Kuramoto–Sivashinsky equation explained

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.^{[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 Kuramoto–Sivashinsky equation is known for its chaotic behavior.^{[7] [8]}

Definition

The 1d version of the Kuramoto–Sivashinsky equation is

u _{t} + u _{xx} + u _{xxxx} + (1)/(2) u ^{2} _{x} = 0 An alternate form isv _{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.^[9]

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 + (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 < inf ty) that depends continuously on the initial data.^[10]

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.^[11] 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.^[11]

Solutions

Solutions of the Kuramoto–Sivashinsky equation possess rich dynamical characteristics.^{[11] [12]} 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.

Modified Kuramoto–Sivashinsky equation

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,^[13] is given by^[14]

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^[15] 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^[16]

\begin{align} u _{t} + qu _{xx} + u _{xxxx} - u _{xxxxxx} + uu _{x} & = 0 , q> 0 ,\\ u _{t} + u _{xx} - u _{xxxxxx} + uu _{x} & = 0 , \\ u _{t} + qu _{xx} - u _{xxxx} - u _{xxxxxx} + uu _{x} & = 0 , 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,^{[17] [18] [19]} 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.^{[9] [20]}

See also

• Michelson–Sivashinsky equation
• List of nonlinear partial differential equations
• List of chaotic maps
• Clarke's equation
• Laminar flame speed

Notes and References

1. Kuramoto. Yoshiki. Diffusion-Induced Chaos in Reaction Systems. Progress of Theoretical Physics Supplement. 64. 1978. 346–367. 0375-9687. 10.1143/PTPS.64.346. free.
2. Sivashinsky. G.I.. Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations. Acta Astronautica. 4. 11–12. 1977. 1177–1206. 0094-5765. 10.1016/0094-5765(77)90096-0.
3. Sivashinsky. G. I.. On Flame Propagation Under Conditions of Stoichiometry. SIAM Journal on Applied Mathematics. 39. 1. 1980. 67–82. 0036-1399. 10.1137/0139007.
4. Book: Homsy. G. M.. Model equations for wavy viscous film flow . Nonlinear Wave Motion . Newell, A. . Lectures in Applied Mathematics. 15. 191–194 . Providence . American Mathematical Society . 1974 . 1974LApM...15.....N.
5. 10.1007/BF01025515 . Stability of wavy conditions in a film flowing down an inclined plane . 1975 . Nepomnyashchii . A. A. . Fluid Dynamics . 9 . 3 . 354–359 .
6. 10.1103/PhysRevLett.34.391 . Nonlinear Saturation of the Trapped-Ion Mode . 1975 . Laquey . R. E. . Mahajan . S. M. . Rutherford . P. H. . Tang . W. M. . Physical Review Letters . 34 . 7 . 391–394 .
7. Pathak. Jaideep. Hunt. Brian. Girvan. Michelle. Lu. Zhixin. Ott. Edward. 2018. Model-Free Prediction of Large Spatiotemporally Chaotic Systems from Data: A Reservoir Computing Approach. Physical Review Letters. en. 120. 2. 024102. 10.1103/PhysRevLett.120.024102. 29376715 . 0031-9007. free.
8. Vlachas. P.R.. Pathak. J.. Hunt. B.R.. Sapsis. T.P.. Girvan. M.. Ott. E.. Koumoutsakos. P.. 2020-03-21. Backpropagation algorithms and Reservoir Computing in Recurrent Neural Networks for the forecasting of complex spatiotemporal dynamics. Neural Networks. 126. 191–217. 10.1016/j.neunet.2020.02.016. 32248008 . 211146609 . 0893-6080. free. 1910.05266.
9. Kalogirou. A.. Keaveny. E. E.. Papageorgiou. D. T.. An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 471. 2179. 2015. 20140932. 1364-5021. 10.1098/rspa.2014.0932. 26345218 . 4528647.
10. Tadmor. Eitan. The Well-Posedness of the Kuramoto–Sivashinsky Equation. SIAM Journal on Mathematical Analysis. 17. 4. 1986. 884–893. 0036-1410. 10.1137/0517063. 1903/8432 . free.
11. Cvitanović. Predrag. Davidchack. Ruslan L.. Siminos. Evangelos. Predrag Cvitanović. On the State Space Geometry of the Kuramoto–Sivashinsky Flow in a Periodic Domain. SIAM Journal on Applied Dynamical Systems. 9. 1. 2010. 1–33. 1536-0040. 10.1137/070705623. 0709.2944 . 17048798 .
12. Michelson. Daniel. Steady solutions of the Kuramoto-Sivashinsky equation. Physica D: Nonlinear Phenomena. 19. 1. 1986. 89–111. 0167-2789. 10.1016/0167-2789(86)90055-2.
13. Topper. J.. Kawahara. T.. Approximate equations for long nonlinear waves on a viscous fluid. Journal of the Physical Society of Japan. 44. 2. 1978. 663–666. 10.1143/JPSJ.44.2003. free.
14. Chang. H. C.. Demekhin. E. A. . Kopelevich. D. I.. Laminarizing effects of dispersion in an active-dissipative nonlinear medium. Physica D: Nonlinear Phenomena. 63. 3–4. 1993. 299–320. 1872-8022. 10.1016/0167-2789(93)90113-F.
15. Akrivis, G., Papageorgiou, D. T., & Smyrlis, Y. S. (2012). Computational study of the dispersively modified Kuramoto–Sivashinsky equation. SIAM Journal on Scientific Computing, 34(2), A792-A813.
16. Rajamanickam. P.. Daou. J.. Tricritical point as a crossover between type-Is and type-IIs bifurcations. Progress in Scale Modeling . 4. 1. 2023. 2. 2693-969X. 10.13023/psmij.2023.04-01-02. free.
17. 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 20–22, 1988, Berkeley, California (pp. 210-221). Berlin, Heidelberg: Springer Berlin Heidelberg.
18. Tribelsky, M. I., & Tsuboi, K. (1996). New scenario for transition to turbulence?. Physical review letters, 76(10), 1631.
19. Matthews, P. C., & Cox, S. M. (2000). One-dimensional pattern formation with Galilean invariance near a stationary bifurcation. Physical Review E, 62(2), R1473.
20. Cuerno. Rodolfo. Barabási. Albert-László. Dynamic Scaling of Ion-Sputtered Surfaces. Physical Review Letters. 74. 23. 1995. 4746–4749. 0031-9007. 10.1103/PhysRevLett.74.4746 . 10058588 . cond-mat/9411083 . 18148655 .