In this paper, the optimal homotopy analysis method is applied to find the solitary wave solutions of the Kuramoto-Sivashinsky equation. With three auxiliary convergence-control parameters, whose possible optimal values can be obtained by minimizing the averaged residual error, the method used here provides us with a simple way to adjust and control the convergence region of the solution. Compa...

This work focuses on linear finite-dimensional output feedback control of the Kuramoto–Sivashinsky equation (KSE) with periodic boundary conditions. Under the assumption that the linearization of the KSE around the zero solution is controllable and observable, linear finite-dimensional output feedback controllers are synthesized that achieve stabilization of the zero solution, for any value of ...

We study the dynamics of a ferrofluid thin film confined in Hele-Shaw cell, and subjected to tilted nonuniform magnetic field. It is shown that interface between an inviscid outer fluid (air) supports traveling waves, governed by novel modified Kuramoto--Sivashinsky-type equation derived under long-wave approximation. The balance energy production dissipation this equations allows for existence...

In this article, the robust Stackelberg controllability (RSC) problem is studied for a nonlinear fourth-order parabolic equation, namely Kuramoto–Sivashinsky equation. When three external sources are acting into system, RSC consists essentially in combining two subproblems: first one saddle point among sources. Such called “follower control” and its associated “disturbance signal.” This procedu...

Constitutive equations for a one-dimensional, active, polar, viscoelastic liquid are derived by treating the strain field as a slow hydrodynamic variable. Taking into account the couplings between strain and polarity allowed by symmetry, the hydrodynamics of an active, polar, viscoelastic body include an evolution equation for the polarity field that generalizes the damped Kuramoto-Sivashinsky ...

We present a heuristic model for the energy spectrum of the one-dimensional phase turbulence in the steady state of the Kuramoto-Sivashinsky equation. Our model contains an energy transfer mechanism from low- to high-wave-vector modes. The energy transfer is written as the sum of local and nonlocal interactions. Our analytical results show good agreement with numerical simulations, particularly...

Abstract: We study a dynamic system described by the Kuramoto-Sivashinsky equation with an external excitation f posed on a finite domain. Firstly it shows that under the given boundary feedback conditions it admits a unique solution and the solution is stable. Secondly it proves that if the external excitation f is a time periodic function, then the system under the boundary conditions admits ...

In this article, we study the boundary controllability of the linear Kuramoto-Sivashinsky equation on a bounded interval. The control acts on the first spatial derivative at the left endpoint. First, we prove that this control system is null controllable. It is done using a spectral analysis and the method of moments. Then, we introduce a boundary feedback law stabilizing to zero the solution o...