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Saxena S, Kosterlitz JM. Dynamics of noise-induced wave-number selection in the stabilized Kuramoto-Sivashinsky equation. Phys Rev E 2021; 103:012205. [PMID: 33601618 DOI: 10.1103/physreve.103.012205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/24/2020] [Accepted: 12/08/2020] [Indexed: 06/12/2023]
Abstract
We revisit the question of wave-number selection in pattern-forming systems by studying the one-dimensional stabilized Kuramoto-Sivashinsky equation with additive noise. In earlier work, we found that a particular periodic state is more probable than all others at very long times, establishing the critical role of noise in the selection process. However, the detailed mechanism by which the noise picked out the selected wave number was not understood. Here, we address this issue by analyzing the noise-averaged time evolution of each unstable mode from the spatially homogeneous state, with and without noise. We find drastic differences between the nonlinear dynamics in the two cases. In particular, we find that noise opposes the growth of Eckhaus modes close to the critical wave number and boosts the growth of Eckhaus modes with wave numbers smaller than the critical wave number. We then hypothesize that the main factor responsible for this behavior is the excitation of long-wavelength (q→0) modes by the noise. This hypothesis is confirmed by extensive numerical simulations. We also examine the significance of the magnitude of the noise.
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Affiliation(s)
- S Saxena
- Department of Physics, Brown University, Providence, Rhode Island 02912, USA
| | - J M Kosterlitz
- Department of Physics, Brown University, Providence, Rhode Island 02912, USA
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Global potential, topology, and pattern selection in a noisy stabilized Kuramoto-Sivashinsky equation. Proc Natl Acad Sci U S A 2020; 117:23227-23234. [PMID: 32917812 DOI: 10.1073/pnas.2012364117] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
We formulate a general method to extend the decomposition of stochastic dynamics developed by Ao et al. [J. Phys. Math. Gen. 37, L25-L30 (2004)] to nonlinear partial differential equations which are nonvariational in nature and construct the global potential or Lyapunov functional for a noisy stabilized Kuramoto-Sivashinsky equation. For values of the control parameter where singly periodic stationary solutions exist, we find a topological network of a web of saddle points of stationary states interconnected by unstable eigenmodes flowing between them. With this topology, a global landscape of the steady states is found. We show how to predict the noise-selected pattern which agrees with those from stochastic simulations. Our formalism and the topology might offer an approach to explore similar systems, such as the Navier Stokes equation.
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Saxena S, Kosterlitz JM. Wave-number selection in pattern-forming systems. Phys Rev E 2019; 100:022223. [PMID: 31574763 DOI: 10.1103/physreve.100.022223] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2019] [Indexed: 06/10/2023]
Abstract
Wave-number selection in pattern-forming systems remains a long-standing puzzle in physics. Previous studies have shown that external noise is a possible mechanism for wave-number selection. We conduct an extensive numerical study of the noisy stabilized Kuramoto-Sivashinsky equation. We use a fast spectral method of integration, which enables us to investigate long-time behavior for large system sizes that could not be investigated by earlier work. We find that a state with a unique wave number has the highest probability of occurring at very long times. We also find that this state is independent of the strength of the noise and initial conditions, thus making a convincing case for the role of noise as a mechanism of state selection.
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Affiliation(s)
- S Saxena
- Department of Physics, Brown University, Providence, Rhode Island 02912, USA
| | - J M Kosterlitz
- Department of Physics, Brown University, Providence, Rhode Island 02912, USA
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Qiao L, Zheng Z, Cross MC. Minimum-action paths for wave-number selection in nonequilibrium systems. Phys Rev E 2016; 93:042204. [PMID: 27176290 DOI: 10.1103/physreve.93.042204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2014] [Indexed: 06/05/2023]
Abstract
The problem of wave-number selections in nonequilibrium pattern-forming systems in the presence of noise is investigated. The minimum-action method is proposed to study the noise-induced transitions between the different spatiotemporal states by generalizing the traditional theory previously applied in low-dimensional dynamical systems. The scheme is shown as an example in the stabilized Kuramoto-Sivashinsky equation. The present method allows us to conveniently find the unique noise selected state, in contrast to previous work using direct simulations of the stochastic partial differential equation, where the constraints of the simulation only allow a narrow band to be determined.
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Affiliation(s)
- Liyan Qiao
- Department of Physics, Hangzhou Dianzi University, Hangzhou 310018, China
| | - Zhigang Zheng
- College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China
| | - M C Cross
- Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
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Gomes SN, Pradas M, Kalliadasis S, Papageorgiou DT, Pavliotis GA. Controlling spatiotemporal chaos in active dissipative-dispersive nonlinear systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:022912. [PMID: 26382481 DOI: 10.1103/physreve.92.022912] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2014] [Indexed: 06/05/2023]
Abstract
We present an alternative methodology for the stabilization and control of infinite-dimensional dynamical systems exhibiting low-dimensional spatiotemporal chaos. We show that with an appropriate choice of time-dependent controls we are able to stabilize and/or control all stable or unstable solutions, including steady solutions, traveling waves (single and multipulse ones or bound states), and spatiotemporal chaos. We exemplify our methodology with the generalized Kuramoto-Sivashinsky equation, a paradigmatic model of spatiotemporal chaos, which is known to exhibit a rich spectrum of wave forms and wave transitions and a rich variety of spatiotemporal structures.
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Affiliation(s)
- S N Gomes
- Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
| | - M Pradas
- Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, United Kingdom
| | - S Kalliadasis
- Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, United Kingdom
| | - D T Papageorgiou
- Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
| | - G A Pavliotis
- Department of Mathematics, Imperial College London, London, SW7 2AZ, United Kingdom
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Elliott D, Vasquez DA. Convection in stable and unstable fronts. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:016207. [PMID: 22400643 DOI: 10.1103/physreve.85.016207] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/10/2011] [Revised: 10/20/2011] [Indexed: 05/31/2023]
Abstract
Density gradients across a reaction front can lead to convective fluid motion. Stable fronts require a heavier fluid on top of a lighter one to generate convective fluid motion. On the other hand, unstable fronts can be stabilized with an opposing density gradient, where the lighter fluid is on top. In this case, we can have a stable flat front without convection or a steady convective front of a given wavelength near the onset of convection. The fronts are described with the Kuramoto-Sivashinsky equation coupled to hydrodynamics governed by Darcy's law. We obtain a dispersion relation between growth rates and perturbation wave numbers in the presence of a density discontinuity accross the front. We also analyze the effects of this density change in the transition to chaos.
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Affiliation(s)
- Drew Elliott
- Department of Physics, Indiana University Purdue University Fort Wayne, Fort Wayne, Indiana 46805, USA
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Gomez H, París J. Numerical simulation of asymptotic states of the damped Kuramoto-Sivashinsky equation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 83:046702. [PMID: 21599329 DOI: 10.1103/physreve.83.046702] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/13/2010] [Revised: 02/21/2011] [Indexed: 05/30/2023]
Abstract
The damped Kuramoto-Sivashinsky equation has emerged as a fundamental tool for the understanding of the onset and evolution of secondary instabilities in a wide range of physical phenomena. Most existing studies about this equation deal with its asymptotic states on one-dimensional settings or on periodic square domains. We utilize a large-scale numerical simulation to investigate the asymptotic states of the damped Kuramoto-Sivashinsky equation on annular two-dimensional geometries and three-dimensional domains. To this end, we propose an accurate, efficient, and robust algorithm based on a recently introduced numerical methodology, namely, isogeometric analysis. We compared our two-dimensional results with several experiments of directed percolation on square and annular geometries, and found qualitative agreement.
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Affiliation(s)
- Hector Gomez
- University of A Coruña, Campus de Elviña s/n, 15071, A Coruña, Spain.
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Pradas M, Tseluiko D, Kalliadasis S, Papageorgiou DT, Pavliotis GA. Noise induced state transitions, intermittency, and universality in the noisy Kuramoto-Sivashinksy equation. PHYSICAL REVIEW LETTERS 2011; 106:060602. [PMID: 21405452 DOI: 10.1103/physrevlett.106.060602] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/05/2010] [Indexed: 05/30/2023]
Abstract
Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.
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Affiliation(s)
- M Pradas
- Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, United Kingdom
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