Kuramoto–Sivashinsky equation



In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation or flame equation) is a fourth-order nonlinear partial differential equation. It is named after Yoshiki Kuramoto and Gregory Sivashinsky, who derived the equation in the late 1970s to model the diffusive–thermal instabilities in a laminar flame front. The equation was independently derived by G. M. Homsy and A. A. Nepomnyashchii in 1974, in connection with the stability of liquid film on an inclined plane and by R. E. LaQuey et. al. in 1975 in connection with trapped-ion instability. The Kuramoto–Sivashinsky equation is known for its chaotic behavior.

Definition
The 1d version of the Kuramoto–Sivashinsky equation is
 * $$u_t + u_{xx} + u_{xxxx} + \frac{1}{2}u_x^2 = 0$$

An alternate form is
 * $$v_t + v_{xx} + v_{xxxx} + v v_x = 0$$

obtained by differentiating with respect to $$x$$ and substituting $$v = u_x$$. This is the form used in fluid dynamics applications.

The Kuramoto–Sivashinsky equation can also be generalized to higher dimensions. In spatially periodic domains, one possibility is
 * $$u_t + \Delta u + \Delta^2 u + \frac{1}{2} |\nabla u|^2 = 0,$$

where $$\Delta$$ is the Laplace operator, and $$\Delta^2$$ is the biharmonic operator.

Properties
The Cauchy problem for the 1d Kuramoto–Sivashinsky equation is well-posed in the sense of Hadamard—that is, for given initial data $$u(x, 0)$$, there exists a unique solution $$u(x, 0 \leq t < \infty)$$ that depends continuously on the initial data.

The 1d Kuramoto–Sivashinsky equation possesses Galilean invariance—that is, if $$u(x,t)$$ is a solution, then so is $$u(x-ct, t) - c$$, where $$c$$ is an arbitrary constant. Physically, since $$u$$ is a velocity, this change of variable describes a transformation into a frame that is moving with constant relative velocity $$c$$. On a periodic domain, the equation also has a reflection symmetry: if $$u(x,t)$$ is a solution, then $$-u(-x, t)$$ is also a solution.

Solutions
Solutions of the Kuramoto–Sivashinsky equation possess rich dynamical characteristics. Considered on a periodic domain $$0 \leq x \leq L $$, the dynamics undergoes a series of bifurcations as the domain size $$L$$ is increased, culminating in the onset of chaotic behavior. Depending on the value of $$L$$, solutions may include equilibria, relative equilibria, and traveling waves—all of which typically become dynamically unstable as $$L$$ is increased. In particular, the transition to chaos occurs by a cascade of period-doubling bifurcations.

Dispersive Kuramoto–Sivashinsky equations
A third-order derivative term represneting dispersion of wavenumbers are often encountered in many applications. The disperseively modified Kuramoto–Sivashinsky equation, which is often called as the Kawahara equation, is given by


 * $$u_t + u_{xx} + \delta_3 u_{xxx}+ u_{xxxx} + uu_x = 0$$

where $$\delta_3$$ is real parameter. A fifth-order derivative term is also often included, which is the modified Kawahara equation and is given by


 * $$u_t + u_{xx} + \delta_3 u_{xxx}+ u_{xxxx} + \delta_5 u_{xxxxx} + uu_x = 0.$$

Sixth-order equations
Three forms of the sixth-order Kuramoto–Sivashinsky equations are encountered in applications involving tricritical points, which are given by


 * $$\begin{align}

u_t + qu_{xx} + u_{xxxx} - u_{xxxxxx} + uu_x &= 0, \quad q>0,\\ u_t + u_{xx}-u_{xxxxxx} + uu_x &= 0, \\ u_t + qu_{xx} - u_{xxxx} - u_{xxxxxx} + uu_x &= 0, \quad q>-1/4\\ \end{align}$$

in which the last equation is referred to as the Nikolaevsky equation, named after V. N. Nikolaevsky who introudced the equation in 1989,  whereas the first two equations has been introduced recently in the context of transitions near tricritical points, i.e., change in the sign of the fourth derivative term with the plus sign approaching a Kuramoto–Sivashinsky type and the minus sign approaching a Ginzburg–Landau type.

Applications
Applications of the Kuramoto–Sivashinsky equation extend beyond its original context of flame propagation and reaction–diffusion systems. These additional applications include flows in pipes and at interfaces, plasmas, chemical reaction dynamics, and models of ion-sputtered surfaces.