Mei对称性，给出其定义、确定方程、限制方程和附加限制方程.研究奇异系统Hamilton正则方程的Mei对称性与Noether对称性、Lie对称性之间的关系，寻求系统的守恒量.给出一个例子说明结果的应用. 关键词： 奇异系统 Hamilton正则方程 约束 对称性 守恒量     

Global Structure Stability of Riemann Solution of Quasilinear Hyperbolic System of Conservation Law in Presence of Boundary: Shocks and Contact Discontinuities

In this paper, we investigate a class of mixed initial-boundary value problems for a kind of n×n quasilinear hyperbolic systems of conservation laws on the quarter plan. We show that the structure of the piecewise C¹ solution u=u(t,x) of the problem, which can be regarded as a perturbation of the corresponding Riemann problem, is globally similar to that of the solution u=U(x/t) of the corresponding Riemann problem. The piecewise C¹ solution u=u(t,x) to this kind of problems is globally structure-stable if and only if it contains only non-degenerate shocks and contact discontinuities, but no rarefaction waves and other weak discontinuities.     

Integrating factors and conservation theorems for Hamilton‘s canonical equations of motion of variable mass nonholonmic nonconservative dynamical systems

We present a general approach to the construction of conservation laws for variable mass noholonmic nonconservative systems.First,we give the definition of integrating factors,and we study in detail the necessary conditions for the existence of the conserved quantities,Then,we establish the conservatioin theorem and its inverse theorem for Hamilton‘s canonical equations of motion of variable mass nonholonomic nonocnservative dynamical systems.Finally,we give an example to illustrate the application of the results.     

An integrable Hamiltonian hierarchy, a high-dimensional loop algebra and associated integrable coupling system

A subalgebra of loop algebra ?_2 is established. Therefore, a new isospectral problem is designed. By making use of Tu's scheme, a new integrable system is obtained, which possesses bi-Hamiltonian structure. As its reductions, a formalism similar to the well-known Ablowitz－Kaup－Newell－Segur (AKNS) hierarchy and a generalized standard form of the Schr?dinger equation are presented. In addition, in order for a kind of expanding integrable system to be obtained, a proper algebraic transformation is supplied to change loop algebra ?_2 into loop algebra ?_1. Furthermore, a high-dimensional loop algebra is constructed, which is different from any previous one. An integrable coupling of the system obtained is given. Finally, the Hamiltonian form of a binary symmetric constrained flow of the system obtained is presented.     

Lie symmetry analysis,optimal system and conservation laws of a new(2+1)-dimensional KdV system

In this paper, Lie point symmetries of a new(2+1)-dimensional KdV system are constructed by using the symbolic computation software Maple. Then, the one-dimensional optimal system,associated with corresponding Lie algebra, is obtained. Moreover, the reduction equations and some explicit solutions based on the optimal system are presented. Finally, the nonlinear selfadjointness is provided and conservation laws of this KdV system are constructed.