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

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

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.

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

Regularized extremal shift in problems of stable control

We discuss a technical approach, based on the method of regularized extremal shift (RES), intended to help solve problems of stable control of uncertain dynamical systems. Our goal is to demonstrate the essence and abilities of the RES technique; for this purpose we construct feedback controller for approximate tracking a prescribed trajectory of an inaccurately observed system described by a parabolic equation. The controller is "resource-saving" in a sense that control resource spent for approximate tracking do not exceed those needed for tracking in an "ideal" situation where the current values of the input disturbance are fully observable. © 2013 IFIP International Federation for Information Processing.German Sci. Found. (DFG) Eur. Sci. Found. (ESF);Natl. Inst. Res. Comput. Sci. Control France (INRIA);DFG Research Center MATHEON;Weierstrass Institute for Applied Analysis and Stochastics (WIAS);European Patent Offic

Conditional Lie-B\"acklund symmetry and reduction of evolution equations.

We suggest a generalization of the notion of invariance of a given partial differential equation with respect to Lie-B\"acklund vector field. Such generalization proves to be effective and enables us to construct principally new Ans\"atze reducing evolution-type equations to several ordinary differential equations. In the framework of the said generalization we obtain principally new reductions of a number of nonlinear heat conductivity equations $u_t=u_{xx}+F(u,u_x)$ with poor Lie symmetry and obtain their exact solutions. It is shown that these solutions can not be constructed by means of the symmetry reduction procedure.Comment: 12 pages, latex, needs amssymb., to appear in the "Journal of Physics A: Mathematical and General" (1995

Flame fronts in Supernovae Ia and their pulsational stability

The structure of the deflagration burning front in type Ia supernovae is considered. The parameters of the flame are obtained: its normal velocity and thickness. The results are in good agreement with the previous works of different authors. The problem of pulsation instability of the flame, subject to plane perturbations, is studied. First, with the artificial system with switched-off hydrodynamics the possibility of secondary reactions to stabilize the front is shown. Second, with account of hydrodynamics, realistic EOS and thermal conduction we can obtain pulsations when Zeldovich number was artificially increased. The critical Zeldovich numbers are presented. These results show the stability of the flame in type Ia supernovae against pulsations because its effective Zeldovich number is small.Comment: 12 pages, 11 figure