Stability of the pulselike solutions of the quintic complex Ginzburg Landau equation

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1 Soto-Crespo et al. Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1439 Stability of the pulselike solutions of the quintic complex Ginzburg Lanau equation J. M. Soto-Crespo Instituto e Óptica, Consejo Superior e Investigaciones Científicas, Serrano 11, 8006 Mari, Spain N. N. Akhmeiev an V. V. Afanasjev Optical Sciences Centre, Institute of Avance Stuies, The Australian National University, Canberra, 000, Australia Receive July 31, 1995; revise manuscript receive November 7, 1995 We performe a etaile investigation of the stability of analytic pulselike solutions of the quintic complex Ginzburg Lanau equation that escribes the ynamics of the fiel in a passively moe-locke laser. We foun that in general they are unstable except in a few special cases. We also obtaine regions in the parameter space in which stable pulse solutions exist. These stable solutions o not have analytical expressions an must be calculate numerically. We compare an connecte the regions in which stable solitonlike solutions exist with the lines for which we ha analytical solutions Optical Society of America 1. INTRODUCTION The cubic-quintic complex Ginzburg Lanau equation (CGLE) is a continuous approximation to the ynamics of the fiel in a passively moe-locke laser. 1 3 It has also been proven to be useful in escribing important phenomena such as ultrashort pulse propagation in optical transmission lines with spectral filtering 4 9 an erbium-ope fiber amplifiers. 10 Earlier, the CGLE was use successfully in moeling of other nonequilibrium processes in physics The quintic terms in the equation escribe important physics, which is lacking in other moels in the literature. 1,14 The most important, the quintic CGLE, has stable pulselike solutions. We recall that pulselike solutions of the cubic CGLE are unstable in general. 15 Of crucial importance is to know for which values of the coefficients the quintic CGLE has stable pulselike solutions. 16 The best way to answer this question is to fin its exact solutions an to stuy their stability. However, exact solutions have been foun analytically only when special relations between the parameters of the equations are satisfie. Otherwise, the solutions must be stuie numerically or with certain approximations. 17,18 Exact solutions of the quintic CGLE have been stuie by van Saarloos an Hohenberg 19,0 an in a recent stuy by Marcq, Chaté, an Conte. The metho use in Refs. 19 an 0 is the reuction of the original partial ifferential equations into a set of three orinary ifferential equations that amits exact solutions. However, solutions have not been written explicitly. The authors of Ref. 1 use Painlevé analysis an symbolic computations. Thus far this stuy constitutes the most comprehensive mathematical treatment of the quintic CGLE. The general approach, use in Ref. 1, consists of the reuction of the ifferential equation to a purely algebraic problem. However, this technique assumes that analytical results can be obtaine in a reasonable time only by use of computers. More importantly, the final formula for the pulselike solutions in Ref. 1 has parameters that are expresse implicitly through the coefficients of the CGLE, an thus some work is still neee if we are to calculate the pulse shapes numerically. In this paper we reerive exact pulselike solutions of the quintic CGLE in a simpler way, obtaining these solutions in analytical form, an, in aition, we stuy their stability. Note that the stability of the pulselike solutions of the quintic CGLE is still an open question. Only for the case of the cubic CGLE has it been investigate. 3 We foun that exact analytic solutions of the quintic CGLE are unstable everywhere in the region of parameters in which they exist. An exception occurs when the pulse is in the vicinity of the zone in which it is transforme into two zero-velocity fronts. The solution then looks like a flat-top soliton. As a next step, we investigate numerically the whole range of parameters in which we woul expect that stable solitonlike pulses exist. As a result we foun the area of global stable pulse propagation, i.e., the region in the parameter space in which a broa class of initial conitions converge to a stationary pulse, which therefore represents a stable pulselike solution of the quintic CGLE. This shoul be very useful for experimental researchers working in this fiel. In principle, the CGLE has ifferent types of solutions, incluing pulses, fronts, sinks, an sources. 1,19 In this paper we restrict ourselves to pulselike solutions, as they are the most important ones for optical applications. In the fiel of nonlinear optics the quintic CGLE is usually written in the following form: ic z 1 1 c tt 1 jcj c ic 1 iejcj c1ibc tt 1 imjcj 4 cnjcj 4 c, (1) where t is the retare time, z is the propagation istance, is the linear excess gain at the carrier frequency, /96/ $ Optical Society of America

2 1440 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 Soto-Crespo et al. b escribes spectral filtering (b.0), e accounts for nonlinear gain/absorption processes, m represents a higherorer correction to the nonlinear amplification/absorption, an n is a higher-orer correction term to the nonlinear refractive inex. All coefficients in Eq. (1) are real constants (we o not require them to be small), an c is the complex envelope of the electric fiel. Equation (1) is written in such a way that, if the right-han sie of it is set to zero, we obtain the stanar nonlinear Schröinger equation. Note that b must be nonnegative in orer to stabilize the pulse in the frequency omain. If the coefficients, b, e, an n on the right-han sie are small an n 0, then solitonlike solutions of Eq. (1) can be stuie by application of perturbative theory to the soliton solutions of the nonlinear Schröinger equation. 1,4,5 This approach, however, can give neither all the relevant properties of solitonlike pulses nor the regions in the parameter space in which they exist. Fining exact solutions is an important step for unerstaning the full range of properties of the complex CGLE, thus helping us to preict the behavior resulting from an arbitrary initial conition. The structure of the rest of the paper is the following. In Section we reerive exact solutions of the quintic CGLE, whose stability is analyze in Section 3, in which we also show some examples of propagation of ifferent inputs. In Section 4 we stuy numerically the regions in the parameter space in which stable stationary pulses are possible. The form of these pulses cannot be expresse analytically yet, but there is a clear relation between the subset of parameters for which we obtaine analytical solutions an that in which stable pulses are possible. This inicates that, although the analytical solutions are efine in a subspace of the full parameter space, they play an important role in unerstaning the ynamics of a general solution. This is ealt with in Section 5. Finally, Section 6 summarizes our results.. ANALYTICAL PROCEDURES In this stuy we are intereste in pulselike solutions of Eq. (1) that have zero transverse velocity. Hence we look for a solution of the form c t, z A t exp ivz, () where v is a real constant. The complex function A t can always be written in an explicit form as A t a t exp if t, (3) where a an f are real functions of t. By inserting Eqs. () an (3) into Eq. (1) an separating real an imaginary terms, we obtain v 1 f0 1bf 00 a 1 bf 0 a a00 1 a 3 1na 5 0, 1 bf f00 a 1f 0 a 0 ba 00 ea 3 ma 5 0, where a prime stans for ifferentiation with respect to t. (4) Let us now assume that f t f 0 1 ln a t, (5) where is the chirp parameter an f 0 is an arbitrary phase; we suppose f 0 0 for simplicity. Equation (5) is, obviously, a restriction impose on f t because the chirp coul have a more general functional epenence on t. However, this restriction allows us to fin some families of solutions in analytical form. The solutions stuie in this section are only those that can be represente in the form of Eqs. (3) an (5). Equations (4) then become! 1 va 1 1b a 00 1 b!! a 1 b a b! a 0 a 1 a3 1na 5 0, a 0 a ea3 ma 5 0. Now, we have two secon-orer orinary ifferential equations relative to the same epenent variable, a t. To have a common solution, the two equations must be compatible. In general, this is not the case. However, for this particular system, they can be mae compatible by a proper choice of parameters. To fin the conitions of compatibility, we use the following proceure. We eliminate the first erivatives from the set of Eqs. (6) an integrate the resulting equation, obtaining b a0 a 1 1! 1b 1eb e a!! n 3 1b 1m b 5 a 4 1 v 11b 1 b (6)! 0. (7) The integration constant is zero for solutions ecreasing to zero at infinity. In contrast, we can eliminate the secon erivative from Eqs. (6), obtaining! b b a0 a 1 e eb a 4 n!1m b b1 1! 3 5 a 4 1vb v b 0. (8) These last two equations must coincie. Hence the following set of three algebraic equations must be satisfie: n 4 1 b 6b 1m8b 13 0, (9a) 3 1 b 4b 16eb 1 e e 0, (9b) v b1b b 0. (9c) Equations (9) are the conitions of compatibility for Eqs. (6).

3 Soto-Crespo et al. Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1441 If both coefficients m an n are nonzero, then Eq. (9a) gives the relation between the four parameters e, b, m, an n when the solution exists in the form of Eqs. (3) an (5). The parameter can be foun from Eq. (9b): eb 6 p eb 1 8 e b. eb (10) This is an important result, which shows that can be foun in terms of b an e only. From Eq. (9c) we obtain for v: v 1 1 4b b1b. (11) Now, taking into account Eqs. (9a), (10), an (11), an after some cumbersome transformations, we can rewrite Eq. (8) [or Eq. (7)] in the form a 0 a 1 n b e 8b 1 3 a b a 0. (1) b1b The coefficient in front of a 4 can equally be written in the form n 8b 1 3 m 3b b. (13) By using the substitution f a, we can rewrite Eq. (1) in the form f 0 f 1 8n 8b 1 3 f 8 b e b f 4 0. (14) b1b It is important to note that Eq. (14) is the consequence of set (6), an its solutions are equivalent to the solutions of set (6). Equation (14) is an elliptic equation, an its solutions can be foun relatively easily. Boune solitonlike solutions exist only if an f 1 an f are the roots of the equation n 8b 1 3 f 1 namely, b e 3 114b f b1b 0, (19) f 1, s 18 n 1 1 4b b e 6 be 1 8b 1 3 b1b 6n 114b 3 8b 13, (0) When n 0, this expression must be replace by f 1, s b e 6 be 9 m 114b 1 3bb b1b 3m 114b 3 3bb (1) Apart from the necessary conition (15), the above soliton solution (17) exists if one of the roots f 1, is positive. Taking these two conitions in min, we can obtain the region of existence of these solutions very easily numerically. The soliton solutions of the quintic CGLE exist for a wie range of values of the coefficients b, e, m, an n. The ansatz (5) is the conition that restricts this range by imposing the relation [Eq. (9a)] on them. The general solution exists for both signs in expression (10) for. This means that solutions must be analyze separately in the two cases (15) b1b If we take, which as is shown below is the most interesting value, the enominator in this expression is positive below the curve S, given by e S b 3p 1 1 4b 1, (16) b an is negative above it (see the soli curve in Fig. 1). Hence if is negative, we can have a solution above curve (16), an below it if is positive. The positive solution of Eq. (14) is 3 f t f 1 f f 1 1 f f 1 f cosh a p f 1 jf jt, (17) where v u É É v É É a t n u t m, (18) 8b 1 3 3b b Fig. 1. Curves elimiting the regions on the (b, e) plane for which the stationary solutions [Eq. (1)] stuie in this paper exist. The otte ashe an the otte curves are obtaine when an n 0.5 an when 0.1 an 0.5, respectively. The region in which the solution exists is the area above the curves. The ashe curve is for 1 an for negative values of an n. The allowe region is in this last case the area below the curve. The soli curve represents the curve given by Eq. (16).

4 144 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 Soto-Crespo et al. As an illustrative example, Fig. 1 shows three curves that elimit the region of allowe values of the parameters (b, e) in which the solution exists for given values of n an. The otte ashe an the otte curves are obtaine for n 0.5,, an for 0.1 an 0.5, respectively. The region above each curve represents the allowe values of the parameters (b, e) in which the solution exists in each case. an n being both negative, the corresponing curve is the same if the prouct n is kept constant. The ashe curve shows the same, but now for 1. In this case the region of allowe values of (b, e) is the area locate below the curve, an as far as the signs of n or o not change, it epens only on prouct n. Given an arbitrary selection of three parameters (let us choose b, n, an ), the stationary solution (1) exists for a certain range of values of e. The parameter m is then etermine by Eq. (9a) to have two possible values corresponing to the two possible values of 6. Figure shows the epenence of m with respect to e as given by Eq. (9a) for b 0.5 an n 0.5. The continuous curve is the one obtaine by choice of 1 in Eq. (10), an the otte curve is for. The horizontal lines mark the allowe values of e at which the solution exists for a fixe value of, which is state on the lines together with the corresponing choice for. Similar couples of curves m e for an 1 can be plotte for other values of b an n. Then the value of elimits the values of e at which the analytical solution exists. eigenvalue, which we call the perturbation growth rate. The gri size was chosen to have as many as 8000 points in the t-axis (usually 000), an the step size in the z irection was typically Dz 0.1, but was chosen much smaller when the resulting growth rate was high. Numerical accuracy was checke by repetition of the simulations for ifferent gri an step sizes. A typical example of the calculate perturbation eigenmoe for a given set of parameters is shown in Fig. 3. Specifically the chosen parameters were b 0.5, n 0.5, e 0.6, 0.1, an. The figure shows the real an the imaginary parts of the eigenfunction; the ashe curve shows the moulus of the stationary but unstable pulse itself. In this case the eigenmoe with the highest growth rate is an even function of t. To corroborate the finings of the linear stability analysis, we also solve the nonlinear propagation Eq. (1) by 3. LINEAR STABILITY ANALYSIS To investigate the stability of any of the solutions, we seek a perturbe solution of the form c t, z A 0 t 1gg t,z exp ivz, () where A 0 t is the stationary solution uner stuy [see Eq. (3)], g is a small parameter, an g z, t is a perturbation function. Inserting Eq. () into Eq. (1) an linearizing in the small parameter g, we obtain! 1 ig z 1vg1 ib g tt 1 ja 0 j 1 ie g 1A 0 1 ie g 1 n im 3jA 0 j 4 g 1 ja 0 j A 0 g 0. (3) This equation for the perturbation function g has potentially many possible types of solutions. Moreover, because the linear operator in Eq. (3) is not Hermitian, its eigenvalues are complex in general. For our purposes here we specifically want to fin those solutions, if any, that isplay exponential growth in the z irection (sometimes accompanie with sinusoial oscillations) an that are therefore unstable. For this task we followe the metho escribe in Ref. 4 by using a Crank Nicholson scheme. Namely, we solve Eq. (3) taking an arbitrary initial conition, which is assume to contain any perturbation eigenmoe. However, for large propagation istances, the perturbation eigenmoe with the largest growth rate will ominate since the growth is exponential. In this way we are able to fin the most unstable eigenfunction an the real part of its corresponing Fig.. Depenence of m on e [Eq. (9a)] that must be satisfie as a necessary conition for the general solution to exist when b 0.5 an n 0.5. The soli curve is for 1 in Eq. (10), an the otte curve is for. The horizontal lines mark the intervals at which the solution given by Eqs. () (17) exists for a given value of, which is written on the lines. Fig. 3. Fiel amplitue of the exact solution (ashe curve) for b 0.5, n 0.5, e 0.6, an 0.1; the real (soli curve) an the imaginary (otte curve) parts of the perturbation eigenmoe associate with this solution.

5 Soto-Crespo et al. Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1443 using a split-step Fourier metho for several initial conitions. The agreement between the two approaches was total in all the cases we compare. As an example, Fig. 4 shows the nonlinear evolution of the unstable pulse represente in Fig. 3. In these numerical simulations, instea of the exact eigenmoe of perturbation, we ae a ranom perturbation to the stationary solution an took this perturbe solution as the initial conition. Clearly, the ranom perturbation contains the eigenmoe that is shown in Fig. 3, but with unknown sign. Depening on this sign, we can observe two types of evolution. They are shown in Figs. 4(a) an 4(b), respectively. Only these two types of evolution were observe, whatever the perturbation was. When, instea of a ranom perturbation, we either ae to or subtracte from the stationary solution its corresponing eigenmoe of perturbation multiplie by a small factor, we obtaine exactly the same two types of evolution that Fig. 4 shows. After propagating a certain istance unaltere, the pulse either increases its energy an transforms into uniformly translating fronts [Fig. 4(a)] or ecreases an finally isappears [Fig. 4(b)]. This is a typical example of the unstable pulse evolution. In principle, this behavior may slightly change epening on the values of the parameters of the quintic CGLE, but there are always two types of evolution of the unstable exact solutions. In this sense we can consier the exact solutions as bounaries between two ifferent types of behavior of arbitrary pulses. This makes them important in a general analysis. We now present the results of the linear perturbative analysis applie systematically to the analytical solutions. We consier separately the cases with negative n from those with positive n, as we notice that they exhibite ifferent characteristics. A. Case i: n,0 Figure 5 shows in a logarithmic scale the perturbation growth rate as a function of e for b, n 0.5, 0.5, an for 60.1 an Figure 5(a) is for the cases with negative, an Fig. 5(b) is for the positive ones. The value of is written near the respective curves. For each value of there are two curves corresponing to the two possible solutions associate with the two values of. For jj we have the soli ( ) an the otte curves ( 1 ), an for jj 0.1 the otte ashe an the ashe curves correspon to an 1, respectively. Clearly, in all cases the solutions obtaine for 1 have much higher instability growth rates than those associate with. Consequently these last solutions are, in principle, of more interest from a practical point of view. Moreover, as given below, those for negative are connecte with the stable solutions. For positive an for the solutions obtaine taking, the figure shows that the perturbation growth rate is exactly equal to ; that is, their instability has its origin solely in the instability of the uniform backgroun c 0. The result shows that the pulse itself is stable. This is exactly the case of what takes place in optical transmission lines. 5 The excess gain is usually positive in orer to amplify the pulse itself, but it is kept small to avoi an appreciable growth of the backgroun. What Fig. 5(a) inicates then is that we can have stable propagation of these states for long istances as far as the excess linear gain is low enough. Figure 6 proves this last assertion. We took as an initial conition the following: c t, 0 A 0 t 11G t, (4) where A 0 t is the corresponing stationary solution [Eq. (3)] for the following coefficients: b 0.5, n 0.5, e 0.1, 0.005, an! m 0.089, an G is a uniform ranom noise obeying q,g t. 0,, jg t j. 0.. (5) The noise term is intene to see any latent instability of the system, as it surely contains all the possible exponentially growing eigenfunctions of the linearize problem. Figure 6 shows that the solution reaches very quickly the profile corresponing to the stationary solution, getting ri of the ranom fluctuations, an then remains with it uring long istances. After propagating a istance of z 1000, the profile has not experience any relevant change. Farther on, the growth of the backgroun starts to be perceptible. For negative (an ) at any given e the perturbation growth rate ecreases as jj ecreases. Another interesting feature of the growth rate curves in Fig. 5(a) is that at the solution becomes more stable as we move to its smallest allowe value of e. This happens when f 1 becomes close to f. Figure 7 shows an example of the stable propagation foun from the perturbative analysis for these cases. We took as initial conitions Eqs. (4) an (5), with A 0 t being the stationary solution for the following values of the coefficients: b 0.5, n 0.5, 0.1, e , an Fig. 4. Two possible scenarios of evolution of the stationary pulse represente in Fig. 3.

6 1444 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 Soto-Crespo et al. B. Case ii: n.0 We also consiere the solutions for positive n. Figure 9(a) shows the epenence of m with respect to e for n 0.5 an b 0.1 (lower curve) an b 0.5 (upper curve). For 0.01 an the solution exists only for positive values of e, which in aition enforces positive values for m. That is, the nonlinear gain compensates the linear losses. The regions in which the solution exists is marke in Fig. 9(a) with a soli curve for b 0.5 an with a ashe curve for b 0.1. Figure 9(b) shows the perturbation growth rates corresponing to these cases. Two aitional curves in Fig. 9(b) are for n 0.1. For positive n, solutions also exist when we take 1 in Eq. (10), but for very high values of jej, which also prouce high values for m. For instance, for b 0.5, n 0.5, an 0.01, the solution obtaine taking Fig. 5. Growth rate of the preominant perturbation eigenmoe associate with the soliton solution as a function of e for b 0.5, n 0.5, (a),0, an (b).0: In both cases the soli curve is for jj an, the otte ashe curve is for jj 0.1 an, the otte curve is for jj an 1, an the ashe curve is for jj 0.1 an 1. For clarity, the value of is on the corresponing curves.! m 0.7. For this specific case, f an f Initially the solution recovers its unperturbe shape an then propagates uring all the istance that we monitore (z max 10000) without moifying its profile. We conclue that solutions obtaine for negative an are stable when we are in the vicinity of having f 1 f (otte ashe an otte curves in Fig. 1). The transformation of the pulses into a pair of zero-velocity fronts happens on these curves. We obtaine similar results on the stability of these solutions for other values of b an n. Figure 8 shows the perturbation growth rate for ifferent solutions with n 0.1 an an with b 0.1 an b 0.5. The corresponing value of b is written on the curves. The curves for the solution obtaine taking 1 are plotte with otte an ashe curves, an those for are represente in continuous an in otte ashe curves. The curves exhibit the same qualitative features as those of Fig. 5. The general behavior of the perturbation growth rate for other values of b an n, 0 is similar to that shown in Figs. 5 an 8. Fig. 6. Evolution of the solution for b 0.5, e 0.1, n 0.5, , an m This stationary solution is initially perturbe as inicate by Eqs. (4) an (5). Fig. 7. Evolution of the solution for b 0.5, e , n 0.5, 0.1, an m 0.7. This stationary solution is initially perturbe as inicate by Eqs. (4) an (5).

7 Soto-Crespo et al. Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1445 of the parameter space) for which stable pulses exist. We fin stable pulses in a certain region an compare it with another (of a lower imensionality) in which the analytical solutions given by Eq. (17) exist. Let us first fix some limits in the parameter space in which to look for stable pulses. The parameter b clearly must be nonnegative in orer to stabilize the soliton in the frequency omain. The linear gain coefficient must be zero or negative to provie the stability of the backgroun. In this case, for n 0, stable pulses can exist only for e above the line S. We choose m,0to stabilize the pulse against the collapse. The parameter n can have either sign. With these restrictions in min we chose a given value for the coefficients an solve numerically the propagation Eq. (1) taking as the initial conition a Gaussian pulse of arbitrary amplitue an with. The shape of the initial pulse appeare of minor importance. Then we Fig. 8. Growth rate of the preominant perturbation eigenmoe associate with the soliton solution as a function of e for n 0.1, b 0.5, b 0.1, (a) 0.01, an (b) The ifferent curves correspon to (i) b 0.5 an (soli curves), (ii) b 0.1 an (otte ashe curves), (iii) b 0.5 an 1 (ashe curves), (iv) b 0.1 an 1 (otte curves). For clarity we wrote the values of b in parentheses near the corresponing curves. 1 exists in the interval (`, 9.93), an, for example, for e 1, m becomes These values are quite unrealistic. Moreover, we foun that the corresponing solutions are highly unstable. The general conclusion from the above stability analysis is that, although an exact solution (17) to the quintic CGLE can be foun when a specific relation between the parameters in Eq. (9a) is satisfie, all of them are unstable. An exception appears in the vicinity of the bounary that separates pulses from pairs of fronts. The perturbation growth rate of these soliton solutions falls to zero when we ten to this limit. These stable solutions have a flat top, announcing the transition from pulses to fronts. 4. REGIONS IN THE PARAMETER SPACE IN WHICH STABLE PULSES EXIST In this section we obtain numerically the values of the coefficients (, b, e, m, n) of the quintic CGLE (subspace Fig. 9. (a) Depenence of m on e [Eq. (9a)] that must be satisfie as a necessary conition for the general solution Eq. (1) to exist when n 0.5 an b 0.5 (upper curve) an b 0.1 (lower curve). The soli an the ashe parts of these curves show the regions in which the solutions actually exists for (b) The growth rate of the preominant perturbation eigenmoe associate with the soliton solution as a function of e for 0.01 an b, n equal to (i) (0.5, 0.1) (soli curve), (ii) (0.5, 0.5) (otte curve), (iii) (0.1, 0.1) (long-ashe curve), an (iv) (0.1, 0.5) (short-ashe curve).

8 1446 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 Soto-Crespo et al. observe its evolution upon propagation. If the solution converge to a stationary one, we consiere that we ha obtaine a stable solution an that the chosen values of the parameters belong to the class of those that permit the existence of solitons. Figure 10 shows three examples of the soliton solutions foun with the above metho. The corresponing values of the coefficients are b 0.5, n m 0.1, e 0.38 (soli curve), e 0.5 (otte curve), an e 0.66 (ashe curve). Repeating these calculations systematically for other sets of parameters, we were able to construct the areas in the parameter space in which stable propagation of boune solutions is possible. The process of convergence to a stable solution was usually quite slow, but we foun that it coul be accelerate in the following manner. After having foun a stable solution, we took it as the initial conition for the next case, in which we kept constant all the parameters except one (usually e), which was slightly change. The point is that, as expecte, for a similar set of parameters we obtaine similar solutions, an therefore we ha a rapi convergence. Figure 11 shows the areas in the (b, e) plane where soliton solutions were foun numerically. The ifferently hatche areas correspon to ifferent values of the parameter m. The lower curve (ashe) represents the curve S [Eq. (16)], an it is plotte to allow us to make some comparisons with the conclusions we obtaine concerning the analytic solutions. First, the region of stable pulses is always above the curve S, an the lower bounary of the stability region (soli curve) is approximately parallel to the curve S. The istance between this lower bounary an the curve S epens on, m, an n. For small m, n, an this istance is small. This result cannot be obtaine by application of perturbative theory to the soliton solutions of the nonlinear Schröinger equations, because b an e are not small. For given values of n an the hatche regions become wier as jmj increases, an its lower bounary becomes higher. For given values of n an m the lower bounary approaches the curve S as goes to zero. We woul expect that at zero the curve S woul be the onset of instability. Figures 11(a) an 11(b) give a rough iea of how the regions of existence of stable pulses in the plane (b, e) change when m an are change. We now consier other planes in the parameter space in which we foun stable pulselike solutions. Figure 1 shows the region of stable pulses in the plane (n, e) for fixe values of m,, an b as written on the figure. The plot shows that the with of the stripe in Fig. 10 increases largely as n increases. The ashe curve in Fig. 1 shows where the exact analytical solutions are locate for the same set of parameters. Interestingly enough, this curve is also almost parallel to the upper borer of the area of stable pulses but is locate some istance from it. This shows that the analytical solutions are beyon that region an are therefore in total agreement with the stability analysis, unstable. It is also interesting to stuy how the region in which stable pulses exist epens on. Figure 13 shows this epenence for fixe values of m, n, an b. Specifically, n m 0.1 an b 0.5. As the linear excess gain ecreases, the interval of allowe values of e increases, an its central value increases as increases, which logically means that larger linear losses must be compensate (if the rest of the parameters are constant) by an increase in the thir-orer nonlinear gain. For the above values of n, m, an b, Eqs. (9) give e 1; that is, as expecte, above the hatche region. Finally, Fig. 14 shows the area of stable pulses in the plane (m, e) for fixe values of n,, an b. As jmj increases, the interval of allowe values of e becomes wier, an its central value becomes larger. This last observation is also expecte, as it inicates that larger fifth-orer nonlinear losses must be compensate by an increase in the thir-orer nonlinear gain. The with of the stripe becomes infinitesimally small at m The ashe curve represents the points at which the exact analytical solutions are locate for the chosen values of, n, b 0.1, 0.1, 0.5. Again, it can be seen that they are out of the area of stable pulses. However, in this case the istance between the region of stable pulses an the exact analytical solutions increases with m an goes to zero at m! The instability growth rate of the corresponing analytical solution for this value of m becomes negligible. Fig. 10. Numerically foun soliton solutions for b 0.5, n m 0.1, e 0.38 (soli curves), e 0.5 (otte curves) an e 0.66 (ashe curves). (a) Amplitue profile jcj, (b) phase profile arg(c). The circle, the iamon, an the triangle symbols associate with the cases e equal to 0.38, 0.5, an 0.66, respectively, are use in the following figures to locate these solutions in the parameter space.

9 Soto-Crespo et al. Vol. 13, No. 7 /July 1996/J. Opt. Soc. Am. B 1447 In all these simulations the curves in the parameter space at which the analytical solutions exist were outsie the region in which we numerically foun stable pulses. This agrees with our stability analysis of the exact analytical solutions in Section 3 an inicates the nee of further analytical evelopments. Our exact analytical solutions can serve as a basis for this task. Further generalizations can be one exactly with a more general ansatz for the pulse chirp than Eq. (5) or approximately with our solutions as zeroth-orer approximation. The first observation of strictly stable pulselike solutions has been reporte by Thual an Fauve. 5 They foun some points in the parameter space at which stable pulses exist. Rough estimates of the location of the bounaries between fronts an pulselike solutions of CGLE have been one by Hakim, Jacobsen, an Pomeau. 6 However, there is not a sharp bounary between the two classes of solutions. It has been foun by Fig. 11. Region in the (b e) plane in which stable pulselike solutions are foun. Differently hatche areas are for ifferent values of m, which are written in the corresponing regions. All these areas are locate above the curves S (ashe curves). (a) 0.01, n 0.1, an (b) n 0.1. Fig. 1. Region in the plane (n, e) for which stable pulses are possible. b 0.5 an m 0.1. The ashe curve represents the points at which the analytical solution given by Eq. (17) exists. The three symbols (circle, iamon, an triangle) show the locations of the solutions represente in Fig DISCUSSION The propagation ynamics outsie the region of existence of stable pulses epens on the parameters of the equation as well as on the initial pulse shape. Its etaile stuy is beyon the scope of this paper. However, we can extract some general behaviors an explain them in relation to our above results. Clearly, stationary pulses must balance losses an gain. For systems whose parameters are locate below the lower bounary of our hatche regions, pulses amp own as they propagate. The energy flux provie to the initial pulse owing to positive e is less than the energy ecrease owing to linear (b.0,,0) an nonlinear (m,0) losses. The physical processes on the upper bounary are ifferent. Generally the upper bounary in the (b, e) iagram coincies with the line at which fronts have zero velocity. Above this curve, two fronts of the wie pulse iverge from each other, an below the curve two fronts converge to each other, forming a stable pulse at the en of this process. Hence, stable pulses can exist only below this curve. Fig. 13. Region in the plane (, e) in which stable pulses are possible. b 0.5 an n m 0.1. For these parameters the analytical solution exists at e 1. The three symbols (circle, iamon, an triangle) show the locations of the solutions represente in Fig. 10.

10 1448 J. Opt. Soc. Am. B/Vol. 13, No. 7/July 1996 Soto-Crespo et al. REFERENCES Fig. 14. Region in the plane (m, e) in which stable pulses are foun. b 0.5 an n 0.1. The ashe line represents the points at which the analytical solution exists. van Saarlos an Hohenberg 0 that, for given values of the parameters, a variety of fronts an pulselike solutions exist. This agrees with our observations, in which we foun that pulses an positive velocity fronts coexist in some tiny region near the upper bounary. Clearly, stable pulses are the most interesting objects in optics. They are prouce by laser systems an they constitute bits of information, transmitte in optical fiber systems. Hence it is important to know where in the parameter space we can expect stable pulses. In this stuy we fin those regions an, aitionally, show that exact analytic solutions known to ate are beyon these regions. 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