Kuramoto–Sivashinsky equation

A spatiotemporal plot of a simulation of the 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.^[1][2][3] It was later and 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

${\displaystyle u_{t}+u_{xx}+u_{xxxx}+{\frac {1}{2}}u_{x}^{2}=0}$

An alternate form is

${\displaystyle v_{t}+v_{xx}+v_{xxxx}+vv_{x}=0}$

obtained by differentiating with respect to ${\displaystyle x}$ and substituting ${\displaystyle v=u_{x}}$. 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

${\displaystyle u_{t}+\Delta u+\Delta ^{2}u+{\frac {1}{2}}|\nabla u|^{2}=0,}$

where ${\displaystyle \Delta }$ is the Laplace operator, and ${\displaystyle \Delta ^{2}}$ 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 ${\displaystyle u(x,0)}$, there exists a unique solution ${\displaystyle u(x,0\leq t<\infty )}$ that depends continuously on the initial data.^[10]

The 1d Kuramoto–Sivashinsky equation possesses Galilean invariance—that is, if ${\displaystyle u(x,t)}$ is a solution, then so is ${\displaystyle u(x-ct,t)-c}$, where ${\displaystyle c}$ is an arbitrary constant.^[11] Physically, since ${\displaystyle u}$ is a velocity, this change of variable describes a transformation into a frame that is moving with constant relative velocity ${\displaystyle c}$. On a periodic domain, the equation also has a reflection symmetry: if ${\displaystyle u(x,t)}$ is a solution, then ${\displaystyle -u(-x,t)}$ is also a solution.^[11]

Solutions

A converged relative periodic orbit for the KS equation with periodic boundary conditions for a domain size ${\displaystyle L=35}$. 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 Kuramoto–Sivashinsky equation possess rich dynamical characteristics.^[11][12][13] Considered on a periodic domain ${\displaystyle 0\leq x\leq L}$, the dynamics undergoes a series of bifurcations as the domain size ${\displaystyle L}$ is increased, culminating in the onset of chaotic behavior. Depending on the value of ${\displaystyle L}$, solutions may include equilibria, relative equilibria, and traveling waves—all of which typically become dynamically unstable as ${\displaystyle L}$ is increased. In particular, the transition to chaos occurs by a cascade of period-doubling bifurcations.^[13]

Modified Kuramoto–Sivashinsky equation

Dispersive Kuramoto–Sivashinsky equations

A third-order derivative term representing dispersion of wavenumbers are often encountered in many applications. The disperseively modified Kuramoto–Sivashinsky equation, which is often called as the Kawahara equation,^[14] is given by^[15]

${\displaystyle u_{t}+u_{xx}+\delta _{3}u_{xxx}+u_{xxxx}+uu_{x}=0}$

where ${\displaystyle \delta _{3}}$ is real parameter. A fifth-order derivative term is also often included, which is the modified Kawahara equation and is given by^[16]

${\displaystyle u_{t}+u_{xx}+\delta _{3}u_{xxx}+u_{xxxx}+\delta _{5}u_{xxxxx}+uu_{x}=0.}$

Sixth-order equations

Three forms of the sixth-order Kuramoto–Sivashinsky equations are encountered in applications involving tricritical points, which are given by^[17]

${\displaystyle {\begin{aligned}u_{t}+qu_{xx}+u_{xxxx}-u_{xxxxxx}+uu_{x}&=0,\quad q>0,\\u_{t}+u_{xx}-u_{xxxxxx}+uu_{x}&=0,\\u_{t}+qu_{xx}-u_{xxxx}-u_{xxxxxx}+uu_{x}&=0,\quad q>-1/4\\\end{aligned}}}$

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

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][21]

See also

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

References

1. Kuramoto, Yoshiki (1978). "Diffusion-Induced Chaos in Reaction Systems". Progress of Theoretical Physics Supplement. 64: 346–367. doi:10.1143/PTPS.64.346. ISSN 0375-9687.
2. Sivashinsky, G.I. (1977). "Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations". Acta Astronautica. 4 (11–12): 1177–1206. doi:10.1016/0094-5765(77)90096-0. ISSN 0094-5765.
3. Sivashinsky, G. I. (1980). "On Flame Propagation Under Conditions of Stoichiometry". SIAM Journal on Applied Mathematics. 39 (1): 67–82. doi:10.1137/0139007. ISSN 0036-1399.
4. 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. 191–194. Bibcode:1974LApM...15.....N.
5. Nepomnyashchii, A. A. (1975). "Stability of wavy conditions in a film flowing down an inclined plane". Fluid Dynamics. 9 (3): 354–359. doi:10.1007/BF01025515.
6. Laquey, R. E.; Mahajan, S. M.; Rutherford, P. H.; Tang, W. M. (1975). "Nonlinear Saturation of the Trapped-Ion Mode". Physical Review Letters. 34 (7): 391–394. doi:10.1103/PhysRevLett.34.391.
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. 120 (2): 024102. doi:10.1103/PhysRevLett.120.024102. ISSN 0031-9007. PMID 29376715.
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. arXiv:1910.05266. doi:10.1016/j.neunet.2020.02.016. ISSN 0893-6080. PMID 32248008. S2CID 211146609.
9. Kalogirou, A.; Keaveny, E. E.; Papageorgiou, D. T. (2015). "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): 20140932. doi:10.1098/rspa.2014.0932. ISSN 1364-5021. PMC 4528647. PMID 26345218.
10. Tadmor, Eitan (1986). "The Well-Posedness of the Kuramoto–Sivashinsky Equation". SIAM Journal on Mathematical Analysis. 17 (4): 884–893. doi:10.1137/0517063. hdl:1903/8432. ISSN 0036-1410.
11. Cvitanović, Predrag; Davidchack, Ruslan L.; Siminos, Evangelos (2010). "On the State Space Geometry of the Kuramoto–Sivashinsky Flow in a Periodic Domain". SIAM Journal on Applied Dynamical Systems. 9 (1): 1–33. arXiv:0709.2944. doi:10.1137/070705623. ISSN 1536-0040. S2CID 17048798.
12. Michelson, Daniel (1986). "Steady solutions of the Kuramoto-Sivashinsky equation". Physica D: Nonlinear Phenomena. 19 (1): 89–111. doi:10.1016/0167-2789(86)90055-2. ISSN 0167-2789.
13. Papageorgiou, D.T.; Smyrlis, Y.S. (1991), "The route to chaos for the Kuramoto-Sivashinsky equation", Theoret. Comput. Fluid Dynamics, 3: 15–42, doi:10.1007/BF00271514, hdl:2060/19910004329, ISSN 1432-2250, S2CID 116955014
14. Topper, J.; Kawahara, T. (1978). "Approximate equations for long nonlinear waves on a viscous fluid". Journal of the Physical Society of Japan. 44 (2): 663–666. doi:10.1143/JPSJ.44.2003.
15. 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 (3–4): 299–320. doi:10.1016/0167-2789(93)90113-F. ISSN 1872-8022.
16. 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.
17. Rajamanickam, P.; Daou, J. (2023). "Tricritical point as a crossover between type-Is and type-IIs bifurcations". Progress in Scale Modeling. 4 (1): 2. doi:10.13023/psmij.2023.04-01-02. ISSN 2693-969X.
18. 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.
19. Tribelsky, M. I., & Tsuboi, K. (1996). New scenario for transition to turbulence?. Physical review letters, 76(10), 1631.
20. Matthews, P. C., & Cox, S. M. (2000). One-dimensional pattern formation with Galilean invariance near a stationary bifurcation. Physical Review E, 62(2), R1473.
21. Cuerno, Rodolfo; Barabási, Albert-László (1995). "Dynamic Scaling of Ion-Sputtered Surfaces". Physical Review Letters. 74 (23): 4746–4749. arXiv:cond-mat/9411083. doi:10.1103/PhysRevLett.74.4746. ISSN 0031-9007. PMID 10058588. S2CID 18148655.

External links

• Weisstein, Eric W. "Kuramoto-Sivashinsky Equation". MathWorld.