We describe a Lohner-type algorithm for rigorous integration of dissipative PDEs. Using it for the Kuramoto-Sivashinsky PDE on the line with odd and periodic boundary conditions we give a computer assisted proof the existence of multiple periodic orbits.

ABSTRACT. For nonlinear parabolic evolution equations, it is proved that, under the assumptions oflocal Lipschitz continuity of nonlinearity and the dissipativity of semiflows, there exist approximate inertial manifolds (AIM) in the energy space and that the approximate inertial manifolds are constructed as the graph of the steady-state determining mapping based on the spectral decomposition. I...

The large scale properties of spatiotemporal chaos in the 2D Kuramoto-Sivashinsky equation are studied using an explicit coarse-graining scheme. A set of intermediate equations are obtained which describe interactions between the small scale structures and the hydrodynamic degrees of freedom. Possible forms of the effective large scale hydrodynamics are constructed and examined. Although a numb...

Abstract In this work, we propose a computationally efficient method for the solution of dynamic constraint optimization problems arising in the context of spatiallydistributed processes governed by highly-dissipative nonlinear partial differential equations (PDEs). The method is based on spatial discretization using combination of the method of weighted residuals with spatially-global basis fu...

When pseudospectral approximations are used for space derivatives, one often encounters spurious eigenvalues. These can lead to severe time stepping difficulties for PDEs. This is especially the case for equations with high order derivatives in space, requiring multiple conditions at one or both boundaries. We note here that a very simple-to-implement fictitious point approach circumvents most ...

We analyze coherent structures in nonlocal dispersive active-dissipative nonlinear systems, using as a prototype the Kuramoto–Sivashinsky (KS) equation with an additional nonlocal term that contains stabilizing/destabilizing and dispersive parts. As for the local generalized Kuramoto–Sivashinsky (gKS) equation (see, e.g., [T. Kawahara and S. Toh, Phys. Fluids, 31 (1988), pp. 2103–2111]), we sho...

We report the observation of defects with fractional topological charges (disclinations) in square and hexagonal patterns as numerical solutions of several generic equations describing many pattern-forming systems: Swift-Hohenberg equation, damped Kuramoto-Sivashinsky equation, as well as nonlinear evolution equations describing large-scale Rayleigh-Benard and Marangoni convection in systems wi...

In this paper, we employ Nevanlinna’s value distribution theory to investigate the existence of meromorphic solutions of some algebraic differential equations. We obtain the representations of all meromorphic solutions of certain algebraic differential equations with constant coefficients and dominant term. Many results are the corollaries of our result, and we will give the complex method to f...

We have used themodified variational iterationmethod MVIM to find the approximate solutions for some nonlinear initial value problems in the mathematical physics, via the Burgers-Fisher equation, the Kuramoto-Sivashinsky equation, the coupled Schrodinger-KdV equations, and the long-short wave resonance equations together with initial conditions. The results of these problems reveal that the mod...

We present a computational study of a simple finite-dimensional feedback control algorithm for stabilizing solutions of infinite-dimensional dissipative evolution equations such as reaction-diffusion systems, the Navier-Stokes equations and the Kuramoto-Sivashinsky equation. This feedback control scheme takes advantage of the fact that such systems possess finite number of determining parameter...