The Kuramoto-Sivashinsky equation is a dissipative evolution equation in one space dimension which, despite its apparent simplicity, gives rise to a very rich dynamical behavior, as evidenced for instance by the study in [16], of its complicated set of stationary solutions and stationary and Hopf bifurcations. The large time behavior of the solutions is usually embodied by the attractor and the...

Initial boundary value problems for the three-dimensional Kuramoto-Sivashinsky equation posed on unbounded 3D grooves (that may serve as mathematical models wildfires) were considered. The existence and uniqueness of global strong solutions well their exponential decay have been established.

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 ...

In this paper, we use a methodology that was recently proposed by Antoniades and Christofides to compute the optimal actuator/sensor locations for the stabilization, via nonlinear static output feedback control, of the zero solution of the Kuramoto–Sivashinsky equation (KSE) for values of the instability parameter for which this solution is unstable. The theoretical results are illustrated thro...

We study the step meandering instability on a surface characterized by the alternation of terraces with different properties, as in the case of Si(001). The interplay between diffusion anisotropy and step stiffness induces a finite wavelength instability corresponding to a meandering mode. The instability sets in beyond a threshold value which depends on the relative magnitudes of the destabili...

We consider a system formed by an infinite viscous liquid layer with a constant horizontal temperature gradient, and a basic nonlinear bulk velocity profile. In the limit of long-wavelength and large nondimensional surface tension, we show that hydrothermal surface-wave instabilities may give rise to disturbances governed by the Kuramoto-Sivashinsky equation. A possible connection to hot-wire e...

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...

Phase transitions can be modeled by the motion of an interface between two locally stable phases. A modiied Kuramoto-Sivashinsky equation, h t + r 2 h + r 4 h = (1 ?)jrhj 2 (r 2 h) 2 + (h xx h yy ? h 2 xy); describes near planar interfaces which are marginally long-wave unstable. We study the question of nite-time singularity formation in this equation in one and two space dimensions on a perio...

Phase transitions can be modeled by the motion of an interface between two locally stable phases. A modified Kuramoto-Sivashinsky equation, h, + 72h + Wh = (1 a)lVhl ~ ± X(V~h): + ~A(hx~hyy h~y) , describes near planar interfaces which are marginally long-wave unstable. We study the question of finite-time singularity formation in this equation in one and two space dimensions on a periodic doma...

We report the results of extensive numerical experiments on the Kuramoto-Sivashinsky equation in the strongly chaotic regime as the viscosity parameter is decreased and increasingly more linearly unstable modes enter the dynamics. General initial conditions are used and evolving states do not assume odd-parity. A large number of numerical experiments are employed in order to obtain quantitative...