约束系统的变换和推广的Killing方程

本文从约束系统的作用量和约束方程的普遍变换性质出发,得到了约束系统的推广Killing方程组,此方程组的解所生成的变换可产生经典Noether定理的守恒量。讨论了连续系统的时空变换和内禀变换,在变换保持约束方程不变时,指出此变换能导致守恒流的充要条件,给出了对不可压缩连续介质的应用,讨论了对广义力学和经典力学的应用,并给出了推广Killing方程组解的某些实例,将Poincare不变量推广到了受约束广义力学系统。 关键词：     

Breaking soliton equations and negative-order breaking soliton equations of typical and higher orders

We develop breaking soliton equations and negative-order breaking soliton equations of typical and higher orders. The recursion operator of the KdV equation is used to derive these models. We establish the distinct dispersion relation for each equation. We use the simplified Hirota’s method to obtain multiple soliton solutions for each developed breaking soliton equation. We also develop generalized dispersion relations for the typical breaking soliton equations and the generalized negative-order breaking soliton equations. The results provide useful information on the dynamics of the relevant nonlinear negative-order equations.     

Conditional Symmetry Groups of Nonlinear Diffusion Equations with x-Dependent Convection and Absorption

The generalized conditional symmetry and sign-invariant approaches are developed to study the nonlinear diffusion equations with x-dependent convection and source terms. We obtain conditions under which the equations admit the second-order generalized conditional symmetries and the first-order sign-invariants on the solutions. Several types of different generalized conditional symmetries and first-order sign-invariants for the equations with diffusion of power law are obtained. Exact solutions to the resulting equations are constructed.     

Nonlinear mean field Fokker-Planck equations. Application to the chemotaxis of biological populations

We study a general class of nonlinear mean field Fokker-Planck equations in relation with an effective generalized thermodynamical (E.G.T.) formalism. We show that these equations describe several physical systems such as: chemotaxis of bacterial populations, Bose-Einstein condensation in the canonical ensemble, porous media, generalized Cahn-Hilliard equations, Kuramoto model, BMF model, Burgers equation, Smoluchowski-Poisson system for self-gravitating Brownian particles, Debye-Hückel theory of electrolytes, two-dimensional turbulence... In particular, we show that nonlinear mean field Fokker-Planck equations can provide generalized Keller-Segel models for the chemotaxis of biological populations. As an example, we introduce a new model of chemotaxis incorporating both effects of anomalous diffusion and exclusion principle (volume filling). Therefore, the notion of generalized thermodynamics can have applications for concrete physical systems. We also consider nonlinear mean field Fokker-Planck equations in phase space and show the passage from the generalized Kramers equation to the generalized Smoluchowski equation in a strong friction limit. Our formalism is simple and illustrated by several explicit examples corresponding to Boltzmann, Tsallis, Fermi-Dirac and Bose-Einstein entropies among others.     

Generalizations of the finite nonperiodic Toda lattice and its Darboux transformation

In this paper, we construct Hamiltonian systems for 2 N particles whose force depends on the distances between the particles. We obtain the generalized finite nonperiodic Toda equations via a symmetric group transformation. The solutions of the generalized Toda equations are derived using the tau functions. The relationship between the generalized nonperiodic Toda lattices and Lie algebras is then be discussed and the generalized Kac-van Moerbeke hierarchy is split into generalized Toda lattices, whose integrability and Darboux transformation are studied.     

Geodesic Motion on Extended Taub-NUT Spinning Space

In this paper we investigate the geodesic motion of the pseudo-classical spinning particle for the extended Taub-NUT metric. The generalized equations for spinning space are investigated and the constants of motion are derived in terms of the solutions of these equations. We find only two types of extended Taub-NUT metrics with Kepler type symmetry admitting Killing-Yano tensors. The solutions for the lowest components of generalized Killing equations are presented for a particular form of extended Taub-NUT metric.     

On the nonequilibrium statistical mechanics of a binary mixture. II. The transport coefficients

This paper is devoted to the study of the hydrodynamic stage of a two-component dense fluid. Starting from the BBGKY hierarchy obtained earlier, we first derive the expressions for the generalized fluxes. We proceed to set up the generalized kinetic equations, using Bogoliubov's functional assumption. Then we solve these equations by means of a Chapman-Enskog method. The generalized expressions for the transport coefficients are thus obtained. All our results are independent of the existence of density expansions of the relevant quantities.     

Noether-type theory for discrete mechanico-electrical dynamical systems with nonregular lattices

We investigate Noether symmetries and conservation laws of the discrete mechanico-electrical systems with nonregular lattices.The operators of discrete transformation and discrete differentiation to the right and left are introduced for the systems.Based on the invariance of discrete Hamilton action on nonregular lattices of the systems with the dissipation forces under the infinitesimal transformations with respect to the time,generalized coordinates and generalized charge quantities,we work out the discrete analog of the generalized variational formula.From this formula we derive the discrete analog of generalized Noether-type identity,and then we present the generalized quasi-extremal equations and properties of these equations for the systems.We also obtain the discrete analog of Noether-type conserved laws and the discrete analog of generalized Noether theorems for the systems.Finally we use an example to illustrate these results.     

Pseudodifferential Operators of Several Variables and Baker Functions

The KP hierarchy consists of an infinite system of nonlinear partial differential equations and is determined by Lax equations, which can be constructed using pseudodifferential operators. The KP hierarchy and the associated Lax equations can be generalized by using pseudodifferential operators of several variables. We construct Baker functions associated to those generalized Lax equations of several variables and prove some of the properties satisfied by such functions.     

Hurst exponents for interacting random walkers obeying nonlinear Fokker-Planck equations

Anomalous diffusion of random walks has been extensively studied for the case of non-interacting particles. Here we study the evolution of nonlinear partial differential equations by interpreting them as Fokker-Planck equations arising from interactions among random walkers. We extend the formalism of generalized Hurst exponents to the study of nonlinear evolution equations and apply it to several illustrative examples. They include an analytically solvable case of a nonlinear diffusion constant and three nonlinear equations which are not analytically solvable: the usual Fisher equation which contains a quadratic nonlinearity, a generalization of the Fisher equation with density-dependent diffusion constant, and the Nagumo equation which incorporates a cubic rather than a quadratic nonlinearity. We estimate the generalized Hurst exponents.     

Collective Excitations in Spin-2 Bose-Einstein Condensates

The Green＇s functions and the correlation functions in spin-2 Bose-Einstein condensates at finite temperature are defined and the generalized Dyson-Beliaev equations are introduced. We discuss the spin conservation in z direction and decouple the Green＇s functions and the generalized Dyson-Beliaev equations according to different spin conservations in z direction. The anomalous vertex functions are introduced and the self-energies are separated into the proper self-energies and the improper self-energies. The generalized Dyson-Beliaev equations are decoupled according to separation of the self-energies. We calculate the Green＇s functions step by step in the Bogoliubov approximation and discuss the collective excitations in spin-2 Bose-Einstein condensates in the polar, ferromagnetic, and cyclic cases, respectively.     

Initial-value Problems for Extended KdV-Burgers Equations via Generalized Conditional Symmetries

We classify initial-value problems for extended KdV-Burgers equations via generalized conditional symmetries. These equations can be reduced to Cauchy problems for some systems of first-order ordinary differential equations. The obtained reductions cannot be derived within the framework of the standard Lie approach.     

Perturbed beta–gamma systems and complex geometry

We consider the equations, arising as the conformal invariance conditions of the perturbed curved beta–gamma system. These equations have the physical meaning of Einstein equations with a B-field and a dilaton on a Hermitian manifold, where the B-field 2-form is imaginary and proportional to the canonical form associated with Hermitian metric. We show that they decompose into linear and bilinear equations and lead to the vanishing of the first Chern class of the manifold where the system is defined. We discuss the relation of these equations to the generalized Maurer–Cartan structures related to BRST operator. Finally we describe the relations of the generalized Maurer–Cartan bilinear operation and the Courant/Dorfman brackets.     

Generalized linear Schrödinger equations and their Darboux transformations

We construct explicit Darboux transformations of arbitrary order for a class of generalized, linear Schrödinger equations. Our construction contains the well-known Darboux transformations for Schrödinger equations with position-dependent mass, Schrödinger equations coupled to a vector potential and Schrödinger equations for weighted energy.     

The Boltzmann equation from quantum field theory

We show from first principles the emergence of classical Boltzmann equations from relativistic nonequilibrium quantum field theory as described by the Kadanoff–Baym equations. Our method applies to a generic quantum field, coupled to a collection of background fields and sources, in a homogeneous and isotropic spacetime. The analysis is based on analytical solutions to the full Kadanoff–Baym equations, using the WKB approximation. This is in contrast to previous derivations of kinetic equations that rely on similar physical assumptions, but obtain approximate equations of motion from a gradient expansion in momentum space. We show that the system follows a generalized Boltzmann equation whenever the WKB approximation holds. The generalized Boltzmann equation, which includes off-shell transport, is valid far from equilibrium and in a time dependent background, such as the expanding universe.     

Light-Induced Hofstadter＇s Butterfly Spectrum of Ultracold Atoms on the Two-Dimensional Kagomé Lattice

We investigate the energy spectrum of ultracold atoms on the two-dimensional Kagomé optical lattice under an effective magnetic field, which can be realized with laser beams. We derive the generalized Harper＇s equations from the Schr？dinger equation. The energy spectrum with a fractal band structure is obtained by numerically solving the generalized Harper＇s equations. We analyze the properties of the Hofstadter＇s butterfly spectrum and discuss its observability.     

Spinning Particles in Curved Spacetime

We briefly discuss the relevant equations for the motion of spinning particles in curved spacetime. We describe the generalized Killing equations for spinning spaces and derive the constants of motion. We apply the formalism to solve for the motion of a pseudoclassical spinning particle in Schwarzschild–de Sitter spacetime.     

Extension of Variable Separable Solutions for Nonlinear Evolution Equations

We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting equations.     

Linearizability conditions for the Rayleigh-like oscillators

In this work we consider a family of nonlinear oscillators that is cubic with respect to the first derivative. Particular members of this family of equations often appear in numerous applications. We solve the linearization problem for this family of equations, where as equivalence transformations we use generalized nonlocal transformations. We explicitly find correlations on the coefficients of the considered family of equations that give the necessary and sufficient conditions for linearizability. We also demonstrate that each linearizable equation from the considered family admits an autonomous Liouvillian first integral, that is Liouvillian integrable. Furthermore, we demonstrate that linearizable equations from the considered family does not possess limit cycles. Finally, we illustrate our results by two new examples of the Liouvillian integrable nonlinear oscillators, namely by the Rayleigh–Duffing oscillator and the generalized Duffing–Van der Pol oscillator.