Kuramoto–Sivashinsky 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.
A. Nepomnyashchii[5] in 1974, in connection with the stability of liquid film on an inclined plane and by R. E. LaQuey et.
[7][8] The 1d version of the Kuramoto–Sivashinsky equation is An alternate form is obtained by differentiating with respect to
[9] The Kuramoto–Sivashinsky equation can also be generalized to higher dimensions.
The Cauchy problem for the 1d Kuramoto–Sivashinsky equation is well-posed in the sense of Hadamard—that is, for given initial data
is a velocity, this change of variable describes a transformation into a frame that is moving with constant relative velocity
On a periodic domain, the equation also has a reflection symmetry: if
[11] Solutions of the Kuramoto–Sivashinsky equation possess rich dynamical characteristics.
, the dynamics undergoes a series of bifurcations as the domain size
is increased, culminating in the onset of chaotic behavior.
In particular, the transition to chaos occurs by a cascade of period-doubling bifurcations.
[13] A third-order derivative term representing dispersion of wavenumbers are often encountered in many applications.
A fifth-order derivative term is also often included, which is the modified Kawahara equation and is given by[16] Three forms of the sixth-order Kuramoto–Sivashinsky equations are encountered in applications involving tricritical points, which are given by[17] in which the last equation is referred to as the Nikolaevsky equation, named after V. N. Nikolaevsky who introudced the equation in 1989,[18][19][20] whereas the first two equations has been introduced by P. Rajamanickam and J. Daou in the context of transitions near tricritical points,[17] 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 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.
