High Frequency Conductivity in the Quantum Hall Regime

We have measured the complex conductivity $\sigma_{xx}$ of a two-dimensional electron system in the quantum Hall regime up to frequencies of 6 GHz at electron temperatures below 100 mK. Using both its imaginary and real part we show that $\sigma_{xx}$ can be scaled to a single function for different frequencies and for all investigated transitions between plateaus in the quantum Hall effect. Additionally, the conductivity in the variable-range hopping regime is used for a direct evaluation of the localization length $\xi$. Even for large filing factor distances $\delta \nu$ from the critical point we find $\xi \propto \delta \nu^{-\gamma}$ with a scaling exponent $\gamma=2.3$

Hopping conductivity in the quantum Hall effect -- revival of universal scaling

We have measured the temperature dependence of the conductivity $\sigma_{xx}$ of a two-dimensional electron system deep into the localized regime of the quantum Hall plateau transition. Using variable-range hopping theory we are able to extract directly the localization length $\xi$ from this experiment. We use our results to study the scaling behavior of $\xi$ as a function of the filling factor distance $|\delta \nu|$ to the critical point of the transition. We find for all samples a power-law behavior $\xi\propto|\delta\nu|^{-\gamma}$ with a universal scaling exponent $\gamma = 2.3$ as proposed theoretically

Stability of monotone solutions for the shadow Gierer-Meinhardt system with finite diffusivity

We consider the following shadow system of the Gierer-Meinhardt model: \left\{\begin{array}{l} A_t= \epsilon^2 A_{xx} -A +\frac{A^p}{\xi^q},\, 00,\\ \tau \xi_t= -\xi + \xi^{-s} \int_0^1 A^2 \,dx,\\ A>0,\, A_x (0,t)= A_x(1, t)=0, \end{array} \right. where 1<p<+\infty,\, \frac{2q}{p-1} >s+1,\, s\geq 0, and \tau >0. It is known that a nontrivial monotone steady-state solution exists if and only if \ep < \frac{\sqrt{p-1}}{\pi}. In this paper, we show that for any \ep < \frac{\sqrt{p-1}}{\pi}, and p=2 or p=3, there exists a unique \tau_c>0 such that for \tau\tau_c it is linearly unstable. (This result is optimal.) The transversality of this Hopf bifurcation is proven. Other cases for the exponents as well as extensions to higher dimensions are also considered. Our proof makes use of functional analysis and the properties of Weierstrass functions and elliptic integrals

Optical and Hall conductivities of a thermally disordered two-dimensional spin-density wave: two-particle response in the pseudogap regime of electron-doped high-TcT_c superconductors

We calculate the longitudinal ($\sigma_{xx}$) and Hall ($\sigma_{xy}$) optical conductivities for two-dimensional metals with thermally disordered antiferromagnetism using a generalization of an approximation introduced by Lee, Rice and Anderson for the self energy. The conductivities are calculated from the Kubo formula, with current vertex function treated in a conserving approximation satisfying the Ward identity. In order to obtain a finite DC limit, we introduce phenomenologically impurity scattering, with relaxation time $\tau$. $\sigma_{xx}(\Omega)$ satisfies the $f$-sum rule. For the infinitely peaked spin correlation function, $\chi(\mathbf{q})\propto \delta(\mathbf{q}-\mathbf{Q})$, we recover the expressions for the conductivities in the mean-field theory of the ordered state. When the spin correlation length $\xi$ is large but finite, both $\sigma_{xx}$ and $\sigma_{xy}$ show behaviors characteristic of the state with long-range order. The calculation runs into difficulty for $\Omega\lesssim 1/\tau$. The difficulties are traced to an inaccurate treatment of the very low energy density of states within the Lee-Rice-Anderson approximation. The results for $\sigma_{xx}(\Omega)$ and $\sigma_{xy}(\Omega)$ are qualitatively consistent with data on electron-doped cuprates when $\Omega>1/\tau$

Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations

In this article, we study the self-similar solutions of the 2-component Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}% \rho_{t}+u\rho_{x}+\rho u_{x}=0 m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation} with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation method, we can obtain a class of blowup or global solutions for $\sigma=1$ or $-1$. In particular, for the integrable system with $\sigma=1$, we have the global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}% \rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right) }{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi} 0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right. ,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}% >0,\text{ }\overset{\cdot}{a}(0)=a_{1} f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right) ^{2}}% \end{array} \right. \end{equation} where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are arbitrary constants.\newline Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.Comment: 5 more figures can be found in the corresponding journal paper (J. Math. Phys. 51, 093524 (2010) ). Key Words: 2-Component Camassa-Holm Equations, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundar

Self-Similar Blowup Solutions to the 2-Component Degasperis-Procesi Shallow Water System

In this article, we study the self-similar solutions of the 2-component Degasperis-Procesi water system:% [c]{c}% \rho_{t}+k_{2}u\rho_{x}+(k_{1}+k_{2})\rho u_{x}=0 u_{t}-u_{xxt}+4uu_{x}-3u_{x}u_{xx}-uu_{xxx}+k_{3}\rho\rho_{x}=0. By the separation method, we can obtain a class of self-similar solutions,% [c]{c}% \rho(t,x)=\max(\frac{f(\eta)}{a(4t)^{(k_{1}+k_{2})/4}},\text{}0),\text{}u(t,x)=\frac{\overset{\cdot}{a}(4t)}{a(4t)}x \overset{\cdot\cdot}{a}(s)-\frac{\xi}{4a(s)^{\kappa}}=0,\text{}a(0)=a_{0}% \neq0,\text{}\overset{\cdot}{a}(0)=a_{1} f(\eta)=\frac{k_{3}}{\xi}\sqrt{-\frac{\xi}{k_{3}}\eta^{2}+(\frac{\xi}{k_{3}}\alpha) ^{2}}% where $\eta=\frac{x}{a(s)^{1/4}}$ with $s=4t;$ $\kappa=\frac{k_{1}}{2}+k_{2}-1,$ $\alpha\geq0,$ $\xi<0$, $a_{0}$ and $a_{1}$ are constants. which the local or global behavior can be determined by the corresponding Emden equation. The results are very similar to the one obtained for the 2-component Camassa-Holm equations. Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems. With the characteristic line method, blowup phenomenon for $k_{3}\geq0$ is also studied.Comment: 13 Pages, Key Words: 2-Component Degasperis-Procesi, Shallow Water System, Analytical Solutions, Blowup, Global, Self-Similar, Separation Method, Construction of Solutions, Moving Boundary, 2-Component Camassa-Holm Equation