STATIONARY LOCALIZED SOLUTIONS IN THE SUBCRITICAL COMPLEX GINZBURG LANDAU EQUATION

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1 International Journal of Bifurcation and Chaos, Vol. 12, No. 11 (2002) c World Scientific Publishing Comany STATIONARY LOCALIZED SOLUTIONS IN THE SUBCRITICAL COMPLEX GINZBURG LANDAU EQUATION O. DESCALZI Facultad de Ingeniería, Universidad de los Andes, Av. San Carlos de Aoquindo 2200, Santiago-Chile M. ARGENTINA and E. TIRAPEGUI Deartamento de Física, F.C.F.M. Universidad de Chile, Casilla 487-3, Santiago-Chile Centro de Física No Lineal y Sistemas Comlejos de Santiago, Casilla 27122, Santiago-Chile Received February 23, 2002; Revised March 18, 2002 It is shown that ulses in the comlete quintic one-dimensional Ginzburg Landau equation with comlex coefficients aear through a saddle-node bifurcation which is determined analytically through a suitable aroximation of the exlicit form of the ulses. The results are in excellent agreement with direct numerical simulations. Keywords: Ginzburg Landau equation; saddle-node bifurcation; localized structures. 1. Introduction The discovery of confined traveling waves in convection in binary fluids [Heinrichs et al., 1987; Kolodner et al., 1987; Kolodner et al., 1988; Niemela et al., 1990; Moses et al., 1987] has been a motivation for theoretical work on localized solutions of amlitude equations [Afanasjev et al., 1996; Akhmediev et al., 1996; Deissler & Brand, 1990, 1994, 1995; Fauve & Thual, 1990; Hakim & Pomeau, 1991; Malomed & Neomnyashchy, 1990; Marcq et al., 1994; Soto-Creso et al., 1997; Thual & Fauve, 1988; van Saarloos & Hohenberg, 1990; van Saarloos & Hohenberg, 1992]. Nevertheless most of this work has been devoted to numerical analysis. The aim of this article is to give an analytical aroach to the study of stationary localized solutions in the subcritical comlex Ginzburg Landau equation. When the equilibrium state of an extended system loses stability through a subcritical Hof bifurcation it is described near threshold by its normal form [Elhick et al., 1987] which is the quintic Ginzburg Landau equation with comlex coefficients for a comlex amlitude A(x, t): t A = µa +(β r + iβ i ) A 2 A +(γ r + iγ i ) A 4 A +(D r + id i ) xx A. (1) 2. Stationary Localized Solutions We look for localized solutions making the Ansatz A = R 0 (x) ex{i(ωt + θ 0 (x))}, (2) where Ω is an unknown arameter. Relacing (2) in(1)weobtain 0= µr 0 + β r R γ rr D r(r 0xx R 0 θ 2 0x ) D i (2R 0x θ 0x + R 0 θ 0xx ) (3) 2459

2 2460 O. Descalzi et al. and ΩR 0 = β i R γ i R D r (2R 0x θ 0x + R 0 θ 0xx ) + D i (R 0xx R 0 θ 2 0x ), (4) where the indices x stand for derivatives with resect to the variable x. After some simle algebra these equations can bewrittenintheform and 0=µ + R 0 + β + R γ + R R 0xx R 0 θ 2 0x. (5) µ R 0 = β R γ R R 0x θ 0x + R 0 θ 0xx, (6) where µ + = D r µ D i Ω D 2 ; β + = D rβ r + D i β i D 2 ; γ + = D rγ r + D i γ i D 2 µ = D i µ + D r Ω D 2 ; β = D rβ i D i β r D 2 ; γ = D rγ i D i γ r D 2 (7) (8) and D 2 = Dr 2 + Di 2. We remark that (5) and (6) have the same form as the equations obtained with a similar Ansatz from (1) when disersive effects are neglected, i.e. D i = 0. We have studied these equations in [Descalzi et al., 2002] and we can see that the only effect of utting D i 0 is to renormalize the coefficients which take here the values given by (7) and (8). In order to solve aroximately (5) and (6) we use then the same strategy which consists in dividing the x axis into a core region R 1 and a region R 2 outside the core of the ulse. In R 1 we aroximate the functions R 0 (x) andθ 0x by their first terms in a Taylor exansion writing R 0 (x) =R m εx 2, θ 0x = αx, (9) where (R m,ε,α) are unknown quantities. Relacing (9) in (5) and (6) we obtain ε and α in terms of (Ω, R m ) and the arameters of (1) [see (7) and (8)]. One has ε = 1 2 (µ +R m + β + R 3 m + γ +R 5 m ) (10) α = β R 2 m + γ R 4 m µ. (11) We see in (9) that in the core R 1 we are aroximating the modulus of the ulse by a arabola with amaximumr m and θ 0x = dθ 0 /dx by a straight line with sloe α. Outside the core, in the region R 2,we aroximate the hase θ 0x (x) utting θ 0x (x) = for x<0andθ 0x (x) = for x>0. Assuming that R 0 (x) vanishes exonentially for x we obtain asymtotically for x that Eqs. (5) and (6) reduce to 2R 0x = µ R 0, R 0xx = R 0 ( µ ). (12) The second equation gives R 0 (x) =C ex{ µ x}, then R 0x /R 0 = µ for big x and from the first equation we obtain µ =2 µ (13) From this exression we can obtain Ω in terms of : Ω= 1 Dr 2 ( D i ( µd r 2 2 D 2 ) +2 D 2 µd r + 2 D 2 ). (14) In fact we can integrate exlicitly Eq. (5) in R 2 utting θ 2 0x = 2. The result is 2b 1 4 ex { µ + + R 0 (x)= 2 ( x +x 0 )} ( ex {2 µ ( x +x 0 )}+ a ), 2 4 b (15) where a = 3β + /2γ +, b = 3( µ )/γ + and x 0 is a constant to be determined. We remark that at this oint we know the arameters (ε, α, Ω) as functions of (R m,) through Eqs. (10), (11) and (14). The next ste is to match R 0 (x) in regions R 1 and R 2. Thisisdoneattheoint (x,r c ) ( (/α), R m εx 2 ) which has an obvious geometrical interretation: r c is the value of R 0 (x) at the matching oint x = (/α) whichis such that θ 0x = (x<x ), θ 0x = (x> x ); [ θ 0x = αx, x α, ]. α (16) Using (15) we obtain ( ab u 2 = 2 ) b ( rc 2 + a b 2 ) 2 ( b a 2 ) rc 2 b 4, (17)

3 Stationary Localized Solutions in the Subcritical Comlex Ginzburg Landau Equation 2461 where u =ex{ µ (x x 0 )}. Then: x 0 = x + ln u µ (18) After using the exressions (10) and (11) for ε and α, the last two Eqs. (17) and (18) determine x 0 in terms of (R m,). We match now the derivative dr 0 (x)/dx at the same oint x. From R 0 (x) outside the core in region R 2 [see (15)] we obtain ( ) dr0 (x) dx x=x 0 = µ r c (u a ) b r c 2 b. (19) Inside the core the derivative is calculated from (9) and one has ( ) dr0 (x) = 2εx. (20) dx x=x +0 Equalizing (19) and (20) we get the relation f(r m,) γ + 3 r c rc 4 arc 2 + b +2εx =0. (21) 3 R0 6 dx = γ + 16( 4b +(a + bu 2 ) 2 ) 2 A second relation between R m and is obtained multilying Eq. (6) by R 0 (x) and integrating on the real axis. Since R 0 (x) is a symmetric function the result can be written as g(r m,) µ β R0 4 dx γ R0dx 2 R 6 0 dx R 2 0dx =0. (22) All the integrals in (22) can be calculated and one obtains R0 2 dx = 1 3 a + b(u 2 +2) ln 2 γ + a + b(u 2 2) R 2 mx R mεx 3. (23) R0dx 4 = 3 (a 2 4b) b + abu 2 γ + ( 4b +(a + bu 2 )2 ) + a 3 a + b(u 2 +2) ln 4 γ + a + b(u 2 2) R 4 m x R3 m εx3. (24) ( 12a b(a 2 4b) 2 4bu 2 (a2 4b)(9a 2 +4b) 12a(3a 2 4b)b 3 2 u 4 +4b 2 ( 3a 2 +4b)u 6 +(3a 2 4b)( 4b +(a + bu 2 ) 2 ) 2 ln a + b(u 2 ) +2) a + b(u 2 2) R 6 m x +2R 5 m εx3. 3. Saddle-Node Bifurcation In order to see that the aearence of ulses in the comlex Ginzburg Landau equation is related to a saddle-node bifurcation we shall study in the two dimensional sace (R m,) the intersections of the two curves f(r m,) = 0 and g(r m,) = 0 [see Eqs. (21) and (22)]. We scale first time, sace and the amlitude A in (1) utting t = σt, x = λx, A(x, t) =νa (x,t ). In our case β r > 0, D r > 0, γ r < 0 (we want coexistence of homogeneous (25) attractors), and choosing ν = β r / γ r, σ = γ r /β 2 r, λ = D r γ r /β r, Eq. (1) becomes (omitting the rimes) t A = µa +(1+iβ) A 2 A (1 + iγ) A 4 A +(1+iα) xx A, (26) with µ = µ( γ r /βr 2 ), α = D i /D r, β = β i /β r, γ = γ i /γ r. We fix now β = 0.2, γ = 0.15, α = 0.1, and study Eq. (26) varying µ. For

4 2462 O. Descalzi et al. µ = µ c1 = the curves f(r m,)=0(continuous line) and g(r m,) = 0 (dashed line) cut tangently giving origin to a saddle-node bifurcation [Fig. 1]. Above µ c1 these curves cut at two oints giving origin to a air of ulses [Fig. 1]. One of them stable and the other one unstable. Under µ = µ c2 = the intersection of the curves f(r m,)=0andg(r m,) = 0 still redicts Fig. 1. For µ = µ c1 = the curves f = g = 0 cut tangently. For µ = >µ c1 the curves f = g =0cut at two oints Fig. 2. For µ = µ c2 = the intersection between the curves f = g = 0 redicts an unstable ulse. The intersection redicting a stable ulse.

5 Stationary Localized Solutions in the Subcritical Comlex Ginzburg Landau Equation Fig. 3. For µ = >µ c2 the intersection between the curves f = g = 0 redicts an unstable ulse. The curves f = g = 0 do not intersect forbidding the existence of a stable ulse. the existence of two ulses. In Fig. 2 the intersection redicts one unstable ulse and in Fig. 2 one stable ulse. Above µ = µ c2 = Fig. 3 shows that there still exist an intersection between the curves f = g = 0 redicting an unstable ulse [Fig. 3], but one does not get an intersection corresonding to the stable ulse [Fig. 3]. In Fig. 4 we see exlicitely the character of the saddle-node bifurcation. We comuted Ω as a function of µ using the exression given by (14) (thick continuous line stands for stable ulses and thick dashed line stands for unstable ulses) and we comare with the curve obtained using the bifurcation software auto 2000 [Doedel et al., 1991] (thin continuous line). The software shows that near µ = and µ = 0.11 the system undergoes a saddle-node bifurcation. Both critical values of µ are in very well agreement with the theoretical redicted values for µ c1 and µ c2. Moreover both curves show that the branch corresonding to the unstable ulses ersists u to µ =0. A direct numerical simulation of Eq. (26), with arameters µ = 0.13, β =0.2, γ = 0.15, α = 0.1 has been carried out. Owing to the fact that the curves f(r m,)=0andg(r m,)=0cut at two oints, namely, = ; R m = and = ; R m = , we redict two Ω Fig. 4. Bifurcation diagram for ulses. The theoretical bifurcation curve is drawn with thick line (continuous line stands for stable ulses and dashed line stands for unstable ulses). The bifurcation curve comuted with software auto 2000 is drawn with thin continuous line. ulses. In Fig. 5 we show the shaes of the ulses obtained with our analytical aroach. The continuous line corresonds to the stable ulse and the dashed line to the unstable one. The ulse µ

6 2464 O. Descalzi et al. θx x Fig. 5. Shae of the stable and unstable ulses redicted by the analytical aroach (continuous and dashed lines). The numerical result is reresented by unctured line. The gradient of the hase. obtained from the direct numerical simulation is drawn with the unctured line. The values of R m and the asymtotical value of the hase gradient agree within 1% of our analytical aroach. In Fig. 5 we show the gradient of the hase for the three above mentioned cases. Acknowledgments E. Tiraegui and O. Descalzi wish to thank Fondo de Ayuda a la Investigación of the U. de los Andes (Project ICIV ) and FONDECYT (P ). M. Argentina acknowledges the suort from FONDECYT (P ). We would like to thank Dr. Marcel Clerc (Universidad de Chile) and Prof. Helmut Brand (Universitaet Bayreuth, Germany) for many useful discussions. The numerical simulations have been erformed using the software develoed in the Institute Nonlinéaire de Nice, France. We are indebted to Prof. P. Coullet (Nice) for allowing us to use this software. References Afanasjev, V. V., Akhmediev, N. & Soto-Creso, J. M. [1996] Three forms of localized solutions of the quintic comlex Ginzburg Landau equation, Phys. Rev. E53, Akhmediev, N., Afanasjev, V. V. & Soto-Creso, J. M. [1996] Singularities and secial soliton solutions of the cubic-quintic comlex Ginzburg Landau equation, Phys. Rev. E53, Deissler, R. J. & Brand, H. R. [1990] The effect of nonlinear gradient terms on localized states near a weakly inverted bifurcation, Phys. Lett. A146, Deissler, R. J. & Brand, H. R. [1994] Periodic, quasieriodic, and chaotic localized solutions of the quintic comlex Ginzburg Landau equation, Phys. Rev. Lett. 72, Deissler, R. J. & Brand, H. R. [1995] Two-dimensional localized solutions for the subcritical bifurcations in systems with broken rotational symmetry, Phys. Rev. E51, R852 R855. Descalzi, O., Argentina, M. & Tiraegui, E. [2002] Saddle-node bifurcation: An aearance mechanism of ulses in the subcritical comlex Ginzburg Landau equation, submitted to Phys. Rev. E. Doedel, E. J., Keller, H. B. & Kernévez, J. P. [1991] Numerical analysis and control of bifurcation roblems, Part II: Bifurcation in infinite dimensions, Int. J. Bifurcation and Chaos 1(4), Elhick, E., Tiraegui, E., Brachet, M. E., Coullet, P. & Iooss, G. [1987] A simle global characterization for normal forms of singular vector field, Physica D29, Fauve, S. & Thual, O. [1990] Solitary waves generated by subcritical instabilities in dissiative systems, Phys. Rev. Lett. 64,

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