The tacnode Riemann-Hilbert problem

The tacnode Riemann-Hilbert problem is a 4 x 4 matrix valued RH problem that appears in the description of the local behavior of two touching groups of non-intersecting Brownian motions. The same RH problem was also found by Duits and Geudens to describe a new critical regime in the two-matrix model. Delvaux gave integral representations for some of the entries of the 4 x 4 matrix. We complement this work by presenting integral representations for all of the entries. As a consequence we give an explicit formula for the Duits-Geudens critical kernel.Comment: 29 pages, 5 figure

Delta-shocks and vacuums in zero-pressure gas dynamics by the flux approximation

In this paper, firstly, by solving the Riemann problem of the zero-pressure flow in gas dynamics with a flux approximation, we construct parameterized delta-shock and constant density solutions, then we show that, as the flux perturbation vanishes, they converge to the delta-shock and vacuum state solutions of the zero-pressure flow, respectively. Secondly, we solve the Riemann problem of the Euler equations of isentropic gas dynamics with a double parameter flux approximation including pressure. Further we rigorously prove that, as the two-parameter flux perturbation vanishes, any Riemann solution containing two shock waves tends to a delta shock solution to the zero-pressure flow; any Riemann solution containing two rarefaction waves tends to a two-contact-discontinuity solution to the zero-pressure flow and the nonvacuum intermediate state in between tends to a vacuum state.Comment: 17 pages, 4 figures, accepted for publication in SCIENCE CHINA Mathematic