Modified Kuramoto-Sivashinsky Equation Research Articles

Predicting chaotic statistics with unstable invariant tori.

It has recently been speculated that long-time average quantities of hyperchaotic dissipative systems may be approximated by weighted sums over unstable invariant tori embedded in the attractor, analogous to equivalent sums over periodic orbits, which are inspired by the rigorous periodic orbit theory and which have shown much promise in fluid dynamics. Using a new numerical method for converging unstable invariant two-tori in a chaotic partial differential equation (PDE), and exploiting symmetry breaking of relative periodic orbits to detect those tori, we identify many quasiperiodic, unstable, invariant two-torus solutions of a modified Kuramoto-Sivashinsky equation. The set of tori covers significant parts of the chaotic attractor and weighted averages of the properties of the tori-with weights computed based on their respective stability eigenvalues-approximate average quantities for the chaotic dynamics. These results are a step toward exploiting higher-dimensional invariant sets to describe general hyperchaotic systems, including dissipative spatiotemporally chaotic PDEs.

Classifying algebraic invariants and algebraically invariant solutions

Abstract A concept of algebraic invariants and algebraically invariant solutions for autonomous ordinary differential equations and systems of autonomous ordinary differential equations is considered. A variety of known exact solutions of autonomous ordinary differential equations and a great number of traveling wave solutions of famous partial differential equations are in fact algebraically invariant solutions. A method, which can be used to find all irreducible algebraic invariants, is introduced. As an example, all irreducible algebraic invariants for the traveling wave reductions of the dispersive Kuramoto–Sivashinsky equation and the modified Kuramoto–Sivashinsky equation are classified. Novel solutions of the modified Kuramoto–Sivashinsky equation are obtained.

On the dynamics of traveling fronts arising in nanoscale pattern formation

We study the stability and dynamics of traveling-front solutions of a modified Kuramoto–Sivashinsky equation arising in the modeling of nanoscale ripple patterns that form when a nominally flat solid surface is bombarded with a broad ion beam at an oblique angle of incidence. Structurally, the linearized operators associated with these fronts have unstable essential spectrum – corresponding to instability of the spatially asymptotic states – and stable point spectrum – corresponding to stability of the transition profile of the front. We show that these waves are linearly orbitally asymptotically stable in appropriate exponentially weighted spaces. While the technical device of exponential weights allows us to accommodate the unstable essential spectrum of individual waves in our linear analysis, it does not shed light on the long-time pattern formation that is observed experimentally and in numerical simulations. To begin to address this issue, we consider a periodic array of unstable front and back solutions. While not an exact solution of the governing equation, this periodic pattern mimics experimentally observed phenomena. Our numerical experiments suggest that the convecting instabilities associated with each individual wave are damped as they pass through transition layers and that this stabilization mechanism underlies the pattern formation seen in experiments.

Longwave nonlinear theory for chemically active droplet division instability

It has been suggested recently that growth and division of a protocell could be modeled by a chemically active droplet with simple chemical reactions driven by an external fuel supply. This model is called the continuum model. Indeed its numerical simulation reveals a shape instability which results in droplet division into two smaller droplets of equal size resembling cell division (Zwicker et al., 2017). In this paper, we investigate the reduced version of the continuum model, which is called the effective model. This model is studied both in the linear and nonlinear regime. First, we perform a linear stability analysis for the flat interface, and then we develop a nonlinear theory using the longwave approach. We find that the interface at the leading order is governed by the modified Kuramoto–Sivashinsky equation. Therefore the interface is subject to a logarithmic blow up after a finite time. In addition, an expression for the interface local velocity is obtained.

Two scenarios of advective washing-out of localized convective patterns under frozen parametric disorder

The effect of spatial localization of states in distributed parameter systems under frozen parametric disorder is well known as the Anderson localization and thoroughly studied for the Schrödinger equation and linear dissipation-free wave equations. Some similar (or mimicking) phenomena can occur in dissipative systems such as the thermal convection ones. Specifically, many of these dissipative systems are governed by a modified Kuramoto–Sivashinsky equation, where the frozen spatial disorder of parameters has been reported to lead to excitation of localized patterns. Imposed advection in the modified Kuramoto–Sivashinsky equation can affect the localized patterns in a non-trivial way; it changes the localization properties and suppresses the pattern. The latter effect is considered in this paper by means of both numerical simulation and model reduction, which turns out to be useful for a comprehensive understanding of the bifurcation scenarios in the system. Two possible bifurcation scenarios of advective suppression (‘washing-out’) of localized patterns are revealed and characterized.

Exhaustive Classification of the Invariant Solutions for a Specific Nonlinear Model Describing Near Planar and Marginally Long-Wave Unstable Interfaces for Phase Transition

Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained. Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore, the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.

Spatially Inhomogeneous Solutions for a Modified Kuramoto–Sivashinsky Equation

We study the periodic boundary value problem for a modified Kuramoto–Sivashinsky equation which can serve as a mathematical model describing formation of nanorelief on the surface of a planar target under the action of ion flux. We show that, as in the case of the traditional Kuramoto–Sivashinsky equation, it is possible to obtain spatially inhomogeneous solutions under the condition that the homogeneous equilibrium states can change the stability. We consider local bifurcations. We find sufficient conditions for the existence of shortwave solutions. Bibliography: 11 titles.

Linearly implicit schemes for multi-dimensional Kuramoto–Sivashinsky type equations arising in falling film flows

This study introduces, analyses and implements space-time discretizations of two-dimensional active dissipative partial differential equations such as the Topper–Kawahara equation; this is the two-dimensional extension of the dispersively modified Kuramoto–Sivashinsky equation found in falling film hydro-dynamics. The spatially periodic initial value problem is considered as the size of the periodic box increases. The schemes utilized are implicit–explicit multistep (BDF) in time and spectral in space. Numerical analysis of these schemes is carried out and error estimates, in both time and space, are derived. Preliminary numerical experiments provided strong evidence of analyticity, thus yielding a practical rule-of-thumb that determines the size of the truncation in Fourier space. The accuracy of the BDF schemes (of order 1–6) is confirmed through computations. Extensive computations into the strongly chaotic regime (as the domain size increases), provided an optimal estimate of the size of the absorbing ball as a function of the size of the domain; this estimate is found to be proportional to the area of the periodic box. Numerical experiments were also carried out in the presence of dispersion. It is observed that sufficient amounts of dispersion reduce the complexity of the chaotic dynamics, and can organize solution into nonlinear travelling wave pulses of permanent form.

Fourth order evolution equations which describe pseudospherical surfaces

Differential equations that describe pseudospherical surfaces are considered. These equations are equivalent to the structure equations of a metric with Gaussian curvature K=−1. They can also be described as the compatibility condition of an associated linear problem also referred to as a zero curvature representation. A complete and explicit classification of a class of fourth order evolution equations is given. The classification provides four huge classes (referred to as Types I–IV) of fourth order evolution equations that describe pseudospherical surfaces, together with the associated one (or more) parameter linear problems. The differential equations of each type are determined by choosing certain arbitrary differentiable functions. Fourth-order member of the Burgers hierarchy and a modified Kuramoto–Sivashinsky equation are examples of equations described by Types I and IV, respectively. Many other explicit examples are presented.

Least squares shadowing sensitivity analysis of a modified Kuramoto–Sivashinsky equation

Computational methods for sensitivity analysis are invaluable tools for scientists and engineers investigating a wide range of physical phenomena. However, many of these methods fail when applied to chaotic systems, such as the Kuramoto–Sivashinsky (K–S) equation, which models a number of different chaotic systems found in nature. The following paper discusses the application of a new sensitivity analysis method developed by the authors to a modified K–S equation. We find that least squares shadowing sensitivity analysis computes accurate gradients for solutions corresponding to a wide range of system parameters.

Localization and advectional spreading of convective currents under parametric disorder

We address a problem which is mathematically reminiscent of that of Anderson localization, although it is related to a strongly dissipative dynamics. Specifically, we study thermal convection in a horizontal porous layer heated from below in the presence of a parametric disorder; physical parameters of the layer are time-independent and randomly inhomogeneous in one of the horizontal directions. Under such a frozen parametric disorder, spatially localized flow patterns appear. We focus our study on their localization properties and the effect on these properties of an imposed advection along the layer. Our interpretation of the results of the linear theory is underpinned by numerical simulation for the nonlinear problem. Weak advection is found to lead to an upstream delocalization of localized current patterns. Due to this delocalization, the transition from a set of localized patterns to a ‘global’ flow that is intense nearly everywhere can be observed, under conditions where the disorder-free system would be not far below the instability threshold. The results presented are derived for a physical system which is mathematically described by a modified Kuramoto–Sivashinsky equation and therefore they are expected to be relevant for a broad variety of dissipative media where pattern selection occurs.

Computational Study of the Dispersively Modified Kuramoto–Sivashinsky Equation

We analyze and implement fully discrete schemes for periodic initial value problems for a general class of dispersively modified Kuramoto-Sivashinsky equations. Time discretizations are constructed using linearly implicit schemes and spectral methods are used for the spatial discretization. The general case analyzed covers several physical applications arising in multiphase hydrodynamics and the emerging dynamics arise from a competition of long-wave instability (negative diffusion), short-wave damping (fourth order stabilization), nonlinear saturation (Burgers nonlinearity), and dispersive effects. The solutions of such systems typically converge to compact absorbing sets of finite dimension (i.e., global attractors) and are characterized by chaotic behavior. Our objective is to employ schemes which capture faithfully these chaotic dynamics. In the general case the dispersive term is taken to be a pseudodifferential operator which is allowed to have higher order than the familiar fourth order stabilizing term in the Kuramoto-Sivashinsky equation. In such instances we show that first and second order time-stepping schemes are appropriate and provide convergence proofs for the schemes. In physical situations when the dispersion is of lower order than the fourth order stabilization term (for example, a hybrid Kuramoto-Sivashinsky-Korteweg-deVries equation also known as the Kawahara equation in hydrodynamics), higher order time-stepping schemes can be used and we analyze and implement schemes of order six or less. We derive optimal order error estimates throughout and utilize the schemes to compute the long time dynamics and to characterize the attractors. Various numerical diagnostic tools are implemented, such as the projection of the infinite-dimensional dynamics to one-dimensional return maps that enable us to probe the geometry of the attractors quantitatively. Such results are possible only if computations are carried out for very long times (we provide examples where integrations are carried out for $10^8$ time units), and it is shown that the schemes used here are very well suited for such tasks. For illustration, computations are carried out for third order dispersion (the Kawahara equation) as well as fifth order dispersion (the Benney-Lin equation) but the methods developed here are applicable for rather general dispersive terms with similar accuracy characteristics.

Dynamic bifurcation of a modified Kuramoto-Sivashinsky equation with higher-order nonlinearity

Under the periodic boundary condition, dynamic bifurcation and stability in the modified Kuramoto-Sivashinsky equation with a higher-order nonlinearity μ(ux)puxx are investigated by using the centre manifold reduction procedure. The result shows that as the control parameter crosses a critical value, the system undergoes a bifurcation from the trivial solution to produce a cycle consisting of locally asymptotically stable equilibrium points. Furthermore, for cases in which the distances to the bifurcation points are small enough, one-order approximations to the bifurcation solutions are obtained.

Thermocapillary instability of core-annular flows

Thermocapillary instability of a core-annular flow is asymptotically examined in the thin annulus limit. Two sets of scalings are established to study the interplays between base flows, interfacial tension, and thermocapillary effects. For each scaling case, an interfacial evolution equation is derived for describing the leading order stability of the system. Both linear and weakly nonlinear stabilities are examined. When the core fluid is warmer (cooler) than the wall, thermocapillarity linearly stabilizes (destabilizes) the system, and hence suppresses (promotes) the capillary instability. For a moderate thermocapillary force and a strong capillary force, the linear instability can be arrested within the weakly nonlinear regime. For a weak thermocapillary force and a moderately strong interfacial tension, the weakly nonlinear evolution is governed by a modified Kuramoto-Sivashinsky equation. The influence of thermocapillarity on the route to chaos is discussed.

On a modified Kuramoto-Sivashinsky equation

Results on blow-up and global solvability of the modified Kuramoto-Sivashinsky equation proposed in [1] are presented.

Sputterring and smoothing of metal surface with energetic gas cluster beams

Sputtering is a process of surface erosion by energetic ions. One of the remarkable effects of sputtering with cluster ions is surface smoothing. To understand its mechanism we have developed a computational model, which utilizes molecular dynamics (MD) to simulate rapid atomic collisions in the central impact zone, and a finite-difference method to account for processes occurring on a longer time scale over a wider target area. A case of gas cluster of a few hundred Ar atoms impacting a Cu target with energy up to 20 keV has been considered. The MD simulation has revealed that the atoms ejected from the surface have a significant lateral momentum component (parallel to the surface) which may have a major effect on surface morphology. Evolution of surface morphology under cluster ion irradiation was described by the modified Kuramoto-Sivashinsky equation. Comparison of the simulations with experimental data shows qualitative agreement.

Localized Flickering Cellular Flames

We illustrate the formation and evolution of cellular flames described by localized, ordered solutions of a modified Kuramoto--Sivashinsky equation (MKSE) derived for flames stabilized on a burner. The MKSE equation accounts for a slight curvature of the flat front due to the slowing of the gas flow near the burner rim. We describe (i) pinned flickering cellular flames, in which there is a cellular array with the center of each cell fixed in space, though the cell width and temperature distribution pulsate in time, and (ii) rotating flickering cellular flames, in which there is a roughly circular array of cells, with each cell both rotating and pulsating.

Studies of phase turbulence in the one-dimensional complex Ginzburg-Landau equation

The phase-turbulent (PT) regime for the one-dimensional complex Ginzburg-Landau equation (CGLE) is carefully studied, in the limit of large systems and long integration times, using an efficient integration scheme. Particular attention is paid to solutions with a nonzero phase gradient. For fixed control parameters, solutions with conserved average phase gradient \ensuremath{\nu} exist only for |\ensuremath{\nu}| less than some upper limit. The transition from phase to defect turbulence happens when this limit becomes zero. A Lyapunov analysis shows that the system becomes less and less chaotic for increasing values of the phase gradient. For high values of the phase gradient a family of nonchaotic solutions of the CGLE is found. These solutions consist of spatially periodic or aperiodic waves traveling with constant velocity. They typically have incommensurate velocities for phase and amplitude propagation, showing thereby a type of quasiperiodic behavior. The main features of these traveling wave solutions can be explained through a modified Kuramoto-Sivashinsky equation that rules the phase dynamics of the CGLE in the PT phase. The latter explains also the behavior of the maximal Lyapunov exponents of chaotic solutions.

Order Parameter for the Transition from Phase to Amplitude Turbulence.

The maximal conserved phase gradient is introduced as an order parameter to characterize the transition from phase to defect turbulence in the complex Ginzburg-Landau equation. It has a finite value in the phase-turbulent regime and decreases to zero when the transition to defect turbulence is approached. Solutions with a nonzero phase gradient are studied via a Lyapunov analysis. The degree of “chaoticity” decreases for increasing values of the phase gradient and finally leads to stable traveling wave solutions. A modified Kuramoto-Sivashinsky equation for the phase dynamics is able to reproduce the main features of the stable waves and to explain their origin. [S0031-9007(96)00860-5]

Singularities in a modified Kuramoto-Sivashinsky equation describing interface motion for phase transition

Abstract Phase transitions can be modeled by the motion of an interface between two locally stable phases. A modified Kuramoto-Sivashinsky equation, h t + ∇ 2 h + ∇ 4 h = (1 − λ )|∇ h | 2 ± λ (∇ 2 h ) 2 + δλ ( h xx h yy − h xy 2 ), describes near planar interfaces which are marginally long-wave unstable. We study the question of finite-time singularity formation in this equation in one and two space dimensions on a periodic domain. Such singularity formation does not occur in the Kuramoto-Sivashinsky equation ( λ = 0). For all 1 ≥ λ > 0 we provide sufficient conditions on the initial data and size of the domain to guarantee a finite-time blow up in which a second derivative of h becomes unbounded. Using a bifurcation theory analysis, we show a parallel between the stability of steady periodic solutions and the question of finite-time blow up in one dimension. Finally, we consider the local structure of the blow up in the one-dimensional case via similarity solutions and numerical simulations that employ a dynamically adaptive self-similar grid. The simulations resolve the singularity to over 25 decades in |h xx | l ∞ and indicate that the singularities are all locally described by a unique self-similar profile in h xx . We discuss the relevance of these observations to the full intrinsic equations of motion and the associated physics.