共查询到20条相似文献,搜索用时 31 毫秒
1.
傅里叶变换是现代光学发展的重要理论工具。自1991年Caola首次定义傅里叶自函数以来1,它在光学领域的应用研究日趋活跃。本文首先对傅里叶自函数定义进行扩展,再讨论其维格纳分布函数及其矩,研究它们在光学中的应用。最后推导出傅里叶自函数应用于光学变换器成象时的变换矩阵。 相似文献
2.
《Journal of Nonlinear Mathematical Physics》2013,20(3-4):330-335
Abstract Nonclassical infinitesimal weak symmetries introduced by Olver and Rosenau and partial symmetries introduced by the author are analyzed. For a family of nonlinear heat equations of the form u t = (k(u) u x)x + q(u), pairs of functions (k(u), q(u)) are pointed out such that the corresponding equations admit nontrivial two-dimensional modules of partial symmetries. These modules yield explicit solutions that look like u(t, x) = F (θ(t) x + φ(t)) or u(t, x) = G(f(x) + g(t)). 相似文献
3.
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
4.
Effat A. Saied 《Journal of statistical physics》1996,82(3-4):951-962
We consider the way in which a solution to a class of nonlinear partial differential equationsS(u)u
t=(K(u)ux)x approaches the similarity form. The problem we solve is chosen for two main reasons: first the equation above is of widespread use in modeling physical situations and second it provides a tractable but significant example of a free boundary problem. 相似文献
5.
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
6.
Boris Dubrovin 《Communications in Mathematical Physics》2006,267(1):117-139
Hamiltonian perturbations of the simplest hyperbolic equation u
t
+ a(u) u
x
= 0 are studied. We argue that the behaviour of solutions to the perturbed equation near the point of gradient catastrophe of the unperturbed one should be essentially independent on the choice of generic perturbation neither on the choice of generic solution. Moreover, this behaviour is described by a special solution to an integrable fourth order ODE. 相似文献
7.
R. N. Garifullin B. I. Suleimanov 《Journal of Experimental and Theoretical Physics》2010,110(1):133-146
An analysis is presented of the effect of weak dispersion on transitions from weak to strong discontinuities in inviscid fluid
dynamics. In the neighborhoods of transition points, this effect is described by simultaneous solutions to the Korteweg—de
Vries equation u
t
′ + uu
x
″ + u
xxx
‴ = 0 and fifth-order nonautonomous ordinary differential equations. As x2 + t
2 →∞, the asymptotic behavior of these simultaneous solutions in the zone of undamped oscillations is given by quasi-simple
wave solutions to Whitham equations of the form r
i(t, x) = tli x/t2. 相似文献
8.
W. Strampp 《Physica D: Nonlinear Phenomena》1982,6(1):113-118
A class of Bäcklund transformations is deduced for the diffusion equation (S(u))t = (C(u,ux))x by taking advantage of the conservative form of the equation. The transformations are used for obtaining new relations connecting various equations of the type ut = (uaux)x and ut = uauxx. 相似文献
9.
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
10.
We prove that the set of solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in two dimensions, (u
t+(um+1)x+uxxx)x=uyy is stable for 0<m<4/3. 相似文献
11.
We demonstrate that for the systems of equations, which are invariant under a point group or possess conservation laws of the zeroth or first order, a nontrivial extension of the module of invertible transformations is possible. That simplifies greatly a classification of the integrable systems of equations. Here we present an exhaustive list and a classification of the second order systems of the formu
t
=u
xx
+f(u, v, u
x
v
x
), –v
t=v
xx
+g(u, v, u
x
,v
x
), which possess the conservation laws of higher order. The reduction group approach allows us to define the Lax type representations for some new equations of our list. 相似文献
12.
13.
The nonclassical symmetries method is applied to a class of reaction-diffusion equations with nonlinear source, i.e. u t =u xx +cu x +R(u, x). Several cases are obtained by using suitable solutions of the heir-equations as described in [M.C. Nucci, Nonclassical symmetries as special solutions of heir-equations, J. Math. Anal. Appl. 279 (2003) 168–179]. 相似文献
14.
We analyze the long time behavior of solutions of the Schrödinger equation ${i\psi_t=(-\Delta-b/r+V(t,x))\psi}We analyze the long time behavior of solutions of the Schr?dinger equation iyt=(-D-b/r+V(t,x))y{i\psi_t=(-\Delta-b/r+V(t,x))\psi},
x ? \mathbbR3{x\in\mathbb{R}^3}, r = |x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t, x) = V(t + 2π/ω, x) with zero time average. 相似文献
15.
Alan Weinstein 《Communications in Mathematical Physics》1985,99(3):385-388
Nontrivial solutions of the equationu
tt=u
xx–g(u) which are 2-periodic int and which decay asx are shown to exist ifg(a)=0 andg(0)>1. Breather-like solutions, which also decay asx –, can be interpreted as homoclinic solutions in thex-dynamics; their existence is still in question for generalg. 相似文献
16.
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
17.
This paper studies variable separation of the evolution equations via the generalized conditional symmetry. To illustrate, we classify the extended nonlinear wave equation utt=Au,uxuxx+Bu,ux,ut which admits the derivative-dependent functional separable solutions DDFSSs). We also extend the concept of the DDFSS to cover other variable separation approaches. 相似文献
18.
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
19.
In a recent paper we proved that for certain class of perturbations of the hyperbolic equation u
t
= f (u)u
x
, there exist changes of coordinate, called quasi-Miura transformations, that reduce the perturbed equations to the unperturbed
one. We prove in the present paper that if in addition the perturbed equations possess Hamiltonian structures of certain type,
the same quasi-Miura transformations also reduce the Hamiltonian structures to their leading terms. By applying this result,
we obtain a criterion of the existence of Hamiltonian structures for a class of scalar evolutionary PDEs and an algorithm
to find out the Hamiltonian structures.
相似文献
20.
This article is a study of the mapping from a potentialq(x) onR
3 to the backscattering amplitude associated with the Hamiltonian –+q(x). The backscattering amplitude is the restriction of the scattering amplitudea(, , k), (, , k)S
2×S
2×+, toa(,–, k). We show that in suitable (complex) Banach spaces the map fromq(x) toa(x/|x|, –x/|x|, |x|) is usually a local diffeomorphism. Hence in contrast to the overdetermined problem of recoveringq from the full scattering amplitude the inverse backscattering problem is well posed. 相似文献