SKYRMIONS IN GROSS-PITAEVSKII FUNCTIONALS

In Bose-Einstein condensates （BECs）, skyrmions can be characterized by pairs of linking vortex rings coming from two-component wave functions. Here we construct skyrmions by studying critical points of Gross-Pitaevskii functionals with two-component wave functions. Using localized energy method, we rigorously prove the existence, and describe the configurations of skyrmions in such BECs.

Numerical Simulation of Free Surface by an Area-Preserving Level Set Method

We apply in this study an area preserving level set method to simulate gas/water interface flow.For the sake of accuracy,the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the fifthorder accurate upwinding combined compact difference(UCCD)scheme.This scheme development employs two coupled equations to calculate the first-and second-order derivative terms in the momentum equations.For accurately predicting the level set value,the interface tracking scheme is also developed to minimize phase error of the first-order derivative term shown in the pure advection equation.For the purpose of retaining the long-term accurate Hamiltonian in the advection equation for the level set function,the time derivative term is discretized by the sixth-order accurate symplectic Runge-Kutta scheme.Also,to keep as a distance function for ensuring the front having a finite thickness for all time,the re-initialization equation is used.For the verification of the optimized UCCD scheme for the pure advection equation,two benchmark problems have been chosen to investigate in this study.The level set method with excellent area conservation property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam-break,Rayleigh-Taylor instability,two-bubble rising in water,and droplet falling problems.

Development of an Explicit Symplectic Scheme that Optimizes the Dispersion-Relation Equation of the Maxwell’s Equations

In this paper an explicit finite-difference time-domain scheme for solving the Maxwell’s equations in non-staggered grids is presented.The proposed scheme for solving the Faraday’s and Amp`ere’s equations in a theoretical manner is aimed to preserve discrete zero-divergence for the electric and magnetic fields.The inherent local conservation laws in Maxwell’s equations are also preserved discretely all the time using the explicit second-order accurate symplectic partitioned Runge-Kutta scheme.The remaining spatial derivative terms in the semi-discretized Faraday’s and Amp`ere’s equations are then discretized to provide an accurate mathematical dispersion relation equation that governs the numerical angular frequency and the wavenumbers in two space dimensions.To achieve the goal of getting the best dispersive characteristics,we propose a fourth-order accurate space centered scheme which minimizes the difference between the exact and numerical dispersion relation equations.Through the computational exercises,the proposed dual-preserving solver is computationally demonstrated to be efficient for use to predict the long-term accurate Maxwell’s solutions.