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Discrete & Continuous Dynamical Systems - B

2003 , Volume 3 , Issue 1

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Positive entropic schemes for a nonlinear fourth-order parabolic equation
José A. Carrillo , Ansgar Jüngel and  Shaoqiang Tang
2003, 3(1): 1-20 doi: 10.3934/dcdsb.2003.3.1 +[Abstract](61) +[PDF](600.5KB)
A finite-difference scheme with positivity-preserving and entropy-decreasing properties for a nonlinear fourth-order parabolic equation arising in quantum systems and interface fluctuations is derived. Initial-boundary value problems, the Cauchy problem and a rescaled equation are discussed. Based on this scheme we elucidate properties of the long-time asymptotics for this equation.
Instability of planar traveling waves in bistable reaction-diffusion systems
Masaharu Taniguchi
2003, 3(1): 21-44 doi: 10.3934/dcdsb.2003.3.21 +[Abstract](36) +[PDF](279.4KB)
This paper is concerned with the stability of a planar traveling wave in a cylindrical domain. The equation describes activator-inhibitor systems in chemistry or biology. The wave has a thin transition layer and is constructed by singular perturbation methods. Let $\varepsilon$ be the width of the layer. We show that, if the cross section of the domain is narrow enough, the traveling wave is asymptotically stable, while it is unstable if the cross section is wide enough by studying the linearized eigenvalue problem. For the latter case, we study the wavelength associated with an eigenvalue with the largest real part, which is called the fastest growing wavelength. We prove that this wavelength is $O(\varepsilon^{1/3})$ as $\varepsilon$ goes to zero mathematically rigorously. This fact shows that, if unstable planar waves are perturbed randomly, this fastest growing wavelength is selectively amplified with as time goes on. For this analysis, we use a new uniform convergence theorem for some inverse operator and carry out the Lyapunov-Schmidt reduction.
Existence result for a mixture of non Newtonian flows with stress diffusion using the Cahn-Hilliard formulation
L. Chupin
2003, 3(1): 45-68 doi: 10.3934/dcdsb.2003.3.45 +[Abstract](32) +[PDF](267.6KB)
We consider a model of mixture of non-newtonian fluids described with an order parameter defined by the volume fraction of one fluid in the mixture, a mean-velocity field and an extra-stress tensor field. The evolution of the order parameter is given by a Cahn-Hilliard equation, while the velocity satisfies the classical Navier-Stokes equation with non constant viscosity. The non-newtonian extra-stress tensor, which is symmetric, evolves through a constitutive law with time relaxation of Oldroyd type. We derive at first a physical model for incompressible flows (with free-divergence property for the velocity). In fact, the model we consider contains an additional stress diffusion, which derives from a microscopic dumbbell model analysis. The main result of this paper concerns the existence and uniqueness of a local regular solution for this model.
Global periodicity in a class of reaction-diffusion systems with time delays
Wei Feng and  Xin Lu
2003, 3(1): 69-78 doi: 10.3934/dcdsb.2003.3.69 +[Abstract](42) +[PDF](445.6KB)
In this paper we study a class of reaction-diffusion systems modelling the dynamics of "food-limited" populations with periodic environmental data and time delays. The existence of a global attracting positive periodic solution is first established in the model without time delay. It is further shown that as long as the magnitude of the instantaneous self-limitation effects is larger than that of the time-delay effects, the positive periodic solution is also the global attractor in the time-delay system. Numerical simulations for both cases (with or without time delays) demonstrate the same asymptotic behavior (extinction or converging to the positive $T$-periodic solution, depending on the growth rate of the species).
Traveling waves for a diffusive Lotka-Volterra competition model I: singular perturbations
Yuzo Hosono
2003, 3(1): 79-95 doi: 10.3934/dcdsb.2003.3.79 +[Abstract](41) +[PDF](212.2KB)
This paper concerns traveling wave solutions for a two species competition-diffusion model with the Lotka-Volterra type interaction. We assume that the corresponding kinetic system has only one stable steady state that one of species is existing and the other is extinct, and that the rate $\epsilon_{2}$ of diffusion coefficients of the former species over the latter is small enough. By singular perturbations, we prove the existence of traveling waves for each $c \ge c(\epsilon)$ and discuss the minimal wave speed.
Center manifold of unstable periodic orbits of helium atom: numerical evidence
A. Carati
2003, 3(1): 97-104 doi: 10.3934/dcdsb.2003.3.97 +[Abstract](36) +[PDF](247.6KB)
An original numerical method is introduced for the calculation of orbits on the center manifold of an unstable periodic orbit. The method is implemented for some unstable periodic orbits of the helium atom, and the dynamics on the corresponding center manifold is exhibited.
Geometric solitary waves in a 2D mass-spring lattice
Gero Friesecke and  Karsten Matthies
2003, 3(1): 105-144 doi: 10.3934/dcdsb.2003.3.105 +[Abstract](37) +[PDF](248.0KB)
The existence of longitudinal solitary waves is shown for the Hamiltonian dynamics of a 2D elastic lattice of particles interacting via harmonic springs between nearest and next nearest neighbours. A contrasting nonexistence result for transversal solitary waves is given. The presence of the longitudinal waves is related to the two-dimensional geometry of the lattice which creates a universal overall anharmonicity.
Characteristics and the initial value problem of a completely integrable shallow water equation
Roberto Camassa
2003, 3(1): 115-139 doi: 10.3934/dcdsb.2003.3.115 +[Abstract](54) +[PDF](586.3KB)
The initial value problem for a completely integrable shallow water wave equation is analyzed through its formulation in terms of characteristics. The resulting integro-differential equations give rise to finite dimensional projections onto a class of distributional solutions of the equation, equivalent to taking the Riemann sum approximation of the integrals. These finite dimensional projections are then explicitly solved via a Gram-Schmidt orthogonalization procedure. A particle method based on these reductions is implemented to solve the wave equation numerically.

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