共查询到19条相似文献,搜索用时 140 毫秒
1.
研究一类具扰动的Sine-Gordon方程utt-uxx+αsinu-βuxxt=g(u),t>0,-∞< x< ∞的周期初值问题,提出了谱方法,并用先验估计方法作了误差估计,证明了近似方法的收敛性,并得到了该问题广义解的存在、唯一性。 相似文献
2.
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4.
本文利用广义条件对称方法对带源项的变系数非线性反应扩散方程 f(x)ut=(g(x)D(u)ux)x+h(x)P(u)ux+q(x)Q(u)进行研究. 当扩散项D(u)取um (m≠-1,0,1)和eu两种重要情形时, 对该方程进行对称约化,得到了具有广义泛函分离变量形式的精确解. 这些精确解包含了该方程对应常系数情况下的解.
关键词:
广义条件对称
精确解
非线性反应扩散方程 相似文献
5.
利用定量相场模型, 以Mg-0.5 wt.%Al合金为例模拟了基面((0001)面)内镁基合金的等温自由枝晶生长过程. 通过研究该合金体系数值模拟的收敛性, 获得了最优化值耦合参数λ = 5.5及网格宽度Δx/W0 = 0.4, 并在该参数下系统研究了各向异性强度和过饱和度对枝晶尖端生长速度、尖端曲率半径、Péclet数及稳定性常数σ* 的影响. 结果表明, 由微观可解性理论得到的稳定性系数σ* 与ε6 拟合值σ*≅ ε6 1.81905, 更接近理想值σ * (ε6) ≅ε6 1.75. 此外, 当过饱和度Ω < 0.6时, 稳定性系数σ * 不随ε6 的变化而变化, 而当Ω > 0.6时, 稳定性系数σ * 随着ε6 的增加而减小. 这反映了枝晶的生长由扩散控制向动力学控制的转变. 随着过饱和度的增加, 枝晶形貌由雪花状枝晶向圆状枝晶转变. 相似文献
6.
基于一维分子晶体相邻分子间静态作用势和分子间的(电)偶极-偶极相互作用,采用分子投影算符表示一维分子晶体激子系统的模型哈密顿量.在谐振近似下,根据激子运动学和动力学非线性效应的理解,推导了晶格运动和激子-孤子运动的非线性Klein-Gordon(K-G)耦合运动方程组.发现激子运动学和动力学非线性效应不但对孤子波函数的3,2{2}\{x2}有重要贡献,且导致重要的高阶非线性项,分别对5非线性和7非线性方程给出了解析解.详细分析非线性方程的Bell型孤子和Kink型孤子解结果,发现激子运动学和动力学非线性效应对激子的有效质量m有显著增加贡献,对激子-孤子能量(Ω)有更负的修正,孤子局域范围更小.对Bell型孤子以超声速(v>cs)沿一维键传播,而Kink型孤子以亚声速传播(v<cs),它们分别出现在激子能带底部和顶部.
关键词:
一维分子晶体
激子-孤立子
运动学和动力学非线性效应
非线性Klein-Gordon方程 相似文献
7.
本文构造了解色散方程u1=auxxx的若干三层恒稳的半显式差分格式。第Ⅰ、Ⅱ类格式的局部截断误差的阶为O(τ2+h2+(τ2)/(h3));而第Ⅲ、Ⅳ类格式的局部截断误差的阶为O(τ2+h4+((τ)/(h))2+τh)。用判别稳定性的Von Neumann准则可证明:第Ⅰ、Ⅱ类格式及当参数α≤1时的第Ⅲ、Ⅳ类格式都是无条件稳定的,并且当必须的边界条件给定时它们可以显式地进行计算。 相似文献
8.
采用简立方格点上的Monte Carlo模拟,研究一端被无限大不可穿透平面壁吸附的高分子链的均方末端距<R2>,以及高分子链的质量中心到平面吸附壁的平均距离<Z>,与链长N、参数u(u=e-ε/kT,ε是链骨架原子间的相互作用能量,k是玻耳兹曼常数,T是热力学温度)的关系。结果表明:<R2>和<Z>都服从标度律,<R2>=αNγ,<Z>=βNη,其中,γ、η、α、β都是u的函数;u从1减小到0.5,则γ从1.01增大到1.19,η从0.51增大到0.60. 相似文献
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考虑一类演化方程ut=au∂2k+1(其中a是常数,u∂2k+1=∂2k+1u/∂x2k+1,k=1,2……)的有限差分解法。构造了两类具有高稳定性的显式差分格式。并用引入耗散项的方法建立了两类半显式差分格式,它们是无条件稳定的且可显式地进行计算。 相似文献
11.
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
12.
L. Friedlander 《Communications in Mathematical Physics》1985,98(1):1-16
Numerical studies of the initial boundary-value problem of the semilinear wave equationu
tt
–u
xx
+u
3=0 subject to periodic boundary conditionsu(t, 0)=u(t, 2),u
t
(t, 0)=u
t
(t, 2) and initial conditionsu(0,x)=u
0(x),u
t(0,x)=v
0(x), whereu
0(x) andv
0(x) satisfy the same periodic conditions, suggest that solutions ultimately return to a neighborhood of the initial stateu
0(x),v
0(x) after undergoing a possibly chaotic evolution. In this paper an appropriate abstract space is considered. In this space a finite measure is constructed. This measure is invariant under the flow generated by the Hamiltonian system which corresponds to the original equation. This enables one to verify the above returning property. 相似文献
13.
This paper studies an initial boundary value problem for a one-dimensional isentropic model system of compressible viscous gas with large external forces, represented by v
t
–u
x
=0,u
t
+(av
–)
x
=(u
x
/v)
x
+f(
0
x
vdx,t), with (v(x, 0),u(x, 0))= (v
0(x),u
0(x)),u(0,t)=u(1,t)=0. Especially, the uniform boundedness of the solution in time is investigated. It is proved that for arbitrary large initial data and external forces, the problem uniquely has an uniformly bounded, global-in-time solution with also uniformly positive mass density, provided the adiabatic constant (>1) is suitably close to 1. The proof is based on L
2-energy estimates and a technique used in [9]. 相似文献
14.
The stability problems for the Korteweg-de Vries equation, where linear stability fails, are investigated by the inverse scattering method. A rather general solution u(t, x) of the K-dV equation is shown to depend, for fixed time t, continuously on the initial condition u(0, x). For a continuum solution uc(t, x), this continuity holds uniformly in t (stability), but for a soliton solution this is not true. A soliton solution can be uniquely decomposed into a continuum and discrete (soliton) part: u(t, x) = ue(t, x) + ud(t, x). Then the perturbed solution u is close to u after a suitable t-dependent “shift” of the soliton part (form stability). 相似文献
15.
Fraydoun Rezakhanlou 《Communications in Mathematical Physics》2000,211(2):413-438
We study the asymptotic behavior of , where u solves the Hamilton–Jacobi equation u
t
+H(x,u
x
) ≡ 0 with H a stationary ergodic process in the x-variable. It was shown in Rezakhanlou–Tarver [RT] that u
ɛ converges to a deterministic function provided H(x,p) is convex in p and the convex conjugate of H in the p-variable satisfies certain growth conditions. In this article we establish a central limit theorem for the convergence by
showing that for a class of examples, u
ɛ(x,t) can be (stochastically) represented as
, where Z(x,t) is a suitable random field. In particular we establish a central limit theorem when the dimension is one and , where ω is a random function that enjoys some mild regularity.
Received: 15 February 1999 / Accepted: 14 December 1999 相似文献
16.
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
17.
Spyridon Kamvissis Dmitry Shepelsky Lech Zielinski 《Journal of Nonlinear Mathematical Physics》2013,20(3):448-473
We consider the initial boundary value (IBV) problem for the focusing nonlinear Schrödinger equation in the quarter plane x>0, t >0 in the case of periodic initial data, u(x,0) = α exp(?2iβx) (or asymptotically periodic, u(x, 0) =α exp(?2iβx)→0 as x→∞), and a Robin boundary condition at x = 0: ux(0, t)+qu(0, t) = 0, q ≠ 0. Our approach is based on the unified transform (the Fokas method) combined with symmetry considerations for the corresponding Riemann-Hilbert (RH) problems. We present the representation of the solution of the IBV problem in terms of the solution of an associated RH problem. This representation also allows us to determine an initial value (IV) problem, of a shock type, a solution of which being restricted to the half-line x > 0 is the solution of the original IBV problem. In the case β < 0, the large-time asymptotics of the solution of the IBV problem is presented in the “rarefaction” sector, demonstrating, in particular, an oscillatory behavior of the boundary values in the case q > 0, contrary to the decay to 0 in the case q < 0. 相似文献
18.
Kenji Yajima 《Communications in Mathematical Physics》1987,110(3):415-426
We study the existence, uniqueness and regularity of the solution of the initial value problem for the time dependent Schrödinger equationiu/t=(–1/2)u+V(t,x)u,u(0)=u
0. We provide sufficient conditions onV(t,x) such that the equation generates a unique unitary propagatorU(t,s) and such thatU(t,s)u
0C
1(,L
2) C
0(H
2(
n
)) foru
0H
2(
n
). The conditions are general enough to accommodate moving singularities of type x–2+(n4) or x–n/2+(n3). 相似文献
19.
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