In mathematics, the Kuramoto–Sivashinsky equation (also called the KS equation) is a partial differential equation used to model complex patterns and chaotic behavior in physical systems. It is one of the simplest PDEs known to exhibit chaos. The fourth-order equation was first derived in the late 1970s by Yoshiki Kuramoto and Gregory Sivashinsky to describe the instabilities of a laminar flame front. It has since been found to apply to other systems, such as the flow of a thin liquid film down an inclined plane and trapped-ion instability in plasmas.
Definition The 1d version of the Kuramoto–Sivashinsky equation is
∂ t u + ∂ x 2 u + ∂ x 4 u + 1 2 ( ∂ x u ) 2 = 0 {\displaystyle \partial _{t}u+\partial _{x}^{2}u+\partial _{x}^{4}u+{\tfrac {1}{2}}(\partial _{x}u)^{2}=0}
An alternate form is
∂ t v + ∂ x 2 v + ∂ x 4 v + v ∂ x v = 0 {\displaystyle \partial _{t}v+\partial _{x}^{2}v+\partial _{x}^{4}v+v\,\partial _{x}v=0}
obtained by differentiating with respect to x {\displaystyle x} and substituting v = ∂ x u {\displaystyle v=\partial _{x}u} . 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
∂ t u + Δ u + Δ 2 u + 1 2 | ∇ u | 2 = 0 , {\displaystyle \partial _{t}u+\Delta u+\Delta ^{2}u+{\tfrac {1}{2}}\left|\nabla u\right|^{2}=0,}
where Δ {\displaystyle \Delta } is the Laplace operator, and Δ 2 {\displaystyle \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 ) {\displaystyle u(x,0)} , there exists a unique solution u ( x , 0 ≤ t < ∞ ) {\displaystyle 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 ) {\displaystyle u(x,t)} is a solution, then so is u ( x − c t , t ) − c {\displaystyle u(x{-}ct,t)-c} , where c {\displaystyle c} is an arbitrary constant. Physically, since u {\displaystyle u} is a velocity, this change of variable describes a transformation into a frame that is moving with constant relative velocity c {\displaystyle c} . On a periodic domain, the equation also has a reflection symmetry: if u ( x , t ) {\displaystyle u(x,t)} is a solution, then − u ( − x , t ) {\displaystyle -u(-x,t)} is also a solution.
Solutions
Solutions of the Kuramoto–Sivashinsky equation possess rich dynamical characteristics. Considered on a periodic domain 0 ≤ x ≤ L {\displaystyle 0\leq x\leq L} , the dynamics undergoes a series of bifurcations as the domain size L {\displaystyle L} is increased, culminating in the onset of chaotic behavior. Depending on the value of L {\displaystyle L} , solutions may include equilibria, relative equilibria, and traveling waves—all of which typically become dynamically unstable as L {\displaystyle L} is increased. In particular, the transition to chaos occurs by a cascade of period-doubling bifurcations.
Modified Kuramoto–Sivashinsky equation
Dispersive Kuramoto–Sivashinsky equations A third-order derivative term representing dispersion of wavenumbers are often encountered in many applications. The dispersively modified Kuramoto–Sivashinsky equation, which is often called as the Kawahara equation, is given by
… excerpt ends here. Continue reading the full article.
