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By making use of the decomposition of U(1) gauge potential theory and the -mapping method we discuss a mixture of interacting neutral and charged Bose condensates, which is supposed being realized in the interior of neutron stars in the form of a coexistent neutron superfluid and protonic superconductor. We propose that this system possesses vortex lines and two classes of knotted solitons. The topological charge of the vortex lines are characterized by the Hopf indices and the Brower degrees of -mapping, and the knotted solitons are described by nontrivial Hopf invariant and the BF action respectively.

By making use of the -mapping topological current theory, this paper shows that the Gauss?Bonnet?Chern density (the Euler?Poincar? characteristic ?(M) density) can be expressed in terms of a smooth vector field and take the form of ?(), which means that only the zeros of contribute to ?(M). This is the elementary fact of the Hopf theorem. Furthermore, it presents that a new topological tensor current of -branes can be derived from the Gauss?Bonnet?Chern density. Using this topological current, it obtains the generalized Nambu action for multi -branes.

By making use of the U(1) gauge potential decomposition theory and the -mapping topological current theory, we investigate the Schrodinger–Chern–Simons model in the thin-film superconductor system and obtain an exact Bogomolny self-dual equation with a topological term. It is revealed that there exist self-dual vortices in the system. We study the inner topological structure of the self-dual vortices and show that their topological charges are topologically quantized and labeled by Hopf indices and Brouwer degrees. Furthermore, the vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the vector field .

We study the properties of the Ginzburg–Landau model at the dual point for the superconductors. By making use of the U(1) gauge potential decomposition and the -mapping theory, we investigate the topological inner structure of the Bogomol'nyi equations and deduce a modified decoupled Bogomol'nyi equation with a nontrivial topological term, which is ignored in conventional model. We find that the nontrivial topological term is closely related to the N-vortex, which arises from the zero points of the complex scalar field. Furthermore, we establish a relationship between Ginzburg–Landau free energy and the winding number.

In this paper, a novel decomposition expression for the U(1) gauge field in liquid crystals (LCs) is derived. Using this decomposition expression and the -mapping topological current theory, we investigate the topological structure of the vortex lines in LCs in detail. A topological invariant, i.e., the Chern–Simons (CS) action for the knotted vortex lines is presented, and the CS action is shown to be the total sum of all the self-linking and linking numbers of the knot family. Moreover, it is pointed out that the CS action is preserved in the branch processes of the knotted vortex lines.

A topological theory of liquid crystal films in the presence of defects is developed based on the phi-mapping topological current theory. By,generalizing the free-energy density in "one-constant" approximation, a covariant free-energy density is obtained, from which the U(1) gauge field and the unified topological current for monopoles and strings in liquid crystals are derived. The inner topological structure of these topological defects is characterized by the winding numbers of phi-mapping.

In this paper, using the -mapping theory, it is shown that two kinds of topological defects, i.e., the vortex lines and the monopoles exist in the helical configuration of magnetic field in triplet superconductors. And the inner topological structure of these defects is studied. Because the knot solitons in the triplet superconductors are characterized by the Hopf invariant, we also establish a relationship between the Hopf invariant and the linking number of knots family, and reveal the inner topological structure of the Hopf invariant.

We quantitatively investigate the third-order optical nonlinear response of Co-doped ZnO thin films prepared by magnetron sputtering using the Z-scan method. The two-photon absorption and optical Kerr effect are revealed to contribute to the third-order nonlinear response of the Co-doped ZnO films. The nonlinear absorption coefficient β is determined to be approximately 8.8 × 10−5 cm/W and the third-order nonlinear susceptibility χ(3) is 2.93 × 10−6 esu. The defect-associated energy levels within the band gap are suggested to be responsible for the enhanced nonlinear response observed in Co-doped ZnO films.

By making use of the decomposition of U(1) gauge potential theory and the o-mapping method we discuss a mixture of interacting neutral and charged Bose condensates, which is supposed to be realized in the interior of neutron stars in the form of a coexistent neutron superfluid and protonic superconductor. We propose that this system possesses vortex line knotted solitons and the topological charges of vortex lines are characterized by the winding numbers of the o-mapping. Furthermore the spatial bifurcation of vortices is also discussed.

We investigate the spin relaxation time of holes in an ultrathin neutral InAs monolayer (1.5 ML) and compare with that of electrons, using polarization-dependent time-resolved photoluminescence (TRPL) experiments. With excitation energies above the GaAs gap, we observe a rather slow relaxation of holes (tau(1h) = 196 +/- 17 ps) that is in the magnitude similar to electrons (tau(1e) = 354 +/- 32 ps) in this ultrathin sample. The results are in good agreement with earlier theoretical prediction, and the phonon scattering due to spin-orbit coupling is realized to play a dominant role in the carrier spin kinetics.