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1.
We consider front solutions of the Swift–Hohenberg equation ∂
t
u= -(1+ ∂
x
2)2
u + ɛ2
u -u
3. These are traveling waves which leave in their wake a periodic pattern in the laboratory frame. Using renormalization techniques
and a decomposition into Bloch waves, we show the non-linear stability of these solutions. It turns out that this problem
is closely related to the question of stability of the trivial solution for the model problem ∂
t
u(x,t) = ∂
x
2
u (x,t)+(1+tanh(x-ct))u(x,t)+u(x,t)
p
with p>3. In particular, we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive
stabilization which sets in once the perturbation has hit the bulk behind the front.
Received: 23 February 2001 / Accepted: 27 August 2001 相似文献
2.
Two metastable states of a multilayer Ge/p-Ge1−x
Six heterosystem with wide (∼ 35 nm) potential wells (Ge) are observed in strong magnetic fields B at low temperatures. In the first state, the Hall resistivity exhibits an inflection near the value ρxy=h/e
2 scaled to one Ge layer. The longitudinal magnetoresistivity ρxx(B) possesses a minimum in the range of fields where this inflection occurs. The temperature evolution of the inflection in
ρxy(B), the minimum of ρ xx(B), and the value of ρxy at the inflection indicates a weakly expressed state of the quantum Hall effect with a uniform current distribution over
the layers. In the second metastable state, an unusually wide plateau near h/2e
2 with a very weak field dependence is observed in ρxy(B). Estimates show that in these samples the Fermi level lies below but close to the top of the inflection in the bottom of
the well. For this reason, the second state can be explained by separation of a hole gas in the Ge layers into two sublayers,
and the saturation of ρxy(B) near h/2e
2 can be explained by the formation of a quantum Hall insulator state.
Pis’ma Zh. éksp. Teor. Fiz. 70, No. 4, 290–297 (25 August 1999) 相似文献
3.
A. S. Demidov 《Russian Journal of Mathematical Physics》2010,17(2):145-153
For a given domain ω ⋐ ℝ2 with boundary γ = ∂ω, we study the cardinality of the set $
\mathfrak{A}_\eta \left( \Phi \right)
$
\mathfrak{A}_\eta \left( \Phi \right)
of pairs of numbers (a, b) for which there is a function u = u
(a,b): ω → ℝ such that ∇2
u(x) = au(x) + b ⩾ 0 for x ∈ ω, u|
γ
= 0, and ||∇u(s)| − Φ(s) ⩽ η for s ∈ γ. Here η ⩾ 0 stands for a very small number, Φ(s) = |∇(s)| / ∫
γ
|∇v| d
γ, and v is the solution of the problem ∇2
v = a
0
v + 1 ⩾ 0 on ω with v|
γ
= 0, where a
0 is a given number. The fundamental difference between the case η = 0 and the physically meaningful case η > 0 is proved. Namely, for η = 0, the set $
\mathfrak{A}_\eta \left( \Phi \right)
$
\mathfrak{A}_\eta \left( \Phi \right)
contains only one element (a, b) for a broad class of domains ω, and a = a
0. On the contrary, for an arbitrarily small η > 0, there is a sequence of pairs (a
j
, b
j
) ∈ $
\mathfrak{A}_\eta \left( \Phi \right)
$
\mathfrak{A}_\eta \left( \Phi \right)
and the corresponding functions u
j
such that ‖f
u
j+1‖ − ‖f
u
j
‖ > 1, where ‖f
u
j
= max
x∈ω
|f
u
j
(x)| and f
u
j
(x) = a
j
u
j
(x) + b
j
. Here the mappings f
u
j
: ω → ℝ necessarily tend as j → ∞ to the δ-function concentrated on γ. 相似文献
4.
Pablo A. Ferrari Beat M. Niederhauser Eugene A. Pechersky 《Journal of statistical physics》2007,128(5):1159-1176
We consider the Harmonic crystal, a measure on
with Hamiltonian H(x)=∑
i,j
J
i,j
(x(i)−x(j))2+h∑
i
(x(i)−d(i))2, where x, d are configurations, x(i), d(i)∈ℝ, i,j∈ℤ
d
. The configuration d is given and considered as observations. The ‘couplings’ J
i,j
are finite range. We use a version of the harness process to explicitly construct the unique infinite volume measure at finite
temperature and to find the unique ground state configuration m corresponding to the Hamiltonian. 相似文献
5.
We study shock statistics in the scalar conservation law ∂
t
u+∂
x
f(u)=0, x∈ℝ, t>0, with a convex flux f and spatially random initial data. We show that the Markov property (in x) is preserved for a large class of random initial data (Markov processes with downward jumps and derivatives of Lévy processes
with downward jumps). The kinetics of shock clustering is then described completely by an evolution equation for the generator
of the Markov process u(x,t), x∈ℝ. We present four distinct derivations for this evolution equation, and show that it takes the form of a Lax pair. The Lax
equation admits a spectral parameter as in Manakov (Funct. Anal. Appl. 10:328–329, 1976), and has remarkable exact solutions for Burgers equation (f(u)=u
2/2). This suggests the kinetic equations of shock clustering are completely integrable. 相似文献
6.
V. P. Ruban 《Journal of Experimental and Theoretical Physics》2010,111(5):776-785
We consider slow, compared to the speed of sound, motions of an ideal compressible fluid (gas) in a gravitational field in
the presence of two isentropic layers with a small specific-entropy difference between them. Assuming the flow to be potential
in each of the layers (v
1, 2 = ▿ϕ1, 2) and neglecting the acoustic degrees of freedom (div($
\bar \rho
$
\bar \rho
(z)▿ϕ1, 2) ≈ 0, where $
\bar \rho
$
\bar \rho
(z) is the average equilibrium density), we derive the equations of motion for the boundary in terms of the shape of the surface
z = η(x, y, t) itself and the difference between the boundary values of the two velocity field potentials: ψ(x, y, t) = ψ1 − ψ2. We prove the Hamilto nian structure of the derived equations specified by a Lagrangian of the form ℒ = ∫$
\bar \rho
$
\bar \rho
(η)η
t
ψdxdy − ℋ{η, ψ}. The system under consideration is the simplest theoretical model for studying internal waves in a sharply stratified
atmosphere in which the decrease in equilibrium gas density due to gas compressibility with increasing height is essentially
taken into account. For plane flows, we make a generalization to the case where each of the layers has its own constant potential
vorticity. We investigate a system with a model dependence $
\bar \rho
$
\bar \rho
(z) ∝ e
−2αz
with which the Hamiltonian ℋ{η, ψ} can be represented explicitly. We consider a long-wavelength dynamic regime with dispersion
corrections and derive an approximate nonlinear equation of the form u
t
+ auu
x
− b[−$
\hat \partial _x^2
$
\hat \partial _x^2
+ α2]1/2
u
x
= 0 (Smith’s equation) for the slow evolution of a traveling wave. 相似文献
7.
The current-voltage characteristics of granular YBa2Cu3O6.95 high-temperature superconductor samples have been measured at a temperature of 77.3 K in external transverse magnetic fields
H
ext with a strength of up to H
ext ≈ 500 Oe for low transport current densities (0.1 A/cm2 ≤ j ≤ 0.6 A/cm2). The current-voltage characteristics obtained have been used to construct dependences of the magnetoresistance ρ on the
quantities j (ρ(j)
Hext=const) and H
ext(ρ(H
ext)
j = const). It has been revealed that the current and field dependences of the magnetoresistance exhibit anomalies at H
ext ≥ H
c1g
, where H
c1g
is the lower critical field of superconducting grains. A comparative analysis of the dependences ρ(j)H
ext = const and ρ(H
ext)
j = const has made it possible to develop concepts regarding the influence of the processes of redistribution of the magnetic field
between grain boundaries and superconducting grains on the transport and galvanomagnetic properties of granular high-temperature
superconductors. It has been established that the field dependences of the magnetoresistance exhibit specific features associated
with the beginning of penetration of Josephson vortices into grain boundaries in the magnetic field H
c1J
and with the breaking of a continuous chain of Josephson junctions in the magnetic field H
c2J
. 相似文献
8.
C?t?lin I. Carstea 《Communications in Mathematical Physics》2010,300(2):487-528
The existence of co-rotational finite time blow up solutions to the wave map problem from ${\mathbb{R}^{2+1} \to N}The existence of co-rotational finite time blow up solutions to the wave map problem from
\mathbbR2+1 ? N{\mathbb{R}^{2+1} \to N} , where N is a surface of revolution with metric d
ρ
2 + g(ρ)2
dθ2, g an entire function, is proven. These are of the form u(t,r)=Q(l(t)t)+R(t,r){u(t,r)=Q(\lambda(t)t)+\mathcal{R}(t,r)} , where Q is a time independent solution of the co-rotational wave map equation −u
tt
+ u
rr
+ r
−1
u
r
= r
−2
g(u)g′(u), λ(t) = t
−1-ν, ν > 1/2 is arbitrary, and R{\mathcal{R}} is a term whose local energy goes to zero as t → 0. 相似文献
9.
Vladimir Rabinovich 《Russian Journal of Mathematical Physics》2012,19(1):107-120
The propagation of electromagnetic waves issued by modulated moving sources of the form j( t,x ) = a( t )e - iw0 t [(x)\dot]0 ( t )d( x - x0 ( t ) )j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right) is considered, where j(t, x) stands for the current density vector, x = (x
1, x
2, x
3) ∈ ℝ3 for the space variables, t ∈ ℝ for time, t → x
0(t) ∈ ℝ3 for the vector function defining the motion of the source, ω
0 for the eigenfrequency of the source, a(t) for a narrow-band amplitude, and δ for the standard δ function. Suppose that the media under consideration are dispersive. This means that the electric and magnetic permittivity
ɛ(ω), μ(ω) depends on the frequency ω. We obtain a representation of electromagnetic fields in the form of time-frequency oscillating integrals whose phase contains
a large parameter λ > 0 characterizing the slowness of the change of the amplitude a(t) and the velocity [(x)\dot]0 ( t )\dot x_0 \left( t \right) and a large distance between positions of the source and the receiver. Applying the two-dimensional stationary phase method
to the integrals, we obtain explicit formulas for the electromagnetic field and for the Doppler effects. As an application
of our approach, we consider the propagation of electromagnetic waves produced by moving source in a cold nonmagnetized plasma
and the Cherenkov radiation in dispersive media. 相似文献
10.
Changjiang Zhu 《Communications in Mathematical Physics》2010,293(1):279-299
In this paper, we study the one-dimensional Navier-Stokes equations connecting to vacuum state with a jump in density when
the viscosity depends on the density. Precisely, when the viscosity coefficient μ(ρ) is proportional to ρ
θ
with θ > 0, where ρ is the density, we give the asymptotic behavior and the decay rate of the density function ρ(x, t). Furthermore, the behavior of the density function ρ(x, t) near the interfaces separating the gas from vacuum and the expanding rate of the interfaces are also studied. The analysis
is based on some new mathematical techniques and some new useful estimates. This fills a final gap on studying Navier-Stokes
equations with the viscosity coefficient μ(ρ) dependent on the density ρ. 相似文献
11.
Lei Gao Yanyan Huang 《The European Physical Journal B - Condensed Matter and Complex Systems》2003,33(2):165-171
The effective linear and nonlinear optical properties of metal/dielectric composite media, in which ellipsoidal metal inclusions
are distributed in shape, are investigated. The shape distribution function P(L
x, L
y) is assumed to be 2Δ-2θ(L
x - 1/3 + Δ/3)θ(L
y - 1/3 + Δ/3)θ(2/3 + Δ/3 - L
x - L
y), where θ( . . . ) is the Heaviside function, Δ is the shape variance and Li are the depolarization factors of the ellipsoidal inclusions along i-symmetric axes (i = x, y). Within the spectral representation, we adopt Maxwell-Garnett type approximation to study the effect of shape variance Δ
on the effective nonlinear optical properties. Numerical results show that both the effective linear optical absorption α
∼ ωIm() and the modulus of the effective third-order optical nonlinearity enhancement |χ(3)
e|/χ(3)
1 exhibit the nonmonotonic behavior with Δ. Moreover, with increasing Δ, the optical absorption and the nonlinearity enhancement
bands become broad, accompanied with the decrease of their peaks. The adjustment of Δ from 0 to 1 allows us to examine the
crossover behavior from no separation to large separation between optical absorption and nonlinearity enhancement peaks. As
Δ → 0, i.e., the ellipsoidal shape deviates slightly from the spherical one, the dependence of |χ(3)
e|/χ(3)
1 on Δ becomes strong first and then weak with increasing the imaginary part of inclusions' dielectric constant. In the dilute
limit, the exact formula for the effective optical nonlinearity is derived, and the present approximation characterizes the
exact results better than old mean field one does.
Received 10 December 2002 Published online 4 June 2003
RID="a"
ID="a"e-mail: lgaophys@pub.sz.jsinfo.net 相似文献
12.
In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + t
n
f(t, u(t)), where A is the generator of a C
0-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C
0-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class S
p-A defined similarly to the case of S
p-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to A
u:= A ∩ BUC(ℝ, X) if n = 0 and to t
n
A
u ⊕ w
n
C
0 (ℝ, X) if n ∈ ℕ, where w
n(t) = (1 + |t|)n. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0.
Dedicated to the memory of B. M. Levitan 相似文献
13.
In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic
equation u
t
=−(u
u
xxx
)
x
, modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u
0(x)∈H
1(ℝ), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L
1-norm of the strong solution u(x,t) towards the unique (among source type solutions) strong source type solution of the equation with the same mass. The method
we use is based on the study of the time decay of the entropy introduced in [13] for the porous medium equation, and uses
analogies between the thin film equation and the porous medium equation.
Received: 2 February 2001 / Accepted: 7 October 2001 相似文献
14.
The character of the evolution of a system of weak links in granular high-temperature superconductors under the action of
an external magnetic field H
ext has been studied by measuring the current-voltage characteristics E(j)Hext = constE{(j)_{{H_{ext}} = const}} of YBa2Cu3O7 − δ (δ ≈ 0.05) ceramic samples. The measurements have been performed at T = 77.3 K in a range of very weak magnetic fields 0 < H
ext ≲ 0.5H
c2J, where H
c2J is the upper critical field of the Josephson weak links. The results have been used to construct the field dependences of
the magnetoresistance Δρ(H
ext) of the superconducting ceramics. It has been established that the parameters of the power equation E = A(j − j
cJ)ν and the magnetoresistance Δρ are nonmonotonic functions of the external magnetic field. The presence of extrema in the curves
A(H
ext), j
cJ(H
ext), ν(H
ext), and Δρ(H
ext) indicates that different systems of weak links between grain boundaries, which are capable of forming extended Josephson
contacts, undergo sequential transitions to a resistive state with an increase in H
ext. 相似文献
15.
Tu Gue-Zhang 《Letters in Mathematical Physics》1979,3(5):387-393
Let (E): u
t=H(u) denote the KdV, MKdV or Burgers equation, and U(s)=(Dj
s)/u
j, where D=d/dx, u
i=Di
u, s=s(u, u
1, ..., u
n) is a polynomial of u
i with constant coefficients, be the generator of invariant group of equation (E). We prove in this paper that all such generators form a commutative Lie algebra, from which it follows that for any symmetry s(u, ..., u
n) of (E), the evolution equation u
t=s(u, ..., u
n) possesses an infinite number of symmetries (or conservation laws in the case of KdV and MKdV equations). 相似文献
16.
The HERA data on the proton structure function F
2(x,Q
2) at very small x and Q
2 show a dramatic departure of the logarithmic slope ∂F
2/∂log Q
2 from theoretical predictions based on the DGLAP evolution. We show that the running BFKL approach provides a quantitative
explanation for the observed x and/or Q
2 dependence of ∂F
2/∂log Q
2.
Pis’ma Zh. éksp. Teor. Fiz. 69, No. 2, 92–97 (25 January 1999)
Published in English in the original Russian journal. Edited by Steve Torstveit. 相似文献
17.
Asymptotic behaviors of zero modes of the massless Dirac operator H = α · D + Q(x) are discussed, where α = (α1, α2, α3) is the triple of 4 × 4 Dirac matrices, , and Q(x) = (q
jk
(x)) is a 4 × 4 Hermitian matrix-valued function with | q
jk
(x) | ≤ C 〈x〉−ρ, ρ > 1. We shall show that for every zero mode f, the asymptotic limit of |x|2
f (x) as |x| → + ∞ exists. The limit is expressed in terms of the Dirac matrices and an integral of Q(x) f (x).
相似文献
18.
The paper answers a question debated by physicists for many years. It is proved that, for almost equal gradients of the magnetic
flux u at its zero-level curve ∂ω, which is the piecewise smooth boundary of a simply-connected domain ω ⋐ ℝ2, the inverse problem for the Grad-Shafranov equation of plasma equilibrium in a tokamak (in the cylindrical approximation)
admits essentially different profiles of distributions f
u
: ω ∋ (x, y) ↦ f(u(x, y)) = u
xx
(x, y) + u
yy
(x, y) ⩾ 0 in the class of third-order polynomials f(u) = Σ
m=03
a
m
u
m
. 相似文献
19.
There are various situations in which it is natural to ask whether a given collection of k functions, ρ
j
(r
1,…,r
j
), j=1,…,k, defined on a set X, are the first k correlation functions of a point process on X. Here we describe some necessary and sufficient conditions on the ρ
j
’s for this to be true. Our primary examples are X=ℝ
d
, X=ℤ
d
, and X an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently
small densities ρ
1(r). Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when
X is a finite set, the existence of a realizing Gibbs measure with k body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example
in which a uniform density ρ and translation invariant ρ
2 are specified on ℤ; there is a gap between our best upper bound on possible values of ρ and the largest ρ for which realizability can be established. 相似文献
20.
Saleh B. Al-Ruwaili Harry A. Mavromatis 《International Journal of Theoretical Physics》1996,35(10):2207-2212
In the course of inverting the partial-wave Born approximation, a new expression for the inverse function ofj
l
2
(ρ) was obtained. Using this result, one can also derive two expressions involving the binomial coefficients. Finally, a particular
differential operator whose effect onj
l
2
(ρ) was previously investigated by Mavromatis and Jalal is shown to have similar effects onn
l
2
(ρ) andn
l
(ρ)j
l
(ρ). 相似文献