Adaptive frame methods for nonlinear variational problems

In this paper we develop adaptive numerical solvers for certain nonlinear variational problems. The discretization of the variational problems is done by a suitable frame decomposition of the solution, i.e., a complete, stable, and redundant expansion. The discretization yields an equivalent nonlinear problem on the space of frame coefficients. The discrete problem is then adaptively solved using approximated nested fixed point and Richardson type iterations. We investigate the convergence, stability, and optimal complexity of the scheme. A theoretical advantage, for example, with respect to adaptive finite element schemes is that convergence and complexity results for the latter are usually hard to prove. The use of frames is further motivated by their redundancy, which, at least numerically, has been shown to improve the conditioning of the discretization matrices. Also frames are usually easier to construct than Riesz bases. We present a construction of divergence-free wavelet frames suitable for applications in fluid dynamics and magnetohydrodynamics. M. Fornasier acknowledges the financial support provided through the Intra-European Individual Marie Curie Fellowship Programme, under contract MOIF-CT-2006-039438. All of the authors acknowledge the hospitality of Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università di Roma “La Sapienza”, Italy, during the early preparation of this work. The authors want to thank Daniele Boffi, Dorina Mitrea, and Karsten Urban for the helpful and fruitful discussions on divergence-free function spaces.     

Electrical and magnetic characterization of hot-extruded BSCCO tape samples

Summary Ag/BSCCO-2223 phase tapes were prepared with a new plastic deformation cycle. Starting from the hot extrusion of composite billets, hundreds of metres of tape 0.15×3 mm were obtained by drawing and rolling the extruded bars. The cross-section, monofilamentary with a central Ag insert, is quite regular along all the length. Samples from 30 cm to 100 cm in length were heat-treated at different temperatures between 830°C and 480°C in Ar+10% O₂ atmosphere. Electrical-transport properties were tested at 77K.J _c of 3000 A/cm² were usually achieved after first heat treatment. The superconducting properties were also tested by magnetic measurements using a vibrating sample magnetometer. Paper presented at the ?VII Congresso SATT?, Torino, 4–7 October 1994.     

Bosonization theory for tunneling spectra in smooth edges of quantum Hall systems

We calculate the spectral function of a smooth edge of a quantum Hall system in the lowest Landau level by means of a bosonization technique. We obtain a general relationship between the one electron spectral function and the dynamical structure factor. The resulting – characteristics exhibit, at low voltage and temperature, power law scaling, generally different from the one predicted by the chiral Luttinger liquid theory, and in good agreement with recent experimental results.     

Dynamic exchange-correlation potentials for the 2D electron gas

Recent progress in the formulation of a fully dynamical local approximation to time-dependent density functional theory (TD-DFT) appeals to the longitudinal and transverse components of the long-wavelength exchange and correlation kernel in the homogeneous electron gas, . We extend to the two-dimensional longitudinal and transverse case our work on the 3D [J. Phys.: Condens. Matter 9 (1997) 475], which accounts for two-pair excitations through an approximate decoupling of the equation of motion for the current–current response function. We present numerical results and compare with asymptotic behaviours and previous approximations.     

Photonic crystal wires for optical parametric oscillators in isotropic media

We investigate four wave mixing in photonic crystal wire microresonators realized in an isotropic medium. One-dimensional optical parametric oscillators are numerically analyzed by solving Maxwells equations in all dimensions and including material dispersion as well as nonlinear polarization. PACS 42.65.Yj; 42.70.Qs; 42.82.Gw     

Route to nonlocality and observation of accessible solitons

We develop a general theory of spatial solitons in a liquid crystalline medium exhibiting a nonlinearity with an arbitrary degree of effective nonlocality. The model accounts the observability of accessible solitons and establishes an important link with parametric solitons.     

X waves generated at the second harmonic

The process of optical frequency doubling in the undepleted regime can lead to the generation of an X-wave envelope with group velocity locked to the pump beam. Its parameters and its angular spectrum are directly related to the zeroth- and first-order dispersive features of the nonlinear process. This constitutes a novel mechanism for spatiotemporal localization of light.     

Interplay between nonlocality and nonlinearity in nematic liquid crystals

Spatial nonlocality has a crucial role in the optical response of bulk liquid crystals. With specific reference to one-dimensional modulation instability, we demonstrate the tuning of transverse nonlocality in the nonlinear response of nematics.