Symbolic algorithms for the Painlevé test, special solutions, and recursion operators for nonlinear PDEs

This paper discusses the algorithms and implementations of three MATHEMATICA packages for the study of integrability and the computation of closed-form solutions of nonlinear polynomial PDEs. The first package, PainleveTest.m, symbolically performs the Painlevé integrability test. The second package, PDESpecialSolutions.m, computes exact solutions expressible in hyperbolic or elliptic functions. The third package, PDERecursionOperator.m, generates and tests recursion operators

Multiquantum well structure with an average electron mobility of 4.0×10^6 cm^2/V s

We report a modulation-doped multiquantum well structure which suppresses the usual ambient light effect associated with modulation doping. Ten GaAs quantum wells 300-Å wide are symmetrically modulation doped using Si δ doping at the center of 3600-Å-wide Al0.1Ga0.9As barriers. The low field mobility of each well is 4.0×10^6 cm/V s at a density of 6.4×10^10 cm^−2 measured at 0.3 K either in the dark, or during, or after, exposure to light. This mobility is an order of magnitude improvement over previous work on multiwells

Phase Diagrams for the ν\nu = 1/2 Fractional Quantum Hall Effect in Electron Systems Confined to Symmetric, Wide GaAs Quantum Wells

We report an experimental investigation of fractional quantum Hall effect (FQHE) at the even-denominator Landau level filling factor $\nu$ = 1/2 in very high quality wide GaAs quantum wells, and at very high magnetic fields up to 45 T. The quasi-two-dimensional electron systems we study are confined to GaAs quantum wells with widths $W$ ranging from 41 to 96 nm and have variable densities in the range of $\simeq 4 \times 10^{11}$ to $\simeq 4 \times 10^{10}$ cm$^{-2}$. We present several experimental phase diagrams for the stability of the $\nu=1/2$ FQHE in these quantum wells. In general, for a given $W$, the 1/2 FQHE is stable in a limited range of intermediate densities where it has a bilayer-like charge distribution; it makes a transition to a compressible phase at low densities and to an insulating phase at high densities. The densities at which the $\nu=1/2$ FQHE is stable are larger for narrower quantum wells. Moreover, even a slight charge distribution asymmetry destabilizes the $\nu=1/2$ FQHE and turns the electron system into a compressible state. We also present a plot of the symmetric-to-antisymmetric subband separation ($\Delta_{SAS}$), which characterizes the inter-layer tunneling, vs density for various $W$. This plot reveals that $\Delta_{SAS}$ at the boundary between the compressible and FQHE phases increases \textit{linearly} with density for all the samples. Finally, we summarize the experimental data in a diagram that takes into account the relative strengths of the inter-layer and intra-layer Coulomb interactions and $\Delta_{SAS}$. We conclude that, consistent with the conclusions of some of the previous studies, the $\nu=1/2$ FQHE observed in wide GaAs quantum wells with symmetric charge distribution is stabilized by a delicate balance between the inter-layer and intra-layer interactions, and is very likely described by a two-component ($\Psi_{311}$) state.Comment: Accepted for publication in Phys. Rev.

Locally Optimally-emitting Clouds and the Narrow Emission Lines in Seyfert Galaxies

The narrow emission line spectra of active galactic nuclei are not accurately described by simple photoionization models of single clouds. Recent Hubble Space Telescope images of Seyfert 2 galaxies show that these objects are rich with ionization cones, knots, filaments, and strands of ionized gas. Here we extend to the narrow line region the ``locally optimally emitting cloud'' (LOC) model, in which the observed spectra are predominantly determined by powerful selection effects. We present a large grid of photoionization models covering a wide range of physical conditions and show the optimal conditions for producing many of the strongest emission lines. We show that the integrated narrow line spectrum can be predicted by an integration of an ensemble of clouds, and we present these results in the form of diagnostic line ratio diagrams making comparisons with observations. We also predict key diagnostic line ratios as a function of distance from the ionizing source, and compare these to observations. The predicted radial dependence of the [O III]/[O II] ratio may be matched to the observed one in NGC4151, if the narrow line clouds see a more intense continuum than we see. The LOC scenario when coupled with a simple Keplerian gravitational velocity field will quite naturally predict the observed line width versus critical density relationship. The influence of dust within the ionized portion of the clouds is discussed and we show that the more neutral gas is likely to be dusty, although a high ionization dust-free region is most likely present too. This argues for a variety of NLR cloud origins.Comment: 29 pages plus 16 figures, accepted for publication in Ap

Spectrophotometry of 2 complete samples of flat radio spectrum quasars

Spectrophotometry of two complete samples of flat-spectrum radio quasars show that for these objects there is a strong correlation between the equivalent width of the CIV wavelength 1550 emission line and the luminosity of the underlying continuum. Assuming Friedmann cosmologies, the scatter in this correlation is a minimum for q (sub o) is approximately 1. Alternatively, luminosity evolution can be invoked to give compact distributions for q (sub o) is approximately 0 models. A sample of Seyfert galaxies observed with IUE shows that despite some dispersion the average equivalent width of CIV wavelength 1550 in Seyfert galaxies is independent of the underlying continuum luminosity. New redshifts for 4 quasars are given

Formation of a high quality two-dimensional electron gas on cleaved GaAs

We have succeeded in fabricating a two-dimensional electron gas (2DEG) on the cleaved (110) edge of a GaAs wafer by molecular beam epitaxy (MBE). A (100) wafer previously prepared by MBE growth is reinstalled in the MBE chamber so that an in situ cleave exposes a fresh (110) GaAs edge for further MBE overgrowth. A sequence of Si-doped AlGaAs layers completes the modulation-doped structure at the cleaved edge. Mobilities as high as 6.1×10^5 cm^2/V s are measured in the 2DEG at the cleaved interface

Symbolic computation of exact solutions expressible in hyperbolic and elliptic functions for nonlinear PDEs

Algorithms are presented for the tanh- and sech-methods, which lead to closed-form solutions of nonlinear ordinary and partial differential equations (ODEs and PDEs). New algorithms are given to find exact polynomial solutions of ODEs and PDEs in terms of Jacobi's elliptic functions. For systems with parameters, the algorithms determine the conditions on the parameters so that the differential equations admit polynomial solutions in tanh, sech, combinations thereof, Jacobi's sn or cn functions. Examples illustrate key steps of the algorithms. The new algorithms are implemented in Mathematica. The package DDESpecialSolutions.m can be used to automatically compute new special solutions of nonlinear PDEs. Use of the package, implementation issues, scope, limitations, and future extensions of the software are addressed. A survey is given of related algorithms and symbolic software to compute exact solutions of nonlinear differential equations.Comment: 39 pages. Software available from Willy Hereman's home page at http://www.mines.edu/fs_home/whereman

Multicomponent fractional quantum Hall states with subband and spin degrees of freedom

In wide GaAs quantum wells where two electric subbands are occupied we apply a parallel magnetic field or increase the electron density to cause a crossing of the two $N=0$ Landau levels of these subbands and with opposite spins. Near the crossing, the fractional quantum Hall states in the filling factor range $1<\nu<3$ exhibit a remarkable sequence of pseudospin polarization transitions resulting from the interplay between the spin and subband degrees of freedom. The field positions of the transitions yield a new and quantitative measure of the composite Fermions' discrete energy level separations. Surprisingly, the separations are smaller when the electrons have higher spin-polarization