This paper presents a complete finite volume method for the Cahn–Hilliard and Kuramoto– Sivashinsky type of equations. The spatial discretization is high-order accurate and suitable for general unstructured grids. The time integration is addressed by means of implicit an implicit–explicit fourth order Runge–Kutta schemes, with error control and adaptive time-stepping. The outcome is a practical...

The work proposes and studies a model for one-dimensional spatially extended systems, which involve nonlocal interactions and finite propagation speed. It shows that the general reaction-diffusion equation, the Swift-Hohenberg equation, and the general Kuramoto-Sivashinsky equation represent special cases of the proposed model for limited spatial interaction ranges and for infinite propagation ...

We present an approach to the design of feedback control laws that stabilize the relative equilibria of general nonlinear systems with continuous symmetry. Using a templatebased method, we factor out the dynamics associated with the symmetry variables and obtain evolution equations in a reduced frame that evolves in the symmetry direction. The relative equilibria of the original systems are fix...

Nonlinear partial differential equations are widely used to describe complex phenomena in various fields of science, for example the Korteweg-de Vries-Kuramoto-Sivashinsky equation (KdV-KS equation) and the Ablowitz-Kaup-Newell-Segur shallow water wave equation (AKNS-SWW equation). To our knowledge the exact solutions for the first equation were still not obtained and the obtained exact solutio...

Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained...

This work focuses on control of surface roughness in sputtering processes including two surface micro-processes, diffusion and erosion. The fluctuation of surface height of such sputtering processes can be described by the stochastic Kuramoto–Sivashinsky equation (KSE), a fourth-order stochastic partial differential equation (PDE). Specifically, we consider sputtering processes, including surfa...

We present a new topological method for the study of the dynamics of dissipative PDE’s. The method is based on the concept of the selfconsistent apriori bounds, which allows to justify rigorously the Galerkin projection. As a result we obtain a low-dimensional system of ODE’s subject to rigorously controlled small perturbation from the neglected modes. To this ODE’s we apply the Conley index to...

This paper is devoted to the study of the local rapid exponential stabilization problem for a controlled Kuramoto–Sivashinsky equation on a bounded interval. We build a feedback control law to force the solution of the closed-loop system to decay exponentially to zero with arbitrarily prescribed decay rates, provided that the initial datum is small enough. Our approach uses a method we introduc...