Dispersive Shock Wave, Generalized Laguerre Polynomials and Asymptotic Solitons of the Focusing Nonlinear Schr\"odinger Equation

We consider dispersive shock wave to the focusing nonlinear Schr\"odinger equation generated by a discontinuous initial condition which is periodic or quasi-periodic on the left semi-axis and zero on the right semi-axis. As an initial function we use a finite-gap potential of the Dirac operator given in an explicit form through hyper-elliptic theta-functions. The paper aim is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in \cite{KK86} and \cite{K91} using Marchenko's inverse scattering technics. We investigate this problem exceptionally using the Riemann-Hilbert problems technics that allow us to obtain explicit formulas for the asymptotic solitons themselves that in contrast with the cited papers where asymptotic formulas are obtained only for the square of absolute value of solution. Using transformations of the main RH problems we arrive to a model problem corresponding to the parametrix at the end points of continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials which are naturally appeared after appropriate scaling of the Riemann-Hilbert problem in a small neighborhoods of the end points of continuous spectrum. Further asymptotic analysis give an explicit formula for solitons at the edge of dispersive wave. Thus, we give the complete description of the train of asymptotic solitons: not only bearing envelope of each asymptotic soliton, but its oscillating structure are found explicitly. Besides the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock wave.Comment: 36 pages, 5 figure

Color Effects Associated with the 1999 Microlensing Brightness Peaks in Gravitationally Lensed Quasar Q2237+0305

Photometry of the Q2237+0305gravitational lens in VRI spectral bands with the 1.5-m telescope of the high-altitude Maidanak observatory in 1995-2000 is presented. Monitoring of Q2237+0305 in July-October 2000, made at nearly daily basis, did not reveal rapid (night-to-night and intranight) variations of brightness of the components during this time period. Rather slow changes of magnitudes of the components were observed, such as 0.08 mag fading of B and C components and 0.05 mag brightening of D in R band during July 23 - October 7, 2000. By good luck three nights in 1999 were almost at the time of the strong brightness peak of image C, and approximately in the middle of the ascending slope of the image A brightness peak. The C component was the most blue one in the system in 1998 and 1999, having changed its (V-I) color from 0.56 mag to 0.12 mag since August 1997, while its brightness increased almost 1.2 mag during this time period. The A component behaved similarly between August 1998 and August 2000, having become 0.47 mag brighter in R, and at the same time, 0.15 mag bluer. A correlation between the color variations and variations of magnitudes of the components is demonstrated to be significant and reaches 0.75, with a regression line slope of 0.33. A color (V-I) vrs color (V-R) plot shows the components settled in a cluster, stretched along a line with a slope of 1.31. Both slopes are noticeably smaller than those expected if a standard galactic interstellar reddening law were responsible for the differences between the colors of images and their variations over time. We attribute the brightness and color changes to microlensing of the quasar's structure, which we conclude is more compact at shorter wavelengths, as predicted by most quasar models featuring an energizing central source.Comment: 14 pages, 7 figures, LaTeX, submitted to A&

Time delays in PG1115+080: new estimates

We report new estimates of the time delays in the quadruple gravitationally lensed quasar PG1115+080, obtained from the monitoring data in filter R with the 1.5-m telescope at the Maidanak Mountain (Uzbekistan, Central Asia) in 2004-2006. The time delays are 16.4 days between images C and B, and 12 days between C and A1+A2, with image C being leading for both pairs. The only known estimates of the time delays in PG1115 are those based on observations by Schechter et al. (1997) -- 23.7 and 9.4 days between images C and B, C and A1+A2, respectively, as calculated by Schechter et al., and 25 and 13.3 days as revised by Barkana (1997) for the same image components with the use of another method. The new values of time delays in PG 1115+080 may be expected to provide larger estimates of the Hubble constant thus decreasing a diversity between the H_0 estimates taken from gravitationally lensed quasars and with other methods.Comment: 5 pages, 2 figures, Accepted for publication in MNRAS Letter

Magnetic properties of colloidal suspensions of interacting magnetic particles

We review equilibrium thermodynamic properties of systems of magnetic particles like ferrofluids in which dipolar interactions play an important role. The review is focussed on two subjects: ({\em i}) the magnetization with the initial magnetic susceptibility as a special case and ({\em ii}) the phase transition behavior. Here the condensation ("gas/liquid") transition in the subsystem of the suspended particles is treated as well as the isotropic/ferromagnetic transition to a state with spontaneously generated long--range magnetic order.Comment: Review. 62 pages, 4 figure