arXiv subjects
Publications and source records attributed to Ke Wu.
The integrable inhomogeneous extension of the Lakshmanan-Myrzakulov equation is constructed by using the prolongation structure theory. The corresponding L-equivalent counterpart is also given, which is the (2+1)-dimensional generalized NLSE.
By using the prolongation structure theory proposed by Morris, we give a (2+1)-dimensional integrable inhomogeneous Heisenberg Ferromagnet models, namely, the inhomogeneous Myrzakulov I equation. Through the motion of space curves endowed with an additional spatial variable, its geometrical equivalent counterpart is also presented.
We study the general representations of positive partial transpose (PPT) states in ${\cal C}^K \otimes {\cal C}^M \otimes {\cal C}^N$. For the PPT states with rank-$N$ a canonical form is obtained, from which a sufficient separability condition is presented.
We propose the difference discrete variational principle in discrete mechanics and symplectic algorithm with variable step-length of time in finite duration based upon a noncommutative differential calculus established in this paper. This approach keeps both symplicticity and energy conservation discretely. We show that there exists the discrete version of the Euler-Lagrange cohomology in these discrete systems. We also discuss the solution existence in finite time-length and its site density in continuous limit, and apply our approach to the pendulum with periodic perturbation. The numerical results are satisfactory.
We give a canonical form of PPT states in ${\cal C}^2 \otimes {\cal C}^2\otimes {\cal C}^2 \otimes {\cal C}^N$ with rank=$N$. From this canonical form a necessary separable condition for these states is presented.
We consider the separability of rank two quantum states on multiple quantum spaces with different dimensions. The sufficient and necessary conditions for separability of these multiparty quantum states are explicitly presented. A nonseparability inequality is also given, for the case where one of the eigenvectors corresponding to nonzero eigenvalues of the density matrix is maximally entangled.
The definition and properties of the Euler-Lagrange cohomology groups $H^{2k-1}$, $1 \leqslant k \leqslant n$, on a symplectic manifold $({\cal M}^{2n},ω)$ are given and studied. For $k = 1$ and $k = n$, they are isomorphic to the corresponding de Rham cohomology groups $H_{dR}^1({\cal M}^{2n})$ and $H_{dR}^{2n-1}({\cal M}^{2n})$, respectively. The other Euler-Lagrange cohomology groups are different from either the de Rham cohomology groups or the harmonic cohomology groups on $({\cal M}^{2n},ω)$, in general. The general volume-preserving equations on $({\cal M}^{2n},ω)$ are also presented from cohomological point of view. In the special cases, these equations become the ordinary canonical equations in the Hamilton mechanics. Therefore, the Hamilton mechanics has been generalized via the cohomology.
We briefly introduce the conception on Euler-Lagrange cohomology groups on a symplectic manifold $(\mathcal{M}^{2n}, ω)$ and systematically present the general form of volume-preserving equations on the manifold from the cohomological point of view. It is shown that for every volume-preserving flow generated by these equations there is an important 2-form that plays the analog role with the Hamiltonian in the Hamilton mechanics. In addition, the ordinary canonical equations with Hamiltonian $H$ are included as a special case with the 2-form $\frac{1}{n-1} H ω$. It is studied the other volume preserving systems on $({\cal M}^{2n}, ω)$. It is also explored the relations between our approach and Feng-Shang's volume-preserving systems as well as the Nambu mechanics.
In this letter, we present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M, ω). It is shown that every volume-preserving flow has some 2-forms acting the role of the Hamiltonian functions in the Hamiltonian mechanics and the ordinary Hamilton equations are included as a special case with a 2-form \frac{1}{n-1} H ωwhere H is the corresponding Hamiltonian.
Explicit sufficient and necessary conditions for separability of $N$-dimensional rank two multiparty quantum mixed states are presented. A nonseparability inequality is also given, for the case where one of the eigenvectors corresponding to nonzero eigenvalues of the density matrix is maximally entangled.
Discrete Mechanics based on finite element methods is presented in this paper. We also explore the relationship between this discrete mechanics and Veselov discrete mechanics. High order discretizations are constructed in terms of high order interpolations.
The Noether current and its variation relation with respect to diffeomorphism invariance of gravitational theories have been derived from the horizontal variation and vertical-horizontal bi-variation of the Lagrangian, respectively. For Einstein's GR in the stationary, axisymmetric black holes, the mass formula in vacuum can be derived from this Noether current although it definitely vanishes. This indicates that the mass formula of black holes is a vanishing Noether charge in this case. The first law of black hole thermodynamics can also be derived from the variation relation of this vanishing Noether current.
Some problems on variations are raised for classical discrete mechanics and field theory and the difference variational approach with variable step-length is proposed motivated by Lee's approach to discrete mechanics and the difference discrete variational principle for difference discrete mechanics and field theory on regular lattice. Based upon Hamilton's principle for the vertical variations and double operation of vertical exterior differential on action, it is shown that for both continuous and variable step-length difference cases there exists the nontrivial Euler-Lagrange cohomology as well as the necessary and sufficient condition for symplectic/multi-symplectic structure preserving properties is the relevant Euler-Lagrange 1-form is closed in both continuous and difference classical mechanics and field theory. While the horizontal variations give rise to the relevant identities or relations of the Euler-Lagrange equation and conservation law of the energy/energy-momentum tensor for continuous or discrete systems. The total variations are also discussed. Especially, for those discrete cases the variable step-length of the difference is determined by the relation between the Euler-Lagrange equation and conservation law of the energy/energy-momentum tensor. In addition, this approach together with difference version of the Euler-Lagrange cohomology can be applied not only to discrete Lagrangian formalism but also to the Hamiltonian formalism for difference mechanics and field theory.
We present a discrete total variation calculus in Hamiltonian formalism in this paper. Using this discrete variation calculus and generating functions for the flows of Hamiltonian systems, we derive two-step symplectic-energy integrators of any finite order for Hamiltonian systems from a variational perspective. The relationship between symplectic integrators derived directly from the Hamiltonian systems and the variationally derived symplectic-energy integrators is explored.
A discrete total variation calculus with variable time steps is presented in this letter. Using this discrete variation calculus, we generalize Lee's discrete mechanics and derive variational symplectic-energy-momentum integrators by Kane, Marsden and Ortiz. The relationship among discrete total variation, Lee's discrete mechanics and Kane-Marsden-Ortiz's integrators is explored.
By the simple finite element method, we study the symplectic, multisymplectic structures and relevant preserving properties in some semi-linear elliptic boundary value problem in one-dimensional and two-dimensional spaces respectively. We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimentional case respectively. These results are in fact the intrinsic reason that the numerical experiments indicate that such finite element schemes are accurate in practice.
We find that with uniform mesh, the numerical schemes derived from finite element method can keep a preserved symplectic structure in one-dimensional case and a preserved multisymplectic structure in two-dimentional case in certain discrete version respectively. These results are in fact the intrinsic reason that the numerical experiments indicate that such finite element algorithms are accurate in practice.
The axial anomaly in lattice gauge theories has a topological nature when the Dirac operator satisfies the Ginsparg-Wilson relation. We study the axial anomaly in Abelian gauge theories on an infinite hypercubic lattice by utilizing cohomological arguments. The crucial tool in our approach is the non-commutative differential calculus~(NCDC) which makes the Leibniz rule of exterior derivatives valid on the lattice. The topological nature of the ``Chern character'' on the lattice becomes manifest in the context of NCDC. Our result provides an algebraic proof of Lüscher's theorem for a four-dimensional lattice and its generalization to arbitrary dimensions.