Modified Kuramoto-Sivashinsky equation: Stability of stationary solutionsand the consequent dynamics

Paolo Politi1,* and Chaouqi Misbah2,†

1Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy2Laboratoire de Spectrométrie Physique, CNRS, Université J. Fourier, Grenoble 1, BP87, F-38402 Saint Martin d’Hères, France

�Received 25 October 2006; published 7 February 2007�

We study the effect of a higher-order nonlinearity in the standard Kuramoto-Sivashinsky equation: �xG̃�Hx�.We find that the stability of steady states depends on dv /dq, the derivative of the interface velocity on the wave

vector q of the steady state. If the standard nonlinearity vanishes, coarsening is possible, in principle, only if G̃is an odd function of Hx. In this case, the equation falls in the category of the generalized Cahn-Hilliard

equation, whose dynamical behavior was recently studied by the same authors. Alternatively, if G̃ is an evenfunction of Hx, we show that steady-state solutions are not permissible.

DOI: 10.1103/PhysRevE.75.027202 PACS number�s�: 05.45.�a, 82.40.Ck, 02.30.Jr

I. INTRODUCTION

One of the most prominent and generic equations thatarises in nonequilibrium systems is the Kuramoto-Sivashinsky �KS� �1–3� equation

Ht + Hxxxx + Hxx + HHx = 0, �1�

where H is some scalar function �like the slope of a one-dimensional growing front� and differentiations are sub-scripted. The linear stability analysis of the KS equation�by looking for solutions in the form of eiqx+�t� yields�=q2−q4. The �linearly� fastest-growing mode has a wavenumber given by qu=1/�2 �obtained from �q�=0�. For alarge box size the KS equation is known to exhibit spa-tiotemporal chaos. The chaotic pattern statistically selects alength scale which is close to 2� /qu: in fact, the structure

factor ��H̃q�2�, where �¯� designates the average over many

runs and H̃q is the Fourier transform of H, exhibits a maxi-mum around q=qu. Other nonequlibrium equations areknown, however, to exhibit different dynamical behaviors:just to limit ourselves to one-dimensional systems, we mayhave coarsening, a diverging amplitude with a fixed wave-length, a frozen pattern, traveling waves, and so on.

An important issue is the recognition of general criteriathat enable one to predict whether or not coarsening takesplace within a class of nonlinear equations, without having toresort to a forward time-dependent calculation. In recentworks �4,5� we have considered several classes of one-dimensional partial differential equations �PDE’s�, havingthe form Ht=N�H�, where N is a nonlinear operator actingon the spatial variable x.

Sometimes, even in the presence of strong nonlinearities,the search for steady states reduces to solving a Newton-type

equation Hxx* +V�H̃*�=0, where H* is some function of H. In

these cases, ��A�, giving the dependence of the wavelength �of the steady state on its amplitude A, is a one-value function

�see Fig. 1, solid lines�, and the criterion for the existence ofcoarsening is expressed in terms of the derivative ���A�.

It has been shown that the sign of −���A� is the same asthat of the phase diffusion coefficient. Thus ���A��0 corre-sponds to a branch which is unstable with respect to thephase of the pattern, entailing thus coarsening. The situationis more complicated when ��A� exhibits a fold �see Fig. 1,dashed line�. This event occurs, e.g., in the KS equation andin the Swift-Hohenberg equation. As for the KS equation,which is the topic of this paper, Nepomnyashchii �1� hasshown that the forearm part of the curve ��A� with positiveslope, ���A��0, is an unstable branch. This result holds forthe pure KS equation, however �6�.

The aim of this paper is the following. �i� First, we shallextend the result of Nepomnyashchii �1� to a generalizedform of the KS equation, which includes higher-order non-linearities. The KS equation has been derived in a number ofphysical problems, including hydrodynamics, crystal growth,chemical waves, etc. �for a review, see Ref. �7��. The KSequation is valid to leading order in the expansion. Nextnonlinear terms may affect dynamics. Indeed, the next lead-

*Electronic address: paolo.politi@isc.cnr.it†Electronic address: chaouqi.misbah@ujf-grenoble.fr

FIG. 1. The periodic stationary solutions of some classes ofnonlinear equations that satisfy a Newton-type equationHxx

* +V�H*�=0, and the resulting ��A� is a single-value function.According to the explicit form of V�H*�, we may have the differentcurves shown as solid lines. The dashed line, which displays a fold,corresponds to a non-single-value function. It comes out in theKuramoto-Sivashinsky and Swift-Hohenberg equations. Units onboth axes are arbitrary.

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ing term in the KS equation �HxHxx� has been analyzed in�8�, and it has been shown that this term significantly affectsthe dynamics; for example, the profile may exhibit deepgrooves. We shall consider a more general form of the modi-

fied KS equation by adding a term like �xG̃�Hx� �this in-cludes as a particular case the term HxHxx�. We find that thestability of the steady-state solutions depends on v��q�,where v has, in a front problem, the meaning of the averageinterface velocity. �ii� If the standard KS nonlinearity van-

ishes �no HHx term�, then there is coarsening if G̃ is an oddfunction. In this case a mapping of the equation onto a gen-

eralized Cahn-Hilliard equation is straightforward. �iii� If G̃is even �still in the absence of the standard nonlinearity�, weshow that there exists no steady-state periodic solution, as

attested to by numerical simulations for G̃=Hx2 �9�.

II. MODIFIED KS EQUATION

A. The method

We study the following equation:

Ht + c4Hxxxx + c2Hxx + �HHx + ��xG̃�Hx� = 0, �2�

which reduces to the standard Kuramoto-Sivashinsky equa-

tion when G̃=0. A rescaling of x, t, and H always allows oneto reduce the equation to a one-parameter equation, which

can be absorbed into a redifinition of G̃. However, for thesake of clarity, we do not get rid of �, so that we write

Ht + Hxxxx + Hxx + �HHx + �xG̃�Hx� = 0. �3�

It is also useful to rewrite Eq. �3� using the variable u,with ux=H:

ut + uxxxx + uxx +�

2ux

2 + G̃�uxx� = 0, �4�

where the integration constant v0 can be canceled out by thetransformation u�x , t�→u�x , t�−v0t.

Within the H formulation, the average d�H� /dt vanishes,because Eq. �3� has the conserved form Ht=−�x�¯�. Withinthe u formulation,

d�u�dt

= − �

2�ux

2� + �G̃�uxx�� = v . �5�

For interfacial problems u is the interface profile, and there-fore v represents the average speed of the front. For steady-state solutions v is related nonlinearly to the front profileu0�x�; see Eq. �5�.

We start from a stationary solution of period q, H�x�, andperturb it by adding h�x�exp�−�t�. The function h�x� there-fore satisfies the linear equation

hxxxx + hxx + ��hH�x + „hxG�Hx�…x = �h , �6�

where G= G̃� and whose coefficients are periodic with period�=2� /q. According to the Floquet-Bloch theorem, the solu-tion has the form h�x�=exp�iKx�F�x�, where F�x� has thesame period as H�x�.

We are interested in weak modulations of longwavelength—i.e., with K�q. It is therefore convenient tointroduce the reduced wave vector Q=K /q and the phase=qx. The equation determining the steady state H�� is

q4H + q2H + �qHH + q2HG�qH� = 0, �7�

and Eq. �6� reads

q4h + q2h + q��hH� + q2„hG�qH�… � Lh = �h ,

�8�

with h��=exp�iQ�F��.The final step is to expand both � and F�� in powers of

Q,

F�� = F0�� + QF1�� + Q2F2�� + ¯ , �9�

� = �0 + Q�1 + Q2�2 + ¯ , �10�

and to solve Eq. �8� at the lowest orders in Q.

B. Zeroth order

The differential equation determining the steady stateH�x� does not depend explicitely on x, so that H�x+x0� is asolution as well. This symmetry implies that Lh=�h �seeEq. �8�� is solved by h=H and �=0, as can be easilychecked by taking the derivative of Eq. �7�. Since thezeroth-order equation is simply

LF0 = 0, �11�

we obtain F0=H.

C. First order

The equation for F1 reads

LF1 = �1H� − iq�4q3H�� + 2q�1 + G�H�

+ ��H + q2H�G��H�� , �12�

where we have used the shorthand notation H�=H, H�=H , . . .. If we differentiate Eq. �7� with respect to q, we geta similar equation,

LHq = − �4q3H�� + 2qH� + �HH� + 3�q2H�H�� , �13�

where Hq is the derivative of H with respect to q.The comparison of Eqs. �12� and �13� suggests one to

look for F1 in the form F1= iqHq+c, where c is a constant.We easily find c=�1 / ��q�, so that

F1 =�1

�q+ iqHq. �14�

This result shows that in the absence of the standard nonlin-

earity, �=0, �1 should vanish whatever G̃ is.We might have started with an even more general Equa-

tion �3�, replacing the standard nonlinearity ��HHx�=�x� �

2 H2� with �xP̃�H�. The term �HH� on the right-handsides of Eqs. �12� and �13� would be replaced by P�H�H�,

with P= P̃�. However, in the general case P� is not a con-

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stant: therefore, it is not possible to look for a solution F1= iqHq+c.

D. Second order

The equation for F2 has the form

LF2 + q2�1 + G��− F0 + 2iF1�� + ��qH + q3H�G��iF1

= �2F0 + �1F1 �15�

or

LF2 = �2H� + iq�1Hq + q2H� + 2q3Hq� − i�1H

− i��1/��q2H�G��qH�� + �12/��q� + q2G�qH��H�

+ 2q3G�qH��Hq� + �q2HHq + q4H�G��qH��Hq.

�16�

Now we take the 2� average of the previous equation,getting

�12

�q+ q2�G�qH��H�� + 2q3�G�q�H��Hq��� + �q2�HHq�

+ q4�H�G��qH��Hq� = 0. �17�

Finally, we obtain

�12 = − �q3 d

dq�

2�H2� + �G̃�qH��� � �q3 d

dqv . �18�

This result proves that �1=0 if �=0, whatever the func-

tion G̃ is. Consider the case ��0 �we are at liberty to choose��0�. Since �0 signals an instability, one sees that ifd

dqv�0, the periodic solution is unstable, because there is areal solution �10. This generalizes the result of �1�, ob-tained for the pure KS equation, to the higher-order KS equa-tion. It is only in the pure KS limit that the spectrum ofstability is related to the slope of the steady amplitude. In thehigher-order equation, however, this ceases to be the case.Instead we should replace the amplitude by the drift velocity,a quantity which can be still obtained from pure steady-stateconsiderations. It is to be noted that the criterion regardingthe slope of v, d

dqv�0, is a sufficient condition for instabil-ity. In the opposite limit no conclusion can be drawn, sincethere is a need to push the calculation to the next order. Thishas not been checked yet, but it is likely that the analysis istoo much involved in order to lend itself to analytical tracta-bility.

E. Determination of �2

The determination of �2 implies the resolution of the dif-ferential equation

L†u = 0, �19�

where

Lu = uxxxx + uxx + ��Hu�x + �G�H��ux�x �20�

and

L†u = uxxxx + uxx − �Hux + �G�H��ux�x. �21�

The equation Lu=0 is solved by u=H�, but we do notknow the solution of L†u=0. L†�L because of the � term.If �=0, such a term is absent and L†=L. This case is treatedin the next subsection.

F. The case �=0

Stationary solutions are determined �see Eq. �3�� from thecondition

�xHxxx + Hx +�

2H2 + G̃�Hx� = 0, �22�

which can be integrated once, giving

Hxxx + Hx +�

2H2 + G̃�Hx� = C , �23�

where C is a constant.If �=0, we can set h=Hx and stationary solutions are

given by the equation

hxx = − h − G̃�h� + C . �24�

The equation for H�x� admits periodic solutions only ifh�x� itself is periodic and has zero average for any initialcondition �such that the solution is bounded�. This requires

that C=0 and G̃�h� be an odd function.In the same limit �=0, the full PDE �3� is written

Ht = − �x�Hxxx + Hx + G̃�Hx�� , �25�

and taking the spatial derivative of both terms, we get

ht = − �xx�hxx + h + G̃�h�� , �26�

where h=Hx, as before. We have therefore obtained a gener-alized Cahn-Hilliard equation, whose dynamical behavior isknown to show coarsening if and only if the wavelength � ofsteady states is an increasing function of their amplitude A�5�. We have reobtained the same result following themethod discussed in this section �calculations are notshown�.

III. FINAL REMARKS

The present and recent work �4,5� has the main objectiveto find general criteria to understand and anticipate the dy-namics of nonlinear systems by the analysis of steady-statesolutions only. For some important classes of PDE’s, whichhave the common feature of a single-value ��A� function, thecriterion is based on the sign of the derivative ���A�. In thisBrief Report we have considered a modified, generalizedKuramoto-Sivashisnky equation, where the ��A� curve is notsingle value, because it displays a fold. We have thereforeestablished a different criterion, based on dv /dq, the deriva-tive of the interface velocity on the wave vector q of thesteady state.

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If the standard KS nonlinearity is absent, we can saymore. If the nonlinearity �xG̃�Hx� corresponds to an evenfunction G̃, the equation does not support periodic stationarysolutions and this prevents coarsening in principle: we havea pattern of fixed wavelength and diverging amplitude.Instead, if G̃ is an odd function, the equation falls into a

previously studied class, the generalized Cahn-Hilliardequation, which can show different behaviors accordingto the form of ��A�. Finally, it is an important task for fu-ture investigations to see whether or not information ofthe types presented here and in �4,5� has analogs in higherdimensions.

�1� A. A. Nepomnyashchii, Fluid Dyn. 9, 586 �1974�.�2� Y. Kuramoto, Chemical Oscillations, Waves and Turbulence

�Springer, Berlin, 1984�.�3� G. I. Sivashinsky, Acta Astronaut. 4, 1177 �1977�.�4� P. Politi and C. Misbah, Phys. Rev. Lett. 92, 090601

�2004�.�5� P. Politi and C. Misbah, Phys. Rev. E 73, 036133 �2006�.

�6� The same criterion is indeed valid for a convective Cahn-Hilliard equation. See A. A. Golovin, A. A. Nepomnyashchy,S. H. Davis, and M. A. Zaks, Phys. Rev. Lett. 86, 1550 �2001�.

�7� C. Misbah and A. Valance, Phys. Rev. E 49, 166 �1994�.�8� T. Ihle and H. Müller-Krumbhaar, J. Phys. I 6, 949 �1996�.�9� Z. Csahók, C. Misbah, and A. Valance, Physica D 128, 87

�1999�.

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