33 research outputs found
Multiresolution weighted norm equivalences and applications
Get PDFSummary.: We establish multiresolution norm equivalences in weighted spaces L 2 w ((0,1)) with possibly singular weight functions w(x)≥0 in (0,1). Our analysis exploits the locality of the biorthogonal wavelet basis and its dual basis functions. The discrete norms are sums of wavelet coefficients which are weighted with respect to the collocated weight function w(x) within each scale. Since norm equivalences for Sobolev norms are by now well-known, our result can also be applied to weighted Sobolev norms. We apply our theory to the problem of preconditioning p-Version FEM and wavelet discretizations of degenerate elliptic and parabolic problems from financ
Soliton compression and supercontinuum spectra in nonlinear diamond photonics
Get PDFWe numerically explore synthetic crystal diamond for realizing novel light
sources in ranges which are up to now difficult to achieve with other
materials, such as sub-10-fs pulse durations and challenging spectral ranges.
We assess the performance of on-chip diamond waveguides for controlling light
generation by means of nonlinear soliton dynamics. Tailoring the cross-section
of such diamond waveguides allows to design dispersion profiles with custom
zero-dispersion points and anomalous dispersion ranges exceeding an octave.
Various propagation dynamics, including supercontinuum generation by soliton
fission, can be realized in diamond photonics. In stark contrast to usual
silica-based optical fibers, where such processes occur on the scale of meters,
in diamond millimeter-scale propagation distances are sufficient. Unperturbed
soliton-dynamics prior to soliton fission allow to identify a pulse
self-compression scenario that promises record-breaking compression factors on
chip-size propagation lengths
