Inner topological structure of Hopf invariant

In light of $\phi$-mapping topological current theory, the inner topological structure of Hopf invariant is investigated. It is revealed that Hopf invariant is just the winding number of Gauss mapping. According to the inner structure of topological current, a precise expression for Hopf invariant is also presented. It is the total sum of all the self-linking and all the linking numbers of the knot family.Comment: 13pages, no figure. Accepted by J.Math.Phy

Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 Taylor & Francis.In this paper, a numerical implementation of a direct united boundary-domain integral equation (BDIE) related to the Neumann boundary value problem for a scalar elliptic partial differential equation with a variable coefficient is discussed. The BDIE is reduced to a uniquely solvable one by adding an appropriate perturbation operator. The mesh-based discretization of the BDIEs with quadrilateral domain elements leads to a system of linear algebraic equations (discretized BDIE). Then, the system is solved by LU decomposition and Neumann iterations. Convergence of the iterative method is discussed in relation to the distribution of eigenvalues of the corresponding discrete operators calculated numerically.The work was supported by the grant EP/H020497/1 "Mathematical analysis of localised boundary-domain integral equations for BVPs with variable coefficients" of the EPSRC, UK

Modular Solutions to Equations of Generalized Halphen Type

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus zero and have $q$--series with integral coefficients. Rational maps relating these functions are derived, implying subgroup relations between their automorphism groups, as well as symmetrization maps relating the associated differential systems.Comment: PlainTeX 36gs. (Formula for Hecke operator corrected.

Characteristic Lie rings, finitely-generated modules and integrability conditions for 2+1 dimensional lattices

Characteristic Lie rings for Toda type 2+1 dimensional lattices are defined. Some properties of these rings are studied. Infinite sequence of special kind modules are introduced. It is proved that for known integrable lattices these modules are finitely generated. Classification algorithm based on this observation is briefly discussed.Comment: 11 page

Supersymmetric Biorthogonal Quantum Systems

We discuss supersymmetric biorthogonal systems, with emphasis given to the periodic solutions that occur at spectral singularities of PT symmetric models. For these periodic solutions, the dual functions are associated polynomials that obey inhomogeneous equations. We construct in detail some explicit examples for the supersymmetric pairs of potentials V_{+/-}(z) = -U(z)^2 +/- z(d/(dz))U(z) where U(z) = \sum_{k>0}u_{k}z^{k}. In particular, we consider the cases generated by U(z) = z and z/(1-z). We also briefly consider the effects of magnetic vector potentials on the partition functions of these systems.Comment: Changes are made to conform to the published version. In particular, some errors are corrected on pp 12-1

Linear stability analysis of resonant periodic motions in the restricted three-body problem

The equations of the restricted three-body problem describe the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$, $0\leq \mu \leq 1/2$, that circle each other with period equal to $2\pi$. When $\mu=0$, the problem admits orbits for the massless particle that are ellipses of eccentricity $e$ with the primary of mass 1 located at one of the focii. If the period is a rational multiple of $2\pi$, denoted $2\pi p/q$, some of these orbits perturb to periodic motions for $\mu > 0$. For typical values of $e$ and $p/q$, two resonant periodic motions are obtained for $\mu > 0$. We show that the characteristic multipliers of both these motions are given by expressions of the form $1\pm\sqrt{C(e,p,q)\mu}+O(\mu)$ in the limit $\mu\to 0$. The coefficient $C(e,p,q)$ is analytic in $e$ at $e=0$ and C(e,p,q)=O(e^{\abs{p-q}}). The coefficients in front of e^{\abs{p-q}}, obtained when $C(e,p,q)$ is expanded in powers of $e$ for the two resonant periodic motions, sum to zero. Typically, if one of the two resonant periodic motions is of elliptic type the other is of hyperbolic type. We give similar results for retrograde periodic motions and discuss periodic motions that nearly collide with the primary of mass $1-\mu$

Vector and Spinor Decomposition of SU(2) Gauge Potential, their quivalence and Knot Structure in SU(2) Chern-Simons Theory

In this paper, spinor and vector decomposition of SU(2) gauge potential are presented and their equivalence is constructed using a simply proposal. We also obtain the action of Faddeev nonlinear O(3) sigma model from the SU(2) massive gauge field theory which is proposed according to the gauge invariant principle. At last, the knot structure in SU(2) Chern-Simons filed theory is discussed in terms of the $\phi$--mapping topological current theory. The topological charge of the knot is characterized by the Hopf indices and the Brouwer degrees of $\phi$-mapping.Comment: 10 pages, ni figur

Magnetic Monopoles in Ferromagnetic Spin-Triplet Superconductors

Using the $\phi$-mapping method, we argue that ferromagnetic spin-triplet superconductors allow formation of unstable magnetic monopoles. In particular, we show that the limit points and the bifurcation points of the $\phi$-mapping will serve as the interaction points of these magnetic monopoles.Comment: 4 pages, no figure

Topological Properties of Spatial Coherence Function

Topology of the spatial coherence function is considered in details. The phase singularity (coherence vortices) structures of coherence function are classified by Hopf index and Brouwer degree in topology. The coherence flux quantization and the linking of the closed coherence vortices are also studied from the topological properties of the spatial coherence function.Comment: 9 page

Topological Excitation in Skyrme Theory

Based on the $\phi$-mapping topological current theory and the decomposition of gauge potential theory, we investigate knotted vortex lines and monopoles in Skyrme theory and simply discuss the branch processes (splitting, merging and intersection) during the evolution of the monopoles.Comment: 10 pages, 0 figure