We present a computer assisted proof of the existence of several attracting fixed points for the Kuramoto–Sivashinsky equation ut = (u )x − uxx − νuxxxx, u(x, t) = u(x+ 2π, t), u(x, t) = −u(−x, t), where ν > 0. The method is general and can be applied to other dissipative PDEs.

We report the first observations of numerical "hopping" cellular flame patterns found in computer simulations of the Kuramoto-Sivashinsky equation. Hopping states are characterized by nonuniform rotations of a ring of cells, in which individual cells make abrupt changes in their angular positions while they rotate around the ring. Until now, these states have been observed only in experiments b...

We report numerical simulations of one-dimensional cellular solutions of the stabilized Kuramoto-Sivashinsky equation. This equation offers a range of generic behavior in pattern-forming instabilities of moving interfaces, such as a host of secondary instabilities or transition toward disorder. We compare some of these collective behaviors to those observed in experiments. In particular, destab...

We consider the solutions lying on the global attractor of the two-dimensional Navier–Stokes equations with periodic boundary conditions and analytic forcing. We show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely, proving a conjecture due to Foias and Temam. Our results also hold for the complex Ginzburg–Landau equation, the ...

The Kuramoto-Sivashinsky equation with fixed boundary conditions is numerically studied. Shocklike structures appear in the time-averaged patterns for some parameter range of the boundary values. Effective diffusion constant is estimated from the relation of the width and the height of the shock structures.

We present a method of self-consistent a-priori bounds, which allows to study rigorously dynamics of dissipative PDEs. As an application present a computer assisted proof of an existence of a periodic orbit for the Kuramoto-Sivashinsky equation ut = (u )x− uxx− νuxxxx, u(t, x) = u(t, x + 2π), u(t, x) = −u(t,−x),

The Kuramoto-Sivashinsky equation arises in a variety of applications, among which are modeling reaction-diffusion systems, flame-propagation and viscous flow problems. It is considered here, as a prototype to the larger class of generalized Burgers equations: those consist of quadratic nonlinearity and arbitrary linear parabolic part. We show that such equations are well-posed, thus admitting ...

The generalized Kuramoto–Sivashinsky equation arises frequently in engineering, physics, biology, chemistry, and applied mathematics, because of its extensive applications, this important model has received much attention regarding obtaining numerical solutions. This article introduces a new hybrid technique based on nonpolynomial splines finite differences for solving the approximately. Specif...

This article is concerned with a Kuramoto–Sivashinsky-Korteweg-de Vries equation in bounded interval. The as well one of the boundary conditions are supposed to be subject presence parameter $ \nu> 0 $. Moreover, this specific condition has time-delay effect. As \nu tends zero, we show that can obtain findings [4,58] concerning two Korteweg–de equations. Indeed, able retrieve well-posedness ...

A simple model for dendritic growth is given by S2d'" + 9' — cos(9). For S ss 1 we prove that there is no bounded, monotonic solution which satisfies d(-oo) = -7t/2 and Q(oo) = n/2. We also investigate the existence of bounded, monotonic solutions of an equation derived from the Kuramoto-Sivashinsky equation, namely y" + y = 1 y1 /2. We prove that there is no monotonic solution which satisfies ...