A spatiotemporal plot of a simulation of the KuramotoâSivashinsky equation
In
mathematics, the KuramotoâSivashinsky 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
diffusiveâthermal instabilities in a
laminar flame front.[1][2][3] The equation was independently derived by G. M. Homsy[4] and A. A. Nepomnyashchii[5] in 1974, in connection with the stability of liquid film on an inclined plane and by R. E. LaQuey et. al.[6] in 1975 in connection with trapped-ion instability. The KuramotoâSivashinsky equation is known for its
chaotic behavior.[7][8]
Definition
The 1d version of the KuramotoâSivashinsky equation is
An alternate form is
obtained by differentiating with respect to and substituting . This is the form used in
fluid dynamics applications.[9]
The KuramotoâSivashinsky equation can also be generalized to higher dimensions. In spatially periodic domains, one possibility is
The
Cauchy problem for the 1d KuramotoâSivashinsky equation is
well-posed in the sense of Hadamardâthat is, for given initial data , there exists a unique solution that depends continuously on the initial data.[10]
The 1d KuramotoâSivashinsky equation possesses
Galilean invarianceâthat is, if is a solution, then so is , where is an arbitrary constant.[11] Physically, since 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 is a solution, then is also a solution.[11]
Solutions
A converged relative periodic orbit for the KS equation with periodic boundary conditions for a domain size . After some time the system returns to its initial state, only translated slightly (~4 units) to the left. This particular solution has three unstable directions and three marginal directions.
Solutions of the KuramotoâSivashinsky equation possess rich dynamical characteristics.[11][12][13] Considered on a periodic domain , the dynamics undergoes a series of
bifurcations as the domain size is increased, culminating in the onset of
chaotic behavior. Depending on the value of , solutions may include equilibria, relative equilibria, and
traveling wavesâall of which typically become dynamically unstable as is increased. In particular, the transition to chaos occurs by a cascade of
period-doubling bifurcations.[13]
Modified KuramotoâSivashinsky equation
Dispersive KuramotoâSivashinsky equations
A third-order derivative term represneting dispersion of wavenumbers are often encountered in many applications. The disperseively modified KuramotoâSivashinsky equation, which is often called as the Kawahara equation,[14] is given by[15]
where is real parameter. A fifth-order derivative term is also often included, which is the modified Kawahara equation and is given by[16]
Sixth-order equations
Three forms of the sixth-order KuramotoâSivashinsky 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 recently 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 KuramotoâSivashinsky type and the minus sign approaching a
GinzburgâLandau type.
Applications
Applications of the KuramotoâSivashinsky equation extend beyond its original context of flame propagation and
reactionâdiffusion systems. These additional applications include flows in pipes and at interfaces,
plasmas, chemical reaction dynamics, and models of ion-sputtered surfaces.[9][21]
^Sivashinsky, G.I. (1977). "Nonlinear analysis of hydrodynamic instability in laminar flamesâI. Derivation of basic equations". Acta Astronautica. 4 (11â12): 1177â1206.
doi:
10.1016/0094-5765(77)90096-0.
ISSN0094-5765.
^Sivashinsky, G. I. (1980). "On Flame Propagation Under Conditions of Stoichiometry". SIAM Journal on Applied Mathematics. 39 (1): 67â82.
doi:
10.1137/0139007.
ISSN0036-1399.
^Homsy, G. M. (1974). "Model equations for wavy viscous film flow". In Newell, A. (ed.). Nonlinear Wave Motion. Lectures in Applied Mathematics. Vol. 15. Providence: American Mathematical Society. pp. 191â194.
Bibcode:
1974LApM...15.....N.
^Nepomnyashchii, A. A. (1975). "Stability of wavy conditions in a film flowing down an inclined plane". Fluid Dynamics. 9 (3): 354â359.
doi:
10.1007/BF01025515.
^Chang, H. C.; Demekhin, E. A.; Kopelevich, D. I. (1993). "Laminarizing effects of dispersion in an active-dissipative nonlinear medium". Physica D: Nonlinear Phenomena. 63 (3â4): 299â320.
doi:
10.1016/0167-2789(93)90113-F.
ISSN1872-8022.
^Akrivis, G., Papageorgiou, D. T., & Smyrlis, Y. S. (2012). Computational study of the dispersively modified KuramotoâSivashinsky equation. SIAM Journal on Scientific Computing, 34(2), A792-A813.
^Nikolaevskii, V. N. (1989). Dynamics of viscoelastic media with internal oscillators. In Recent Advances in Engineering Science: A Symposium dedicated to A. Cemal Eringen June 20â22, 1988, Berkeley, California (pp. 210-221). Berlin, Heidelberg: Springer Berlin Heidelberg.
^Tribelsky, M. I., & Tsuboi, K. (1996). New scenario for transition to turbulence?. Physical review letters, 76(10), 1631.
^Matthews, P. C., & Cox, S. M. (2000). One-dimensional pattern formation with Galilean invariance near a stationary bifurcation. Physical Review E, 62(2), R1473.