Three-Dimensional Multi-mesh Material Point Method for Solving Collision Problems

A new multi-mesh contact algorithm for three-dimensional material point method is presented. The contact algorithm faithfully recovers the opposite acting forces between colliding bodies. Collision procedures between regular bodies and/or rigid bodies are treated within the same framework. Multi-value of momentum and mass are defined on every node to describe the contact/sliding/separation procedure. Both normal and tangential velocities of each particle at the contact surface are calculated in respective individual mesh. A Coulomb friction is applied to describe the sliding or slipping between the contacting bodies. The efficiency of the contact algorithm is linearly related to the number of the contacting bodies because the overlapped nodes are labeled by sweeping the material particles of all bodies when the nodal momentum and mass are formed at every time step. Numerical simulation shows that our contact algorithm possesses high accuracy and low numerical energy dissipation, which is very important for solving collision problems.     

Analytical investigation velocity fields on mean and turbulent of a plane jet

This paper analyses the downstream developments of the mean and the turbulent velocity fields of a plane jet. Based on the conservation of mass and the conservation of momentum, the mean-velocity half width （reflecting the jet spread rate） and the relative mass flow rate （jet entrainment） are related to the decay rate of the centreline mean velocity. These relations are not subject to self-preservation. Both analytical and experimental results suggest that the jet spread rate （K1） and the entrainment rate （K3） （and thus the decay rate K2） can be well estimated from the centreline velocity, i.e., K1 ≈ 0.6K2 and K3 ∝K2. The effect of initial mean velocity and RMS velocity profiles on the downstream mean velocity field appears to be embodied in the constants K1 K2 and K3. The analytical relationship for the self-preserving Reynolds shear stress, obtained for the first time, works well.     

What do we learn from the ρ-π puzzle

The experimental observation indicates that the branching ratio of ψ＇→ρπ is very small while the ρ-π channel is a main one in J/ψ decays. To understand the puzzle, various interpretations have been proposed. Meanwhile according to the hadronic helicity selection rule, this decay mode should be suppressed. Numerical calculations are needed to determine how it is suppressed. We calculate the branching ratios of J/ψ→ρπ and ππ in the framework of QCD. The results show that the branching ratios are proportional to （mu＋md/MJ/ψ）^2 for the ρπ mode and （mu-md/MJ/ψ）^2 for the ππ mode which is isospin violated. The theoretical prediction of the ratio of J/ ψ→ρπ is smaller than data, but not too small to invoke a completely new mechanism. Thus the puzzle is still standing even though we learn much knowledge towards the puzzle and this will help to finally interpret the puzzle and then gain a deeper insight to the heavy quarkonia.     

Collisions of Two Soatial Solitons in Inhomogeneous Nonlinear Media

Collisions of spatial solitons occurring in the nonlinear Schroeinger equation with harmonic potential are studied, using conservation laws and the split-step Fourier method. We find an analytical solution for the separation distance between the spatial solitons in an inhomogeneous nonlinear medium when the light beam is self-trapped in the transverse dimension. In the self-focusing nonlinear media the spatial solitons can be transmitted stably, and the interaction between spatial solitons is enhanced due to the linear focusing effect （and also diminished for the linear defocusing effect）. In the self-defocusing nonlinear media, in the absence of self-trapping or in the presence of linear self-defocusing, no transmission of stable spatial solitons is possible. However, in such media the linear focusing effect can be exactly compensated, and the spatial solitons can propagate through.     

Conservation Laws and Symmetries of the Levi Equation

The conservation laws of the Levi equation are presented. Two types of symmetry of the Levi equation hierarchy are deduced, Further it is proved that these symmetries construct an infinite-dimensional Lie algebra.     

Lie symmetry and Mei continuum conservation law of system＊

Lie symmetry and Mei conservation law of continuum Lagrange system are studied in this paper. The equation of motion of continuum system is established by using variational principle of continuous coordinates. The invariance of the equation of motion under an infinitesimal transformation group is determined to be Lie-symmetric. The condition of obtaining Mei conservation theorem from Lie symmetry is also presented. An example is discussed for applications of the results.     

Isospectral and Nonisospectral Lattice Hierarchies Associated with a Discrete Spectral Problem and Their Infinitely Many Conservation Laws

In this paper, based on the discrete zero curvature representation, isospectrai and nonisospectrai lattice hierarchies are proposed. By means of solving corresponding discrete spectral equations, we demonstrate the existence of infinitely many conservation laws for this two hierarchies and obtain the formulae of the corresponding conserved densities and associated fluxes.     

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.     

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 pieeewise C^1 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 C^1 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.     

Integrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem

Based on a new discrete three-by-three matrix spectral problem, a hierarchy of integrable lattice equations with three potentials is proposed through discrete zero-curvature representation, and the resulting integrable lattice equation reduces to the classical Toda lattice equation. It is shown that the hierarchy possesses a HamiItonian structure and a hereditary recursion operator. Finally, infinitely many conservation laws of corresponding lattice systems are obtained by a direct way.     

Riccati-type Baicklund transformations of nonisospectral and generalized variable-coefficient KdV equations

We extend the method of constructing Bgcklund transformations for integrable equations through Riccati equations to the nonisospectral and the variable-coefficient equations. By taking nonisospectral and generalized variable-coefficient Korteweg-de Vries （KdV） equations as examples, their Backlund transformations are obtained under a more generalized constrain condition. In addition, the Lax pairs and infinite numbers of conservation laws of these equations are given. Es- pecially, some classical equations such as the cylindrical KdV equation are just the special cases of the constrain condition.     

A note on teleparallel Killing vector fields in Bianchi type Ⅷ and Ⅸ space-times in teleparallel theory of gravitation

In this paper we classify Bianchi type Ⅷ and Ⅸ space-times according to their teleparallel Killing vector fields in the teleparallel theory of gravitation by using a direct integration technique.It turns out that the dimensions of the teleparallel Killing vector fields are either 4 or 5.From the above study we have shown that the Killing vector fields for Bianchi type VIII and IX space-times in the context of teleparallel theory are different from that in general relativity.     

A Super Extension of Kaup-Newell Hierarchy

With the help of the zero-curvature equation and the super trace identity, we derive a super extension of the Kaup-Newell hierarchy associated with a 3 × 3 matrix spectral problem and establish its super bi-Hamiltonian structures. Furthermore, infinite conservation laws of the super Kaup Newell equation are obtained by using spectral parameter expansions.     

A General Approach to the Construction of Conservation Laws for Birkhoffian Systems in Event Space

For a Birkhoffing system in the event space, a general approach to the construction of conservation laws is presented. The conservation laws are constructed by finding corresponding integrating factors for the parametric equations of the system. First, the parametric equations of the Birkhoffian system in the event space are established, and the definition of integrating factors for the system is given; second the necessary conditions for the existence of conservation laws are studied in detail, and the relation between the conservation laws and the integrating factors of the system is obtained and the generalized Killing equations for the determination of the integrating factors are given; finally, the conservation theorem and its inverse for the system are established, and an example is given to illustrate the application of the results.     

How Will a Particle Detector Respond in Massless Complex Scalar Field？

I study the response of a particle detector coupled to quantized massless complex scalar field in four dimensional Minkowski spacetime through nonlinear Lagrangian. I find that as in the real scalar field： the particle detector will not respond when it is in inertial motion; If accelerated in its own frame reference, it does respond and feel the same temperature. But different from the real scalar field case, the detector＇s transition amplitude is concerned with particle-antiparticle creation, and the response of the detector is （1/α^2 ＋ ε^2）/24π^2 times of that in real scalar field, with 1/α the accelerator of the detector and e the energy gap between the detector＇s two energy level. It is due to the nonlinear property of the coupling Lagrangian. Whether the total charge of the system constructed by the particle detector and vacuum is conserved is also considered and analyzed.     

An Extension of Algorithm on Symbolic Computations of Conserved Densities for High-Dimensional Nonlinear Systems

An improved algorithm for symbolic computations of polynomial-type conservation laws （PCLaws） of a general polynomial nonlinear system is presented. The algorithm is implemented in Maple and can be successfully used for high-dimensional models. Furthermore, the algorithm discards the restriction to evolution equations. The program can also be used to determine the preferences for a given parameterized nonlinear systems. The code is tested on several known nonlinear equations from the soliton theory.     

General Covariant Conservation Law of Energy-Momentum in f（R） Gravity

In the light of the local Lorentz transformations and the general Noether theorem, a new formulate of the general covariant energy-momentum conservation law in f（R） gravity is obtained, which does not depend on the coordinative choice.     

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.     

Conservation Laws and Analytic Soliton Solutions for Coupled Integrable Dispersionless Equations with Symbolic Computation

Under investigation in this paper are two coupled integrable dispersionless （CID） equations modeling the dynamics of the current-fed string within an external magnetic field. Through a set of the dependent variable transformations, the bilinear forms for the CID equations are derived. Based on the Hirota method and symbolic computation, the analytic N-soliton solutions are presented. Infinitely many conservation laws for the CID equations are given through the known spectral problem. Propagation characteristics and interaction behaviors of the solitons are analyzed graphically.     

On a New Three-Component Camassa-Holm Equation with Peakons

A new three-component Camassa-Holm equation is introduced. This system is endowed with a structure similar to the Camassa Holm equation. It has peakon solitons and conserves H^1-norm conservation law.