K(m,n)方程的对称性约化

利用对称性约化的直接法,给出了具有非线性色散情况下的K(m,n)模型的所有对称性约化.从第一种约化方程的Painlev啨性质分析可知,K(m,n)模型仅当m=n+1和m=n+2时是可积的.特殊情况下(行波约化),这种约化的解可用一个积分表示.给出了K(m,1)和K(m,m)的一般孤波解的明显表达式. 关键词：     

一类非线性方程的compacton解及其移动compacton解

研究一类非线性方程,即广义Camassa-Holm方程C(n):ut+kux+β1u\{xxt\}+β2u\{n+1\}x+β3uxun\{xx\}+β4uun\{xxx\}=0.通过四种拟设得到丰富的精确解，特别是当k≠0时得到了com pacton解,当k=0时得到了移动compacton解.最后利用线 性化的方法得到了其他形式的广义Camassa-Holm方程的compacton解. 关键词： 广义Camassa-Holm方程 compacton解 移动compacton解     

具有物理背景的高维Painlevé可积模型

提出了一种求解任意维数非线性模型的“M?bious”变换下不变的渐进展开方法,并可同时获得许多新的与原模型有着相同维数的Painlevé可积模型.取(2+1)维KdV-Burgers(KdVB)方程和Kadomtsev-Petviashvili(KP)方程为具体例子,获得了一些新的具有Painlevé性质的高维“M?bious”变换下不变的方程及原模型的近似解.在某些特殊情况下,某些近似解可以成为精确解 关键词： 高维可积模型 “M?bious”不变 近似方法     

用格子Boltzmann方法模拟MKDV方程

用精确到0(ε)的5速格子Boltzmann模型模拟MKDV方程:u_t+6u²u_x+u_xxx=0,并与MKDV方程的孤子解比较,二者精确吻合. 关键词： 格子Boltzmann方法 MKDV方程 孤子解     

单一多体不可逆聚集方程的解

本文研究的是单一多体不可逆聚集方程的解,从广义Smoluchovski方程出发,分别讨论凝结核为K(i₁,i₂,…,i_n)=const和凝结核为K(i₁,i₂,…,i_n)=sumfrom i=1 to n(i_l)的精确解,求出集团的体积分布C_m(t)。并且还讨论它们的长时行为。 关键词：     

一类非线性色散方程中的新型奇异孤立波

研究一类非线性色散广义DGH方程的新型奇异孤立波及其Painlevé可积性.利用Painlevé分析发现当对流项强度m=2时广义DGH方程是可积的,这是一个新的可积方程.通过构造新的变量代换以及auto-Backlund变换获得该方程丰富的奇异孤立波解,如紧孤立波（compacton）、尖峰孤立波（peakon）、新型带尖点的双孤立波和带爆破点的双孤立波等. 关键词： 非线性色散方程 可积性 奇异孤立波     

(2+1)维Camassa-Holm方程的相似约化与解析解

将Clarkson和Krushal引入的直接约化方法推广并应用到(2+1)维CamassaHolm方程组,获得了该方程的若干相似约化和解析解,其中包括Logistic方程和Bernoulli方程.约化结果得到了Peakon解、Cuspon解和关于时间t的奇异解.该方法也适用于其他有重要物理背景的非线性演化方程 关键词： Camassa-Homl方程 相似约化 直接方法 解析解     

高维微分-差分模型的Virasoro对称子代数，多线性变量分离解和局域激发模式

寻找高维可积模型是非线性科学中的重要课题.利用无穷维Virasoro对称子代数［σ(f₁),σ(f₂)］=σ(f′₁f₂-f′₂f₁)和向量场的延拓结构理论，能够得到各种高维模型.选取一些特殊的实现，可以给出具有无穷维Virasoro对称子代数意义下的高维微分可积模型.把该方法推广到微分-差分模型上，构造出具有弱多线性变量分离可解性的(3+1)维类Toda晶格.另外，该模型的一个约化方程为具有多线性变量分离可解性的(2+1)维特殊Toda晶格.连续运用对称约化方法可以得到此特殊Toda晶格的一个(1+1)维约化方程具有多线性变量分离可解性.因为得到的精确解里含有低维任意函数，从而可以构造出丰富地局域激发模式，如dromion解，lump解，环孤子解，呼吸子解，瞬子解，混沌斑图和分形斑图等等. 关键词： Virasoro代数 微分-差分模型 变量分离 局域激发模式     

推广的KdV方程的孤波解

本文研究了推广的KdV方程 u_t+2μuu_x+v_3x+δu_5x=0(μvδ≠0) (1)的精确孤子解,得到了(1)式的一些新的孤波解,对文献[10]的若干结论作了补充与修正。 关键词：     

带源项的变系数非线性反应扩散方程的精确解

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)u_t=(g(x)D(u)u_x)_x+h(x)P(u)u_x+q(x)Q(u)进行研究. 当扩散项D(u)取u^m (m≠-1,0,1)和e^u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     

Lattice Boltzmann method for the generalized Kuramoto-Sivashinsky equation

In this paper, a lattice Boltzmann model with an amending function is proposed for the generalized Kuramoto-Sivashinsky equation that has the form u_t+uu_x+αu_xx+βu_xxx+γu_xxxx=0. With the Chapman-Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. It is found that the numerical results agree well with the analytical solutions.     

Products of distributions and collision of a δ-wave with a δ′-wave in a turbulent model

We study the possibility of collision of a δ-wave with a stationary δ′-wave in a model ruled by equation f (t)u _t+[u²?β(x?γ(t))u]_x = 0, where f, β and γ are given real functions and u = u(x, t) is the state variable. We adopt a solution concept which is a consistent extension of the classical solution concept. This concept is defined in the setting of a distributional product, which is not constructed by approximation processes. By a convenient choice of f, β and γ, we are able to distinguish three distinct dynamics for that collision, to which correspond phenomena of solitonic behaviour, scattering, and merging. Also, as a particular case, taking f = 2 and β = 0 we prove that the referred collision is impossible to arise in the setting of the inviscid Burgers equation. To show how this framework can be applied to other physical models, we included several results already obtained.     

Robin boundary condition and shock problem for the focusing nonlinear Schrödinger equation

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: u_x(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.     

Die EINSTEIN-Gruppe und die Symmetrie-Gruppen der Gravitationstheorie

The transformations x^?α = x^?α(x^β, g_μn?) in the function space of the g_μn?(x^λ) are corresponding to the coordinate transformations x^α = x^α(x^β) with some non-covariant conditions on the g_μn?(x^λ). Therefore, the transformations in the function space are corresponding to subgroups of the EINSTEIN group. The conditions for the g_μn? may be given in the space- time V₄ or on submanifolds (points, curves, surfaces and hypersurfaces) of the V₄. – A special case of the last problem is given by the CAUCHY conditions or by the DIRAC constraints for a special choice of the coordinates on a CAUCHY hypersurface x⁰ = 0. Then, the transformations x^?α = x^?α(x^l, g_rs, p^mn) in the phase space are EINSTEIN transformations preserving the synchronicity for x⁰ → 0.     

Dynamical Localization of Quantum Walks in Random Environments

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.     

Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation

We prove that the set of solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in two dimensions, (u _t+(u^m+1)_x+u_xxx)_x=u_yy is stable for 0<m<4/3.     

Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation

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     

The Painlevé property and a partial differential equation with an essential singularity

It is shown that the equation u²_t = 2uu²_x - (1 + u²)u_xx possesses the Painlevé property for partial differential equations as defined by Weiss, Tabor and Carnevale, yet does not satisfy the necessary conditions of the Painlevé conjecture to be completely integrable since it is reducible, via a similarity reduction, to an ordinary differential equation which has a movable essential singularity. It is further shown that in a more general sense, the equation does not possess the Painlevé property for partial differential equations.     

无扩散相变中界面动力学的唯象理论及其应用

固态相变中相界面或畴界面的平均运动速度V与有效相变驱动力△G＇(相变驱动力△G与相界面运动阻力△G_R之差)之间的关系可表示为V=φ(△G—△G_R)。当有单向变化的外场(场强为ξ,变化速率为ξ)作用于相变系统并能诱导相界面运动时,就会产生母相/新相间的转变。在相变过程中同时叠加一个交变应力时,则可计算得界面动力学关系V=φ(△G—△G_R)与相变过程内耗Q^-1、相关的模量亏损(△M/M)、相变速率dF/dξ、相变应变ε₀间的关系为 [d lnφ(△G-△G_R)/d(△G-△G_R)]= Q^-1ω/n²M(dF/dξ)ξ = (△M/M)ω/nMε₀(dF/dξ)ξ, 以及 (△M/M)/Q^-1=ε₀/n。此处ω为交变应力的圆频率,M为与振动模式有关的弹性模量,n为应力与界面运动的耦合因子。因此,界面动力学关系式的通解为 V = ∑(±n)/(α≠-1) A_α exp{[(△G-△G_R)/△G_α^*]^{α+ 1}/(α+ 1)} +∑(m)/(β₀) A_β[(△G-△G_R)/△G_β^*]^β 此处n,m为正整数。上式中的各项参数可由实验数据确定。此外,(△M/M)/Q^-1的等式还可用于判别相变过程的模量亏损中有无声子模软化的贡献。 关键词：