The generalized Kolmogorov-Petrovskii-Piskunov equation

Nonlinear nonlocal equations of mathematical physics such as the K.P.P. equation, the generalized nonlinear Schrödinger equation, the Witham equation for water waves et al. are solved by decomposition.     

Rotating bunsen flames as solutions of the Kuramoto-Sivashinsky equation

Steadily rotating solutions of the Kuramoto-Sivashinsky equationu _t + ² u++¦u¦ ² =c ² are studied. These solutions bifurcate from the steady radial solution of the above equation. For large values ofc and angular velocities such thatN¦¦<2c<(N+1)¦¦, we show that there exists a 2N-1 family of bifurcating solutions. The proof is based on a certain generic transversality assumption. A computer-assisted proof of this assumption is given for 1N10.     

A mesh-free numerical method for solution of the family of Kuramoto-Sivashinsky equations

In this paper, radial basis function (RBFs) based mesh-free method is implemented to find numerical solution of the Kuramoto-Sivashinsky equations. This approach has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. The accuracy of the method is assessed in terms of the error norms L₂,L_∞, number of nodes in the domain of influence, free parameter, dependent parameter RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations.     

Spatio-temporal symmetries and bifurcations via bi-orthogonal decompositions

Summary A tool for analyzing spatio-temporal complex physical phenomena was recently proposed by the authors, Aubry et al. [5]. This tool consists in decomposing a spatially and temporally evolving signal into orthogonal temporal modes (temporal “structures”) and orthogonal spatial modes (spatial “structures”) which are coupled. This allows the introduction of a temporal configuration space and a spatial one which are related to each other by an isomorphism. In this paper, we show how such a tool can be used to analyze space-time bifurcations, that is, qualitative changes in both space and time as a parameter varies. The Hopf bifurcation and various spatio-temporal symmetry related bifurcations, such as bifurcations to traveling waves, are studied in detail. In particular, it is shown that symmetry-breaking bifurcations can be detected by analyzing the temporal and spatial eigenspaces of the decomposition which then lose their degeneracy, namely their invariance under the symmetry. Furthermore, we show how an extension of the theory to “quasi-symmetries” permits the treatment of nondegenerate signals and leads to an exponentially decreasing law of the energy spectrum. Examples extracted from numerically obtained solutions of the Kuramoto-Sivashinsky equation, a coupled map lattice, and fully developed turbulence are given to illustrate the theory.     

Remarks on Third-Order ODEs Relevant to the Kuramoto-Sivashinsky Equation

We are interested in the third-order ordinary differential equations (ODEs) which are related to the Kuramoto-Sivashinsky equation. So-called steady solutions of the Kuramoto-Sivashinsky equation are known to admit several types; for example, bounded global solutions or periodic solutions. We show that, in addition to these, there exist solutions which blow up on bounded intervals. Moreover, for certain classes of these ODEs, the nonexistence of nontrivial bounded entire solutions is exhibited.     

Modified method of simplest equation and its application to nonlinear PDEs

We search for traveling-wave solutions of the class of PDEswhere A_p(Q),B_r(Q),C_s(Q),D_u(Q) and F(Q) are polynomials of Q. The basis of the investigation is a modification of the method of simplest equation. The equations of Bernoulli, Riccati and the extended tanh-function equation are used as simplest equations. The obtained general results are illustrated by obtaining exact solutions of versions of the generalized Kuramoto-Sivashinsky equation, reaction-diffusion equation with density-dependent diffusion, and the reaction-telegraph equation.     

Asymptotic Analysis in a Gas-Solid Combustion Model with Pattern Formation

The authors consider a free interface problem which stems from a gas-solid model in combustion with pattern formation. A third-order, fully nonlinear, self-consistent equation for the flame front is derived. Asymptotic methods reveal that the interface approaches a solution to the Kuramoto-Sivashinsky equation. Numerical results which illustrate the dynamics are presented.     

关于Kuramoto-Sivashinsky方程的非线性Galerkin方法之注记*

本文致力于讨论求解Kuramoto-Sivashinsky方程的非线性Galerkin方法,我们采用了s_m个小尺度分量作反馈,并给出了收敛性结果,分析了误差估计．结论表明我们的修正方法是十分有效的．     

An alternative energy bound derivation for a generalized Hasegawa-Mima equation

We present an alternative derivation of the H¹-boundedness of solutions to a generalized Hasegawa-Mima equation, first investigated by Grauer (1998) [2]. We apply a Lyapunov function technique similar to the one used for constructing energy bounds for the Kuramoto-Sivashinsky equation. Different from Grauer (1998) [2], who uses this technique in a Fourier space approach, we employ the physical space construction of the Lyapunov function, as developed in Bronski and Gambill (2006) [11]. Our approach has the advantage that it is more transparent in what concerns the estimates and the dominant terms that are being retained. A key tool of the present work, which replaces the algebraic manipulations on the Fourier coefficients from the other approach, is a Hardy-Rellich type inequality.     

Kuramoto-Sivashinsky与Karda-Parisi-Zhang模型中生长界面分形特性研究

在（2＋1）维情况下，利用数值模拟研究了Kuramoto-Sivashinsky (K-S)与Karda-Parisi-Z hang (KPZ)模型所决定的非平衡态界面生长演化过程.结果表明，KPZ与K-S模型都表现出明 显的时间和空间标度特性.相对于KPZ模型而言，K-S模型所对应的表面具有更明显的颗粒特 征，当生长时间较长时，生长界面呈现蜂窝状结构.通过数值相关分析得到了生长界面的粗 糙度指数、生长指数和动态标度指数等参数.从两种模型对应的表面形貌特征和表面参数来 看，在(2+1)维情况下，KPZ与K-S模型所决定的表面具有完全不同的动态标度行为，属于不 同的两类物理模型. 关键词： Kuramoto-Sivashinsky (K-S)模型 Karda-Parisi-Zhang(KPZ)模型 分形 数值模拟