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1.
Summary Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived
for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that,
near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small—as,
for instance, can be the case in double-layer convection. Based on these assumptions we first derive a singularly perturbed
modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system
to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study
and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions
of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed
system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic
system. Thus, the ‘Landau reduction’ to the nonlocal system has no significant influence on the stationary quasi-periodic
patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed
system. These orbits all correspond to so-called ‘localised structures’ in the underlying system: They connect simple periodic
patterns atx → ± ∞. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal
system destroys a rich and important set of patterns. 相似文献
2.
H. Uecker 《Journal of Nonlinear Science》2001,11(2):89-121
Summary. We show the existence and stability of modulating multipulse solutions for a class of bifurcation problems given by a dispersive
Swift-Hohenberg type of equation with a spatially periodic forcing. Equations of this type arise as model problems for pattern
formation over unbounded weakly oscillating domains and, more specifically, in laser optics. As associated modulation equation,
one obtains a nonsymmetric Ginzburg-Landau equation which possesses exponentially stable stationary n—pulse solutions. The modulating multipulse solutions of the original equation then consist of a traveling pulselike envelope
modulating a spatially oscillating wave train. They are constructed by means of spatial dynamics and center manifold theory.
In order to show their stability, we use Floquet theory and combine the validity of the modulation equation with the exponential
stability of the n—pulses in the modulation equation. The analysis is supplemented by a few numerical computations.
In addition, we also show, in a different parameter regime, the existence of exponentially stable stationary periodic solutions
for the class of systems under consideration.
Received November 30, 1999; accepted December 4, 2000 Online publication March 23, 2001 相似文献
3.
L. A. Petrov 《Functional Analysis and Its Applications》2009,43(4):279-296
We construct a two-parameter family of diffusion processes X
α,θ
on the Kingman simplex, which consists of all nonincreasing infinite sequences of nonnegative numbers with sum less than
or equal to one. The processes on this simplex arise as limits of finite Markov chains on partitions of positive integers.
For α = 0, our process coincides with the infinitely-many-neutral-alleles diffusion model constructed by Ethier and Kurtz (1981)
in population genetics. The general two-parameter case apparently lacks population-genetic interpretation. In the present
paper, we extend Ethier and Kurtz’s main results to the two-parameter case. Namely, we show that the (two-parameter) Poisson-Dirichlet
distribution PD(α,θ) is the unique stationary distribution for the process X
α,θ
and that the process is ergodic and reversible with respect to PD(α, θ). We also compute the spectrum of the generator of X
α,θ
. The Wright-Fisher diffusions on finite-dimensional simplices turn out to be special cases of X
α,θ
for certain degenerate parameter values. 相似文献
4.
J. Haro 《Theoretical and Mathematical Physics》2012,171(1):563-574
Using quantum corrections from massless fields conformally coupled to gravity, we study the possibility of avoiding singularities
that appear in the flat Friedmann-Robertson-Walker model. We assume that the universe contains a barotropic perfect fluid
with the state equation p = ωρ, where p is the pressure and ρ is the energy density. We study the dynamics of the model for
all values of the parameter ω and also for all values of the conformal anomaly coefficients α and β. We show that singularities
can be avoided only in the case where α > 0 and β < 0. To obtain an expanding Friedmann universe at late times with ω > −1 (only a one-parameter family of solutions, but no a general solution, has this behavior at late times), the initial conditions
of the nonsingular solutions at early times must be chosen very exactly. These nonsingular solutions consist of a general
solution (a two-parameter family) exiting the contracting de Sitter phase and a one-parameter family exiting the contracting
Friedmann phase. On the other hand, for ω < −1 (a phantom field), the problem of avoiding singularities is more involved because if we consider an expanding Friedmann phase
at early times, then in addition to fine-tuning the initial conditions, we must also fine-tune the parameters α and β to obtain
a behavior without future singularities: only a oneparameter family of solutions follows a contracting Friedmann phase at
late times, and only a particular solution behaves like a contracting de Sitter universe. The other solutions have future
singularities. 相似文献
5.
Firstly, we analyze a codimension-two unfolding for the Hopf-transcritical bifurcation, and give complete bifurcation diagrams and phase portraits. In particular, we express explicitly the heteroclinic bifurcation curve, and obtain conditions under which the secondary bifurcation periodic solutions and the heteroclinic orbit are stable. Secondly, we show how to reduce general retarded functional differential equation, with perturbation parameters near the critical point of the Hopf-transcritical bifurcation, to a 3-dimensional ordinary differential equation which is restricted on the center manifold up to the third order with unfolding parameters, and further reduce it to a 2-dimensional amplitude system, where these unfolding parameters can be expressed by those original perturbation parameters. Finally, we apply the general results to the van der Pol’s equation with delayed feedback, and obtain the existence of stable or unstable equilibria, periodic solutions and quasi-periodic solutions. 相似文献
6.
A novel approach of using harmonic balance (HB) method is presented to find front, soliton and hole solutions of a modified complex Ginzburg-Landau equation. Three families of exact solutions are obtained, one of which contains two parameters while the others one parameter. The HB method is an efficient technique in finding limit cycles of dynamical systems. In this paper, the method is extended to obtain homoclinic/heteroclinic orbits and then coherent structures. It provides a systematic approach as various methods may be needed to obtain these families of solutions. As limit cycles with arbitrary value of bifurcation parameter can be found through parametric continuation, this approach can be extended further to find analytic solution of complex quintic Ginzburg-Landau equation in terms of Fourier series. 相似文献
7.
We propose a method of obtaining the dispersion equation for normal waves in an orthotropic cylinder from the boundary conditions
on the rigidly clamped boundary using a system of exponential particular solutions of the three-dimensional equations of its
stationary wave motions. We compute the real and imaginary branches of the dispersion spectrum for a waveguide made of monocrystalline
strontium sulfate. On figure. Bibliography: 7 titles.
Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 96–99. 相似文献
8.
N. E. Kulagin L. M. Lerman T. G. Shmakova 《Computational Mathematics and Mathematical Physics》2008,48(4):659-676
The generalized Swift-Hohenberg equation with an additional quadratic term is studied. Time-stable localized stationary solutions of the pulse and front types are found. It is shown that stationary fronts give rise to traveling fronts, whose branches are also obtained. This study combines theoretical methods for dynamical systems (in particular, the theory of homo-and heteroclinic orbits) and numerical simulation. 相似文献
9.
W. Eckhaus 《Journal of Nonlinear Science》1993,3(1):329-348
Summary The Ginzburg-Landau modulation equation arises in many domains of science as a (formal) approximate equation describing the
evolution of patterns through instabilities and bifurcations. Recently, for a large class of evolution PDE's in one space
variable, the validity of the approximation has rigorously been established, in the following sense: Consider initial conditions
of which the Fourier-transforms are scaled according to the so-calledclustered mode-distribution. Then the corresponding solutions of the “full” problem and the G-L equation remain close to each other on compact intervals
of the intrinsic Ginzburg-Landau time-variable. In this paper the following complementary result is established. Consider
small, but arbitrary initial conditions. The Fourier-transforms of the solutions of the “full” problem settle to clustered
mode-distribution on time-scales which are rapid as compared to the time-scale of evolution of the Ginzburg-Landau equation. 相似文献