On the tau-functions of the Degasperis-Procesi equation

The DP equation is investigated from the point of view of determinant-pfaffian identities. The reciprocal link between the Degasperis-Procesi (DP) equation and the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system is used to construct the N-soliton solution of the DP equation. The N-soliton solution of the DP equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations, the identities between determinants and pfaffians, and the $\tau$-functions of the DP equation are obtained from the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system.Comment: 27 pages, 4 figures, Journal of Physics A: Mathematical and Theoretical, to be publishe

Deficient Reasoning for Dark Matter in Galaxies

Astronomers have been using the measured luminosity to estimate the {\em luminous mass} of stars, based on empirically established mass-to-light ratio which seems to be only applicable to a special class of stars---the main-sequence stars---with still considerable uncertainties. Another basic tool to determine the mass of a system of stars or galaxies comes from the study of their motion, as Newton demonstrated with his law of gravitation, which yields the {\em gravitational mass}. Because the luminous mass can at best only represent a portion of the gravitational mass, finding the luminous mass to be different or less than the gravitational mass should not be surprising. Using such an apparent discrepancy as a compelling evidence for the so-called dark matter, which has been believed to possess mysterious nonbaryonic properties and present a dominant amount in galaxies and the universe, seems to be too far a stretch when seriously examining the facts and uncertainties in the measurement techniques. In our opinion, a galaxy with star type distribution varying from its center to edge may have a mass-to-light ratio varying accordingly. With the thin-disk model computations based on measured rotation curves, we found that most galaxies have a typical mass density profile that peaks at the galactic center and decreases rapidly within $\sim 5$ of the cut-off radius, and then declines nearly exponentially toward the edge. The predicted mass density in the Galactic disk is reasonably within the reported range of that observed in interstellar medium. This leads us to believe that ordinary baryonic matter can be sufficient for supporting the observed galactic rotation curves; speculation of large amount of non-baryonic matter may be based on an ill-conceived discrepancy between gravitational mass and luminous mass which appears to be unjustified