We study an extensive sample of 87 GRBs for which there are well sampled and simultaneous optical and X-ray light-curves. We extract the cleanest possible signal of the afterglow component, and compare the temporal behaviors of the X-ray light-curve,
observed by Swift XRT, and optical data, observed by UVOT and ground-based telescopes for each individual burst. Overall we find 62% GRBs that are consistent with the standard afterglow model. When more advanced modeling is invoked, up to 91% of the bursts in our sample may be consistent with the external shock model. A large fraction of these bursts are consistent with occurring in a constant interstellar density medium (ISM) (61%) while only 39% of them occur in a wind-like medium. Only 9 cases have afterglow light-curves that exactly match the standard fireball model prediction, having a single power law decay in both energy bands which are observed during their entire duration. In particular, for the bursts with chromatic behavior additional model assumptions must be made over limited segments of the light-curves in order for these bursts to fully agree with the external shock model. Interestingly, for 54% of the X-ray and 40% of the optical band observations the end of the shallow decay ($t^{sim-0.5}$) period coincides with the jet break ($t^{sim-p}$) time, causing an abrupt change in decay slope. The fraction of the burst that consistent with the external shock model is independent of the observational epochs in the rest frame of GRBs. Moreover, no cases can be explained by the cooling frequency crossing the X-ray or optical band.
The Izergin-Korepin model with general non-diagonal boundary terms, a typical integrable model beyond A-type and without U(1)-symmetry, is studied via the off-diagonal Bethe ansatz method. Based on some intrinsic properties of the R-matrix and the K-
matrices, certain operator product identities of the transfer matrix are obtained at some special points of the spectral parameter. These identities and the asymptotic behaviors of the transfer matrix together allow us to construct the inhomogeneous T-Q relation and the associated Bethe ansatz equations. In the diagonal boundary limit, the reduced results coincide exactly with those obtained via other methods.
We continue our systematic statistical study on optical afterglow data of gamma-ray bursts (GRBs). We present the apparent magnitude distributions of early optical afterglows at different epochs (t= 10^2 s, t = 10^3 s, and 1 hour) for the optical lig
htcurves of a sample of 93 GRBs (the global sample), and for sub-samples with an afterglow onset bump or a shallow decay segment. For the onset sample and shallow decay sample we also present the brightness distribution at the peak time t_{p} and break time t_{b}, respectively. All the distributions can be fit with Gaussian functions. We further perform Monte Carlo simulations to infer the luminosity function of GRB optical emission at the rest-frame time 10^3 seconds, t_{p}, and t_{b}, respectively. Our results show that a single power-law luminosity function is adequate to model the data, with indices -1.40+/-0.10, -1.06+/- 0.16, and -1.54+/- 0.22, respectively. Based on the derived rest-frame 10^3 s luminosity function, we generate the intrinsic distribution of the R-band apparent magnitude M_{R} at the observed time 10^{3} seconds post trigger, which peaks at M_{R}=22.5 mag. The fraction of GRBs whose R-band magnitude is fainter than 22 mag, and 25 mag and at the observer time 10^3 seconds are ~63% and ~25%, respectively. The detection probabilities of the optical afterglows with ground-based robotic telescopes and UVOT onboard {Swift} are roughly consistent with that inferred from this intrinsic M_{R} distribution, indicating that the variations of the dark GRB fraction among the samples with different telescopes may be due to the observational selection effect, although the existence of an intrinsically dark GRB population cannot be ruled out.
We continue our systematic statistical study of various components in gamma-ray burst (GRB) optical lightcurves. We decompose the early onset bump and the late re-brightening bump with empirical fits and analyze their statistical properties. Among th
e 146 GRBs that have well-sampled optical lightcurves, the onset and re-brightening bumps are observed in 38 and 26 GRBs, respectively. It is found that the typical rising and decaying slopes for both the onset and re-brightening bumps are ~1.5 and -1.15, respectively. No early onset bumps in the X-ray band are detected to be associated with the optical onset bumps, while an X-ray re-brightening bump is detected for half of the re-brightening optical bumps. The peak luminosity is anti-correlated with the peak time, L_ppropto t_{p}^{-1.81+/-0.32} and L_ppropto t_{p}^{-0.83+/-0.17} for the onset and re-brightening bumps, respectively. Both L_p and the isotropic energy release of the onset bumps are correlated with E_{gamma, iso}, whereas no similar correlation is found for the re-brightening bumps. Taking the onset bumps as probes for the properties of the fireballs and their ambient medium, we find that the typical power-law index of the relativistic electrons is 2.5 and the medium density profile behaves as npropto r^{-1} within the framework of the synchrotron external shock models. With the medium density profile obtained from our analysis, we also confirm the correlation between initial Lorentz factor (Gamma_0) and E_{gamma, iso} in our previous work. The jet component that produces the re-brightening bump seems to be on-axis and independent of the prompt emission jet component. Its typical kinetic energy budget would be about one order of magnitude larger than the prompt emission component, but with a lower Gamma_0, typically several tens.
Well-sampled optical lightcurves of 146 gamma-ray bursts (GRBs) are compiled from the literature. By empirical fitting we identify eight possible emission components and summarize the results in a synthetic lightcurve. Both optical flare and early sh
allow-decay components are likely related to long-term central engine activities. We focus on their statistical properties in this paper. Twenty-four optical flares are obtained from 19 GRBs. The isotropic R-band energy is smaller than 1% of $E_{gamma, rm iso}$. The relation between isotropic luminosities of the flares and gamma-rays follows $L^{rm F}_{rm R, iso}propto L_{{gamma}, rm iso}^{1.11pm 0.27}$. Later flares tend to be wider and dimmer, i.e., $w^{rm F}sim t^{rm F}_{rm p}/2$ and $L^{rm F}_{rm R, iso}propto [t^{rm F}_{rm p}/(1+z)]^{-1.15pm0.15}$. The detection probability of the optical flares is much smaller than that of X-ray flares. An optical shallow decay segment is observed in 39 GRBs. The relation between the break time and break luminosity is a power-law, with an index of $-0.78pm 0.08$, similar to that derived from X-ray flares. The X-ray and optical breaks are usually chromatic, but a tentative correlation is found. We suggest that similar to the prompt optical emission that tracks $gamma$-rays, the optical flares are also related to the erratic behavior of the central engine. The shallow decay component is likely related to a long-lasting spinning-down central engine or piling up of flare materials onto the blastwave. Mixing of different emission components may be the reason of the diverse chromatic afterglow behaviors.
The Cauchy problem of a multi-dimensional ($dgeqslant 2$) compressible viscous liquid-gas two-phase flow model is concerned in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stabl
e equilibrium and the local in time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.
We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating sub-domain. The g
lobal existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without coriolis forces, the rotating effect causes a coupling between two parts of Hodges decomposition of the velocity vector field, additional regularity is required in order to carry out the Friedrichs regularization and compactness arguments.
In this paper, we establish the global well-posedness of the Cauchy problem for the Gross-Pitaevskii equation with an angular momentum rotational term in which the angular velocity is equal to the isotropic trapping frequency in the space $Real^3$.
In this paper, we establish the global well-posedness of the Cauchy problem for the Gross-Pitaevskii equation with an rotational angular momentum term in the space $Real^2$.
The unipolar and bipolar macroscopic quantum models derived recently for instance in the area of charge transport are considered in spatial one-dimensional whole space in the present paper. These models consist of nonlinear fourth-order parabolic equ
ation for unipolar case or coupled nonlinear fourth-order parabolic system for bipolar case. We show for the first time the self-similarity property of the macroscopic quantum models in large time. Namely, we show that there exists a unique global strong solution with strictly positive density to the initial value problem of the macroscopic quantum models which tends to a self-similar wave (which is not the exact solution of the models) in large time at an algebraic time-decay rate.
