Whitham's Modulation Equations and Stability of Periodic Wave Solutions of the Generalized Kuramoto Sivashinsky

Pascal Noble

Description

Whitham's Modulation Equations and Stability of Periodic Wave Solutions of the Generalized Kuramoto-Sivashinsky Equations Pascal Noble? L.Miguel Rodrigues† Keywords: modulation; wave trains; periodic traveling waves; Kuramoto-Sivashinsky equations, Korteweg-de Vries equations; Bloch decomposition. 2000 MR Subject Classification: 35B35. Abstract We study the spectral stability of periodic wave trains of generalized Kuramoto- Sivashinsky equations which are, among many other applications, often used to describe the evolution of a thin liquid film flowing down an inclined ramp. More precisely, we show that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to side-band perturbations. Here, we use a direct Bloch expansion method and spectral perturbation analysis instead of Evans function computations. We first establish, in our context, the now usual connection between first order expansion of eigenvalues bifurcating from the origin (both eigenvalue 0 and Floquet parameter 0) and the first order Whitham's modulation system: the hyperbolicity of such a system provides a necessary condition of spectral stability. Under a condition of strict hyperbolicity, we show that eigenvalues are indeed analytic in the neighborhood of the origin and that their expansion up to second order is connected to a viscous correction of the Whitham's equations. This, in turn, provides new stability criteria. Finally, we study the Korteweg-de Vries limit: in this case the domain of validity of the previous expansion shrinks to nothing and a new modulation theory is needed.

• low-frequency hyperbolic


• full nonlinear


• order whitham's


• periodic wave


• modulation averaged


• setting ?


• low-floquet stability


• system


• whitham's system


• floquet parameter

Sujets

• Subject
• Jordan
• Université Claude Bernard Lyon 1
• Rodrigues

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 18
• Langue: English

Extrait

Whitham’s Modulation Equations and Stability of Periodic Wave Solutions of the Generalized Kuramoto-Sivashinsky Equations

Pascal Noble ∗ L.Miguel Rodrigues †

Keywords wave trains; periodic traveling waves; Kuramoto-Sivashinsky: modulation; equations, Korteweg-de Vries equations; Bloch decomposition.

2000 MR Subject Classification: 35B35.

Abstract

We study the spectral stability of periodic wave trains of generalized Kuramoto- Sivashinsky equations which are, among many other applications, often used to describe the evolution of a thin liquid film flowing down an inclined ramp. More precisely, we show that the formal slow modulation approximation resulting in the Whitham system accurately describes the spectral stability to side-band perturbations. Here, we use a direct Bloch expansion method and spectral perturbation analysis instead of Evans function computations. We first establish, in our context, the now usual connection between first order expansion of eigenvalues bifurcating from the origin (both eigenvalue 0 and Floquet parameter 0) and the first order Whitham’s modulation system: the hyperbolicity of such a system provides a necessary condition of spectral stability. Under a condition of strict hyperbolicity, we show that eigenvalues are indeed analytic in the neighborhood of the origin and that their expansion up to second order is connected to a viscous correction of the Whitham’s equations. This, in turn, provides new stability criteria. Finally, we study the Korteweg-de Vries limit: in this case the domain of validity of the previous expansion shrinks to nothing and a new modulation theory is needed. The new modulation system consists in the Korteweg-de Vries modulation equations supplemented with a source term: relaxation limit in such a system provides in turn some stability criteria. ∗ venom-bd4311dumilleJorstitutCaRN5S02,8ad,nMUCRni,UonLyde´eitrsnI,InoyLe´tisrevinevU bre 1918, F - 69622 Villeurbanne Cedex, France; [e-mail]: Research of P.N. is partially supported by the French ANR Project no. ANR-09-JCJC-0103-01. † Unerb1nu1emov,408dd3bSR25RMNCnaU,oJdrilletCamtitu,Ins1noyLe´tisrevinUn,yoeLedt´sieriv 1918, F - 69622 Villeurbanne Cedex, France; [e-mail].

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intro

gKS

1 INTRODUCTION

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1 Introduction Coherent structures such as solitary waves, fronts or periodic traveling waves, usually play an essential role as elementary processes in nonlinear phenomena. It is both usual and useful

to try first to analyze the behavior of these elementary structures with canonical models for pattern formation [ C 8 H  we try for such a canonical equation to relate side-band]. Here stability of periodic traveling waves with modulation averaged equations. We focus our attention on a scalar equation of the form ∂ t u+ 6u∂ x u+δ 1 ∂ 2 x u+δ 2 ∂ x 3 u+δ 3 ∂ x 4 u= 0,(x, t)∈R×R ∗ + , whereδ 1 ,δ 2 andδ 3  This kind of equations arises in manyare some constant real numbers. situations as a simplified asymptotic equation. For this purpose, whenδ 1 <0, it is often sufficient to setδ 2 =δ 3 = 0, that is to consider a viscous Burgers’ equation. In the limit caseδ 1 = 0, ifδ 2 6for some purposes, one may also set= 0, δ 3 = 0 and work with the Korteweg-de Vries equation (KdV). However, for well-posedness issues, whenδ 1 <0, one can not stop before a fourth order term 1 , and stop there only ifδ 3 >0. This is this latter case we are interested in here. Therefore the equation we study incorporates nonlinearity, dispersion, and, as far as it is considered about constant states, dissipative instability with respect to low frequency perturbations, stability to high frequencies. We now perform some scaling transformations to make the structure of the equation clearer. First, up to changing (x, u) into (−x,−u), may be assumedδ 2 ≥0. Then ifδ 1 ,δ 2 andδ 3 are positive, up to changest=δ 2 (δ 1 /δ 3 ) 3/2 t,x= (δ 1 /δ 3 ) 1/2 x,u=δ 2 −1 (δ 1 /δ 3 ) −1 u, δ=δ 2 −1 (δ 1 /δ 3 ) −1/2 δ 1 the above equation may be recasted into, (1.1)∂ t u+ 6u∂ x u+∂ 3 x u+δ  ∂ x 2 u+∂ 4 x u  = 0. K No u t r e am th o a t t o- w Si e va h s a h v i e ns r k ul y ed eq o u u a t ti t o h n e . r Y el e e t v , a i n n t r c e a f s e e re δ n 2 t = o 0 tec ,δ h 1 is > ueqa ,0δ t 3 io >erroc,0 on(uati to ,nl s l po qe n e d w in a g c g t 1 K h 1. S e ) the generalized Kuramoto-Sivashinsky equation (gKS) or the Korteweg-de Vries-Kuramoto- Sivashinsky equation. The above scaling was intended to enlighten the crucial role of the aboveδ 1 parameter. Moreover, with pattern formation in mind, we have also ensured that, about any constant state, the linear most unstable mode has frequency±1. Settingδ= 0, the most common form of the Korteweg-de Vries is recovered 2  should be clear now. It that the equation enters naturally in the description of weakly-nonlinear large-scale waves above a threshold where all constant states become unstable to low-frequency perturbations, threshold corresponding toδ= 0. reac T ti h o o n u s g a h nd th fl e a K m u e r s a t m ab o i t li o t - y Si [ K 1 v , 7 a K , s T h , 1 i S n 8 1 s , , k S y 2 2 tiuaeq 2, neebsahno 1qedna]32(noitau g . fi K 1 r S lcadevitsotcyduimehr a e c d n t l s onin,etaru,b)caofngei has been widely used to describe plasma instabilities, flame front propagation, turbulence 1 In some situations one may argue then that a termD∂ x 2 (u 2 ) should be taken into account [ B 1 N ], but we disregard this term here. 2 That is the reason why we have chosen not to eliminate the 6 factor.

1 INTRODUCTION

3

in reaction-diffusion systems, for a short time we now specialize the discussion to the evo- l d u e t t i a o il n ed of d n es o c n r l i i p n t e i a o r n w of av t e h s es i e n a fl p u p i l d ica m ti e o c n h s an m ic a s y ,r b e e fl f e o c u ti n n d g in pe [ C r 6 D sonalinterestfohtaetuohsrA. yasalyonwe],tbauorosdefww the threshold of stability. When analyzing, with free-surface incompressible Navier-Stokes equations, the evolution of a thin fluid flow down an incline plane of a given slope, ap- pears a critical Reynolds number above which flows parallel to the incline become unstable. Above but close to this critical Reynolds number, equation ( g 1 K .1 S ) may be used to describe c t r h i e tic d a y l n v a a m lu ic e s, [ W 2 i 6 w n W.] htiδ h sfreritomlosdunbmtfeheRnyviationoltothede -tniaSehtthwierthraedelodinmstaoiisutfl-wolloweshaesamenth rogpinbenaiortpo Venan [ Y t 2 u ] eq 7 Y . a u n a W g tsnoi m , ree δivtaoiontfehrFuodenumberfromitschtocneesrrndpootstdehelacitir rka value _ that the classic Kuram er o t t i o c - alflolmfi kyShinsi [ vas S e 2 M q 4 , , C u D ,noncoottiaereh, .] d 6 isndere would rather correspond to modeling of avw Related to the instability of constant states is the fact that, for any fixedδ >0, from c T o h n i s s t f a a nt mi s l t y a i t s es en b c if lo urcatesafamilyofperiodictravelingw [ C a 7 D v , K e , ] s BJ t N h R r Z o 3 ughaH ptionof bfporufiitac.no sed by a family of solitary waves. See 2 for a detailed descri phase portraits. 3 nature, all constant states are spectrally unstable, all solitarySince, by waves have unstable essential spectrum and small and large periodic waves are expected to be unstable. Yet it does not forbida prioriperiodic waves of intermediate per K i a o T d o s ,P o S r U s A o s m f e or ar t r h a e ys fo o r f m s e o r li , ta n r o y te w t a h ve a s t t n o u b m e er s i t c a al ble s . tud F i o e r s d [ C i D s K c , u 2 B s ] J s N oi i R n Z n 3 of the latter case, see [15, 20]. 7, deed show that for anyδ, even up to the Kuramoto-Sivashinsky equation, in parameter space a full band of spectrally stable periodic wave trains does exist. Note also that under precise assumptions of diffusive s p p e e r c t t u r r a b l a s ti t o a n bil [ B i 2 J t ] N ( y R s Z e o 3 gyamen [ artcatslorfoepsm enemncou l n i n b o i a t l o s t e y a B n J o ] N n ). R l Z i 4 near stability under localized t of this work in 3 To be more precise, let us say that the spectral stability we discussed above is under arbitrary bounded perturbations. Being related to co-periodic stability, the stability we discuss in the rest of this work – side-band stability – is of weaker nature. Many numerical works and even experiments have been focused on the even weaker requirement of stability under co-periodic perturbations. Yet, to be fully significant from a realistic point of view, these studies should be at least extended to an analysis of stability under close-to-co-periodic he low-Floquet stability ( p s e e r e tu d r e b fi a n t i i t o i n o s n 4 woinsordor,s f y t s h i e b oh t t fe r h a n B a l l o a cmrnisnof2( c . e h 6 l e ) o c , w kg erwhheetoqFlpteumararetesi lysisoftbyanana ξ). This is the well-known issue ofside-band side-band stabilitystability/instability. Determining should tell for which waves stability to co-periodic perturbations may be observed, whereas side-band instability yields in any case spectral instability. Obviously a side-band analysis is required only where co-periodic spectrum intersects the imaginary axis, thus only when 3 We note, by the way, that our analysis is intended to deal with the family of periodic wave trains that persist for all values ofδand not the ones that exist close enough to the Kuramoto-Sivashinsky equation, that is forδlarge enough. 4 frame making the wave both stationary and periodic of periodThat is perturbations given, in a co-moving one, byx7→e iξx ˜u(x) withu˜ co-periodic, square-integrable on [0,1] andξsmall. Recall that Floquet theory tells that, relaxing the smallness ofξtoξ∈[−π, π], the full spectrum is attained with such perturbations. We callξa Floquet or a Bloch parameter.