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Discrete & Continuous Dynamical Systems - B

2002 , Volume 2 , Issue 1

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Convergence of a boundary integral method for 3-D water waves
Thomas Y. Hou and  Pingwen Zhang
2002, 2(1): 1-34 doi: 10.3934/dcdsb.2002.2.1 +[Abstract](54) +[PDF](306.2KB)
Abstract:
We prove convergence of a modified point vortex method for time-dependent water waves in a three-dimensional, inviscid, irrotational and incompressible fluid. Our stability analysis has two important ingredients. First we derive a leading order approximation of the singular velocity integral. This leading order approximation captures all the leading order contributions of the original velocity integral to linear stability. Moreover, the leading order approximation can be expressed in terms of the Riesz transform, and can be approximated with spectral accuracy. Using this leading order approximation, we construct a near field correction to stabilize the point vortex method approximation. With the near field correction, our modified point vortex method is linearly stable and preserves all the spectral properties of the continuous velocity integral to the leading order. Nonlinear stability and convergence with 3rd order accuracy are obtained using Strang’s technique by establishing an error expansion in the consistency error.
Simulation of stationary chemical patterns and waves in ionic reactions
Arno F. Münster
2002, 2(1): 35-46 doi: 10.3934/dcdsb.2002.2.35 +[Abstract](30) +[PDF](252.3KB)
Abstract:
In numerical simulations based on a general model chemical patterns in ionic reaction-advection systems assuming a "self-consistent" electric field are presented. Chemical waves as well as stationary concentration patterns arise due to an interplay of an autocatalytic chemical reaction with diffusion, migration of ions in an applied electric field and hydrodynamic flow. Concentration gradients inside the chemical pattern lead to electric diffusion-potentials which in turn affect the patterns. Thus,the model equations take the general form of the Fokker-Planck equation. The principles of modeling a ionic reaction-diffusion-migration system are applied to a real chemical system, the nonlinear methylene blue-sulfide-oxygen reaction.
Asymptotic behavior of solutions of time-delayed Burgers' equation
Weijiu Liu
2002, 2(1): 47-56 doi: 10.3934/dcdsb.2002.2.47 +[Abstract](43) +[PDF](153.8KB)
Abstract:
In this paper, we consider Burgers' equation with a time delay. By using the Liapunov function method, we show that the delayed Burgers' equation is exponentially stable if the delay parameter is sufficiently small. We also give an explicit estimate of the delay parameter in terms of the viscosity and initial conditions, which indicates that the delay parameter tends to zero if the initial states tend to infinity or the viscosity tends to zero. Furthermore, we present numerical simulations for our theoretical results.
Lyapunov-based transfer between elliptic Keplerian orbits
Dong Eui Chang , David F. Chichka and  Jerrold E. Marsden
2002, 2(1): 57-67 doi: 10.3934/dcdsb.2002.2.57 +[Abstract](35) +[PDF](212.7KB)
Abstract:
We present a study of the transfer of satellites between elliptic Keplerian orbits using Lyapunov stability theory specific to this problem. The construction of Lyapunov functions is based on the fact that a non-degenerate Keplerian orbit is uniquely described by its angular momentum and Laplace (- Runge-Lenz) vectors. We suggest a Lyapunov function, which gives a feedback controller such that the target elliptic orbit becomes a locally asymptotically stable periodic orbit in the closed-loop dynamics. We show how to perform a global transfer between two arbitrary elliptic orbits based on the local transfer result. Finally, a second Lyapunov function is presented that works only for circular target orbits.
Transmission boundary conditions in a model-kinetic decomposition
C. Bourdarias , M. Gisclon and  A. Omrane
2002, 2(1): 69-94 doi: 10.3934/dcdsb.2002.2.69 +[Abstract](29) +[PDF](254.9KB)
Abstract:
This paper deals with the fluid limit using the Perthame-Tadmor model with initial and boundary conditions of transmission type within two positive parameters $\varepsilon_1$ and $\varepsilon_2$ for the kinetic dynamical problem. We show that the kinetic problem is well posed in $L^\infty \bigcap L^1(0,T;L^1(\mathbb{R}^n \times \mathbb{R}_v))$. We also prove a BV estimate which allows us to pass to the limit in each kinetic region or, under restrictive conditions, in a single region. This result can be applied to scalar conservation laws with decomposition domain.
Finite element analysis and approximations of phase-lock equations of superconductivity
Mei-Qin Zhan
2002, 2(1): 95-108 doi: 10.3934/dcdsb.2002.2.95 +[Abstract](37) +[PDF](211.9KB)
Abstract:
In [22], the author introduced the phase-lock equations and established existences of both strong and weak solutions of the equations. We also investigated the relations between phase-lock equations and Ginzburg-Landau equations of Superconductivity. In this paper, we present finite element analysis and computations of phase-lock equations. We derive the error estimates for both semi-discrete and fully discrete equations, including optimal $L^2$ and $H^1$ error estimates. In the fully discrete case, we use backward Euler method to discretize the time variable.
The nonlinear Schrödinger equation as a resonant normal form
Dario Bambusi , A. Carati and  A. Ponno
2002, 2(1): 109-128 doi: 10.3934/dcdsb.2002.2.109 +[Abstract](49) +[PDF](249.8KB)
Abstract:
Averaging theory is used to study the dynamics of dispersive equations taking the nonlinear Klein Gordon equation on the line as a model problem: For approximatively monochromatic initial data of amplitude $\epsilon$, we show that the corresponding solution consists of two non interacting wave packets, each one being described by a nonlinear Schrödinger equation. Such solutions are also proved to be stable over times of order $1/ \epsilon^2$. We think that this approach puts into a new light the problem of obtaining modulations equations for general dispersive equations. The proof of our results requires a new use of normal forms as a tool for constructing approximate solutions.
Identification of modulated rotating waves in pattern-forming systems with O(2) symmetry
A. Palacios
2002, 2(1): 129-147 doi: 10.3934/dcdsb.2002.2.129 +[Abstract](28) +[PDF](390.2KB)
Abstract:
A numerical algorithm for identifying Modulated Rotating Waves in spatially extended systems with O(2) symmetry—the symmetry group of rotations and reflections on the plane, is presented. The algorithm can be applied to numerical simulations of Partial Differential Equations (PDEs) and experimental data obtained in a laboratory. The basic methodology is illustrated with various cellular patterns obtained from video images of a combustion experiment carried out on a circular burner. Rotating waves and modulated rotating waves are successfully identified in the experiment. The algorithm is then validated by comparing the analysis of experimental patterns with the analysis of computational patterns obtained from numerical simulations of a reaction-diffusion PDE model.

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