arXiv subjects

Qing Guo

Publications and source records attributed to Qing Guo.

194 records · Page 11Linked to original sources

arXiv2019

Reconfigurable edge-state engineering in graphene using LaAlO$_3$/SrTiO$_3$ nanostructures

The properties of graphene depend sensitively on doping with respect to the charge-neutrality point (CNP). Tuning the CNP usually requires electrical gating or chemical doping. Here, we describe a technique to reversibly control the CNP in graphene with nanoscale precision, utilizing LaAlO$_3$/SrTiO$_3$ (LAO/STO) heterostructures and conductive atomic force microscope (c-AFM) lithography. The local electron density and resulting conductivity of the LAO/STO interface can be patterned with a conductive AFM tip, and placed within two nanometers of an active graphene device. The proximal LAO/STO nanostructures shift the position of graphene CNP by ~ $10^{12}$ cm$^{-2}$, and are also gateable. Here we use this effect to create reconfigurable edge states in graphene, which are probed using the quantum Hall effect. Quantized resistance plateaus at $h/e^2$ and $h/3e^2$ are observed in a split Hall device, demonstrating edge transport along the c-AFM written edge that depends on the polarity of both the magnetic field and direction of currents. This technique can be readily extended to other device geometries.

cond-mat.mes-hall↗

arXiv2019

Scattering and blow-up criteria for 3D cubic focusing nonlinear inhomogeneous NLS with a potential

In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schrödinger equation with a potential $$ iu_{t}+Δu-Vu+|x|^{-b}|u|^{2}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{3}}, $$ where $0 \mathcal{K}$ if $V$ satisfies some additional assumptions.0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}>\mathcal{K}$ if $V$ satisfies some additional assumptions.

math.AP↗

arXiv2018

Dynamics of the focusing 3D cubic NLS with slowly decaying potential

In this paper, we consider a 3d cubic focusing nonlinear Schrödinger equation (NLS) with slowing decaying potentials. Adopting the variational method of Ibrahim-Masmoudi-Nakanishi \cite{IMN}, we obtain a condition for scattering. It is actually sharp in some sense since the solution will blow up if it's false. The proof of blow-up part relies on the method of Du-Wu-Zhang \cite{DWZ}

math.AP↗

arXiv2018

Scattering and blowup for $L^{2}$-supercritical and $\dot{H}^{2}$-subcritical biharmonic NLS with potentials

We mainly consider the focusing biharmonic Schrödinger equation with a large radial repulsive potential $V(x)$: \begin{equation*} \left\{ \begin{aligned} iu_{t}+(Δ^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{N}}, u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}), \end{aligned}\right. \end{equation*} If $N>8$, \ $1+\frac{8}{N} N+4$, then we firstly prove a global well-posedness and scattering result for the radial data $u_0\in H^2({\bf R}^N)$ which satisfies that $$ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0) \|Q\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|ΔQ\|_{L^{2}}. $$

math.AP↗

arXiv2018

Blow-up for the nonlinear Schrödinger equation with combined nonlinearities

In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schrödinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type. Especially, we find a sequence of initial data with mass approximating that of the ground state from above, the correspondng solution of which blows up. This result partially gives a positive answer to the open problem left by T.Tao, M. Visan and X. Zhang in \cite{TVZ}. After then, we obtain the lower blow-up rate and concentration rate for the blow-up solutions in the same case. Finally, we consider the mass-critical combined with the mass-supercritical power type case, studying the blow-up criteria, blow-up rate and concentration of the blow-up solutions in this case.

math.AP↗

arXiv2018

Orbital stability of solitary waves for generalized derivative nonlinear Schrödinger equations in the endpoint case

We consider the following generalized derivative nonlinear Schrödinger equation \begin{equation*} i\partial_tu+\partial^2_xu+i|u|^{2σ}\partial_xu=0,\ (t,x)\in\mathbb R\times\mathbb R \end{equation*} when $σ\in(0,1)$. The equation has a two-parameter family of solitary waves $$u_{ω,c}(t,x)=Φ_{ω,c}(x)e^{iωt+\frac{ic}2x-\frac i{2σ+2}\int_0^xΦ_{ω,c}(y)^{2σ}dy},$$ with $(ω,c)$ satisfying $ω>c^2/4$, or $ω=c^2/4$ and $c>0$. The stability theory in the frequency region $ω>c^2/4$ was studied previously. In this paper, we prove the stability of the solitary wave solutions in the endpoint case $ω=c^2/4$ and $c>0$.

math.AP↗

arXiv2017

Sharp Criteria of Scattering for the Fractional NLS

In this paper, the sharp threshold of scattering for the fractional nonlinear Schrödinger equation in the $L^2$-supercritical case is obtained, i.e., if $1+\frac{4s}{N} M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s},$$ then the corresponding solution $u(t)$ blows up in finite time, according to Boulenger, Himmelsbach, and Lenzmann's results in [2].

math.AP↗

arXiv2017

Sharp Threshold of Blow-up and Scattering for the fractional Hartree equation

We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0} M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, the solution $u(t)$ blows up in finite time. This condition is sharp in the sense that the solitary wave solution $e^{it}Q(x)$ is global but not scattering, which satisfies the equality in the above conditions. Here, $Q$ is the ground-state solution for the fractional Hartree equation.

math.AP↗

arXiv2016

Method for Transferring High-Mobility CVD-Grown Graphene with Perfluoropolymers

The transfer of graphene grown by chemical vapor deposition (CVD) using amorphous polymers represents a widely implemented method for graphene-based electronic device fabrication. However, the most commonly used polymer, poly(methyl methacrylate) (PMMA), leaves a residue on the graphene that limits the mobility. Here we report a method for graphene transfer and patterning that employs a perfluoropolymer---Hyflon---as a transfer handle and to protect graphene against contamination from photoresists or other polymers. CVD-grown graphene transferred this way onto LaAlO$_3$/SrTiO$_3$ heterostructures is atomically clean, with high mobility (~30,000 cm$^2$V$^{-1}$s$^{-1}$) near the Dirac point at 2 K and clear, quantized Hall and magneto-resistance. Local control of the LaAlO$_3$/SrTiO$_3$ interfacial metal-insulator transition---through the graphene---is preserved with this transfer method. The use of perfluoropolymers such as Hyflon with CVD-grown graphene and other 2D materials can readily be implemented with other polymers or photoresists.

cond-mat.mtrl-sci↗

arXiv2015

Scattering for the focusing $L^{2}$ -supercritical and $\dot H^2$-subcritical biharmonic NLS Equations

We consider the focusing $\dot H^{s_c}$-critical biharmonic Schrödinger equation, and prove a global wellposedness and scattering result for the radial data $u_0\in H^2(\mathbb R^N)$ satisfying $ M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E(Q) $ and $ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{2}\|Δu_{0}\|_{2}<\|Q\|^{\frac{2-s_c}{s_c}}_{2}\|ΔQ\|_{2}, $ where $s_c\in(0,2)$ and $Q$ is the ground state of $Δ^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$.

math.AP↗

arXiv2011

Threshold solutions for the focusing $L^{2}$ -supercritical NLS Equations

We investigate the $L^2$-supercritical and $\dot{H}^1$-subcritical nonlinear Schrödinger equation in $H^1$. In \cite{G1} and \cite{yuan}, the mass-energy quantity $M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ has been shown to be a threshold for the dynamical behavior of solutions of the equation. In the present paper, we study the dynamics at the critical level $M(u)^{\frac{1-s_{c}}{s_{c}}}E(u)=M(Q)^{\frac{1-s_{c}}{s_{c}}}E(Q)$ and classify the corresponding solutions using modulation theory, non-trivially generalize the results obtained in \cite{holmer3} for the 3D cubic Schrödinger equation.

math.AP↗

arXiv2011

Nonscattering solutions to the $L^{2}$-supercritical NLS Equations

We investigate the nonlinear Schrödinger equation $iu_{t}+Δu+|u|^{p-1}u=0$ with $1+\frac{4}{N} \|Q\|_{2}^{\frac{1-s_{c}}{s_{c}}}\|\nabla Q\|_{2},$ then either $u(t)$~blows up in finite forward time, or $u(t)$ exists globally for positive time and there exists a time sequence $t_{n}\rightarrow+\infty$ such that $\|\nabla u(t_{n})\|_{2}\rightarrow+\infty.$ Here $Q$ is the ground state solution of $-Q+ΔQ+|Q|^{p-1}Q=0.$ A similar result holds for negative time. This extend the result of the 3D cubic Schrödinger equation in \cite{holmer10} to the general mass-supercritical and energy-subcritical case .

math.AP↗

arXiv2011

Divergent solutions to the 5D Hartree Equations

We consider the Cauchy problem for the focusing Hartree equation $iu_{t}+Δu+(|\cdot|^{-3}\ast|u|^{2})u=0$ in $\mathbb{R}^{5}$ with the initial data in $H^1$, and study the divergent property of infinite-variance and nonradial solutions. Letting $Q$ be the ground state solution of $-Q+ΔQ+(|\cdot|^{-3}\ast|Q|^{2})Q=0 $ in $ \mathbb{R}^{5}$, we prove that if $u_{0}\in H^{1}$ satisfying $M(u_0) E(u_0) \|\nabla Q\|_{2}\|Q\|_{2} ,$ then the corresponding solution $u(t)$ either blows up in finite forward time, or exists globally for positive time and there exists a time sequence $t_{n}\rightarrow+\infty$ such that $\|\nabla u(t_{n})\|_{2}\rightarrow+\infty.$ A similar result holds for negative time.

math.AP↗

arXiv2006

The discrete fractional random cosine and sine transforms

Based on the discrete fractional random transform (DFRNT), we present the discrete fractional random cosine and sine transforms (DFRNCT and DFRNST). We demonstrate that the DFRNCT and DFRNST can be regarded as special kinds of DFRNT and thus their mathematical properties are inherited from the DFRNT. Numeral results of DFRNCT and DFRNST for one and two dimensional functions have been given.

math-ph↗