PHYSICAL REVIEW E VOLUME 58, NUMBER 5 NOVEMBER 1998 Moving fronts for complex Ginzburg-Lau equation with Raman term Adrian Ankiewicz Nail Akhmediev Optical Sciences Centre, The Australian National University, Canberra, Australian Capital Territory 000, Australia Received 6 January 1998 Moving fronts, or optical shock-type solitons, are discussed for systems with gain loss under the influence of the Raman effect. We present energy momentum segment balance equations establish the exact moving front solutions. We also show here that stationary moving fronts also exist when we allow for various other nonlinear terms in the modified Ginzburg-Lau equation. S1063-651X 98 09611-1 PACS number s :.65.Tg I. INTROUCTION The nonlinear Schrödinger equation NSE is a model equation describing a variety of short-pulse propagation phenomena in optics 1,. The NSE with nonconservative terms added is usually called the complex Ginzburg-Lau equation CGLE. Particular areas of application of the CGLE are all-optical fiber transmission lines passively modelocked fiber solid-state lasers. The NSE can be modified to include the influence of various physical phenomena on short-pulse generation propagation. The behavior of ultra-short pulses changes under the influence of these terms. For example, third-order dispersion results in pulse asymmetry radiation phenomena. Fourth-order dispersion may result in solitons with oscillating tails. Selfsteepening causes the leading edge of a pulse to become more sharp 1. There are various works on optical wave breaking as well. The inclusion of the Raman term results in a continuous downward shift in pulse frequency 3 6.In the time domain, this represents the fact that the glass optical fiber response to the imposed field is not instantaneous. This delay of a few femtoseconds fs can affect propagation of fs signals. Gagnon Belanger 7 showed that the exact form of the soliton self-frequency shift follows from a symmetry analysis of the equation. Kink-type solutions for the NSE with Raman term present were discovered in 8. In nonlinear optics, a kink is a shock wave which propagates undistorted in a dispersive nonlinear medium. Interestingly enough, when gain loss terms are added to the NSE with the Raman term, the kink solution can still exist. In contrast to the NSE case, the front moves with a certain velocity which depends on the parameters of the equation. In the present work we consider moving front solutions for the modified Ginzburg-Lau equation. We have developed a special technique to find the solution in analytic form which is based on energy momentum balance equations 9. In order to do this, we use the ansatz which follows from the point symmetry of the equation use this ansatz in the energy balance equations. Allowing for nonzero velocity generalizes this ansatz to cover the case of moving fronts. This technique allows us to find the coefficients of the solution in terms of the equation coefficients. We consider here two examples which allow us to obtain the solution this way: the CGLE with the Raman term the CGLE with quintic terms. Other cases can also be considered exact solutions can be found. II. FUNAMENTAL EQUATION For a field (z, ) in a fiber with nonlinearity N, we may write the modified NSE in the following form: i z tt N 0. Here, the group delay dispersion coefficient, is positive for anomalous dispersion, negative for normal dispersion. When we introduce the nonlinear nonconservative terms, consider the Raman self-steepening effects, we obtain 10 13 i z i tt 1 i i R t is t i. Here the term with s, which is the self-steepening coefficient, that with R, which is the Raman coefficient, modify the complex Ginzburg-Lau equation. Inthe CGLE, z is the normalized distance, t is the retarded time, is the normalized envelope of the optical field, describes the gain spectrum, is a constant gain or loss if negative, is a nonlinear gain or two-photon absorption if negative. In general, some of these terms, which are additional to those in the nonlinear Schrödinger equation, are needed in the analysis of soliton pulse propagation in optical fibers,1. We seek a suitable ansatz which can be used to solve some cases of the above equation. To do this, we first analyze the special case with 0, then use the same functional form in more general cases. III. ANSATZ Appropriate transformations for finding solutions for these types of equations can be formally described by considering the point-symmetry group of the equation. Each transformation then has an infinitesimal operator or generator associated with it 15,16. Thus by setting U(t)e i z, allowing for possible phase chirp by using U 1 1063-651X/98/58 5 /673 5 /$15.00 PRE 58 673 1998 The American Physical Society
67 ARIAN ANKIEWICZ AN NAIL AKHMEIEV PRE 58 F(t)exp i (t), with F real, we obtain a complex equation. The real part is F tt F t F F R F F t sf 3 t F 5, 3 while the imaginary part is d dt F t 6sF 3 F t 3s d dt F. Equation is nontrivial only if s is nonzero. In that case, we have t 3s F c/f, c is a constant of integration. Thus Then Eq. 3 transforms to 3s F dt c dt F. F tt F 3 F sc R F F t F 5 c F 3, 6 3 s 1 3. We now let F w. This converts Eq. 6 to w tt w t 8w w w sc R ww t w 3 c w. 8 We look for solutions with approaching zero at one end the low end, approaching a nonzero constant at the other high end. If we set the constant c to zero, it is clear that Eq. 8 can be solved using w P 1 tanh( t), P are constants. This is true because w t w 1 tanh t, then each term in Eq. 8 is of the form tanh n ( t) for n 0, 1,, or 3. We could equally well use the mirror image function (t t) for w. Then 3s wdt 3sP t log cosh t. Therefore the suitable ansatz for stationary front solutions is P 1 tanh t e id log cosh t e ikt i z, d k are constants. 5 7 9 We are interested in moving front solutions. Hence, we can allow for solutions to move with velocity V by using t t Vz in Eq. 9. In what follows, we prove that this form also gives the required solution. The real parameters of the solution (d, P,, k, V, will be expressed in terms of the equation parameters (,,,, R,,. To relate the parameters, we will use an original technique of balance equations. IV. SEGMENT ENERGY BALANCE FOR FRONT SOLUTIONS First we consider the balance equations in general form with all the coefficients in Eq. being nonzero. Let us convert to the moving group velocity frame by setting t Vz. Then, using the ansatz,z f exp iz KV, we substitute it into Eq.. The resulting equation is f ivf KV f f f is f f i f i f i f f i f f R f f. 10 By multiplying by f *, taking the complex conjugate, subtracting the two expressions, integrating over, wefind that W f 3 s f V f f d f f 6 d f d, 11 W Im(f f*). This equation is a consequence of the energy balance. We may set y t,, for convenience, set the lower upper limits at 0 y, respectively, both for the evaluations on the left the integrals on the right. The Raman term only causes a frequency shift not a gain or loss of energy, so it does not affect the segment energy balance equation for the front solutions. By using the above ansatz we can thus find the gain/loss contribution of each term. For a moving front solution the net gain must be zero. Following this procedure, the terms contain the independent functions y, tanh(y), tanh (y), log cosh(y). The overall coefficient of each term must therefore be zero. In this case the coefficients for the terms y log cosh(y) are the same: P P k d 0. Equating coefficients of tanh(y) we find k d V d kd 1 P 6 P 3sP, 1 13
PRE 58 MOVING FRONTS FOR A COMPLEX GINZBURG-LANAU... 675 while equating those for tanh (y) we obtain d P d 3 6sP. 1 n 0 ka d 3k k3 R P 3, V. SEGMENT MOMENTUM BALANCE FOR FRONT SOLUTIONS On the other h, by multiplying Eq. 10 by f *, taking the complex conjugate, adding the two expressions, we find that kv f f 1 f W 3 d d f 6 Ws d d f i f f* f* f f W f W R d d f. 15 By integrating with respect to we find kv f f 1 f 3 f 6 Wg d Im f f * d R d d f d, g f f s d d f. 16 This equation is the result of the balance of momentum. We now represent the complex function f using Eq. 9 in the form which allows for nonzero velocity. This leads to five momentum balance equations: n 0 n n 0, n 1 n 3 0, kv kd k P P 3 n n 0, 17 18 19 d kd 1 P P 3 n 3 0, 0 d 1 3 d P dp P s P R 0, 1 n 1 kh a k d k 3d k, n kh 3 h k d ad h R P 3, n 3 h 3 k d d h h 5, n d h 3 d 3 d 1 R P 3. The coefficients used in the above equations are a P P s, h P P, h 3 P P s, h 3 d k k d 1, h 5 d d 3 k 3d 1. These equations allow us to find the coefficients of the solution in terms of the coefficients of the fundamental equation. Henceforth, we set s 0, but of course, this restriction is not necessary when seeking other types of solutions. VI. EXAMPLE 1: MOVING RAMAN FRONT SOLUTIONS As an example, let us consider moving Raman front soliton solutions. Thus we set s 0, but take R 0. Equation then becomes i z i tt 1 i i R t. By using the solution of form 9, we can equate coefficients of the powers of tanh (t Vz) find the required constants. We obtain which is clearly real, d 3, P d 3/8 d / / R. For convenience, we now define r P 1 d / 1 d a 1 3 / d/ 16,
676 ARIAN ANKIEWICZ AN NAIL AKHMEIEV PRE 58 b 1 r d P c 1 r /. Then b 1 b 1 a 1 c 1 / a 1 k r. The only restriction is that must be real, i.e., b 1 a 1 c 1. The velocity is V P k d d. Finally, the frequency shift is P R 8 d k k P. These formulas can be used to present the relation between the equation solution parameters in a simple way. From Eq. 9 we see that when t changes from 1/ to 1/, the front intensity increases from 0.69 of its maximum to 0.731 of its maximum. We thus define the width of the front to be 1/. In fact, at t 1/, the intensity is 0.1 of its maximum, while at t 1/, it is 0.88 of its maximum. One example, giving the Raman kink width, height, velocity versus for a given set of parameters, is shown in Fig. 1. Other dependencies of the soliton parameters versus parameters of the equation can also be presented on similar plots. The stationary (V 0) kink solution, obtained in 8, isa limit of this solution when, 0 0. Then most solution coefficients are zero, but 3 R Thus the above solution reduces to 9 8 R. m 1 tanh mt e im z m e mt sech mt e im z, m 3/ R, in agreement with 8. VII. EXAMPLE : FRONT SOLUTION FOR QUINTIC EQUATION 3 We now consider the front solution for the quintic equation, i.e., we take s R 0 but, 0. A Painleve analysis of the equation for this situation has been presented in 17. We define b ( )/( ). Then d b b 3 FIG. 1. a The width as defined in text, b the height, c the velocity of the kink solution of Eq. as a function of. The parameters of the equation are given in the diagrams. We now introduce P d d 3. g r d, r 8P d 1 d, a 3 d 3 P gr,
PRE 58 MOVING FRONTS FOR A COMPLEX GINZBURG-LANAU... 677 b 3 r g r P, r is as defined in the preceding section. Then 1 a b 3 b 3 a r the velocity is now k r r, V P k d d P, while the frequency offset is 8 d P P k k. The actual solutions in these two sections satisfy both the energy Eqs. 1 1 momentum Eqs. 17 1 balance equations. 0 solution family As pointed out in 17, when 0, there is a oneparameter family of solutions, as long as a consistency condition relating the equation parameters is satisfied. This solution has the same functional form as that above, with solution parameters simplifying to Furthermore, P b / P d/. 1 d 8 d 1 0d. 5 Now V k can be written in terms of the equation parameters. The consistency condition is 3 d P c c 3, c 5 d d 1, c 3 5d d 1. 6 For example, we can specify,,, then use Eq. 6 to find. Then, with V as an arbitrary velocity, we have k V P c c 3, with given by Eq. with 0). Thus moving front solutions exist even when the Raman term is absent. VIII. CONCLUSION We have found moving-front-type solitons of the extended CGLE. We have used the novel energy momentum segment balance equations for analytical calculations. This method shows clearly the contribution of each term to the overall physical balance. In particular, the weighted losses gains must add to zero for a valid front solution. The method allows us to find soliton solutions in a number of cases. In this paper we considered only particular cases of kinks for the equation with the Raman term for the quintic CGLE. Other types of solutions e.g., pulses solutions for the equation with other terms can be found in a similar fashion. 1 G. P. Agrawal, Nonlinear Fiber Optics, nd ed. Academic Press, Inc., New York, 1995. N. N. Akhmediev A. Ankiewicz, Solitons: Nonlinear Pulses Beams Chapman Hall, London, 1997. 3 F. M. Mitschke L. F. Mollenauer, Opt. Lett. 11, 659 1986. E. M. ianov, A. Ya. Karasik, P. V. Mamyshev, A. M. Prokhorov, V. N. Serkin, M. F. Stel makh, A. A. Fomichev, Pis ma Zh. Eksp. Teor. Fiz. 1, 9 1985 JETP Lett. 1, 9 1985. 5 J. P. Gordon, Opt. Lett. 11, 66 1986. 6 K. J. Blow, N. J. oran,. Wood, J. Opt. Soc. Am. B 5, 1301 1988. 7 L. Gagnon P. A. Belanger, Opt. Lett. 15, 66 1990. 8 G. P. Agrawal C. Headley, Phys. Rev. A 6, 1573 199. 9 N. Akhmediev, A. Ankiewicz, J. M. Soto-Crespo, Phys. Rev. Lett. 79, 07 1997. 10 A. Ankiewicz N. N. Akhmediev, Opt. Commun. 1, 95 1996. 11 J. Herrmann, Opt. Commun. 87, 161 199. 1. N. Christodoulides, Opt. Commun. 86, 31 1991. 13 K. Hayata M. Koshiba, J. Opt. Soc. Am. B 11, 61 199. 1 C. Goedde, W. L. Kath, P. Kumar, Opt. Lett. 19, 077 199. 15 M. Florjańczyk L. Gagnon, Phys. Rev. A 1, 78 1990. 16 M. Florjańczyk L. Gagnon, Phys. Rev. A 5, 6881 199. 17 P. Marcq et al., Physica 73, 305 199.