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),

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.

The nondeterministic Kuramoto-Sivashinsky ~KS! equation is solved numerically in 211 dimensions. The simulations reveal the presence of early and late scaling regimes. The initial-time values for the growth exponent b, the roughness exponent a, and the dynamic exponent z are found to be 0.22–0.25, 0.75–0.80, and 3.0–4.0, respectively. For long times, the scaling exponents are notably less than ...

Global well-posedness, existence of globally absorbing sets and existence of inertial manifolds is investigated for a class of diffusive Burgers equations. The class includes diffusive Burgers equation with nontrivial forcing, the Burgers-Sivashinsky equation and the QuasiStedy equation of cellular flames. The global dissipativity is proven in 2D for periodic boundary conditions. For the proof ...