Relevant bibliographies by topics / Kuramoto-Sivashinsky equation

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Academic literature on the topic 'Kuramoto-Sivashinsky equation'

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Published: 4 June 2021
Last updated: 7 September 2023

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Journal articles on the topic "Kuramoto-Sivashinsky equation"

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Kuramoto-Sivashinsky equation"

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Lu, Fei, Kevin K. Lin, and Alexandre J. Chorin. "Data-based stochastic model reduction for the Kuramoto–Sivashinsky equation." ELSEVIER SCI LTD, 2017. http://hdl.handle.net/10150/622792.

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The problem of constructing data-based, predictive, reduced models for the Kuramoto–Sivashinsky equation is considered, under circumstances where one has observation data only for a small subset of the dynamical variables. Accurate prediction is achieved by developing a discrete-time stochastic reduced system, based on a NARMAX (Nonlinear Autoregressive Moving Average with eXogenous input) representation. The practical issue, with the NARMAX representation as with any other, is to identify an efficient structure, i.e., one with a small number of terms and coefficients. This is accomplished here by estimating coefficients for an approximate inertial form. The broader significance of the results is discussed.
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Kent, Philip. "Bifurcations of the travelling-wave solutions of the Kuramoto-Sivashinsky equation." Thesis, Imperial College London, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.515563.

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Falcon, Michael Andrew. "Approximation of the attractor and the inertial manifold of the Kuramoto-Sivashinsky equation." Thesis, University of Bath, 1998. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.268211.

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Roitner, Heinz Helmut. "Applications of the inverse spectral transform to a Korteweg-de Vries equation with a Kuramoto-Sivashinsky-type perturbation." Diss., The University of Arizona, 1991. http://hdl.handle.net/10150/185572.

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In this dissertation, the initial-boundary value problem u(t) - uuₓ + δ²uₓₓₓ + uₓₓ + β²uₓₓₓₓ = 0. u(x + 1) = u(x); u(x,0) = u(I)(x) is studied analytically and numerically. This partial differential equation is a hybrid between the well-known Korteweg-deVries and Kuramoto-Sivashinsky equations. It is shown numerically that this problem has for strong dispersion (δ² ≫ 1) travelling wave attractors which can be constructed as perturbations of cnoidal waves of the Korteweg-deVries equation. The perturbation theory is extended to the spectral structure and the linear stability of these travelling waves. The linear stability theory makes use of the squared eigenfunction basis related to the spectral theory of the Korteweg-deVries equation. This yields better estimates of the linear stability than those previously known. This seems to be the first use of the squared eigenfunction basis in the study of dissipative perturbations of the Korteweg-deVries equation. Next, the equations of motion for the action and angle variables of the KdV-equation are written down for the perturbed flow and the transient and attracting phases of the dynamics of the initial-boundary value problem are interpreted with these equations. A numerical study of the dynamics of these 'spectral coordinates' exhibits a series of interesting phenomena. In certain parameter regions a mode reduction is considered and a perturbation theory of the action and angle variables is applied to the truncated system. Finally, the effects of an additional uniform damping term νu in the initial-boundary value problem are discussed. We also compiled various ideas and concepts for an analytical proof of the existence of travelling wave attractors for strong dispersion. They might serve as guidelines for the actual proof which is still missing. A theoretical appendix presents some proofs and calculations to complement the main text and a numerical appendix describes the computational setup in the numerical study of the initial-boundary value problem.
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Rodrigues, Eduardo Vitral Freigedo. "Formação de nanopadrões em superfícies por sputtering iônico: Estudo numérico da equação anisotrópica amortecida de Kuramoto-Sivashinsky." Universidade do Estado do Rio de Janeiro, 2015. http://www.bdtd.uerj.br/tde_busca/arquivo.php?codArquivo=9215.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Apresenta-se uma abordagemnumérica para ummodelo que descreve a formação de padrões por sputtering iônico na superfície de ummaterial. Esse processo é responsável pela formação de padrões inesperadamente organizados, como ondulações, nanopontos e filas hexagonais de nanoburacos. Uma análise numérica de padrões preexistentes é proposta para investigar a dinâmica na superfície, baseada em ummodelo resumido em uma equação anisotrópica amortecida de Kuramoto-Sivashinsky, em uma superfície bidimensional com condições de contorno periódicas. Apesar de determinística, seu caráter altamente não-linear fornece uma rica gama de resultados, sendo possível descrever acuradamente diferentes padrões. Umesquema semi implícito de diferenças finitas com fatoração no tempo é aplicado na discretização da equação governante. Simulações foram realizadas com coeficientes realísticos relacionados aos parâmetros físicos (anisotropias, orientação do feixe, difusão). A estabilidade do esquema numérico foi analisada por testes de passo de tempo e espaçamento de malha, enquanto a verificação do mesmo foi realizada pelo Método das Soluções Manufaturadas. Ondulações e padrões hexagonais foram obtidos a partir de condições iniciais monomodais para determinados valores do coeficiente de amortecimento, enquanto caos espaço-temporal apareceu para valores inferiores. Os efeitos anisotrópicos na formação de padrões foramestudados, variando o ângulo de incidência.
A numerical approach is presented for amodel describing the pattern formation by ion beam sputtering on a material surface. This process is responsible for the appearance of unexpectedly organized patterns, such as ripples, nanodots, and hexagonal arrays of nanoholes. A numerical analysis of preexisting patterns is proposed to investigate surface dynamics, based on a model resumed in an anisotropic damped Kuramoto-Sivashinsky equation, in a two dimensional surface with periodic boundary conditions. While deterministic, its highly nonlinear character gives a rich range of results, making it possible to describe accurately different patterns. A finite-difference semi-implicit time splitting scheme is employed on the discretization of the governing equation. Simulations were conducted with realistic coefficients related to physical parameters (anisotropies, beam orientation, diffusion). The stability of the numerical scheme is analyzed with time step and grid spacing tests for the pattern evolution, and the Method ofManufactured Solutions has been used to verify the scheme. Ripples and hexagonal patterns were obtained from amonomodal initial condition for certain values of the damping coefficient, while spatiotemporal chaos appeared for lower values. The anisotropy effects on pattern formation were studied, varying the angle of incidence.
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MacKenzie, Tony. "Create accurate numerical models of complex spatio-temporal dynamical systems with holistic discretisation." University of Southern Queensland, Faculty of Sciences, 2005. http://eprints.usq.edu.au/archive/00001466/.

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This dissertation focuses on the further development of creating accurate numerical models of complex dynamical systems using the holistic discretisation technique [Roberts, Appl. Num. Model., 37:371-396, 2001]. I extend the application from second to fourth order systems and from only one spatial dimension in all previous work to two dimensions (2D). We see that the holistic technique provides useful and accurate numerical discretisations on coarse grids. We explore techniques to model the evolution of spatial patterns governed by pdes such as the Kuramoto-Sivashinsky equation and the real-valued Ginzburg-Landau equation. We aim towards the simulation of fluid flow and convection in three spatial dimensions. I show that significant steps have been taken in this dissertation towards achieving this aim. Holistic discretisation is based upon centre manifold theory [Carr, Applications of centre manifold theory, 1981] so we are assured that the numerical discretisation accurately models the dynamical system and may be constructed systematically. To apply centre manifold theory the domain is divided into elements and using a homotopy in the coupling parameter, subgrid scale fields are constructed consisting of actual solutions of the governing partial differential equation(pde). These subgrid scale fields interact through the introduction of artificial internal boundary conditions. View the centre manifold (macroscale) as the union of all states of the collection of subgrid fields (microscale) over the physical domain. Here we explore how to extend holistic discretisation to the fourth order Kuramoto-Sivashinsky pde. I show that the holistic models give impressive accuracy for reproducing the steady states and time dependent phenomena of the Kuramoto-Sivashinsky equation on coarse grids. The holistic method based on local dynamics compares favourably to the global methods of approximate inertial manifolds. The excellent performance of the holistic models shown here is strong evidence in support of the holistic discretisation technique. For shear dispersion in a 2D channel a one-dimensional numerical approximation is generated directly from the two-dimensional advection-diffusion dynamics. We find that a low order holistic model contains the shear dispersion term of the Taylor model [Taylor, IMA J. Appl. Math., 225:473-477, 1954]. This new approach does not require the assumption of large x scales, formerly absolutely crucial in deriving the Taylor model. I develop holistic discretisation for two spatial dimensions by applying the technique to the real-valued Ginzburg-Landau equation as a representative example of second order pdes. The techniques will apply quite generally to second order reaction-diffusion equations in 2D. This is the first study implementing holistic discretisation in more than one spatial dimension. The previous applications of holistic discretisation have developed algebraic forms of the subgrid field and its evolution. I develop an algorithm for numerical construction of the subgrid field and its evolution for 1D and 2D pdes and explore various alternatives. This new development greatly extends the class of problems that may be discretised by the holistic technique. This is a vital step for the application of the holistic technique to higher spatial dimensions and towards discretising the Navier-Stokes equations.
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Belova, Anna. "Computational dynamics – real and complex." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-332280.

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The PhD thesis considers four topics in dynamical systems and is based on one paper and three manuscripts. In Paper I we apply methods of interval analysis in order to compute the rigorous enclosure of rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points which is used to check rationality of the rotation number. In Manuscript II we provide a numerical algorithm for computing critical points of the multiplier map for the quadratic family (i.e., points where the derivative of the multiplier with respect to the complex parameter vanishes). Manuscript III concerns continued fractions of quadratic irrationals. We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. As a corollary we can compute the Lévy constant of any quadratic irrational explicitly in terms of its partial quotients. Finally, in Manuscript IV we develop a method for finding rigorous enclosures of all odd periodic solutions of the stationary Kuramoto-Sivashinsky equation. The problem is reduced to a bounded, finite-dimensional constraint satisfaction problem whose solution gives the desired information about the original problem. Developed approach allows us to exclude the regions in L2, where no solution can exist.
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Al, Jamal Rasha. "Bounded Control of the Kuramoto-Sivashinsky equation." Thesis, 2013. http://hdl.handle.net/10012/8014.

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Feedback control is used in almost every aspect of modern life and is essential in almost all engineering systems. Since no mathematical model is perfect and disturbances occur frequently, feedback is required. The design of a feedback control has been widely investigated in finite-dimensional space. However, many systems of interest, such as fluid flow and large structural vibrations are described by nonlinear partial differential equations and their state evolves on an infinite-dimensional Hilbert space. Developing controller design methods for nonlinear infinite-dimensional systems is not trivial. The objectives of this thesis are divided into multiple tasks. First, the well-posedness of some classes of nonlinear partial differential equations defined on a Hilbert space are investigated. The following nonlinear affine system defined on the Hilbert space H is considered z ̇(t)=F(z(t))+Bu(t), t≥0 z (0) = z0, where z(t) ∈ H is the state vector and z0 is the initial condition. The vector u(t) ∈ U, where U is a Hilbert space, is a state-feedback control. The nonlinear operator F : D ⊂ H → H is densely defined in H and the linear operator B : U → H is a linear bounded operator. Conditions for the closed-loop system to have a unique solution in the Hilbert space H are given. Next, finding a single bounded state-feedback control for nonlinear partial differential equations is discussed. In particular, Lyapunov-indirect method is considered to control nonlinear infinite-dimensional systems and conditions on when this method achieves the goal of local asymptotic stabilization of the nonlinear infinite-dimensional system are given. The Kuramoto-Sivashinsky (KS) equation defined in the Hilbert space L2(−π,π) with periodic boundary conditions is considered. ∂z/∂t =−ν∂4z/∂x4 −∂2z/∂x2 −z∂z/∂x, t≥0 z (0) = z0 (x) , where the instability parameter ν > 0. The KS equation is a nonlinear partial differential equation that is first-order in time and fourth-order in space. It models reaction-diffusion systems and is related to various pattern formation phenomena where turbulence or chaos appear. For instance, it models long wave motions of a liquid film over a vertical plane. When the instability parameter ν < 1, this equation becomes unstable. This is shown by analyzing the stability of the linearized system and showing that the nonlinear C0- semigroup corresponding to the nonlinear KS equation is Fr ́echet differentiable. There are a number of papers establishing the stabilization of this equation via boundary control. In this thesis, we consider distributed control with a single bounded feedback control for the KS equation with periodic boundary conditions. First, it is shown that sta- bilizing the linearized KS equation implies local asymptotical stability of the nonlinear KS equation. This is done by establishing Fr ́echet differentiability of the associated nonlinear C0-semigroup and showing that it is equal to the linear C0-semigroup generated by the linearization of the equation. Next, a single state-feedback control that locally asymptot- ically stabilizes the KS equation is constructed. The same approach to stabilize the KS equation from one equilibrium point to another is used. Finally, the solution of the uncontrolled/state-feedback controlled KS equation is ap- proximated numerically. This is done using the Galerkin projection method to approximate infinite-dimensional systems. The numerical simulations indicate that the proposed Lyapunov-indirect method works in stabilizing the KS equation to a desired state. Moreover, the same approach can be used to stabilize the KS equation from one constant equilibrium state to another.
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Gambill, Thomas Naylor. "Application of uncertainty inequalities to bound the radius of the attractor for the Kuramoto-Sivashinsky equation /." 2006. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3250245.

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Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2006.
Source: Dissertation Abstracts International, Volume: 68-02, Section: B, page: 1010. Adviser: Jared Bronski. Includes bibliographical references (leaves 122-125) Available on microfilm from Pro Quest Information and Learning.
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Roy, Dipankar. "Steady state properties of discrete and continuous models of nonequilibrium phenomena." Thesis, 2020. https://etd.iisc.ac.in/handle/2005/4880.

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The understanding of nonequilibrium phenomena, of fundamental importance in statistical physics, has great implications for many physical, chemical, and biological systems. Such phenomena are observed almost everywhere in the natural world. These phenomena are characterized by complicated spatiotemporal evolution. To explore nonequilibrium phenomena we often study simple model systems that embody their essential characteristics. In this thesis, we report the results of our investigations of the statistically steady state properties of three one-dimensional models: multispecies asymmetric simple exclusion processes, the Kuramoto- Sivashinsky equation, and the Burgers equation. The thesis is divided into two parts: Part I and Part II. In Chapters 2–5 of Part I, we present our results for multispecies exclusion models, principally the phase diagrams and statistical properties of their nonequilibrium steady state (NESS). We list below abstracts of these chapters. • In Chapter 2, we consider a multispecies ASEP (mASEP) on a one-dimensional lattice with semipermeable boundaries in contact with particle reservoirs. The mASEP involves ¹2𝑟 ¸1º species of particles: 𝑟 species of positive charges and their negative counterparts as well as vacancies. At the boundaries, a species can replace or be replaced by its negative counterpart. We derive the exact nonequilibrium phase diagram for the system in the long time limit. We find two new phenomena in certain regions of the phase diagram: dynamical expulsion when the density of a species becomes zero throughout the system, and dynamical localization when the density of a species is nonzero only within an interval far from the boundaries. We give a complete explanation of the macroscopic features of the phase diagram using what we call nested fat shocks. • In Chapter 3, we study an asymmetric exclusion process with two species and vacancies on an open one-dimensional lattice called the left-permeable ASEP (LPASEP). The left boundary is permeable for the vacancies but the right boundary is not. We find a matrix product solution for the stationary state and the exact stationary phase diagram for the densities and currents. By calculating the density of each species at the boundaries, we find further structure in the stationary phases. In particular, we find that the slower species can reach and accumulate at the far boundary, even in phases where the bulk density of these particles approaches zero. • In Chapter 4, we study a multispecies generalization of the model in Chapter 3. We determine all phases in the phase diagram using an exact projection to the LPASEP solved earlier. In most phases, we observe the phenomenon of dynamical expulsion of one or more species. We explain the density profiles in each phase using interacting shocks. This explanation is corroborated by simulations. • In Chapter 5, we investigate a multispecies generalization of the single-species asymmetric simple exclusion process defined on an open one-dimensional, finite lattice connected to particle reservoirs. At the boundaries, a species can be replaced with any other species. We devise an exact projection scheme to find the phase diagram in terms of densities and currents of all species. In most of the phases, one or more species are absent in the system due to dynamical expulsion. We observe shocks as well in some regions of the phase diagram. We explain the density profiles using a generalized shock structure that is substantiated by numerical simulations. In Chapters 7 and 8 of Part II, we study the statistical properties of turbulent, but statistically steady, states of the Kuramoto-Sivashinsky and the Burgers equations in one dimension. Our main results are summarized below. • In Chapter 7, we investigate the long time and large system size properties of the onedimensional Kuramoto-Sivashinsky equation. Tracy-Widom and Baik-Rains distributions appear as universal limit distributions for height fluctuations in the one-dimensional Kardar-Parisi-Zhang (KPZ) stochastic partial differential equation (PDE). We obtain the same universal distributions in the spatiotemporally chaotic, nonequilibrium, but statistically steady state of KS deterministic PDE, by carrying out extensive pseudospectral direct numerical simulations to obtain the spatiotemporal evolution of the KS height profile h(x,t) for different initial conditions. We establish, therefore, that the statistical properties of the one-dimensional (1D) KS PDE in this state are in the 1D KPZ universality class. • In Chapter 8, we study the statistical properties of decaying turbulence in the onedimensional Burgers equation, in the vanishing-viscosity limit; we start with random initial conditions, whose energy spectra have simple functional dependences on the wavenumber k: E_0(k) = A \mathcal{E}(k) exp[ - 2 k^2 / k^2_c ] , where A is a positive real number, and k_c is a cutoff wavenumber. The simplest case is the single-power law \mathcal{E}(k) = k^{n}. We focus here on the case of the Gaussian laws which are characterized by E_0(k) = exp[ - 2 (k-k_c)^2 / k^2_c +2 k^2 / k^2_c]; in addition, we consider initial spectra which are combinations of either two or four single-power law spectral regions. For all these initial conditions, we systematize (a) the temporal decay of the total energy, (b) the rich temporal evolution of the energy spectrum, and (c) the spatiotemporal evolution of the velocity field. We present our results in the context of earlier studies of this problem.
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Books on the topic "Kuramoto-Sivashinsky equation"

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Papageorgiou, Demetrios T. The route to chaos for the Kuramoto-Sivashinsky equation. Hampton, Va: Institute for Computer Applications in Science and Engineering, 1990.

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Papageorgiou, Demetrios T. Modulational stability of periodic solutions of the Kuramoto-Sivashinsky equation. Hampton, Va: Institute for Computer Applications in Science and Engineering, 1993.

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Smyrlis, Yiorgos S. Computational study of chaotic and ordered solutions of the Kuramoto-Sivashinsky equation. Hampton, Va: Langley Research Center, 1996.

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Papageorgiou, Demetris. The route to chaos for the Kuramoto-Sivashinsky equation. Hampton, Va: NASA Langley Research Center, 1990.

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5

T, Papageorgiou Demetrios, and Langley Research Center, eds. Predicting chaos for infinite dimensional dynamical systems: The Kuramoto-Sivashinsky equation, a case study. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1991.

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6

T, Papageorgiou Demetrios, Smyrlis Yiorgos S, and Institute for Computer Applications in Science and Engineering., eds. Nonlinear stability of oscillatory core-annular flow: A generalized Kuramoto-Sivashinsky equation with time periodic coefficients. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1994.

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Nonlinear stability of oscillatory core-annular flow: A generalized Kuramoto-Sivashinsky equation with time periodic coefficients. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1994.

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Book chapters on the topic "Kuramoto-Sivashinsky equation"

1

Constantin, P., C. Foias, B. Nicolaenko, and R. Teman. "Application: The Kuramoto—Sivashinsky Equation." In Applied Mathematical Sciences, 72–81. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3506-4_16.

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2

Manneville, P. "The Kuramoto-Sivashinsky Equation: A Progress Report." In Springer Series in Synergetics, 265–80. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/978-3-642-73861-6_24.

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Michelson, Daniel. "Order and Disorder in the Kuramoto-Sivashinsky Equation." In Progress and Supercomputing in Computational Fluid Dynamics, 331–44. Boston, MA: Birkhäuser Boston, 1985. http://dx.doi.org/10.1007/978-1-4612-5162-0_17.

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Alfaro, C. M., R. D. Benguria, and M. C. Depassier. "The Role Of Dispersion In The Generalized Kuramoto Sivashinsky Equation." In Instabilities and Nonequilibrium Structures IV, 281–87. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-1906-1_27.

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Papageorgiou, Demetrios T., George C. Papanicolaou, and Yiorgos S. Smyrlis. "Modulational stability of periodic solutions of the Kuramoto-Sivashinsky equation." In Singularities in Fluids, Plasmas and Optics, 255–63. Dordrecht: Springer Netherlands, 1993. http://dx.doi.org/10.1007/978-94-011-2022-7_19.

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Shibata, Hiroshi. "Lyapunov Exponent of the System Described by Kuramoto-Sivashinsky Equation." In Statistical Theories and Computational Approaches to Turbulence, 269–73. Tokyo: Springer Japan, 2003. http://dx.doi.org/10.1007/978-4-431-67002-5_19.

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Nicolaenko, Basil. "The Kuramoto-Sivashinsky Equation: Spatio-Temporal Chaos and Intermittencies for a Dynamical System." In NATO ASI Series, 1029–52. Boston, MA: Springer US, 1988. http://dx.doi.org/10.1007/978-1-4613-0707-5_70.

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Elezgaray, J., G. Berkooz, and P. Holmes. "Modelling the coupling between small and large scales in the Kuramoto-Sivashinsky equation." In CRM Proceedings and Lecture Notes, 293–302. Providence, Rhode Island: American Mathematical Society, 1999. http://dx.doi.org/10.1090/crmp/018/23.

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Kaur, Deepti, and R. K. Mohanty. "A Higher Order Finite Difference Method for Numerical Solution of the Kuramoto–Sivashinsky Equation." In Springer Proceedings in Mathematics & Statistics, 217–29. Singapore: Springer Singapore, 2020. http://dx.doi.org/10.1007/978-981-15-5455-1_18.

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Ueno, Kazuto. "Application of the Renormalization Group Analysis to a Noisy Kuramoto–Sivashinsky Equation and its Numerical Simulation." In Frontiers of Computational Science, 231–34. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007. http://dx.doi.org/10.1007/978-3-540-46375-7_31.

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Conference papers on the topic "Kuramoto-Sivashinsky equation"

1

Dubljevic, Stevan. "Optimal boundary control of Kuramoto-Sivashinsky equation." In 2009 American Control Conference. IEEE, 2009. http://dx.doi.org/10.1109/acc.2009.5160231.

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Elder, K. R., Hao-wen Xi, Matt Deans, and J. D. Gunton. "Spatiotemporal chaos in the damped Kuramoto-Sivashinsky equation." In CAM-94 Physics meeting. AIP, 1995. http://dx.doi.org/10.1063/1.48763.

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Xie, Junyao, and Stevan Dubljevic. "Discrete Kalman Filter Design for Kuramoto-Sivashinsky Equation." In 2019 American Control Conference (ACC). IEEE, 2019. http://dx.doi.org/10.23919/acc.2019.8814595.

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al Jamal, Rasha, and Kirsten Morris. "Output feedback control of the Kuramoto-Sivashinsky equation." In 2015 54th IEEE Conference on Decision and Control (CDC). IEEE, 2015. http://dx.doi.org/10.1109/cdc.2015.7402289.

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Byrnes, C. I., D. S. Gilliam, and C. Hu. "Set-point boundary control for a Kuramoto-Sivashinsky equation." In Proceedings of the 45th IEEE Conference on Decision and Control. IEEE, 2006. http://dx.doi.org/10.1109/cdc.2006.377117.

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Qimin Zhang. "Exponential stability of solution for the Kuramoto-Sivashinsky equation." In 2008 7th World Congress on Intelligent Control and Automation. IEEE, 2008. http://dx.doi.org/10.1109/wcica.2008.4594318.

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Li, Cai, and Xie Wenxian. "Efficient lattice Boltzmann method for specialized Kuramoto-Sivashinsky equation." In TENCON 2013 - 2013 IEEE Region 10 Conference. IEEE, 2013. http://dx.doi.org/10.1109/tencon.2013.6718850.

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Liu, Jian-Guo, Zhi-Fang Zeng, and Qing Ye. "New exact solutions for the generalized Kuramoto-Sivashinsky equation." In 2018 Chinese Control And Decision Conference (CCDC). IEEE, 2018. http://dx.doi.org/10.1109/ccdc.2018.8407535.

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al Jamal, Rasha, and Kirsten Morris. "Distributed control of the Kuramoto-Sivashinsky equation using approximations." In 2015 American Control Conference (ACC). IEEE, 2015. http://dx.doi.org/10.1109/acc.2015.7171845.

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Bruzón, M. S., and M. L. Gandarias. "Conservation laws for a Kuramoto-Sivashinsky equation with dispersive effects." In NONLINEAR AND MODERN MATHEMATICAL PHYSICS: Proceedings of the 2nd International Workshop. AIP, 2013. http://dx.doi.org/10.1063/1.4828678.

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