We consider a model for the propagation of a subsonic detonation wave through a porous medium introduced by Sivashinsky [8]. We show that it admits travelling wave solutions that converge in the limit of zero temperature diffusivity to the travelling fronts of a reduced system constructed in [6].

In this article we study the solution of the Kuramoto–Sivashinsky equation on a bounded interval subject to a random forcing term. We show that a unique solution to the equation exists for all time and depends continuously on the initial data.

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),

Spatiotemporally chaotic dynamics of a Kuramoto-Sivashinsky system is described by means of an innnite hierarchy of its unstable spatiotemporally periodic solutions. An intrinsic parametrization of the corresponding invariant set serves as accurate guide to the high-dimensional dynamics, and the periodic orbit theory yields several global averages characterizing the chaotic dynamics.

We introduce a new variational method for finding periodic orbits of flows and spatio-temporally periodic solutions of classical field theories, a generalization of the Newton method to a flow in the space of loops. The feasibility of the method is demonstrated by its application to several dynamical systems, including the Kuramoto-Sivashinsky system.

An investigation of interior crisis of high dimensions in an extended spatiotemporal system exemplified by the Kuramoto-Sivashinsky equation is reported. It is shown that unstable periodic orbits and their associated invariant manifolds in the Poincaré hyperplane can effectively characterize the global bifurcation dynamics of high-dimensional systems.

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.

in this paper we propose a method for solving some well-known classes of lane-emden type equations which are nonlinear ordinary differential equations on the semi-innite domain. the proposed approach is based on an unsupervised combined articial neural networks (ucann) method. firstly, the trial solutions of the differential equations are written in the form of feed-forward neural networks co...