Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.Comment: 31 page

Physical Properties of Wolf-Rayet Stars

The striking broad emission line spectroscopic appearance of Wolf-Rayet (WR) stars has long defied analysis, due to the extreme physical conditions within their line and continuum forming regions. Recently, model atmosphere studies have advanced sufficiently to enable the determination of stellar temperatures, luminosities, abundances, ionizing fluxes and wind properties. The observed distributions of nitrogen (WN) and carbon (WC) sequence WR stars in the Milky Way and in nearby star forming galaxies are discussed; these imply lower limits to progenitor masses of ~25, 40, 75 Msun for hydrogen-depleted (He-burning) WN, WC, and H-rich (H-burning) WN stars, respectively. WR stars in massive star binaries permit studies of wind-wind interactions and dust formation in WC systems. They also show that WR stars have typical masses of 10-25 Msun, extending up to 80 Msun for H-rich WN stars. Theoretical and observational evidence that WR winds depend on metallicity is presented, with implications for evolutionary models, ionizing fluxes, and the role of WR stars within the context of core-collapse supernovae and long-duration gamma ray bursts.Comment: 76 pages, 8 figures. Minor revision to "Annual Review of Astronomy & Astrophysics" review article Volume 45 (2007) following editors comments. Version with full resolution figures is available from ftp://astro1.shef.ac.uk/pub/pac/AnnRev_revised.pd