Ground state solution of fractional Schr\"odinger equations with a general nonlinearity

In this paper, we study the following fractional Schr\"odinger equation: \left\{\begin{gathered} {(- \Delta)^s}u + mu = f(u){\text{in}}{\mathbb{R}^N}, \hfill u \in {H^s}({\mathbb{R}^N}),{\text{}}u > 0{\text{on}}{\mathbb{R}^N}, \hfill \\ \end{gathered} \right. where $m>0$, $N>2s$, ${(- \Delta)^s}$, $s \in (0,1)$ is the fractional Laplacian. Using minimax arguments, we obtain a positive ground state solution under general conditions on $f$ which we believe to be almost optimal

Enhancing and suppressing radiation with some permeability-near-zero structures

Using some special properties of a permeability-near-zero material, the radiation of a line current is greatly enhanced by choosing appropriately the dimension of a dielectric domain in which the source lies and that of a permeability-near-zero shell. The radiation of the source can also be completely suppressed by adding appropriately another dielectric domain or an arbitrary perfect electric conductor (PEC) inside the shell. Enhanced directive radiation is also demonstrated by adding a PEC substrate.Comment: 6 pages, 5 figure

High-subwavelength-resolution imaging of multilayered structures consisting of alternating negative-permittivty and dielectric layers with flattened transmission curves

Multilayered structures consisting of alternating negative-permittivity and dielectric layers are explored to obtain high-resolution imaging of subwavelength objects. The peaks with the smallest |ky| (ky is the transverse wave vector) on the transmission curves, which come from the guided modes of the multilayered structures, can not be completely damped by material loss. This makes the amplitudes of the evanescent waves around these peaks inappropriate after transmitted through the imaging structures, and the imaging quality is not good. To solve such a problem, the permittivity of the dielectric layers is appropriately chosen to make these sharp peaks merge with their neighboring peaks. Wide flat upheavals are then generated on the transmission curves so that evanescent waves in a large range are transmitted through the structures with appropriate amplitudes. In addition, it is found that the sharp peaks with the smallest |ky| can be eliminated by adding appropriate coating layers and wide flat upheavals can also be obtained.Comment: 26 pages, 6 figure

Standing waves for a class of Schr\"odinger-Poisson equations in R3{\mathbb{R}^3} involving critical Sobolev exponents

We are concerned with the following Schr\"odinger-Poisson equation with critical nonlinearity: \left\{\begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u + \psi u = \lambda |u{|^{p - 2}}u + |u{|^4}u{\text{in}}{\mathbb{R}^3}, \hfill - {\varepsilon ^2}\Delta \psi = {u^2}{\text{in}}{\mathbb{R}^3},{\text{}}u > 0,{\text{}}u \in {H^1}({\mathbb{R}^3}), \hfill \end{gathered} \right. where $\varepsilon > 0$ is a small positive parameter, $\lambda > 0$, $3 < p \le 4$. Under certain assumptions on the potential $V$, we construct a family of positive solutions ${u_\varepsilon} \in {H^1}({\mathbb{R}^3})$ which concentrates around a local minimum of $V$ as $\varepsilon \to 0$.Comment: 40 page

Large Margin Softmax Loss for Speaker Verification

In neural network based speaker verification, speaker embedding is expected to be discriminative between speakers while the intra-speaker distance should remain small. A variety of loss functions have been proposed to achieve this goal. In this paper, we investigate the large margin softmax loss with different configurations in speaker verification. Ring loss and minimum hyperspherical energy criterion are introduced to further improve the performance. Results on VoxCeleb show that our best system outperforms the baseline approach by 15\% in EER, and by 13\%, 33\% in minDCF08 and minDCF10, respectively.Comment: submitted to Interspeech 2019. The code and models have been release

Quantum Renormalization Groups Based on Natural Orbitals

We propose a new concept upon the renormalization group (RG) procedure for an interacting many-electron correlated system in the framework of natural orbitals, and formulate an algorithm for this RG approach. To demonstrate its effectiveness, we apply this new approach on a quantum cluster-impurity model with four impurities in comparison with the exact diagonalization method. We also find a shortcut to dramatically improving this RG algorithm. Further discussion is presented with the cluster dynamical mean-field theory and multi-impurity/orbital Kondo problems.Comment: 4.5 pages, 4 figures, and 1 table; 3 pages Supl. with another 2 figures and 4 tables; In this version, we update the Supl. with more discussions and comparisons with DMR

Rational and semi-rational solutions of the nonlocal Davey-Stewartson equations

In this paper, the partially party-time ($PT$) symmetric nonlocal Davey-Stewartson (DS) equations with respect to $x$ is called $x$-nonlocal DS equations, while a fully $PT$ symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely breather, rational and semi-rational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the $x$-nonlocal DS equations, the usual ($2+1$)-dimensional breathers are periodic in $x$ direction and localized in $y$ direction. Nonsingular rational solutions are lumps, and semi-rational solutions are composed of lumps, breathers and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both $x$ and $y$ directions with parallels in profile, but localized in time. Nonsingular rational solutions are ($2+1$)-dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semi-rational solutions describe interactions of line rogue waves and periodic line waves.Comment: 23pages, 12 figures.This is the accepted version by Studies in Applied Mathematic

Interaction-Induced Characteristic Length in Strongly Many-Body Localized Systems

We propose a numerical method for explicitly constructing a complete set of local integrals of motion (LIOM) and definitely show the existence of LIOM for strongly many-body localized systems. The method combines exact diagonalization and nonlinear minimization, and gradually deforms the LIOM for the noninteracting case to those for the interacting case. By using this method we find that for strongly disordered and weakly interacting systems, there are two characteristic lengths in the LIOM. The first one is governed by disorder and is of Anderson-localization nature. The second one is induced by interaction but shows a discontinuity at zero interaction, showing a nonperturbative nature. We prove that the entanglement and correlation in any eigenstate extend not longer than twice the second length and thus the eigenstates of the system are `quasi-product states' with such a localization length.Comment: 5 pages, 2 figures; algorithm improve

The rational solutions of the mixed nonlinear Schr\"odinger equation

The mixed nonlinear Schr\"odinger (MNLS) equation is a model for the propagation of the Alfv\'en wave in plasmas and the ultrashort light pulse in optical fibers with two nonlinear effects of self-steepening and self phase-modulation(SPM), which is also the first non-trivial flow of the integrable Wadati-Konno-Ichikawa(WKI) system. The determinant representation $T_n$ of a n-fold Darboux transformation(DT) for the MNLS equation is presented. The smoothness of the solution $q^{[2k]}$ generated by $T_{2k}$ is proved for the two cases ( non-degeneration and double-degeneration ) through the iteration and determinant representation. Starting from a periodic seed(plane wave), rational solutions with two parameters $a$ and $b$ of the MNLS equation are constructed by the DT and the Taylor expansion. Two parameters denote the contributions of two nonlinear effects in solutions. We show an unusual result: for a given value of $a$, the increasing value of $b$ can damage gradually the localization of the rational solution, by analytical forms and figures. A novel two-peak rational solution with variable height and a non-vanishing boundary is also obtained.Comment: 34 pages, 10 figure

Inverse Transformation Optics and Reflection Analysis for Two-Dimensional Finite Embedded Coordinate Transformation

Inverse transformation optics is introduced, and used to calculate the reflection at the boundary of a transformation medium under consideration. The transformation medium for a practical device is obtained from a two-dimensional (2D) finite embedded coordinate transformation (FECT) which is discontinuous at the boundary. For an electromagnetic excitation of particular polarization, many pairs of original medium (in a virtual space V') and inverse transformation can give exactly the same anisotropic medium through the conventional procedure of transformation optics. Non-uniqueness of these pairs is then exploited for the analysis and calculation of the boundary reflection. The reflection at the boundary of the anisotropic FECT medium (associated with the corresponding vacuum virtual space V) is converted to the simple reflection between two isotropic media in virtual space V' by a selected inverse transformation continuous at the boundary. A reflectionless condition for the boundary of the FECT medium is found as a special case. The theory is verified numerically with the finite element method.Comment: 7 pages, 4 figure