On realization of tangent cones of homologically area-minimizing compact singular submanifolds

We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].Comment: Section 6 got update

How to construct metrics to control distributions of homologically mass-minimizing currents

By Federer and Fleming there exist at least one mass-minimizing normal current in every real-valued homology class of a Riemannian manifold. However the regularity of the mass-minimizing currents and their distributions may generally be quite complicated. In this paper we shall study how to construct nice metrics so that (as functionals over smooth forms) almost all homologically mass-minimizing currents are just linear combinations over smooth submanifolds.Comment: 20 pages, 6 figures. Comments are welcom

On Extending Calibrations

This is a companion note of [Zhaa] (arXiv:1501.01836) where the extension of local calibration pairs of smooth submanifolds is discussed. Here we emphasize on the case of singular submanifolds. More precisely, we study when a calibration pair around the singular set of a submanifold can extend to a local calibration pair about the entire submanifold. Based upon [Zhab] (arXiv:1501.04681) several examples of particular interests under the view of calibrated geometry are considered.Comment: 10 pages, 6 figures, comments are welcom

On non-existence of solutions of the Dirichlet problem for the minimal surface system

Recently we made systematic developments \cite{x-y-z0} regarding Lawson-Osserman constructions in their 1977' Acta Math paper "Non-existence, non-uniqueness and irregularity of solutions to the minimal surface system" in the aspects of non-uniqueness and irregularity. In this note we generalize Lawson-Osserman's result on non-existence.Comment: Updated New Versio

On extending calibration pairs

The paper studies how to extend local calibration pairs to global ones in various situations. As a result, new discoveries involving mass-minimizing properties are exhibited. In particular, we show that a $\mathbb R$-homologically nontrivial connected submanifold $M$ of a smooth Riemannian manifold $X$ is homologically mass-minimizing for some metrics in the same conformal class. Moreover, several generalizations for $M$ with multiple connected components or for a mutually disjoint collection (see {\S}3.5) are obtained. For a submanifold with certain singularities, we also establish an extension theorem for generating global calibration pairs. By combining these results, we find that, in some Riemannian manifolds, there are homologically mass-minimizing smooth submanifolds which cannot be calibrated by any smooth calibration.Comment: Improved Version. arXiv admin note: text overlap with arXiv:1501.0183

Quench Dynamics in a Trapped Bose-Einstein Condensate with Spin-Orbit Coupling

We consider the phase transition dynamics of a trapped Bose-Einstein condensate subject to Raman-type spin-orbit coupling (SOC). By tuning the coupling strength the condensate is taken through a second order phase transition into an immiscible phase. We observe the domain wall defects produced by a finite speed quench is described by the Kibble-Zurek mechanism (KZM), and quantify a power law behavior for the scaling of domain number and formation time with the quench speed.Comment: 6 pages, 4 figure

Whittaker Modules for the Schr\"{o}dinger Algebra

In this paper, the property and the classification the simple Whittaker modules for the schr\"{o}dinger algebra are studied. A quasi-central element plays an important role in the study of Whittaker modules of level zero. For the Whittaker modules of nonzero level, our arguments use the Casimir element of semisimple Lie algebra $sl_2$ and the description of simple modules over conformal Galilei algebras by R. L\"{u}, V. Mazorchuk and K. Zhao

Dynamical spin-density waves in a spin-orbit-coupled Bose-Einstein condensate

Synthetic spin-orbit (SO) coupling, an important ingredient for quantum simulation of many exotic condensed matter physics, has recently attracted considerable attention. The static and dynamic properties of a SO coupled Bose-Einstein condensate (BEC) have been extensively studied in both theory and experiment. Here we numerically investigate the generation and propagation of a \textit{dynamical} spin-density wave (SDW) in a SO coupled BEC using a fast moving Gaussian-shaped barrier. We find that the SDW wavelength is sensitive to the barrier's velocity while varies slightly with the barrier's peak potential or width. We qualitatively explain the generation of SDW by considering a rectangular barrier in a one dimensional system. Our results may motivate future experimental and theoretical investigations of rich dynamics in the SO coupled BEC induced by a moving barrier.Comment: 8 pages, 8 figure

A Complete Representation Theorem for GG-martingales

In this paper we establish a complete representation theorem for $G$-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second order term, which corresponds to the second order derivative in Markovian case. The main ingredient of the paper is a new norm for that second order term, which is based on an operator introduced by Song [26].Comment: 22 page

Global well-posedness of the generalized KP-II equation in anisotropic Sobolev spaces

In this paper, we consider the Cauchy problem for the generalized KP-II equation \begin{eqnarray*} u_{t}-|D_{x}|^{\alpha}u_{x}+\partial_{x}^{-1}\partial_{y}^{2}u+\frac{1}{2}\partial_{x}(u^{2})=0,\alpha\geq4. \end{eqnarray*} The goal of this paper is two-fold. Firstly, we prove that the problem is locally well-posed in anisotropic Sobolev spaces H^{s_{1},\>s_{2}}(\R^{2}) with s_{1}>\frac{1}{4}-\frac{3}{8}\alpha, s_{2}\geq 0 and \alpha\geq4. Secondly, we prove that the problem is globally well-posed in anisotropic Sobolev spaces H^{s_{1},\>0}(\R^{2}) with -\frac{(3\alpha-4)^{2}}{28\alpha}<s_{1}\leq0. and \alpha\geq4. Thus, our global well-posedness result improves the global well-posedness result of Hadac (Transaction of the American Mathematical Society, 360(2008), 6555-6572.) when 4\leq \alpha\leq6.Comment: We correct some misprints. arXiv admin note: substantial text overlap with arXiv:1709.01983, arXiv:1712.0933