Potential equivalence transformations for nonlinear diffusion-convection equations

Potential equivalence transformations (PETs) are effectively applied to a class of nonlinear diffusion-convection equations. For this class all possible potential symmetries are classified and a theorem on connection of them with point ones via PETs is also proved. It is shown that the known non-local transformations between equations under consideration are nothing but PETs. Action of PETs on sets of exact solutions of a fast diffusion equation is investigated.Comment: 10 page

Potential Nonclassical Symmetries and Solutions of Fast Diffusion Equation

The fast diffusion equation $u_t=(u^{-1}u_x)_x$ is investigated from the symmetry point of view in development of the paper by Gandarias [Phys. Lett. A 286 (2001) 153-160]. After studying equivalence of nonclassical symmetries with respect to a transformation group, we completely classify the nonclassical symmetries of the corresponding potential equation. As a result, new wide classes of potential nonclassical symmetries of the fast diffusion equation are obtained. The set of known exact non-Lie solutions are supplemented with the similar ones. It is shown that all known non-Lie solutions of the fast diffusion equation are exhausted by ones which can be constructed in a regular way with the above potential nonclassical symmetries. Connection between classes of nonclassical and potential nonclassical symmetries of the fast diffusion equation is found.Comment: 13 pages, section 3 is essentially revise

Formation of Millisecond Pulsars in Globular Clusters

In this contribution we discuss how neutron stars are produced and retained in globular clusters, outlining the most important dynamical channels and evolutionary events that affect thepopulation of mass-transferring binaries with neutron stars and result in the formation of recycled pulsars. We confirm the importance of electron-capture supernovae in globular clusters as the major supplier of retained neutron stars.By comparing the observed millisecond pulsar population and the results obtained from simulations, we discuss several constraints on the evolution of mass-transferring systems.In particular, we find that in our cluster model the following mass-gaining events create populations of MSPs that do not match the observations (with respect to binary periods and companion masses or the number of produced systems) and therefore likely do not lead to NSs spun up to millisecond periods: (i) accretion during a common envelope event with a NS formed through accretion-induced collapse, and (ii) mass transfer from a WD donor. By restricting ourselves to the evolutionary and dynamical paths that most likely lead to neutron star recycling, we obtain good agreement between our models and the numbers and characteristics of observed millisecond pulsars in the clusters Terzan 5 and 47 Tuc.Comment: 6 pages, 3 figures, to appear in the proceedings of the 40 Years of Pulsars conference held at McGill University in August 200