Chirped Lambert W-kink solitons of the complex cubic-quintic Ginzburg-Landau equation with intrapulse Raman scattering
Nisha, Neetu Maan, Amit Goyal, Thokala Soloman Raju, C.N. Kumar
CChirped Lambert W-kink solitons of the complexcubic-quintic Ginzburg-Landau equation withintrapulse Raman scattering
Nisha a,b , Neetu Maan b , Amit Goyal a, ∗ , Thokala Soloman Raju c , C. N. Kumar b a Department of Physics, GGDSD College, Chandigarh 160030, India b Department of Physics, Panjab University, Chandigarh 160014, India c Indian Institute of Science Education and Research (IISER) Tirupati, Andhra Pradesh517507, India
Abstract
In this paper, an exact explicit solution for the complex cubic-quintic Ginzburg-Landau equation is obtained, by using Lambert W function or omega function.More pertinently, we term them as Lambert W-kink-type solitons, begottenunder the influence of intrapulse Raman scattering. Parameter domains are de-lineated in which these optical solitons exit in the ensuing model. We report theeffect of model coefficients on the amplitude of Lambert W-kink solitons, whichenables us to control efficiently the pulse intensity and hence their subsequentevolution. Also, moving fronts or optical shock-type solitons are obtained as abyproduct of this model. We explicate the mechanism to control the intensityof these fronts, by fine tuning the spectral filtering or gain parameter. It is ex-hibited that the frequency chirp associated with these optical solitons dependson the intensity of the wave and saturates to a constant value as the retardedtime approaches its asymptotic value.
Keywords:
Optical solitons, Frequency chirp, Lambert W function, Movingfronts ∗ Corresponding author
Email addresses: [email protected] (Amit Goyal), [email protected] (Thokala Soloman Raju), [email protected] (C. N. Kumar)
Preprint submitted to Physics Letters A June 16, 2020 a r X i v : . [ n li n . PS ] J un . Introduction The complex Ginzburg-Landau equation is a canonical model for weaklynonlinear, dissipative systems and one of the most-studied nonlinear equationsin the physics community. It can be used to describe a vast variety of nonlin-ear phenomena such as superconductivity [1], Bose-Einstein condensation [2],superfluidity [3], strings in field theory [4], liquid crystals [5] and lasers [6, 7].Exact solitary pulse solutions of complex Ginzburg-Landau equation with thecubic nonlinearity are available, but these pulses are stable only in a regionwhere the background is unstable [8]. However, it is possible to create stablesolitary pulses, in a region where background is also stable, with the inclusion ofan extra term, represents delayed Raman scattering, in the model [9, 10]. Theexistence and stability of solitary wave solutions for cubic complex Ginzburg-Landau equation is also studied in the presence of driven term [11, 12, 13].Recently, lot of attention has been paid to obtain exact analytical solutionsfor complex systems modeled by nonautonomous partial differential equations[14, 15, 16, 17, 18].Soliton propagation in fibers with linear and nonlinear gain and spectral fil-tering [19] or pulse generation in fiber lasers with additive pulse mode-lockingor nonlinear polarization rotation [20, 21] have been studied by consideringcomplex cubic-quintic Ginzburg-Landau equation (CQGLE) as model equation.Also Kengne and Vaillancourt have used modified GinzburgLandau equationthat describes the pulse propagation in a lossy electrical transmission line [22].The CQGLE supports a class of localized solutions such as stationary solitons,sources, sinks, moving solitons and fronts with fixed velocity [23, 24, 25]. Apartfrom these solutions, the CQGLE also possesses the solutions with special prop-agation properties: pulsating, creeping, and erupting solitons [26, 27, 28]. Theerupting solitons are those that periodically exhibits explosive instability. Thesesolitons were found numerically [26] and also experimentally in passively mode-locked lasers [29]. The effect of higher-order terms, namely, third-order disper-sion, self-steepening and intrapulse Raman scattering has been investigated on2rupting solitons and it is found that the explosions of an erupting soliton canbe controlled or even canceled due to the inclusion of one or more higher-orderterms [30, 31, 32, 33, 34, 35]. Recently, work has been done to study the tran-sitions of stationary to pulsating solutions [36] and on the selection mechanismof soliton explosions in CQGLE under the influence of higher-order terms [37].Fac˜ao et al. have studied the effect of intrapulse Raman scattering (IRS) onerupting solitons of CQGLE and numerically shown the propagation of stabletraveling solitons for a specific range of IRS parameter [32, 33]. Although, CQ-GLE is a well studied dynamical system, the exact solutions of CQGLE withIRS term have rarely appeared in the literature. However, in Ref. [38], theauthors have presented the exact stationary front solutions for CQGLE withRaman term and generalized these solutions into moving fronts using energyand momentum balance equations for particular cases by assuming either thequintic or Raman term to be zero.In this work, we consider the CQGLE in the presence of IRS and report theexistence of exact localized solutions in the form of dark and front solitons. Thedark solitons are presented by a new kind of kink solution in terms of LambertW function which we shall refer to as Lambert W-kink solitons. The LambertW function is an implicitly elementary function, also known as the product log-arithm, has rich variety of applications in number of areas of physics, computerscience, pure and applied mathematics and ecology, and is defined as the inverseof f ( W ) = W e W [39, 40]. Several well-known problems in electrostatics andin quantum mechanics can be solved with greater ease using the notation ofLambert W function. Biswas et al. used the notation of Lambert W function toobtain soliton solutions of modified nonlinear Schr¨odinger equation using vari-ational principle [41]. This function is also used as a step potential for whichthe one-dimensional stationary Schr¨odinger equation is exactly solved in termsof the confluent hypergeometric functions [42]. Recently, soliton solution inthe form of Lambert W function was obtained for analytically solvable parity-breaking φ model and the results so obtained were compared with kink of φ theory [43]. Apart from Lambert W-kink solitons, we have also explored moving3ront solitons for this model. The evolution of optical solitons can be controlledby judicious choice of model parameters. The frequency chirp is found to bedirectly proportional to the intensity of the wave and saturates at some finitevalue as t → ±∞ . Frequency chirp is a well-known result of the interaction ofthe group velocity dispersion and the nonlinear self phase modulation. Chirpis very useful in the process of optical pulse compression and found potentialapplications in optical communication systems [44, 45, 46]. A significant workhas been done on the existence of chirped solitons in the context of nonlinearoptics [47, 48, 49, 50].
2. Model Equation
We begin our analysis by considering the complex cubic-quintic Ginzburg-Landau equation (CQGLE) with intrapulse Raman scattering (IRS) term iU z + 12 U tt + γ | U | U = iδU + iβU tt + i(cid:15) | U | U − ν | U | U + iµ | U | U + T r ( | U | ) t U, (1)where U is the normalized envelope of the pulse, z and t are the normalizedpropagation distance and retarded time, respectively. For laser system [21],the physical meaning of various coefficients is the following: δ is a constantgain (or loss if negative), β describes spectral filtering or gain dispersion, (cid:15) represents nonlinear gain (or two-photon absorption if negative), µ representsa higher order correction to the nonlinear amplification or absorption, ν is ahigher order correction term to the nonlinear refractive index, T r represents theIRS coefficient and γ represents positive Kerr effect (or negative Kerr effect ifnegative) .
3. Chirped soliton-like solutions
In order to find the exact solution of Eq. (1), we choose the following ansatz U ( z, t ) = ρ ( ξ ) e i ( φ ( ξ ) − kz ) , (2)4here ξ = t − uz is the traveling coordinate, ρ and φ are real functions of ξ . Here u = v , where v indicates the group velocity of the pulse envelope.The corresponding intensity of the propagating pulse is given by | U ( z, t ) | = | ρ ( ξ ) | . The spectral changes introduced across the pulse at any distance z are a direct consequence of time dependence of nonlinear phase shift. Thefrequency change across the pulse is the time derivative of phase and is given by δω ( z, t ) = − ∂∂t [ φ ( ξ ) − kz ] = − φ (cid:48) ( ξ ). This time dependence of δω is referred to asfrequency chirping. Now, substituting Eq. (2) into Eq. (1), and separating outthe real and imaginary parts of the equation, we obtain the following coupledequations in ρ and φ , uρφ (cid:48) + kρ + 12 ρ (cid:48)(cid:48) − ρφ (cid:48) + γρ = − βρ (cid:48) φ (cid:48) − βρφ (cid:48)(cid:48) − νρ + 2 T r ρ ρ (cid:48) , (3) − uρ (cid:48) + ρ (cid:48) φ (cid:48) + 12 ρφ (cid:48)(cid:48) = δρ + βρ (cid:48)(cid:48) − βρφ (cid:48) + (cid:15)ρ + µρ . (4)Assuming that the qualitative features of frequency chirp depend considerablyon the exact pulse shape through the relation δω ( z, t ) = − φ (cid:48) ( ξ ) = − ( Aρ + B ),where A and B are the nonlinear and constant chirp parameters, respectively,the coupled equations given by Eq. (3) and Eq. (4) reduce to ρ (cid:48)(cid:48) +4(2 βA − T r ) ρ ρ (cid:48) +4 βBρ (cid:48) +(2 uB − B +2 k ) ρ +2( uA + γ − AB ) ρ +(2 ν − A ) ρ = 0 , (5) ρ (cid:48)(cid:48) − Aβ ρ ρ (cid:48) + ( u − B ) β ρ (cid:48) + (cid:18) δβ − B (cid:19) ρ + (cid:18) (cid:15)β − AB (cid:19) ρ + (cid:18) µβ − A (cid:19) ρ = 0 , (6)for β (cid:54) = 0. By assuming the following identifications: M ≡ (cid:18) βA − T r = − Aβ (cid:19) , (7) N ≡ (cid:18) βB = u − Bβ (cid:19) , (8)5 ≡ (cid:18) uB + 2 k − B = δβ − B (cid:19) , (9) R ≡ (cid:18) Au − AB + 2 γ = (cid:15)β − AB (cid:19) , (10) S ≡ (cid:18) ν − A = µβ − A (cid:19) , (11)Eqs. (5) and (6) can be mapped into a single equation ρ (cid:48)(cid:48) + M ρ ρ (cid:48) + N ρ (cid:48) + Qρ + Rρ + Sρ = 0 . (12)Solving Eqs. (7)-(11), we obtain the constraint conditions as A = 2 βT r β , u = 1 A (cid:18) (cid:15) β − γ (cid:19) , B = u β k = δ β − uB, µ = 2 βν. (13)Eq. (12) can be solved to obtain exact localized solution for compatible formof first-order differential equation for the function ρ ( ξ ). In this work, we haveexplored the Lambert W-kink and moving front soliton solutions for this equa-tion. In order to explore exact analytical solution of Eq. (12), use shall be madeof the differential equation ρ (cid:48) = ( a − ρ ) ( a − ρ ) , (14)( a is a real parameter here), that admits Lambert W-kink solution of the form[39, 43] ρ ( ξ ) = a (cid:18) −
21 + W ( e a ξ +1 ) (cid:19) , (15)where W represents Lambert W function. The corresponding second-order dif-ferential equation for ρ ( ξ ) reads ρ (cid:48)(cid:48) − ρ + 5 aρ + 2 a ρ − a ρ + a ρ + a = 0 . (16)6or ρ (cid:48) given by Eq. (14), Eq. (12) is consistent with the Eq. (16) by theidentification of various unknown parameters as M = − , N = a , Q = 2 a , R = − a , S = 2. Solving these conditions along with constraints given by Eq. (13),the model coefficients and solution parameter ‘ a ’ fixed as T r = 54 (cid:0) β (cid:1) , (cid:15) = 11 βγ β ) , ν = 1 + 258 β ,δ = γ (cid:0) β (cid:1) β (4 + 5 β ) , a = (cid:114) − γ β . (17) T r Figure 1: Curves of model coefficients versus spectral filtering term ‘ β ’ for γ = − It should be noted that γ will take only negative values, as solution parameter‘ a ’ should be real, and β can be chosen arbitrarily ( β (cid:54) = 0) while the other modelcoefficients depend on β and γ . In Fig. 1, we have presented the allowed valuesof the model coefficients with β lying in the interval [0 ,
1] for γ = −
1. In Fig. 2,we have shown the amplitude profile of Lambert W-kink solution, given by Eq.(15), for different values of the spectral filtering term ‘ β ’ and γ = −
1. Fromthis plot, one can observe that kink wave has large amplitude and becomes moresteep for small values of β as solution parameter ‘ a ’ is inversely proportional to β . It should be noted that, from Fig. 1, these Lambert W-kink solutions arepossible only for δ > , (cid:15) < γ < - - Ξ A m p lit ud e Figure 2: Amplitude profiles of Lambert W-kink solution for different values of β , β = 0 . β = 0 . β = 0 . solitons, for the model equation Eq. (1), reads I W ( z, t ) = a (cid:18) −
21 + W ( e a ξ +1 ) (cid:19) , (18)with chirping given by δω ( z, t ) = − (cid:34) a β (cid:18) −
21 + W ( e a ξ +1 ) (cid:19) − γ β (4 + 5 β ) (cid:35) . (19)The intensity profile of Lambert W-kink soliton is depicted in the Fig. 3(a,b)for γ = − β , 0 . .
5, respectively. Theseintensity profiles are similar to dark solitons (albeit asymmetric in nature) andshows relative compression of pulses with the modulation of spectral filteringterm ‘ β ’. Fig. 3(c) shows the profiles of corresponding frequency chirp δω across the pulse of Lambert W-kink soliton at z = 0. One can observe thatfrequency chirp saturates to negative value as the retarded time approachesits asymptotic limit and the amplitude of chirp can be controlled for judiciouschoice of parameter ‘ β ’. To exemplify the existence of moving front solitons as exact solutions of thismodel, let us consider the differential equation in ρρ (cid:48) = cρ (cid:18) − ρ b (cid:19) , (20)8 t z I w (a) - t z I w (b) - - - - - t ∆ Ω (c)Figure 3: (a,b) Intensity profile of Lambert W-kink soliton for γ = − β , 0 . .
5, respectively. (c) The corresponding chirp profiles for β = 0 . β = 0 . where b, c are real parameters. The explicit moving front soliton is given by[51, 52] ρ ( ξ ) = (cid:112) b (1 + tanh( cξ )) , (21)for b >
0. The corresponding second-order differential equation for ρ ( ξ ) reads ρ (cid:48)(cid:48) − c b ρ + 2 c b ρ − c ρ = 0 . (22)9 - Ξ A m p lit ud e Figure 4: Amplitude profiles of kink solution for different values of β , β = 0 . β = 0 . β = 0 . (cid:15) = 0 . , γ = 1 , δ = − T r = 0 . Substituting Eq. (20) into Eq. (12) and comparing the resultant equation withEq. (22), the solution parameters found to be c = − N ± (cid:112) N − Q , b = ( N + 4 c ) c R + M c ) , (23)along with constraint on model coefficient ν = A + cM b − c b . Here, theparameters M, N, Q and R can be obtained from Eqs. (7)-(10), using Eq. (13),for different values of the model coefficients β, (cid:15), γ, δ and T r . For illustrativepurpose, we choose the model coefficients (cid:15) > γ > δ <
0, just oppositeto the case of Lambert W-kink solution, to depict the evolution of kink solutions.The amplitude profile of kink solution is shown in Fig. 4 for (cid:15) = 0 . , γ = 1 , δ = − , T r = 0 . β . The expression of the intensity for thesekink solution can be written as I K ( z, t ) = b (1 + tanh( cξ )) . (24)The corresponding chirping is given by δω ( z, t ) = − (cid:18) u + 2 βT r b (1 + tanh( cξ ))1 + 4 β (cid:19) . (25)10 t z I K (a) - t z I K (b) - - - - - t ∆ Ω (c)Figure 5: (a,b) Intensity profile of front soliton for different values of β , 0 . .
5, respec-tively. (c) The corresponding chirp profiles for β = 0 . β = 0 . The intensity and chirp profiles for front solitons are shown in Fig. 5 for differentvalues of β . For β = 0 .
3, the frequency chirp approaches negative value as t → ±∞ while it approaches negative and positive value as t → + ∞ and t → − ∞ , respectively, for β = 0 .
5. 11 . Conclusion
In conclusion, we have shown the existence of exact explicit solution forthe complex cubic-quintic Ginzburg-Landau equation in terms of Lambert Wfunction or omega function, under the influence of of intrapulse Raman scat-tering. Parameter domains are delineated in which these optical solitons exitin the ensuing model. It is observed that these optical solitons are possible fornegative values of nonlinear gain and Kerr effect and positive value of constantgain. Whereas, no such restrictions are imposed on the moving fronts or opticalshock-type solitons that are obtained as a byproduct of this model. We haveobserved that the intensity of these fronts has been doubled with a slight changein the value of the spectral filtering or gain parameter. The frequency chirp as-sociated with these nonlinear waves has been identified. Furthermore, we haveexplicated the pivotal role played by this nonlinear chirp on the intensity ofthese waves. These results may be useful for experimental realization of undis-torted transmission of optical waves in optical fibers and further understandingof their optical transmission properties. Finally, we hope that the exact natureof these nonlinear waves presented here may be profitably exploited in designingthe optimal Raman fiber laser experiments.
5. Acknowledgment
A.G. gratefully acknowledges Science and Engineering Research Board (SERB),Department of Science and Technology, Government of India for the award ofSERB Start-Up Research Grant (Young Scientists) (Sanction No: YSS/2015/001803)during the course of this work. Nisha is thankful to SERB-DST, India for theaward of fellowship during the work tenure. We sincerely thank the referees fortheir useful comments.