Bouncing droplets on a billiard table

In a set of experiments, Couder et. al. demonstrate that an oscillating fluid bed may propagate a bouncing droplet through the guidance of the surface waves. We present a dynamical systems model, in the form of an iterative map, for a droplet on an oscillating bath. We examine the droplet bifurcation from bouncing to walking, and prescribe general requirements for the surface wave to support stable walking states. We show that in addition to walking, there is a region of large forcing that may support the chaotic bouncing of the droplet. Using the map, we then investigate the droplet trajectories for two different wave responses in a square (billiard ball) domain. We show that for waves which are quickly damped in space, the long time trajectories in a square domain are either non-periodic dense curves, or approach a quasiperiodic orbit. In contrast, for waves which extend over many wavelengths, at low forcing, trajectories tend to approach an array of circular attracting sets. As the forcing increases, the attracting sets break down and the droplet travels throughout space

Simulation of stellar instabilities with vastly different timescales using domain decomposition

Strange mode instabilities in the envelopes of massive stars lead to shock waves, which can oscillate on a much shorter timescale than that associated with the primary instability. The phenomenon is studied by direct numerical simulation using a, with respect to time, implicit Lagrangian scheme, which allows for the variation by several orders of magnitude of the dependent variables. The timestep for the simulation of the system is reduced appreciably by the shock oscillations and prevents its long term study. A procedure based on domain decomposition is proposed to surmount the difficulty of vastly different timescales in various regions of the stellar envelope and thus to enable the desired long term simulations. Criteria for domain decomposition are derived and the proper treatment of the resulting inner boundaries is discussed. Tests of the approach are presented and its viability is demonstrated by application to a model for the star P Cygni. In this investigation primarily the feasibility of domain decomposition for the problem considered is studied. We intend to use the results as the basis of an extension to two dimensional simulations.Comment: 15 pages, 10 figures, published in MNRA

Self-Similar Solutions to a Density-Dependent Reaction-Diffusion Model

In this paper, we investigated a density-dependent reaction-diffusion equation, $u_t = (u^{m})_{xx} + u - u^{m}$. This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation which is widely used in the population dynamics, combustion theory and plasma physics. By employing the suitable transformation, this equation was mapped to the anomalous diffusion equation where the nonlinear reaction term was eliminated. Due to its simpler form, some exact self-similar solutions with the compact support have been obtained. The solutions, evolving from an initial state, converge to the usual traveling wave at a certain transition time. Hence, it is quite clear the connection between the self-similar solution and the traveling wave solution from these results. Moreover, the solutions were found in the manner that either propagates to the right or propagates to the left. Furthermore, the two solutions form a symmetric solution, expanding in both directions. The application on the spatiotemporal pattern formation in biological population has been mainly focused.Comment: 5 pages, 2 figures, accepted by Phys. Rev.

Structures and waves in a nonlinear heat-conducting medium

The paper is an overview of the main contributions of a Bulgarian team of researchers to the problem of finding the possible structures and waves in the open nonlinear heat conducting medium, described by a reaction-diffusion equation. Being posed and actively worked out by the Russian school of A. A. Samarskii and S.P. Kurdyumov since the seventies of the last century, this problem still contains open and challenging questions.Comment: 23 pages, 13 figures, the final publication will appear in Springer Proceedings in Mathematics and Statistics, Numerical Methods for PDEs: Theory, Algorithms and their Application

Flux-splitting schemes for parabolic problems

To solve numerically boundary value problems for parabolic equations with mixed derivatives, the construction of difference schemes with prescribed quality faces essential difficulties. In parabolic problems, some possibilities are associated with the transition to a new formulation of the problem, where the fluxes (derivatives with respect to a spatial direction) are treated as unknown quantities. In this case, the original problem is rewritten in the form of a boundary value problem for the system of equations in the fluxes. This work deals with studying schemes with weights for parabolic equations written in the flux coordinates. Unconditionally stable flux locally one-dimensional schemes of the first and second order of approximation in time are constructed for parabolic equations without mixed derivatives. A peculiarity of the system of equations written in flux variables for equations with mixed derivatives is that there do exist coupled terms with time derivatives

Probing the hydrogen melting line at high pressures by dynamic compression

We investigate the capabilities of dynamic compression by intense heavy ion beams to yield information about the high pressure phases of hydrogen. Employing ab initio simulations and experimental data, a new wide range equation of state for hydrogen is constructed. The results show that the melting line up to its maximum as well as the transition from molecular fluids to fully ionized plasmas can be tested with the beam parameters soon to be available. We demonstrate that x-ray scattering can distinguish between phases and dissociation states

Time--space white noise eliminates global solutions in reaction diffusion equations

We prove that perturbing the reaction--diffusion equation $u_t=u_{xx} + (u_+)^p$ ($p>1$), with time--space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.Comment: New results included. To be published in Physica

Symmetry-preserving discrete schemes for some heat transfer equations

Lie group analysis of differential equations is a generally recognized method, which provides invariant solutions, integrability, conservation laws etc. In this paper we present three characteristic examples of the construction of invariant difference equations and meshes, where the original continuous symmetries are preserved in discrete models. Conservation of symmetries in difference modeling helps to retain qualitative properties of the differential equations in their difference counterparts.Comment: 21 pages, 4 ps figure

Mathematical modeling of the electric spark ignition of the coal-water particle fuel under conditions relevant to the internal combustion engines

The results of the mathematical simulation of the coal-water fuel particle in the conditions of electric-spark discharge have been given. It has been found that the CWF particle can ignite even in a relatively low (Tg<500K) ambient temperature. It has been shown that the strength of the current and ambient temperature have a significant effect on the characteristics and conditions of ignition