An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation

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.

[1]  P. Pelcé,et al.  Intrinsic stochasticity with many degrees of freedom , 1984 .

[2]  Jonathan Goodman,et al.  Stability of the kuramoto-sivashinsky and related systems† , 1994 .

[3]  J. Eckmann,et al.  A global attracting set for the Kuramoto-Sivashinsky equation , 1993 .

[4]  Eitan Tadmor,et al.  The well-posedness of the Kuramoto-Sivashinsky equation , 1986 .

[5]  Long Time Behavior for Radially Symmetric Solutions of the Kuramoto-Sivashinsky Equation , 2010 .

[6]  Philip Holmes,et al.  Scale and space localization in the Kuramoto-Sivashinsky equation. , 1999, Chaos.

[7]  Roger Temam,et al.  Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation , 1999 .

[8]  G. Sivashinsky,et al.  Nonlinear analysis of hydrodynamic instability in laminar flames—II. Numerical experiments , 1977 .

[10]  Spatial Analyticity on the Global Attractor for the Kuramoto–Sivashinsky Equation , 2000 .

[11]  Ines Gloeckner,et al.  Order Within Chaos Towards A Deterministic Approach To Turbulence , 2016 .

[12]  Michael Siegel,et al.  Effect of surfactants on the nonlinear interfacial stability of core-annular film flows. , 1990 .

[13]  Anna Kalogirou.  Nonlinear dynamics of surfactant-laden multilayer shear flows and related systems , 2014 .

[14]  G. Sell,et al.  Local dissipativity and attractors for the Kuramoto-Sivashinsky equation in thin 2D domains , 1992 .

[15]  S. Toh,et al.  Statistical Model with Localized Structures Describing the Spatio-Temporal Chaos of Kuramoto-Sivashinsky Equation , 1987 .

[16]  D. Michelson.  Radial asymptotically periodic solutions of the Kuramoto–Sivashinsky equation , 2008 .

[17]  K. Indireshkumar,et al.  Mutually penetrating motion of self-organized two-dimensional patterns of solitonlike structures , 1996, patt-sol/9606002.

[18]  J. París,et al.  Numerical simulation of asymptotic states of the damped Kuramoto-Sivashinsky equation. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[19]  G. Sivashinsky,et al.  On Irregular Wavy Flow of a Liquid Film Down a Vertical Plane , 1980 .

[20]  Jayaprakash,et al.  Universal properties of the two-dimensional Kuramoto-Sivashinsky equation. , 1993, Physical review letters.

[21]  Yiorgos Sokratis Smyrlis,et al.  Computational Study of the Dispersively Modified Kuramoto-Sivashinsky Equation , 2012, SIAM J. Sci. Comput..

[22]  Demetrios T. Papageorgiou,et al.  COMPUTATIONAL STUDY OF CHAOTIC AND ORDERED SOLUTIONS OF THE KURAMOTO-SIVASHINSKY EQUATION , 1996 .

[23]  Georgios Akrivis,et al.  Linearly implicit schemes for multi-dimensional Kuramoto–Sivashinsky type equations arising in falling film flows , 2015 .

[24]  Felix Otto,et al.  Optimal bounds on the Kuramoto–Sivashinsky equation , 2009 .

[25]  Stéphane Zaleski,et al.  Order and complexity in the Kuramoto-Sivashinsky model of weakly turbulent interfaces , 1986 .

[26]  Ju. S. Il'yashenko.  Global analysis of the phase portrait for the Kuramoto-Sivashinsky equation , 1992 .

[27]  Lorenzo Giacomelli,et al.  New bounds for the Kuramoto‐Sivashinsky equation , 2005 .

[28]  S. Kalliadasis,et al.  Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses , 2005 .

[29]  Milton S. Plesset,et al.  Viscous Effects in Rayleigh-Taylor Instability. , 1974 .

[30]  William Tang,et al.  Non-linear saturation of the dissipative trapped-ion mode by mode coupling , 1976 .

[31]  G. Sivashinsky.  On Flame Propagation Under Conditions of Stoichiometry , 1980 .

[32]  Yoshiki Kuramoto,et al.  On the Formation of Dissipative Structures in Reaction-Diffusion Systems Reductive Perturbation Approach , 1975 .

[33]  Uriel Frisch,et al.  Viscoelastic behaviour of cellular solutions to the Kuramoto-Sivashinsky model , 1986, Journal of Fluid Mechanics.

[34]  Toh,et al.  Two-dimensionally localized pulses of a nonlinear equation with dissipation and dispersion. , 1989, Physical review. A, General physics.

[35]  Burt S. Tilley,et al.  Dynamics and rupture of planar electrified liquid sheets , 2001 .

[36]  I. Kukavica.  Oscillations of solutions of the Kuramoto-Sivashinsky equation , 1994 .

[37]  S. Kalliadasis,et al.  Two-dimensional wave dynamics in thin films. II. Formation of lattices of interacting stationary solitary pulses , 2005 .

[38]  Krug,et al.  Anisotropic Kuramoto-Sivashinsky equation for surface growth and erosion. , 1995, Physical review letters.

[39]  William Tang,et al.  Nonlinear Saturation of the Trapped-Ion Mode , 1974 .

[40]  Barabási,et al.  Dynamic scaling of ion-sputtered surfaces. , 1995, Physical review letters.

[41]  Marco Paniconi,et al.  STATIONARY, DYNAMICAL, AND CHAOTIC STATES OF THE TWO-DIMENSIONAL DAMPED KURAMOTO-SIVASHINSKY EQUATION , 1997 .

[42]  A. Nepomnyashchiǐ.  Stability of wavy conditions in a film flowing down an inclined plane , 1974 .

[43]  Demetrios T. Papageorgiou,et al.  The route to chaos for the Kuramoto-Sivashinsky equation , 1990, Theoretical and Computational Fluid Dynamics.

[44]  Y. Kuramoto,et al.  Persistent Propagation of Concentration Waves in Dissipative Media Far from Thermal Equilibrium , 1976 .

[45]  Roger Temam,et al.  Some global dynamical properties of the Kuramoto-Sivashinsky equations: nonlinear stability and attr , 1985 .

[46]  S. Kas-Danouche.  Nonlinear interfacial stability of core-annular film flows in the presence of surfactants , 2002 .

[47]  Trivial stationary solutions to the Kuramoto–Sivashinsky and certain nonlinear elliptic equations , 2006, math/0603723.

[48]  A. P. Hooper,et al.  Nonlinear instability at the interface between two viscous fluids , 1985 .

[49]  Y S Smyrlis,et al.  Predicting chaos for infinite dimensional dynamical systems: the Kuramoto-Sivashinsky equation, a case study. , 1991, Proceedings of the National Academy of Sciences of the United States of America.

[50]  G. I. Siv Ashinsky,et al.  Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations , 1988 .

[51]  Y. Kuramoto,et al.  A Reduced Model Showing Chemical Turbulence , 1976 .

[52]  Pierre Collet,et al.  Analyticity for the Kuramoto-Sivashinsky equation , 1993 .

[53]  L. Molinet.  Local Dissipativity in L2 for the Kuramoto–Sivashinsky Equation in Spatial Dimension 2 , 2000 .