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Journal articles on the topic 'Kuramoto-Sivashinsky equation'

Author: Grafiati

Published: 4 June 2021

Last updated: 7 September 2023

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1

Li, Jin. "Linear barycentric rational interpolation method for solving Kuramoto-Sivashinsky equation." AIMS Mathematics 8, no. 7 (2023): 16494–510. http://dx.doi.org/10.3934/math.2023843.

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<abstract><p>The Kuramoto-Sivashinsky (KS) equation being solved by the linear barycentric rational interpolation method (LBRIM) is presented. Three kinds of linearization schemes, direct linearization, partial linearization and Newton linearization, are presented to get the linear equation of the Kuramoto-Sivashinsky equation. Matrix equations of the discrete Kuramoto-Sivashinsky equation are also given. The convergence rate of LBRIM for solving the KS equation is also proved. At last, two examples are given to prove the theoretical analysis.</p></abstract>

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2

Gao, Peng. "Irreducibility of Kuramoto-Sivashinsky equation driven by degenerate noise." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 20. http://dx.doi.org/10.1051/cocv/2022014.

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In this paper, we study irreducibility of Kuramoto-Sivashinsky equation which is driven by an additive noise acting only on a finite number of Fourier modes. In order to obtain the irreducibility, we first investigate the approximate controllability of Kuramoto-Sivashinsky equation driven by a finite-dimensional force, the proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the continuity of solving operator for deterministic Kuramoto-Sivashinsky equation. Finally, combining the approximate controllability with continuity of solving operator, we establish the irreducibility of Kuramoto-Sivashinsky equation.

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3

XIE, YUANXI, SHUHUA ZHU, and KALIN SU. "SOLVING THE KdV-BURGERS-KURAMOTO EQUATION BY A COMBINATION METHOD." International Journal of Modern Physics B 23, no. 08 (March 30, 2009): 2101–6. http://dx.doi.org/10.1142/s0217979209052017.

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Based on the analysis on the characteristics of the KdV equation, Kuramoto-Sivashinsky equation and KdV-Burgers-Kuramoto equation, a combination method is proposed to construct the explicit exact solutions for the KdV-Burgers-Kuramoto equation by combining with those of the KdV equation and Kuramoto-Sivashinsky equation. As a result, many explicit exact solutions to the KdV-Burgers-Kuramoto equation are successfully derived by this approach.

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4

Mohammed, Wael W., A. M. Albalahi, S. Albadrani, E. S. Aly, R. Sidaoui, and A. E. Matouk. "The Analytical Solutions of the Stochastic Fractional Kuramoto–Sivashinsky Equation by Using the Riccati Equation Method." Mathematical Problems in Engineering 2022 (May 11, 2022): 1–8. http://dx.doi.org/10.1155/2022/5083784.

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In this work, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation using conformable derivative. The Riccati equation method is used to get the analytical solutions to the space-fractional stochastic Kuramoto–Sivashinsky equation. Because this equation has never been examined with space-fractional and multiplicative noise at the same time, we generalize some previous results. Moreover, we display how the multiplicative noise influences on the stability of obtained solutions of the space-fractional stochastic Kuramoto–Sivashinsky equation.

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5

Mohammed, Wael W., Meshari Alesemi, Sahar Albosaily, Naveed Iqbal, and M. El-Morshedy. "The Exact Solutions of Stochastic Fractional-Space Kuramoto-Sivashinsky Equation by Using (G′G)-Expansion Method." Mathematics 9, no. 21 (October 26, 2021): 2712. http://dx.doi.org/10.3390/math9212712.

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In this paper, we consider the stochastic fractional-space Kuramoto–Sivashinsky equation forced by multiplicative noise. To obtain the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation, we apply the G′G-expansion method. Furthermore, we generalize some previous results that did not use this equation with multiplicative noise and fractional space. Additionally, we show the influence of the stochastic term on the exact solutions of the stochastic fractional-space Kuramoto–Sivashinsky equation.

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6

Albosaily, Sahar, Wael W. Mohammed, Ali Rezaiguia, Mahmoud El-Morshedy, and Elsayed M. Elsayed. "The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation." Open Mathematics 20, no. 1 (January 1, 2022): 108–16. http://dx.doi.org/10.1515/math-2022-0012.

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Abstract In this article, we take into account the stochastic Kuramoto-Sivashinsky equation forced by multiplicative noise in the Itô sense. To obtain the exact stochastic solutions of the stochastic Kuramoto-Sivashinsky equation, we apply the G ′ G \frac{{G}^{^{\prime} }}{G} -expansion method. Furthermore, we extend some previous results where this equation has not been previously studied in the presence of multiplicative noise. Also, we show the influence of multiplicative noise on the analytical solutions of the stochastic Kuramoto-Sivashinsky equation.

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7

EDSON, RUSSELL A., J. E. BUNDER, TRENT W. MATTNER, and A. J. ROBERTS. "LYAPUNOV EXPONENTS OF THE KURAMOTO–SIVASHINSKY PDE." ANZIAM Journal 61, no. 3 (July 2019): 270–85. http://dx.doi.org/10.1017/s1446181119000105.

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The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence.

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8

Edson, Russell A., Judith E. Bunder, Trent W. Mattner, and Anthony J. Roberts. "Lyapunov exponents of the Kuramoto--Sivashinsky PDE." ANZIAM Journal 61 (September 8, 2019): 270–85. http://dx.doi.org/10.21914/anziamj.v61i0.13939.

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The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence. doi:10.1017/S1446181119000105

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9

Tilley, B. S., S. H. Davis, and S. G. Bankoff. "Nonlinear long-wave stability of superposed fluids in an inclined channel." Journal of Fluid Mechanics 277 (October 25, 1994): 55–83. http://dx.doi.org/10.1017/s0022112094002685.

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We consider the two-layer flow of immiscible, viscous, incompressible fluids in an inclined channel. We use long-wave theory to obtain a strongly nonlinear evolution equation which describes the motion of the interface. This equation includes the physical effects of viscosity stratification, density stratification, and shear. A weakly nonlinear analysis of this equation yields a Kuramoto–Sivashinsky equation, which possesses a quadratic nonlinearity. However, certain physical situations exist in two-layer flow for which modifications of the Kuramoto–Sivashinsky equation are physically pertinent. In particular, the presence of the second layer can mediate the wave-steepening instability found in single-phase falling films, requiring the inclusion of a cubic nonlinearity in the weakly nonlinear analysis. The introduction of the cubic nonlinearity destroys the symmetry-breaking bifurcations of the Kuramoto–Sivashinsky equation, and new isolated solution branches emerge as the strength of the cubic nonlinearity increases. Bistability between these new solutions and those associated with the Kuramoto–Sivashinsky equation is found, as well as the formation of a hysteresis loop from smaller-amplitude travelling waves to larger-amplitude travelling waves. The physical implications of these dynamics to the phenomenon of laminar flooding in a channel are discussed.

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10

Amali Paul Rose, Gregory, Murugan Suvinthra, and Krishnan Balachandran. "Large deviations for stochastic Kuramoto–Sivashinsky equation with multiplicative noise." Nonlinear Analysis: Modelling and Control 26, no. 4 (July 1, 2021): 642–60. http://dx.doi.org/10.15388/namc.2021.26.24178.

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The Kuramoto–Sivashinsky equation is a nonlinear parabolic partial differential equation, which describes the instability and turbulence of waves in chemical reactions and laminar flames. The aim of this work is to prove the large deviation principle for the stochastic Kuramoto–Sivashinsky equation driven by multiplicative noise. To establish the large deviation principle, the weak convergence approach is used, which relies on proving basic qualitative properties of controlled versions of the original stochastic partial differential equation.

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11

Feng, Dahe. "Exact Solutions of Kuramoto-Sivashinsky Equation." International Journal of Education and Management Engineering 2, no. 6 (June 29, 2012): 61–66. http://dx.doi.org/10.5815/ijeme.2012.06.11.

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12

Duan, Jinqiao, and Vincent J. Ervin. "On the stochastic Kuramoto–Sivashinsky equation." Nonlinear Analysis: Theory, Methods & Applications 44, no. 2 (April 2001): 205–16. http://dx.doi.org/10.1016/s0362-546x(99)00259-x.

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13

Shiraishi, Kenji, and Yukio Saito. "Anisotropy Effect on Kuramoto-Sivashinsky Equation." Journal of the Physical Society of Japan 64, no. 1 (January 15, 1995): 9–13. http://dx.doi.org/10.1143/jpsj.64.9.

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14

Collet, P., J. P. Eckmann, H. Epstein, and J. Stubbe. "Analyticity for the Kuramoto-Sivashinsky equation." Physica D: Nonlinear Phenomena 67, no. 4 (September 1993): 321–26. http://dx.doi.org/10.1016/0167-2789(93)90168-z.

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15

BO, LIJUN, KEHUA SHI, and YONGJIN WANG. "ON A NONLOCAL STOCHASTIC KURAMOTO–SIVASHINSKY EQUATION WITH JUMPS." Stochastics and Dynamics 07, no. 04 (December 2007): 439–57. http://dx.doi.org/10.1142/s0219493707002104.

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In this paper, we study a class of nonlocal stochastic Kuramoto–Sivashinsky equations driven by compensated Poisson random measures and show the existence and uniqueness of the weak solution to the equation. Furthermore, we prove that an invariant measure of the equation indeed exists under some appropriate assumptions.

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16

AKERS, B. F., and D. M. AMBROSE. "EFFICIENT COMPUTATION OF COORDINATE-FREE MODELS OF FLAME FRONTS." ANZIAM Journal 63, no. 1 (January 2021): 58–69. http://dx.doi.org/10.1017/s1446181121000079.

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AbstractWe present an efficient, accurate computational method for a coordinate-free model of flame front propagation of Frankel and Sivashinsky. This model allows for overturned flames fronts, in contrast to weakly nonlinear models such as the Kuramoto–Sivashinsky equation. The numerical procedure adapts the method of Hou, Lowengrub and Shelley, derived for vortex sheets, to this model. The result is a nonstiff, highly accurate solver which can handle fully nonlinear, overturned interfaces, with similar computational expense to methods for weakly nonlinear models. We apply this solver both to simulate overturned flame fronts and to compare the accuracy of Kuramoto–Sivashinsky and coordinate-free models in the appropriate limit.

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17

Akers, Benjamin, and D. M. Ambrose. "Efficient computation of coordinate-free models of flame fronts." ANZIAM Journal 63 (July 30, 2021): 58–69. http://dx.doi.org/10.21914/anziamj.v63.15970.

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We present an efficient, accurate computational method for a coordinate-free model of flame front propagation of Frankel and Sivashinsky. This model allows for overturned flames fronts, in contrast to weakly nonlinear models such as the Kuramoto–Sivashinsky equation. The numerical procedure adapts the method of Hou, Lowengrub and Shelley, derived for vortex sheets, to this model. The result is a nonstiff, highly accurate solver which can handle fully nonlinear, overturned interfaces, with similar computational expense to methods for weakly nonlinear models. We apply this solver both to simulate overturned flame fronts and to compare the accuracy of Kuramoto–Sivashinsky and coordinate-free models in the appropriate limit. doi:10.1017/S1446181121000079

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18

Kudryashov, N. A., P. N. Ryabov, and B. A. Petrov. "Dissipative structures of the Kuramoto–Sivashinsky equation." Automatic Control and Computer Sciences 49, no. 7 (December 2015): 508–13. http://dx.doi.org/10.3103/s0146411615070147.

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19

Kudryashov, N. A., P. N. Ryabov, and B. A. Petrov. "Dissipative Structures of the Kuramoto–Sivashinsky Equation." Modeling and Analysis of Information Systems 22, no. 1 (January 1, 2015): 105–13. http://dx.doi.org/10.18255/1818-1015-2015-1-105-113.

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20

Ong, Kiah Wah. "Dynamic transitions of generalized Kuramoto-Sivashinsky equation." Discrete and Continuous Dynamical Systems - Series B 21, no. 4 (March 2016): 1225–36. http://dx.doi.org/10.3934/dcdsb.2016.21.1225.

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21

Gao, Ping, Chengjian Cai, and Xiaoyi Liu. "Numerical Simulation of Stochastic Kuramoto-Sivashinsky Equation." Journal of Applied Mathematics and Physics 06, no. 11 (2018): 2363–69. http://dx.doi.org/10.4236/jamp.2018.611198.

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22

Biagioni, H. A., and R. J. Iorio Jr. "Generalized solutions to the kuramoto—sivashinsky equation." Integral Transforms and Special Functions 6, no. 1-4 (March 1998): 1–8. http://dx.doi.org/10.1080/10652469808819145.

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23

LI, JI, and LEONARD M. SANDER. "SCALING PROPERTIES OF THE KURAMOTO-SIVASHINSKY EQUATION." Fractals 03, no. 03 (September 1995): 507–14. http://dx.doi.org/10.1142/s0218348x95000436.

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The Kuramoto-Sivishinsky model describes the dynamics of a cellular flame front. It has been known for some time that on scales large compared with the size of a cell the front appears to be a self-affine fractal which has noisy dynamics in 1+1 dimensions. We use the inverse method of Lam and Sander (Phys. Rev. Lett.71, 561 (1993)) to show explicitly how the scaling occurs and how deterministic chaos at small scales develops into noisy dynamics at large scales, and how a small scale pattern becomes a large scale disordered fractal via an intermediate scaling regime.

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24

Kobayashi, M. U., and H. Fujisaka. "Dynamical Correlations for the Kuramoto-Sivashinsky Equation." Progress of Theoretical Physics 118, no. 6 (December 1, 2007): 1043–52. http://dx.doi.org/10.1143/ptp.118.1043.

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25

Bohr, Tomas, and Arcady Pikovsky. "Anomalous diffusion in the Kuramoto-Sivashinsky equation." Physical Review Letters 70, no. 19 (May 10, 1993): 2892–95. http://dx.doi.org/10.1103/physrevlett.70.2892.

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26

Kobayashi, Toshihiro. "Adaptive stabilization of the Kuramoto-Sivashinsky equation." International Journal of Systems Science 33, no. 3 (January 2002): 175–80. http://dx.doi.org/10.1080/00207720110092171.

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27

Zhang, Yindi, Lingyu Song, and Wang Axia. "Dynamical bifurcation for the Kuramoto–Sivashinsky equation." Nonlinear Analysis: Theory, Methods & Applications 74, no. 4 (February 2011): 1155–63. http://dx.doi.org/10.1016/j.na.2010.09.052.

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28

Armaou, Antonios, and Panagiotis D. Christofides. "Feedback control of the Kuramoto–Sivashinsky equation." Physica D: Nonlinear Phenomena 137, no. 1-2 (March 2000): 49–61. http://dx.doi.org/10.1016/s0167-2789(99)00175-x.

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29

Foias, Ciprian, and Igor Kukavica. "Determining nodes for the Kuramoto-Sivashinsky equation." Journal of Dynamics and Differential Equations 7, no. 2 (April 1995): 365–73. http://dx.doi.org/10.1007/bf02219361.

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30

Michelson, Daniel. "Steady solutions of the Kuramoto-Sivashinsky equation." Physica D: Nonlinear Phenomena 19, no. 1 (February 1986): 89–111. http://dx.doi.org/10.1016/0167-2789(86)90055-2.

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31

Blomgren, Peter, Scott Gasner, and Antonio Palacios. "Hopping behavior in the Kuramoto–Sivashinsky equation." Chaos: An Interdisciplinary Journal of Nonlinear Science 15, no. 1 (March 2005): 013706. http://dx.doi.org/10.1063/1.1848311.

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32

Robinson, James C. "Inertial manifolds for the Kuramoto-Sivashinsky equation." Physics Letters A 184, no. 2 (January 1994): 190–93. http://dx.doi.org/10.1016/0375-9601(94)90775-7.

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33

Changpin, Li, and Yang Zhonghua. "Bifurcation of two-dimensional Kuramoto-Sivashinsky equation." Applied Mathematics-A Journal of Chinese Universities 13, no. 3 (September 1998): 263–70. http://dx.doi.org/10.1007/s11766-998-0018-2.

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34

Otto, Felix. "Optimal bounds on the Kuramoto–Sivashinsky equation." Journal of Functional Analysis 257, no. 7 (October 2009): 2188–245. http://dx.doi.org/10.1016/j.jfa.2009.01.034.

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35

Duan, Jinqiao, and Vincent J. Ervin. "Dynamics of a Nonlocal Kuramoto–Sivashinsky Equation." Journal of Differential Equations 143, no. 2 (March 1998): 243–66. http://dx.doi.org/10.1006/jdeq.1997.3371.

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36

Giacomelli, Lorenzo, and Felix Otto. "New bounds for the Kuramoto-Sivashinsky equation." Communications on Pure and Applied Mathematics 58, no. 3 (2004): 297–318. http://dx.doi.org/10.1002/cpa.20031.

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37

Kilicman, Adem, and Rathinavel Silambarasan. "Modified Kudryashov Method to Solve Generalized Kuramoto-Sivashinsky Equation." Symmetry 10, no. 10 (October 21, 2018): 527. http://dx.doi.org/10.3390/sym10100527.

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The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudryashov method for the new exact solutions. The modified Kudryashov method converts the given nonlinear partial differential equation to algebraic equations, as a result of various steps, which upon solving the so-obtained equation systems yields the analytical solution. By this way, various exact solutions including complex structures are found, and their behavior is drawn in the 2D plane by Maple to compare the uniqueness and wave traveling of the solutions.

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38

Kalogirou, A., E. E. Keaveny, and D. T. Papageorgiou. "An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 471, no. 2179 (July 2015): 20140932. http://dx.doi.org/10.1098/rspa.2014.0932.

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The Kuramoto–Sivashinsky equation in one spatial dimension (1D KSE) is one of the most well-known and well-studied partial differential equations. It exhibits spatio-temporal chaos that emerges through various bifurcations as the domain length increases. There have been several notable analytical studies aimed at understanding how this property extends to the case of two spatial dimensions. In this study, we perform an extensive numerical study of the Kuramoto–Sivashinsky equation (2D KSE) to complement this analytical work. We explore in detail the statistics of chaotic solutions and classify the solutions that arise for domain sizes where the trivial solution is unstable and the long-time dynamics are completely two-dimensional. While we find that many of the features of the 1D KSE, including how the energy scales with system size, carry over to the 2D case, we also note several differences including the various paths to chaos that are not through period doubling.

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39

Yang, Desheng. "Kolmogorov equation associated to a stochastic Kuramoto–Sivashinsky equation." Journal of Functional Analysis 263, no. 4 (August 2012): 869–95. http://dx.doi.org/10.1016/j.jfa.2012.05.007.

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40

LI, CHANGPIN, and GUANRONG CHEN. "BIFURCATION ANALYSIS OF THE KURAMOTO–SIVASHINSKY EQUATION IN ONE SPATIAL DIMENSION." International Journal of Bifurcation and Chaos 11, no. 09 (September 2001): 2493–99. http://dx.doi.org/10.1142/s021812740100353x.

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In this Letter, we study the bifurcation of the Kuramoto–Sivashinsky (K–S) equation in one-spatial dimension with three kinds of boundary value conditions. Using the Liapunov–Schmidt reduction technique, the original equation is first reduced to one or two bifurcation equations, so that bifurcation analysis of the original equation can be transformed to that of the reduced-order systems, and can therefore be carried out in detail.

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41

Akgül, Ali, Esra Karatas Akgül, Sahin Korhan, and Mustafa Inc. "Reproducing kernel functions for the generalized Kuramoto-Sivashinsky equation." ITM Web of Conferences 22 (2018): 01028. http://dx.doi.org/10.1051/itmconf/20182201028.

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Reproducing kernel functions are obtained for the solution of generalized Kuramoto–Sivashinsky (GKS) equation in this paper. These reproducing kernel functions are valuable in the reproducing kernel Hilbert space method. They will be useful for interested researchers.

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42

Al-Khaled, Kamel, and Issam Abu-Irwaq. "Computational Sinc-scheme for extracting analytical solution for the model Kuramoto-Sivashinsky equation." International Journal of Electrical and Computer Engineering (IJECE) 9, no. 5 (October 1, 2019): 3720. http://dx.doi.org/10.11591/ijece.v9i5.pp3720-3731.

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The present article is designed to supply two different numerical<br />solutions for solving Kuramoto-Sivashinsky equation. We have made<br />an attempt to develop a numerical solution via the use of<br />Sinc-Galerkin method for Kuramoto-Sivashinsky equation, Sinc<br />approximations to both derivatives and indefinite integrals reduce<br />the solution to an explicit system of algebraic equations. The fixed<br />point theory is used to prove the convergence of the proposed<br />methods. For comparison purposes, a combination of a Crank-Nicolson<br />formula in the time direction, with the Sinc-collocation in the<br />space direction is presented, where the derivatives in the space<br />variable are replaced by the necessary matrices to produce a system<br />of algebraic equations. In addition, we present numerical examples<br />and comparisons to support the validity of these proposed<br />methods.

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43

OKAMOTO, Yuriko, Hidesbi ISHIDA, and Genta KAWAHARA. "307 Modeling of Kuramoto-Sivashinsky Equation Using Stochastic Differential Equations." Proceedings of Conference of Kansai Branch 2011.86 (2011): _3–7_. http://dx.doi.org/10.1299/jsmekansai.2011.86._3-7_.

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44

Zgliczynski, Piotr, and Konstantin Mischaikow. "Rigorous Numerics for Partial Differential Equations: The Kuramoto—Sivashinsky Equation." Foundations of Computational Mathematics 1, no. 3 (July 2001): 255–88. http://dx.doi.org/10.1007/s002080010010.

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45

Varlamov, Vladimir. "On the Kuramoto-Sivashinsky equation in a disk." Annales Polonici Mathematici 73, no. 3 (2000): 227–56. http://dx.doi.org/10.4064/ap-73-3-227-256.

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46

Akrivis, Georgios. "Finite element discretization of the Kuramoto-Sivashinsky equation." Banach Center Publications 29, no. 1 (1994): 155–63. http://dx.doi.org/10.4064/-29-1-155-163.

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47

Gotoda, Hiroshi, Marc Pradas, and Serafim Kalliadasis. "Nonlinear Forecasting of the Generalized Kuramoto–Sivashinsky Equation." International Journal of Bifurcation and Chaos 25, no. 05 (May 2015): 1530015. http://dx.doi.org/10.1142/s0218127415300153.

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The emergence of pattern formation and chaotic dynamics is studied in the one-dimensional (1D) generalized Kuramoto–Sivashinsky (gKS) equation by means of a time-series analysis, in particular, a nonlinear forecasting method which is based on concepts from chaos theory and appropriate statistical methods. We analyze two types of temporal signals, a local one and a global one, finding in both cases that the dynamical state of the gKS solution undergoes a transition from high-dimensional chaos to periodic pulsed oscillations through low-dimensional deterministic chaos while increasing the control parameter of the system. Our results demonstrate that the proposed nonlinear forecasting methodology allows to elucidate the dynamics of the system in terms of its predictability properties.

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48

Ganaie, Ishfaq Ahmad, Shelly Arora, and V. K. Kukreja. "Cubic Hermite collocation solution of Kuramoto–Sivashinsky equation." International Journal of Computer Mathematics 93, no. 1 (January 13, 2015): 223–35. http://dx.doi.org/10.1080/00207160.2014.999052.

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49

Iosevich, Alex, and Judith R. Miller. "DISPERSIVE EFFECTS IN A MODIFIED KURAMOTO–SIVASHINSKY EQUATION." Communications in Partial Differential Equations 27, no. 11-12 (December 31, 2002): 2423–48. http://dx.doi.org/10.1081/pde-120016163.

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50

Hayot, F., C. Jayaprakash, and Ch Josserand. "Long-wavelength properties of the Kuramoto-Sivashinsky equation." Physical Review E 47, no. 2 (February 1, 1993): 911–15. http://dx.doi.org/10.1103/physreve.47.911.

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