Remarks on the formation and decay of multidimensional shock waves

In this paper, we present a formula describing the formation and decay of shock wave type solutions in some special cases.Comment: Latex, 7

Weak asymptotics method

We present a new method for constructing solutions to nonlinear evolutionary equations describing the propagation and interaction of nonlinear waves.Comment: 15 pages, 2 figure

Global in Time Asymptotic Solutions to Kolmogorov--Feller-Type Parabolic Pseudodifferential Equations with a Small Parameter. Forward and Backward in Time Motion

The goal of the present paper is to present a new approach to the construction of asymptotic (approximating) solutions to parabolic PDE by using the characteristics.Comment: Latex, 19 p.,9 fig

A simple proof of associativity and commutativity of LR-coefficients (or the hive ring)

We give a simple bijective proof of associativity and commutativity of the Littlewood-Richardson coefficients or the hive ring. Specifically, we establish existence a polarized polymatroidal discretely concave functions on the tetrahedron with given boundary values at two adjoint faces.Comment: 18 pages, 9 figure

Arrays and the octahedron recurrence

Recently, in papers by Knutson, Tao and Woodward, Henriques and Kamnitzer, Pak and Vallejo have been constructed several interesting bijections of associativity and commutativity. In the first two papers bijections relate special sets of discretely concave functions (hives) on triangular grids and the octahedron recurrence plays the key role for these bijections. Pak and Vallejo related special sets of Young tableaux and constructions of these bijections based on standard algorithms in this theory, jeu de taquen, Schutzenberger involution, tableaux switching, etc. In this paper we investigate these constructions from the third point of view, combinatorics of arrays, theory worked out recently by the authors. Arrays naturally related as well to functions on the lattice of integers as to Young tableaux. In the tensor category of arrays, the bijections of associativity and commutativity arise naturally. We establish coincidence of our bijections with that defined in the first two papers and in the integer-valued set-up with the bijection in the third paper (that is, in particular, a solution of Conjecture 1 by Pak and Vallejo). In order to relate different approaches and to reveal combinatorics of the octahedron recurrence, we, first, show that the octahedron recurrence agrees with discrete convexity and, second, we construct another bijection using the octahedron recurrence, the functional form of the RSK correspondence.Comment: 34 pages, 11 figure

Numerical and analytical investigation of the free boundary confluence for the phase field system

In this paper we numerically research the solutions of the phase field system for the spherically symmetric Stefan-Gibbs-Thomson problem in the case of interaction of the free boundaries. We analyze the effect of the soliton type disturbance of the temperature in the point of the contact of the free boundaries.Comment: Latex, 15p., 14 fig

On the Origin of the Multiplicity Fluctuations in High Energy Heavy Ion Collisions

Multiplicity fluctuations in heavy ion collisions obtain comparable contributions both from initial stage of the collisions, and from final stage interaction. We calculate the former component, using the ``wounded nucleon'' model and standard assumptions about nuclei and NN cross section. Combining it with the second one, calculated previously by Stephanov,Rajagopal and Shuryak, ref.2, we obtain good quantitative description of experimental data (ref.3) from NA49 collaboration at CERN on central PbPb collisions.Comment: 3 pages, 2 figure

Confluence of the nonlinear waves in the Stefan problem with undercooling

We assume that the Stefan problem with undercooling has a classical solution until the moment of contact of free boundaries and the free boundaries have continuous velocities until the moment of contact. Under these assumptions, we construct a smooth approximation of the global solution of the Stefan problem with undercooling, which, until the contact, gives the classical solution mentioned above and, after the contact, gives a solution which is the solution of the heat equation.Comment: 36 pages, Late

Global in Time Madelung Transformation for Kolmogorov-Feller Pseudodifferential Equations

Using an idea going back to Madelung we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov-Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta -shock solution of the continuity equation for a discontinuous velocity field. We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics).Comment: Latex, 21

Delta shock wave formation in the case of triangular hyperbolic system of conservation laws

We describe $\delta$ shock wave arising from continuous initial data in the case of triangular conservation law system arising from "generalized pressureless gas dynamics model". We use the weak asymptotic method.Comment: 22 pages, Late