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...

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 introduce a method of estimating the space analyticity radius of solutions for the Navier Stokes and related equations in terms of L p and L norms of the initial data. The method enables us to express the space analyticity radius for 3D Navier Stokes equations in terms of the Reynolds number of the flow. Also, for the Kuramoto Sivashinsky equation, we give a partial answer to a conjecture th...

We present here different situations in which the filtering of high or low modes is used either for stabilizing semi-implicit numerical schemes when solving nonlinear parabolic equations, building adapted damping operators case dispersive equation. We consider provided by mutigrid-like techniques as well resulting from operator with monotone symbols. Our approach ap...