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1.
61. IntroductionLet us consider the clajss of partial differential equations of the form--div(a.(x, Du.)) ~ f on fi, us E Wt,p(fl), (1.1)where a6 is increasingly oscillating as E -- 0, fi is an open bounded subset of R", 1 < p < cot1/p 1/q = 1 and f E W--"q(fl). The homogenization problem for (l.1) consists of the studyof the asymptotic behavior of solutions net as e - 0. In many important cases nE convergesweakly in WI'p(n) to the solution no of the homogenized problem--div(b(Duo)) = f …  相似文献   

2.
1 IntroductionConsider the parameter dependent equationu"+ (λ+ s(μ) ) f( u) -μsinx =0  in ( 0 ,π)u( 0 ) =u(π) =0 ( 1 .1 )whereλ,μ∈R are parameters and f:R→R and S:R→R are smooth odd functions anda) f′( 0 ) =1 ,   b) f ( 0 )≠ 0 ,   c) s( 0 ) =0 ,   d) s′( 0 ) =1 . ( 1 .2 )Let S:u( x)→ u(π-x) ,Γ ={ S,I} ,then ( 1 .1 ) isΓ -equivariant.The equality ( 1 .2 a) isjust a normalization of f at x=0 .Otherwise,one may reseek the parameter x to ensure( 1 .2 a) .To simplify an…  相似文献   

3.
1 IntroductionConsider tl1e optimizatioll problemndn{f(x): gj(x) 5 0, j e I, x E R"}, j1)where f(x), gj(x): R" - R, j E I = {l,2,...,m}.We know tl1e quasi-Newton meth.d[1]'[9]1[5]1[1O1 is one of the most effective methods to solveproblenl (1) due to its property of superlinear convergence and is still all hot topic at presenttime, which attracts a Iot of authors to make iInprovemellt both in theory a1ld app1ication.Fechinei and Lucidi[3] in 1995 proposed a locally superlinearly convergell…  相似文献   

4.
A GENERALIZATION OF A PROPOSITION ON EXPONENTIAL DICHOTOMY   总被引:2,自引:0,他引:2  
1 Introduction and Statement of TheoremConsider systemx' = A(t)x f(t, x), (l.1)where x E R", A(t) is a continuous matrix function, f: R x R" - R" is acontinuous function.We say that the linear differential equation X' = A(t)x admits an exponential dichotomy, if it has a fundamental matrix X(t) such thatIX(t)PX--'(s)l 5 K' e--a(t--s) for s S t,(1.2)IX(t)(I -- P)X--'(s)I 5 K' e--a(s--t) for s 2 t, (1'2)where P is a projection (P' = P), K and a are positive constants.Remark Without …  相似文献   

5.
This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation {ut+(u^2/2)x+px=εuxx, t〉0,x∈R, -αPxx+P=f(u)+α/2ux^2-1/2u^2, t〉0,x∈R, (E) with the initial data u(0,x)=u0(x)→u±, as x→±∞ (I) Here, u_ 〈 u+ are two constants and f(u) is a sufficiently smooth function satisfying f" (u) 〉 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law It is well known that if u_ 〈 u+, the above Riemann problem admits a unique global entropy solution u^R(x/t) u^R(x/t)={u_,(f′)^-1(x/t),u+, x≤f′(u_)t, f′(u_)t≤x≤f′(u+)t, x≥f′(u+)t. Let U(t, x) be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u0(x) - U(0,x) ∈ H^1(R) and u_ 〈 u+, the above Cauchy problem (E) and (I) admits a unique global classical solution u(t, x) which tends to the rarefaction wave u^R(x/t) as → +∞ in the maximum norm. The proof is given by an elementary energy method.  相似文献   

6.
1 IntroductionIn tl1is paper we coIlsider positive steady-statc solutiolls to the fOllowi1lg weakly-c()uI)ledparabolic systenlwllere fl is a bouuded don1ain in Rn with sufficiently smooth bouudary, t1T = n x [0, T) and0f1T = 0fl x [0, T) l' is the olltward ul1it 11orIIlaJ vector. The col1stants ai, bi, (i = 1. 2, 3, 4, j =l, 2, 3, 4) are positive.The systenl (1.1) describes tlle LotkaVOlterra two-predator, two-prey nlodcl. u1 a11d u2represeIlt tl1e del1sitie8 of twDeprey wllile u3, '('4 …  相似文献   

7.
1. IntroductionConsider the optimization problemmin {f(x): gi(x) 5 0, j E I; x E R"}, (l)where f(x), gi(x): Rad - R, j E I ~ {1, 2,...,m}.It is well known that one of the most effective methods to solve problem (1) is thesequential quadratic programming (i.e., SoP) (see [1--6]), due to its property of superlinearconvergence. Especially in recent years, in order to perfect SoP both in theory and application, there have many papers, such as [7--10], been published. These papers focus mainly…  相似文献   

8.
1. IntroductionWe are concerned with the following variational inequality problem of finding amx E X such thatwhere f: R" - R" is assumed to be a continuously differentiable function, and X g R"is specified bywhere gi: R" -- R and h,-: R" - R are twice continuously differentiable functions.The variational inequality (1.1) is denoted by VI(X, f). An important special case ofVI(X, f) is the so--called nonlinear complementarity problem (NCP(f)) with X ~ R7 {x E R" I x 2 0}. Variational…  相似文献   

9.
1 IntroductionConsider the fOllowing system:t'here E is a small parameter, H(I, y) is a polynomial of degree (n + 1 ) ? f(I, y)and g(x, y) are polynomials of degree S n. The corresponding Abelian integrallswhere b(h) is a compact leve1 curve H--'(h) of (1.l) when E = 0, for h lyingbetween criticaI values of H. The Hilbert-Arnold prob1em (weakend Hilbert16th problern) is to find an upper boulld of the nurnber of zeros of (l.2) forfixed n 2 2 and for al1 H, f and g. In this paper, we tvi11…  相似文献   

10.
Let I be a compact interval of real axis R, and(I, H) be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f : I → I be a continuous multi-valued map. Assume that Pn =(x_0, x_1,..., xn) is a return tra jectory of f and that p ∈ [min Pn, max Pn] with p ∈ f(p). In this paper, we show that if there exist k(≥ 1) centripetal point pairs of f(relative to p)in {(x_i; x_i+1) : 0 ≤ i ≤ n-1} and n = sk + r(0 ≤ r ≤ k-1), then f has an R-periodic orbit, where R = s + 1 if s is even, and R = s if s is odd and r = 0, and R = s + 2 if s is odd and r 0. Besides,we also study stability of periodic orbits of continuous multi-valued maps from I to I.  相似文献   

11.
本文证明了:当Ginzburg-Landau参数足够大时,一维Ginzburg-Landau超导方程组的对称解 是唯一的.该问题的难点在于所考虑的解具有“奇点”:也即,当Ginzburg-Landau参数趋于无穷大 时,解的导数在这些点处趋于无穷.证明的关键是要得到解在这些奇点近旁的精细估计.  相似文献   

12.
In this paper, it is proved that for any given d non-degenerate local minimum points of the renormalized energy of weighted Ginzburg-Landau eqautions, one can find solutions to the Ginzburg-Landau equations whose vortices tend to these d points. This provides the connections between solutions of a class of Ginzburg-Landau equations with weight and minimizers of the renormalized energy.  相似文献   

13.
Traveling waves in the complex Ginzburg-Landau equation   总被引:1,自引:0,他引:1  
Summary In this paper we consider a modulation (or amplitude) equation that appears in the nonlinear stability analysis of reversible or nearly reversible systems. This equation is the complex Ginzburg-Landau equation with coefficients with small imaginary parts. We regard this equation as a perturbation of the real Ginzburg-Landau equation and study the persistence of the properties of the stationary solutions of the real equation under this perturbation. First we show that it is necessary to consider a two-parameter family of traveling solutions with wave speedυ and (temporal) frequencyθ; these solutions are the natural continuations of the stationary solutions of the real equation. We show that there exists a two-parameter family of traveling quasiperiodic solutions that can be regarded as a direct continuation of the two-parameter family of spatially quasi-periodic solutions of the integrable stationary real Ginzburg-Landau equation. We explicitly determine a region in the (wave speedυ, frequencyθ)-parameter space in which the weakly complex Ginzburg-Landau equation has traveling quasi-periodic solutions. There are two different one-parameter families of heteroclinic solutions in the weakly complex case. One of them consists of slowly varying plane waves; the other is directly related to the analytical solutions due to Bekki & Nozaki [3]. These solutions correspond to traveling localized structures that connect two different periodic patterns. The connections correspond to a one-parameter family of heteroclinic cycles in an o.d.e. reduction. This family of cycles is obtained by determining the limit behaviour of the traveling quasi-periodic solutions as the period of the amplitude goes to ∞. Therefore, the heteroclinic cycles merge into the stationary homoclinic solution of the real Ginzburg-Landau equation in the limit in which the imaginary terms disappear.  相似文献   

14.
We study the adiabatic limit in hyperbolic Ginzburg-Landau equations which are the Euler-Lagrange equations for the Abelian Higgs model. By passing to the adiabatic limit in these equations, we establish a correspondence between the solutions of the Ginzburg-Landau equations and adiabatic trajectories in the moduli space of static solutions, called vortices. Manton proposed a heuristic adiabatic principle stating that every solution of the Ginzburg-Landau equations with sufficiently small kinetic energy can be obtained as a perturbation of some adiabatic trajectory. A rigorous proof of this result has been found recently by the first author.  相似文献   

15.
We study the inviscid limit of the complex Ginzburg-Landau equation. We observe that the solutions for the complex Ginzburg-Landau equation converge to the corresponding solutions for the nonlinear Schrödinger equation. We give its convergence rate. We estimate the integral forms of solutions for two equations.  相似文献   

16.
Summary. We consider weakly unstable reaction—diffusion systems defined on domains with one or more unbounded space-directions. In the systems which we have in mind, at criticality, the most unstable eigenvalue belongs to the wave vector zero and possesses a nonvanishing imaginary part. This instability leads to an almost spatially homogeneous Hopf-bifurcation in time. A standard example is the Brusselator in certain parameter ranges. Using multiple scaling analysis we derive a Ginzburg-Landau equation and show that all small solutions develop in such a way that they can be approximated after a certain time by the solutions of the Ginzburg-Landau equation. The proof differs essentially from the case when the bifurcating pattern is oscillatory in space. Our proof is based on normal form methods. As a consequence of the results, the global existence in time of all small bifurcating solutions and the upper-semicontinuity of the original system attractor towards the associated Ginzburg-Landau attractor follows. Original received February 21, 1996; revision accepted April 16, 1997  相似文献   

17.
The Ginzburg-Landau equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolution of small amplitude instabilities near criticality. It is well known that the (cubic) Ginzburg-Landau equation has various unstable solitary pulse solutions. However, such localized patterns have been observed in systems in which there are two competing instability mechanisms. In such systems, the evolution of instabilities is described by a Ginzburg-Landau equation coupled to a diffusion equation. In this article we study the influence of this additional diffusion equation on the pulse solutions of the Ginzburg-Landau equation in light of recently developed insights into the effects of slow diffusion on the stability of pulses. Therefore, we consider the limit case of slow diffusion, i.e., the situation in which the additional diffusion equation acts on a long spatial scale. We show that the solitary pulse solution of the Ginzburg-Landau equation persists under this coupling. We use the Evans function method to analyze the effect of the slow diffusion and to show that it acts as a control mechanism that influences the (in)stability of the pulse. We establish that this control mechanism can indeed stabilize a pulse when higher order nonlinearities are taken into account.  相似文献   

18.
In this paper,we prove the existence of global classical solutions to time-dependent Ginzburg-Landau(TDGL) equations.By the properties of Besov and Sobolev spaces,together with the energy method,we establish the global existence and uniqueness of classical solutions to the initial boundary value problem for time-dependent Ginzburg-Landau equations.  相似文献   

19.
研究了时空分数阶复Ginzburg-Landau方程.首先通过分数阶复变换将时空分数阶复Ginzburg-Landau方程转化为一个常微分方程.然后将常微分方程化为初等积分形式.最后用多项式完全判别系统法求得一系列精确解,其中包含有孤立波解、有理函数解、三角函数周期解、Jacobi椭圆函数双周期解.  相似文献   

20.
We are interested in spatially extended pattern forming systems close to the threshold of the first instability in case when the so-called degenerated Ginzburg-Landau equation takes the role of the classical Ginzburg-Landau equation as the amplitude equation of the system. This is the case when the relevant nonlinear terms vanish at the bifurcation point. Here we prove that in this situation every small solution of the pattern forming system develops in such a way that after a certain time it can be approximated by the solutions of the degenerated Ginzburg-Landau equation. In this paper we restrict ourselves to a Swift-Hohenberg-Kuramoto-Shivashinsky equation as a model for such a pattern forming system.  相似文献   

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