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.     

Equations with Singular Diffusivity

Recently models of faceted crystal growth and of grain boundaries were proposed based on the gradient system with nondifferentiable energy. In this article, we study their most basic forms given by the equations u _t=(u _x/|u _x|) _x and u _t=(1/a)(au _x/|u _x|) _x , where both of the related energies include a |u _x| term of power one which is nondifferentiable at u _x=0. The first equation is spatially homogeneous, while the second one is spatially inhomogeneous when a depends on x. These equations naturally express nonlocal interactions through their singular diffusivities (infinitely large diffusion constant), which make the profiles of the solutions completely flat. The mathematical basis for justifying and analyzing these equations is explained, and theoretical and numerical approaches show how the solutions of the equations evolve.     

Global existence and blow up of solutions for Cauchy problem of generalized Boussinesq equation

We study the Cauchy problem of generalized Boussinesq equation u_tt−u_xx+(u_xx+f(u))_xx=0, where f(u)=±|u|^p or ±|u|^p−1u, p>1. By introducing a family of potential wells we obtain invariant sets, vacuum isolating and threshold result of global existence and nonexistence of solution.     

Letter: Solutions of All One-Dimensional Wave Equations with Time Independent Potential and Separable Variables

Solutions, in terms of special functions, of all wave equations u _xx – u _tt = V(x) u(t,x), characterised by eight inequivalent time independent potentials and by variables separation, have been found. The real valueness and the properties of the solutions produced by computer algebra programs are not always manifest and in this work we provide ready to use solutions. We discuss especially the potential (m ₁ + m ₂ sinh x)cosh^–2x. Such potential approximates the Schwarzschild black hole potential and its use for determining black holes quasi-normal modes is hinted to.     

Bäcklund transformations for diffusion equations

A class of Bäcklund transformations is deduced for the diffusion equation (S(u))_t = (C(u,u_x))_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 u_t = (u^au_x)_x and u_t = u^au_xx.     

Stability for the Korteweg-de Vries equation by inverse scattering theory

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 u_c(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) = u_e(t, x) + u_d(t, x). Then the perturbed solution u is close to u after a suitable t-dependent “shift” of the soliton part (form stability).     

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.     

Weak and Partial Symmetries of Nonlinear PDE in Two Independent Variables

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)).     

Stability and instability of periodic travelling wave solutions for the critical Korteweg-de Vries and nonlinear Schrödinger equations

In this paper we establish new results about the existence, stability, and instability of periodic travelling wave solutions related to the critical Korteweg-de Vries equation

u_t+5u⁴u_x+u_xxx=0,     

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：     

On the Dimensions of the Chiral and Dilatation Symmetry Breaking Hamiltonian Densities

Denoting by l_u and l_δ the dimensions (if any) of the chiral and dilation symmetry breaking Hamiltonian densities u(x) and δ(x), respectively, the property (4 – l_u) (l_u – l_δ) > 0 (which is already known in special cases) is derived for a much larger class of models. We furthermore obtain a simple and general explicit expression for u(x) in terms of the current divergences, and review the physical applications of this (or an analogous, almost obvious) expression.     

The inverse problem for the third order equation uxxx + q(x)ux + r(x)u = −iζ3u

The spectral problem u_xxx + q(x)u_x + r(x)u = ?iξ³u is considered. A set of spectral data which is sufficient for the reconstruction of the potentials q(x) and r(x) is found and the problem of this reconstruction, this inverse problem solved.     

An invariant measure for the equationu tt −u xx +u 3=0

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.     

A Note on a Model System with Sudden Directional Diffusion

We study qualitative properties of solutions to a monodimensional problem

Non-linear Stability of Modulated Fronts¶for the Swift–Hohenberg Equation

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     

傅里叶自函数的维格纳分布及应用

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。     

Conservative Solutions to a Nonlinear Variational Wave Equation

We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation u _tt − c(u)(c(u)u _x ) _x =0, for initial data of finite energy. Here c(·) is any smooth function with uniformly positive bounded values.     

一类广义强阻尼Sine-Gordon方程的整体解

同时考虑黏性效应及外阻尼作用研究了一类广义强阻尼Sine-Gordon方程-利用Galerkin方法,首先证明了该方程在初值u(x,0)∈H¹₀(Ω),u_t(x,0)∈L²(Ω)的条件下初边值问题存在整体弱解u(x,t),并证明了整体弱解关于初始条件具有 关键词： Sine-Gordon型方程 强阻尼 Galerkin方法 整体解     

具有阻尼项的非线性波动方程的相似约化

利用Clarkson和Kruskal引入的直接约化法,给出了具有阻尼项的非线性波动方程u_tt-2bu_xxt+αu_xxxx=β(uⁿ_x)x(α>0,β≠0,n≥2)三种类型的相似约化.从这些约化方程的Painlevé分析表明该方程在Ablowitz的猜测意义下是不可积的.此外还获得了该方程(n=2)的4种精确类孤波解. 关键词： 波动方程 相似约化 Painlevé分析 精确解     

Time-space white noise eliminates global solutions in reaction-diffusion equations

We prove that perturbing the reaction-diffusion equation u_t=u_xx+(u₊)^p (p>1), with time-space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.