839 research outputs found
Heavy ion beam lifetimes at relativistic and ultrarelativistic colliders
The effects of higher order corrections in ultra-relativistic nuclear
collisions are considered. It is found that higher order contributions are
small at low energy, large at intermediate energy and small again at very high
energy. An explanation for this effect is given. This means that the
Weizsacker-Williams formula is a good approximation to use in calculating cross
sections and beam lifetimes at energies relevant to RHIC and LHC.Comment: 10 pages, 2 tables, 4 figure
The Nystrom plus Correction Method for Solving Bound State Equations in Momentum Space
A new method is presented for solving the momentum-space Schrodinger equation
with a linear potential. The Lande-subtracted momentum space integral equation
can be transformed into a matrix equation by the Nystrom method. The method
produces only approximate eigenvalues in the cases of singular potentials such
as the linear potential. The eigenvalues generated by the Nystrom method can be
improved by calculating the numerical errors and adding the appropriate
corrections. The end results are more accurate eigenvalues than those generated
by the basis function method. The method is also shown to work for a
relativistic equation such as the Thompson equation.Comment: Revtex, 21 pages, 4 tables, to be published in Physical Review
Travelling Combustion Waves in a Porous Medium. Part I—Existence
A one-space-dimensional, time-dependent model for travelling combustion waves in a porous medium is analysed. The key variables are the temperature of the solid medium and its density and the temperature of the gaseous phase and its density. The key parameters µ, λ and a are related (respectively) to the driving gas velocity, the specific heat of the combustible solid and the ratio of consumption of oxygen to that of solid. The regions of existence of the different types of combustion waves are found in µ, λ parameter space, with a = 0. The types of combustion wave are classified by the switch mechanism that turns off the combustion, which occurs over a finite, but unknown, interval. Because the model is linear outside the combustion zone, the eigenvalue problem governing the existence of travelling waves may be reformulated as a two-point free boundary problem on a finite domain. Existence and nonexistence theorems are established for this unusual bifurcation problem
Travelling Combustion Waves in a Porous Medium. Part II—Stability
The linear stability properties of the travelling combustion waves found in Part I are examined. The key parameters which determine the stability properties of the waves are found to be the (scaled) driving velocity and the solid specific heat. In particular, the destabilising influence of increasing either of these two parameters is demonstrated. The results indicate that travelling combustion waves whose reaction is turned off because the solid temperature becomes too low are always unstable, whereas travelling waves whose reaction is turned off due to depletion of solid reactant can be stable. Global techniques are employed to prove that, for large enough values of the scaled solid specific heat, combustion cannot be sustained in any form, and all initial conditions lead to extinction
Limit cycles in the presence of convection, a travelling wave analysis
We consider a diffusion model with limit cycle reaction functions, in the
presence of convection. We select a set of functions derived from a realistic
reaction model: the Schnakenberg equations. This resultant form is
unsymmetrical. We find a transformation which maps the irregular equations into
model form. Next we transform the dependent variables into polar form. From
here, a travelling wave analysis is performed on the radial variable. Results
are complex, but we make some simple estimates.
We carry out numerical experiments to test our analysis. An initial `knock'
starts the propagation of pattern. The speed of the travelling wave is not
quite as expected. We investigate further. The system demonstrates distinctly
different behaviour to the left and the right. We explain how this phenomenon
occurs by examining the underlying behaviour.Comment: 20 pages, 5 figure
Existence and stability of singular patterns in a Ginzburg–Landau equation coupled with a mean field
We study singular patterns in a particular system of parabolic partial differential equations which consist of a Ginzburg–Landau equation and a mean field equation. We prove the existence of the three simplest concentrated periodic stationary patterns (single spikes, double spikes, double transition layers) by composing them of more elementary patterns and solving the corresponding consistency conditions. In the case of spike patterns we prove stability for sufficiently large spatial periods by first showing that the eigenvalues do not tend to zero as the period goes to infinity and then passing in the limit to a nonlocal eigenvalue problem which can be studied explicitly. For the two other patterns we show instability by using the variational characterization of eigenvalues
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