Thermodynamic potentials for non-equilibrium systems

Chaos, Solitons and Fractals 12 (2001) 2619±2630

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Thermodynamic potentials for non-equilibrium systems q Orazio Descalzi a,d,*, Servet Martinez b,d, Enrique Tirapegui c,d a b

Facultad de Ingenierõa, Universidad de los Andes, Av. San Carlos de Apoquindo 2200, Santiago, Chile Departamento de Ingenierõa Matem atica, F.C.F.M. Universidad de Chile, Casilla 487-3, Santiago, Chile c Departamento de Fõsica, F.C.F.M. Universidad de Chile, Casilla 487-3, Santiago, Chile d Centro de Fõsica No Lineal y Sistemas Complejos de Santiago, Casilla 27122, Santiago, Chile

Abstract The notion of non-equilibrium potential for systems far from equilibrium is reviewed and the relation to the reversed process is examined. The potential is constructed in the neighborhood of the homogeneous attractors for a nonvariational extended system, namely the subcritical complex Ginzburg±Landau equation. This construction is the second known example of a Lyapunov functional for a non-variational system. Ó 2001 Elsevier Science Ltd. All rights reserved.

1. Introduction In the macroscopic description of physical systems in thermodynamic equilibrium the existence of thermodynamic potentials is of fundamental importance. These potentials lead to the extremum principles of equilibrium thermodynamics which are all consequences of the fact that the entropy of an isolated system is maximal. For systems which are slightly out of equilibrium, more precisely in the linear region where the Onsager reciprocity relations are valid, the extremum principle is that of minimum entropy production as was ®rst shown by Prigogine [1,2]. The problem of ®nding potentials if they exist, for systems far away from equilibrium, is then obviously a relevant problem for our comprehension of non-equilibrium phenomena. The question has been addressed by Graham and collaborators in a series of papers of great interest (see [3] and references quoted therein). The existence of these non-equilibrium potentials is of course intimately related to the symmetry of detailed balance (see [4,5] for a discussion and the relation to the time-reversed process) and to the classi®cation of temporal evolutions as variational and non-variational [6]. We recall brie¯y how the non-equilibrium potential appears in the macroscopic description of a system with a ®nite number of ¯uctuating variables which we modelize with Langevin-type equations of the form q_ l ˆ Al …~ q; r† ‡

m p X g rlj nj …t†;

l ˆ 1; 2; . . . ; n;

…1†

jˆ1

where the dot stands for derivative with respect to time, r for a set of parameters (we shall often omit r in the formulas), ~ q ˆ …q1 ; q2 ; . . . ; qn † is a set of macroscopic variables whose evolution in the absence of ¯uctuations is given by the deterministic dynamical system q_ l ˆ Al …~ q…t††, the rlj are constant quantities, g is a parameter measuring the intensity of j the noise fn …t†g are Gaussian white noises of zero mean and correlations hnj …t†nl …t0 †i ˆ cjl d…t t0 †. De®ning P and l jl m lm g ˆ j;l rj c rl we have that the conditional probability density P …~ q; tjq~0 ; t0 ; r†, t > t0 , of the Markov (di€usion) process de®ned by (1) satis®es the Fokker±Planck or forward Kolmogorov equation (from now on we sum over repeated indices and om  o=oqm ) q *

Presented at the conference on ``Probability and Irreversibility in Quantum Mechanics'' at Les Treilles, France, 4±8 July 1999. Corresponding author.

0960-0779/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 1 ) 0 0 0 7 7 - 7

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ot P …~ q; tjq~0 ; t0 ; r† ˆ ol



 g q; r† ‡ om glm P …~ Al …~ q; tjq~0 ; t0 ; r†: 2

…2†

We assume that the Markov process has a stationary probability q…~ q; r† ˆ P …~ q; tjq~0 ; t0 ; r†, …t t0 † ! 1 (the function 0 0 l 0 ~ P …~ q; tjq ; t ; r† depends only on the di€erence …t t † since A …~ q† is time independent). Putting q…~ q† ˆ exp‰ 1g U…~ q†Š we q† by de®ne Rl …~ Al

Rl ˆ

1 lm g om U  Dl : 2

…3†

Since the stationary probability satis®es   g q† ‡ om glm q…~ q† ˆ 0 ol Al …~ 2

…4†

we have Rl ol U ˆ gol Rl :

…5† l

l

q†. Let us calculate the time derivative of U…~ q…t†† for a solution ~ q…t† of the deterministic dynamical system q_ ˆ A …~ One has dU…~ q…t†† ˆ q_ l ol U ˆ dt

1 lm g ol Uom U ‡ gol Rl ; 2

…6†

where we have used (3) and (5). We assume that the limit g ! 0 of U…~ q; g† exists and we put U…0† ˆ limg!0 U. Then (6) gives dU…0† …~ q…t†† ˆ dt

1 lm g ol U…0† om U…0† < 0; 2

…7†

which shows that U…0† is a Lyapunov functional of the deterministic system (U…0† is bounded from below since exp‰ U…0† =gŠ is the leading term of the stationary probability in the weak noise limit). Taking the limit g ! 0 in (5) we see that U…0† satis®es the equation 1 Al ol U…0† ‡ glm ol U…0† om U…0† ˆ 0: 2

…8†

q† is the non-equilibrium potential and it will be a minimum on point attractors of the system acThis function U…0† …~ cording to (7). From (3) we have Al ˆ Dl ‡ Rl and the equation for the conditional probability (2) will be   g q; tjq~0 ; t0 ; r†: ot P …~ q; tjq~0 ; t0 ; r† ˆ ol …Dl ‡ Rl † ‡ om glm P …~ …9† 2 Let us consider the time-reversed process in the stationary state associated to (9). Its conditional probability density P^…~ q; tjq~0 ; t0 ; r†, t > t0 , will satisfy [4,5]   g q; tjq~0 ; t0 ; r† ˆ ol …Dl Rl † ‡ om glm P^…~ q; tjq~0 ; t0 ; r†: …10† ot P^…~ 2 Since Rl has changed sign it is associated to the reversible part of the drift Al while Dl is associated to the irreversible relaxational part. For a ®nite number of variables ~ q ˆ …q1 ; . . . ; qn † it is known that generically [7±9] the function U…0† is non-di€erentiable near saddles of the deterministic dynamical system and this property has been illustrated in numerous examples. The potential U…0† is however regular near the point attractors [10] although it may not have a polynomial approximation [11,12]. On the other hand in extended systems where the deterministic system will be a partial di€erential equation instead of an ordinary one we have much less information. In fact, apart from the trivial cases in which the deterministic evolution is of the gradient form (variational evolutions) only one example is known and it corresponds to the supercritical complex Ginzburg±Landau equation [21,22,26] where the potential has been calculated in a gradient expansion around the homogeneous attractors. In particular it was shown in [22] that the potential has two branches: one which is valid when the phase is stable and the other which is valid in the regime of phase turbulence. Our central purpose here will be to present a second example of a non-equilibrium potential for an extended system. We shall calculate the potential for the subcritical (quintic) complex Ginzburg±Landau equation which has an important di€erence with the previous case, namely the coexistence of attractors (the zero ®eld and the homogeneous oscillation)

O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

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for a certain region in the space of parameters. This equation is in fact the normal form for a subcritical bifurcation to a spatially homogeneous and time-periodic state, and in one space dimension has the form ot w ˆ aw ‡ bjwj2 w ‡ cjwj4 w ‡ Do2x w  A…w; ox †:

…11†

Here w ˆ r…x; t†eiu…x;t† is a complex ®eld, a real, b ˆ br ‡ ibi , c ˆ cr ‡ ici , D ˆ Dr ‡ iDi , br > 0, and cr < 0 guarantee that the bifurcation is inverted and saturates to quintic order. Eq. (11) admits a class of homogeneous time-periodic solutions 2 4 ‡ ci r1;2 Št ‡ u0 ††; …12† w1;2 ˆ r1;2 exp…i…‰bi r1;2 p   2 ˆ …br  b2r ‡ 4jcr ja†=2jcr j and u0 is an arbitrary phase. The existence of w1;2 requires that a P b2r =4jcr j. where r1;2 However inside this range only w1 is stable against small perturbations. It is easy to see that w0 ˆ 0 is also a solution of Eq. (11) but it is stable only for a < 0. Therefore the stable solutions w0 and w1 coexist for b2r =4jcr j 6 a 6 0. It is well known that if bi ˆ ci ˆ Di ˆ 0 then Eq. (11) can be written as

ot w ˆ

1 dU ; 2 dw

…13†

where w denotes the complex conjugate of w and U…fw; w g† is a real functional  Z  br c U…fw; w g† ˆ 2 ajwj2 ‡ jwj4 ‡ r jwj6 Dr jox wj2 dx: 2 3

…14†

One can show that U…fw; w g† is a Lyapunov functional for Eq. (11) since multiplying Eq. (13) by dU=dw, adding the complex conjugate and integrating we obtain the Lyapunov property  Z  Z dU 2 dU dU ˆ ot w ‡ c:c: dx ˆ …15† dw dx 6 0: dt dw 6 0, Di 6ˆ 0) Eq. (11) cannot be written like a gradient system. It is also well known that in the general case (bi 6ˆ 0, ci ˆ Nevertheless we can try to put Eq. (11) in the following way: ot w ˆ

1 dU ‡ R; 2 dw

…16†

where U is a Lyapunov functional for the subcritical complex Ginzburg±Landau equation and R is a rest which determines the remaining dynamics on the attractors inasmuch as U has relaxed (dU=dw ˆ 0). In this case U is no longer given by (14) and in Eq. (16) U as well as R are unknowns. To get an equation for U we multiply Eq. (16) by dU=dw, add the complex conjugate and integrate to obtain  Z Z  dU 2 dU dU dx ‡ ˆ ‡ c:c: dx: …17† R dw dt dw If we impose  Z  dU ‡ c:c: dx ˆ 0; R dw

…18†

Eq. (17) can be written as Z dU 2 dU ˆ dw dx 6 0: dt

…19†

Therefore U decreases in time. From Eqs. (11) and (16) we obtain R ˆ aw ‡ bjwj2 w ‡ cjwj4 w ‡ Do2x w ‡

1 dU ; 2 dw

and replacing R in Eq. (18) we obtain an equation for U, ( ) 2 Z 1 dU dU 2 4 2 ‡ …a ‡ bjwj ‡ cjwj ‡ Dox †w dx ‡ c:c: ˆ 0; 2 dw dw

…20†

…21†

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which corresponds of course to Eq. (8) for U…0† …~ q†. Therefore if we can ®nd a suitable solution U of Eq. (21) we would have that U is a Lyapunov functional for the one-dimensional subcritical complex Ginzburg±Landau equation provided it is bounded from below. This paper is organized as follows. In Section 2 we show how to solve Eq. (21) around the homogeneous attractors w0 and w1 by means of a systematical gradient expansion [21,22]. In Section 3 we show that one branch of the Lyapunov functional around w1 has a minimum in function space which is degenerate with respect to arbitrary long-wavelength phase variations. The non-linear phase equation describing the dynamics on the minimum set, considering terms up to fourth order in gradient, is for the ®rst time derived in this paper. In Section 4 the minima of the other branch of the Lyapunov functional U are investigated. We found that U has minima in the traveling plane waves which coexist in the space of functions. The second variation of U gives us the stability conditions for the plane waves generalizing the Benjamin±Feir±Newell and the Eckhaus instabilities from the supercritical bifurcation [23±25] to the subcritical case. Finally, Section 5 is devoted to the conclusions and outlook. 2. The Lyapunov functional In this section we show explicitly how to solve Eq. (21). The method consists in making a systematic gradient expansion around the homogeneous attractors w0 and w1 . We assume: (i) The ®eld variable w varies on length scales large compared to the coherence length …Dr =jaj†1=2 . In order to carry out the expansion we consider formally ox as being of order 1=2 . (ii) Periodic boundary conditions. (iii) The functional U is non-singular in the attractors jw0 j ˆ 0, for a < 0, and jw1 j2 ˆ r12 , for a P b2r =4jcr j. After a change of variables …w; w † ! …r; u†, where w ˆ reiu , Eq. (21) takes the following form: ( " 2  2 # Z 1 dU 1 dU dU ‰ar ‡ br r3 ‡ cr r5 ‡ Dr rxx Dr ru2x rDi uxx 2Di ux rx Š dx ‡ 2 ‡ 4 dr r du dr ) 1 dU 3 5 2 ‰rDr uxx ‡ bi r ‡ ci r ‡ 2Dr ux rx ‡ Di rxx Di rux Š ˆ 0; …22† ‡ r du where the subindex x denotes partial derivative with respect to x. We expand now the functional U in powers of : U ˆ U0 ‡ U1 ‡ 2 U2 ‡ 3 U3 ‡




…23†

Inserting this expression for U in Eq. (22) we obtain a hierarchy of equations: 2.1. Order 0 Z

) ( " 2  2 # 1 dU0 1 dU0 dU0 dU0 dx ‡Q ˆ 0; ‡ 2 ‡P 4 r dr du dr du

…24†

where P …r† ˆ ar ‡ br r3 ‡ cr r5 and Q…r† ˆ bi r2 ‡ ci r4 . Due to the invariance of the subcritical complex Ginzburg±Landau Eq. (11) under phase translations u ! u ‡ u0 the functional U does not depend explicitly on the variable u. In the lowest order 0 this just means that dU0 =du ˆ 0 and from (24) we obtain   Z br c …25† U0 ˆ 2 dx ar2 ‡ r4 ‡ r r6 : 2 3 2.2. Order 1 Z

 dU1 dx P dr

Q

dU1 du



Z ˆ4

dx P … Dr rxx ‡ Dr ru2x ‡ rDi uxx ‡ 2Di ux rx †:

Since we assume periodic boundary conditions we can write for U1 the ansatz Z  U1 ˆ dx a1 …r†rx2 ‡ a2 …r†rx ux ‡ a3 …r†u2x ;

…26†

…27†

O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

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where the functions ai …r† …i ˆ 1; . . . ; 3† are only functions of r ˆ jwj. Inserting (27) in (26) we obtain a set of ordinary di€erential equations (primes denote derivatives with respect to r): a01 P ‡ 2a1 P 0 ˆ 4Dr P 0 ‡ a2 Q0 ;

…28†

a02 P ‡ a2 P 0 ˆ 4Di P

…29†

4rDi P 0 ‡ 2a3 Q0 ;

a03 ˆ 4Dr r:

…30†

The solution of (30) is a3 …r† ˆ 2Dr r2 ‡ C0 ;

…31†

where C0 is an integration constant which cannot be determined in this order. It can only be determined in the next order 2 . The integration constants associated to Eqs. (28) and (29) can be determined in this order by the requirement of regularity (iii). In the neighborhood of r ˆ 0 this leads to  
r2 16 D ci r6 ‡ 23 bi D ‡ ci …C0 ci ‡ D † r4 ‡ 23 bi …3C0 ci ‡ D †r2 ‡ C0 b2i 15 ; …32† a1 ˆ 2Dr ‡ …a ‡ br r2 ‡ cr r4 †2 a2 ˆ

2C0 bi r ‡ 2…C0 ci ‡ D †r3 ‡ 83 D r5 ; a ‡ br r2 ‡ cr r4

where D ˆ Dr ci a1 ˆ 2Dr ‡

a2 ˆ

Di cr and D ˆ Dr bi ~ 2 ‡ D† ~ 6 ‡ Br ~ 4 ‡ Cr ~ A…r r22 †2

c2r r2 …r2

…33†

Di br , and in the neighborhood of r ˆ r1 to ;

…34†

8D …r4 ‡ Br2 ‡ D† ; 3cr r…r2 r22 †

…35†

where 3 3r2 3C0 bi 16 …C0 ci ‡ D †; D ˆ r14 ‡ 1 …C0 ci ‡ D † ‡ ; A~ ˆ D ci ;  4D 15 4D 4D 2 2 5b 15 5b r 15r 5bi i i 2 4 1 1 B~ ˆ 2r1 ‡ ‡ …C0 ci ‡ D †; C~ ˆ 3r1 ‡ ‡ …C0 ci ‡ D † ‡  …3C0 ci ‡ D †; 8ci 16D 4ci 8D 8D ci B ˆ r12 ‡

~ 2 ~ ˆ 2Cr D 1

2 ~ 14 ‡ C0 bi ‡ ci C1 ; Br A~ A~

C1 ˆ

2…C0 ci ‡ D †r14

2C0 bi r12

8  6 D r1 : 3

2.3. Order 2 Z

 dU2 dx P dr

Q

dU2 du





Z ˆ

dx

  dU1 1 dU1 ‡ Dr rxx Dr ru2x rDi uxx 2Di ux rx dr 4 dr   1 dU1 1 dU1 ‡ rDr uxx ‡ 2Dr ux rx ‡ Di rxx Di ru2x : ‡ r du 4r du

…36†

The ansatz for U2 contains 10 terms after eliminating surface integrals due to the periodic boundary conditions. Z 2 U2 ˆ dxfb1 u4x ‡ b2 u2xx ‡ b3 rx u3x ‡ b4 rx2 u2x ‡ b5 rx ux uxx ‡ b6 rxx rx ux ‡ b7 rxx uxx ‡ b8 rx3 ux ‡ b9 rx4 ‡ b10 rxx g; …37† where bi …r† are only functions of r. Proceeding in the same way as in the calculation at order 1 we have to replace now expression (37) in (36). This leads to a system of ordinary di€erential equations for the functions bi …r† …i ˆ 1; . . . ; 10†:

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O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

b01

ˆ 0;

b02 P ˆ a2

…38†  aC 3 0 ‡ rDi ‡ 2 ; 4 r

a

2

…39†

b03 P ‡ b3 P 0

1 4b1 Q0 ˆ …Dr a2 ‡ Dr ra02 †; 3

b04 P ‡ 2b4 P 0

3b3 Q0 ˆ Dr ra01 ‡ 2Dr a1

b05 P ‡ b5 P 0 ˆ

4Dr C0 r

b06 P ‡ 2b6 P 0 b07 P ‡ b7 P 0

4D2r ;

…41†

a1  ; 2

…42†  8Dr Di

2b7 P 00 ‡ b5 Q0 ‡ 4b2 Q00 ˆ 4Di a1 2b2 Q0 ˆ a1 a2 ‡ 2Di ra1

b08 P ‡ 3b8 P 0 ˆ 2Di a01 ‡

 8Dr r Dr

…40†

b07 P 00 2Dr a02 r

C0 a2 2a3  a2 ‡ 2r2 r 4r

2a01

a

2

2

‡ rDi



a02 C0 r2

a02 a3 ; r2

…43†

 Di ;

…44†

b7 P 000 ‡ b6 P 00 2b4 Q0 ‡ b05 Q0 ‡ 2b02 Q00 ‡ 48b2 ci r    a0 C d 0  a2 a0 a3 0 a1 ‡ rDi ‡ 2 2 ‡ 2 2 ; dr 2 2r 2r

2 0 00 2 b P b10 P 000 3 10 3  …a0 †2 …a0 †2 1 d 0 a …Dr ˆ 1 ‡ 22 ‡ 3 dr 1 4 4r

b09 P ‡ 4b9 P 0

b010 P ‡ 2b10 P 0

a2 Dr ‡

a2  4r

b7 Q0 ˆ 2a1

a

1

2

…45†

1 1 1 b8 Q0 ‡ b06 Q0 ‡ b6 Q00 ‡ b07 Q00 ‡ 8b7 ci r 3 3 3  a02  a2  Di ; a1 † ‡ r 2r  a a 2 2 Dr ‡ r 4r

…46†

 Di :

…47†

Eq. (39) is equivalent to b02 ˆ

a2 …a2 =4 ‡ rDi † ‡ a3 C0 =r2 : ar ‡ br r3 ‡ cr r5

…48†

The regularity condition in the attractors (iii) implies that the right side of Eq. (48) must exist when r ! 0 and when r ! r1 . In the neighborhood of r ˆ 0 this leads to r!0

b02 !

C02 : ar3

…49†

Therefore C0 ˆ 0:

…50†

In the neighborhood of r ˆ r1 in order to avoid a singularity in Eq. (48) C0 has to satisfy the following quadratic equation: ! ^ 2r12 b^i D 2 …C0 ‡ 2Dr r1 † C0 ‡ ˆ 0; …51† ^2 jbj ^ 2 ˆ b^2 ‡ b^2 and D ^ ˆ Dr b^i where we have introduced the notation b^r ˆ br ‡ 2cr r12 , b^i ˆ bi ‡ 2ci r12 , jbj r i Therefore we have two choices for C0 : …1†

C0 ˆ

2Dr r12

Di b^r . …52†

O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

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and ^ 2r12 b^i D : 2 ^ jbj

…2†

C0 ˆ

…53†

The di€erent values for C0 we found due to the regularity of the limits r ! 0 and r ! r1 determine three di€erent gradient expansions: one is valid only around r ˆ 0 and the other two are valid only around r ˆ r1 . The two possible …1† …2† choices C0 and C0 , valid in the neighborhood of r ˆ r1 , will be interpreted in the next sections as the constants associated to the gradient expansions which are valid beyond the generalized Benjamin±Feir±Newell instability and in the region where the plane waves are stable, respectively. For the purpose of this paper it is not necessary to solve completely Eqs. (38)±(47). Nevertheless we have to know some of the functions bi evaluated at r ˆ r1 . From Eqs. (38) and (39) we see that b1 is a constant and b2 has an integration constant which cannot be ®xed at order 2 and we have to go to the third order to determine these constants. 2.4. Order 3 Z



dU3 dx P dr

dU3 Q du



Z ˆ



   a dU2 a01 2 2 2 r ‡ …a1 Dr †rxx ‡ ‡ rDi uxx ‡ 2Di ux rx rDr ux dx dr 2 x 2  0  a   1 dU2 a2 2  a2 3 rx ‡ Di rxx ‡ rDr uxx ‡ 2Dr ux rx ‡ rDi u2x : ‡ r du 2r 2r r

The ansatz for U3 reads Z U3 ˆ dxfg1 u2x uxx ‡ g2 u2xxx ‡


g;

…55†

where …


† means other terms, which are not relevant for the purpose of this paper. By inserting (55) into (54) we get   a a3  2 Pg10 ˆ 3b3 ; ‡ rDi ‡ …2b5 ‡ b02 †rDr 12b1 Dr 2 r2 Pg20 ˆ 2b2

a

…56†

  a 2 Dr ‡ b7 ‡ rDi : 2

3 r2

…54†

…57† …1†

By imposing regularity conditions at r ˆ r1 in Eqs. (39), (40), (42), (44), (56) and (57) we obtain, for C0 ˆ C0 , b1 ˆ

D2r ; b^r

…58†

b2 …r1 † ˆ

D2i ; b^r

b3 …r1 † ˆ

2Dr …5Dr b^i ‡ Di b^r †; 3r1 b^2

…60†

^‡ 4Dr D ; ^ r1 b2

…61†

…59†

r

b5 …r1 † ˆ

r

b7 …r1 † ˆ

b02 …r1 † ˆ

2Di Dr ; r1 b^r

…62†

^ ‡ ‡ 2Di D ^ 4Dr D ; 2 ^ r1 b r

…63†

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O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

and for C0 ˆ b1 ˆ

…2† C0 ,

^ 2 Dr D ^ ‡ 2Dr r2 c^ …D ^ ‡ b^i ‡ D ^ b^r † b^i jbj 1 : 4 ^ 3b^r jbj

…64†

In this second situation we shall not need the other functions bj …r†. We have used here the abbreviations ^ ‡ ˆ Dr b^r ‡ Di b^i and c^ ˆ cr b^i ci b^r . D Summarizing we have shown in this section that a Lyapunov functional for the subcritical complex Ginzburg± Landau equation can be constructed by means of a systematic gradient expansion around the homogeneous attractors w0 and w1 . Around w0 we have only one branch of the functional but around w1 we get two branches which will be analyzed in the next sections. 3. The generalized Benjamin±Feir instability …1†

In this section we study the branch of the Lyapunov functional around w1 which is associated to C0 ˆ C0 . The extrema of the Lyapunov functional must satisfy dU ˆ 0; dr

…65†

dU ˆ 0: du

…66†

Using expressions (58)±(63) we ®nd that Eq. (65) gives the amplitude r as a functional of the phase u. Taking into account terms up to fourth order in gradient of u we can write r‰uŠ as ! ! Dr 2 Di cr D2r 4 cr D2i b^r jDj2 Dr Di b^i 2Dr Di cr 2D2r b^i 2 2 2 r ‰uŠ ˆ r1 ‡ ux ‡ uxx ‡ uxx u2x uxx ux b^r b^r b^3r b^3r b^3r r12 b^3r r12 b^3r ‡

2Dr Di b^i D2 b^i ux uxxx ‡ i2 3 uxxxx : 2 ^3 r b 2r b^ 1 r

…67†

1 r

To compute dU=du we use the expressions (58)±(63) and r2 ‰uŠ given by Eq. (67), and we get the identity dU  0: du

…68†

That means that this branch of the Lyapunov functional has an extremum set which is degenerate with respect to arbitrary long-wavelength phase variations. In Section 1 we have written the one-dimensional subcritical complex Ginzburg±Landau equation in the following way: 1 dU ‡ R: 2 dw

ot w ˆ

…69†

By writing w ˆ reiu Eq. (69) is equivalent to the two following equations: ot r ˆ

1 dU ‡ Rr ; 4 dr

…70†

ot u ˆ

1 dU ‡ Ri ; 4r2 du

…71†

where R ˆ …Rr ‡ iRi †eiu . Thus, inasmuch as U has relaxed, the remaining dynamics on the extremum set is given by ot r ˆ Rr ;

…72†

ot u ˆ Ri :

…73†

O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

2627

If we write u ˆ Xt ‡ h, as it is usually done, we obtain from Eq. (73) X ˆ bi r12 ‡ ci r14 and the following equation for h: " # ^2 ^2 ^2 2 ^ 2 b^i jDj2 2 ^‡ ^ 2 D2i jbj D D 2Dr Di jbj 2D2r jbj Dr Di jbj c^ hxx hxx ‡ hx ‡ 2 3 hxxxx ‡ hx hxxx ‡ 2 3 hx hxx ‡ ‰Dr h2x ‡ Di hxx Š2 : ot h ˆ 3 3 2 2 2 2 b^r b^r b^3 2r b^ r b^ r b^ r b^ r b^ 1 r

1 r

1 r

1 r

1 r

r

…74† This is the generalized Kuramoto±Sivashinsky equation describing the weak-turbulent dynamics on the extremum set, where terms up to fourth order in gradients have been considered. We have to notice that the term proportional to h4x has no analog in the phase equation corresponding to the supercritical complex Ginzburg±Landau equation [27]. Since b^r is negative a linear analysis of Eq. (74) for h…x; t† shows that the Fourier modes with small wavenumbers become unstable if ^ ‡ ˆ Dr …br ‡ 2cr r2 † ‡ Di …bi ‡ 2ci r2 † > 0: D 1 1

…75†

This condition represents the analogue of the Benjamin±Feir±Newell criterion for the subcritical bifurcation. For the case of the forward bifurcation the condition (75) reduces to [25] D‡  Dr br ‡ Di bi > 0:

…76†

In this case we have to notice that br should be negative in order to guarantee the non-linear saturation to third order. 4. The generalized Eckhaus instability In the previous section we have analyzed one of the branches of the Lyapunov functional constructed around w1 . …2† Here in this section we will study the other branch, which is associated to C0 ˆ C0 . The expressions for the module rk , p br ‡ b2r 4jcr j…Dr k 2 a† rk2 ˆ ; …77† 2jcr j and for the phase u, u…x† ˆ kx ‡ C;

…78†

are solutions of dU=dr ˆ dU=du ˆ 0, where k and C are arbitrary constants. That means plane waves are extrema of this branch of the Lyapunov functional. When U has relaxed the remaining dynamics is given by Eqs. (72) and (73). Explicitly the resulting traveling waves acquire the following form: wk ˆ rk ei…kx‡xk t‡u0 † ;

…79†

where xk ˆ bi rk2 ‡ ci rk4 Di k 2 and u0 is an arbitrary constant. To study the local stability of plane waves we calculate the second variation of the functional U. Plane waves will be stable against small disturbances if the second variation d2 U evaluated at the plane waves is positive de®nite. For this purpose it is sucient to consider U only as a functional of r, rx and ux : Z …80† U ˆ L…r; rx ; ux †dx: U is varied by making r ! r ‡ g…x†;

u ! u ‡ m…x†:

Thus the second variation reads   2   Z 1 o L d o2 L dx g d2 U ˆ 2 or2 dx ororx    2 oL d ‡ mx 2 g : orx oux dx

…81†

o2 L d2 orx2 dx2

d dx



o2 L orx2



 2   d o2 L oL mx m ‡ g 2 g ‡ mx x dx ou2x oroux …82†

2628

O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

By considering U as written in (25), (27) and (37) the expression (82) takes the following form:         Z 1 d2 d d dx g a 2a1 …rk † 2 g ‡ g b a2 …rk † mx ‡ mx b ‡ a2 …rk † g ‡ qm2x ; d2 U ˆ dx 2 dx dx where a ˆ 16…a Dr k 2 † ‡ 8br rk2 ; b ˆ 8Dr rk k; q ˆ 2a3 …rk † ‡ 12b1 k 2 . It is useful to set d2 U in a matrix form   Z g 1 2 d Uˆ ; dx… g mx †L mx 2 where the 2  2 matrix operator L reads   d2 b a2 dxd a 2a1 dx 2 : Lˆ b ‡ a2 dxd q

…83†

…84†

…85†

If the operator L has positive eigenvalues the second variation will be positive de®nite and the plane waves will be stable. The condition is then     g g L ˆk ; …86† mx mx with k positive. Eq. (86) is equivalent to two di€erential equations     d2 d mx ˆ kg; a 2a1 2 g ‡ b a2 dx dx  b ‡ a2

 d g ‡ qmx ˆ kmx : dx

Eliminating mx in terms of g one has  2    b2 a2 d k 2a1 ‡ 2 g ˆ 0: a‡ k q k q dx2 The perturbation g…x† is expressed in Fourier series X nq eiqx : g…x† ˆ

…87†

…88†

…89†

…90†

q

Replacing (90) into (89) one ®nds a relation between q and k: q2 ˆ

k2

k…a ‡ q† ‡ aq b2 : ‡ 2a1 k 2a1 q

a22

…91†

For long-wavelength perturbations …q ! 0† one has that k is positive if aq

b2 > 0:

Replacing a, b and q given by (83) in (92) we ®nd explicitly the condition for the stability of plane waves ! ^ r12 b^i D 2 2 2 …br ‡ 2cr rk † Dr rk ‡ 3b1 k > 2D2r k 2 : ^2 jbj

…92†

…93†

Before analyzing this expression in order to establish the analog of the Eckhaus instability for a weakly inverted bifurcation we can study some known limits. For the one-dimensional subcritical real Ginzburg±Landau equation (bi ˆ ci ˆ Di ˆ 0), the inequality (93) reduces to …br ‡ 2cr rk2 †rk2 > 2Dr k 2 ;

…94†

and using (77) we get

s# " a 3b2r 32jcr ja ‡ 1‡ 1‡ : k < 2Dr 32jcr jDr 9b2r 2

…95†

O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

2629

Inequality (95) gives the stability domain for stable ®nite-amplitude plane-wave solutions of the subcritical real Ginzburg±Landau equation (see formula (2.12) in [13]). The other interesting limit corresponds to the supercritical complex Ginzburg±Landau equation (br < 0; cr ˆ ci ˆ 0). From (93) we obtain the well-known stability condition [24±26] k2 <

abr D‡ : Dr …3D‡ br ‡ 2D bi †

…96†

We study now the inequality (93) for the subcritical complex Ginzburg±Landau equation. Considering terms up to order k 2 we ®nd out two conditions for the stability of long-wavelength plane waves: (i) ^ ‡ < 0; D

…97†

(ii) k2 <

^2 3D2r jbj

^ ‡ r12 b^2r D : ^ ^ 2 †^ ^ ‡ 2cr Dr D ^ ‡ r12 ‡ …2Dr r12 =jbj ^ ‡ ‡ b^r D ^ † bi Dr D c …b^i D

…98†

Condition (i) de®nes the Benjamin±Feir stable regime (see (75)) and condition (ii) represents the analog of the Eckhaus criterion for the stability of plane waves.

5. Conclusions and outlook In this paper we have studied the one-dimensional complex subcritical Ginzburg±Landau equation by means of a Lyapunov functional which has been calculated as a systematical gradient expansion around the two homogeneous attractors w0 and w1 . Around w0 the gradient expansion consists only of one branch but around w1 it splits in two branches. One of them is associated to the Lyapunov functional valid in the Benjamin±Feir unstable regime. A generalization of the non-linear Kuramoto±Sivashinsky phase equation describing weak turbulence is presented for the ®rst time here considering terms up to fourth order in the gradients of the phase. The other branch is the Lyapunov functional valid in the Benjamin±Feir stable regime. Plane waves minimize this functional when their wavenumbers are smaller than a critical value. This represents the analog of the Eckhaus criterion for the stability of plane waves. In the range of parameters where the homogeneous solutions coexist the branches of the functional become singular near the repellor w2 and are then not suitable to study problems such as the stability of pulses [14±20]. To investigate the stability conditions of localized solutions of the subcritical complex Ginzburg±Landau equation we have to build a Lyapunov functional around non-homogeneous pulse-like attractors. This represents a major challenge for the future that requires a generalization of the method presented here and will be discussed in a future publication. We are also making, with very good agreement, numerical simulations to check the validity of the Lyapunov property of our functionals. Similar simulations have been done recently by Montagne et al. [28] for the Lyapunov functional calculated by Descalzi and Graham [21,22] for the supercritical complex Ginzburg±Landau equation.

Acknowledgements E.T. thanks the support of this work by Fondecyt (P. 1990991), FONDAP (P. 11980002), CNRS-CONICYT Project and Catedra Presidencial en Ciencias. S.M. wishes to thank Fondecyt (P. 1970506). O.D. wishes to thank Fondecyt (P. 3940001) and Fondo de Ayuda a la Investigaci on de la Universidad de los Andes (P. ICIV-001-2000). References [1] Prigogine I. Introduction to thermodynamics of irreversible processes. New York: Interscience; 1969. [2] Glansdor€ P, Prigogine I. Physica 1954;20:773. [3] Graham R. In: Moss F, McClintock P, editors. Noise in nonlinear dynamical systems, vol. 1. Cambridge: Cambridge University Press; 1989. [4] Barra F, Clerc M, Huepe C, Tirapegui E. In: Tirapegui E, Zeller W, editors. Instabilities and nonequilibrium structures, vol. V. Dordrecht: Kluwer; 1996.

2630 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

O. Descalzi et al. / Chaos, Solitons and Fractals 12 (2001) 2619±2630

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