Anti-dark and Mexican-hat solitons in the Sasa-Satsuma equation on the continuous wave background
aa r X i v : . [ n li n . S I] F e b Anti-dark and Mexican-hat solitons in the Sasa-Satsuma equationon the continuous wave background
Tao Xu , Min Li and Lu Li
1. College of Science, China University of Petroleum, Beijing 102249, China,E-mail: [email protected]. Department of Mathematics and Physics, North China ElectricPower University, Beijing 102206, China.3. Institute of Theoretical Physics, Shanxi University, Taiyuan, Shanxi 030006, China.
Abstract
In this letter, via the Darboux transformation method we construct new analytic solitonsolutions for the Sasa-Satsuma equation which describes the femtosecond pulses propagation ina monomode fiber. We reveal that two different types of femtosecond solitons, i.e., the anti-dark(AD) and Mexican-hat (MH) solitons, can form on a continuous wave (CW) background, andnumerically study their stability under small initial perturbations. Different from the commonbright and dark solitons, the AD and MH solitons can exhibit both the resonant and elasticinteractions, as well as various partially/completely inelastic interactions which are composedof such two fundamental interactions. In addition, we find that the energy exchange betweensome interacting soliton and the CW background may lead to one AD soliton changing into anMH one, or one MH soliton into an AD one.PACS numbers: 05.45.Yv; 42.65.Tg; 42.81.Dp
Introduction. — As localized wave packets formed by the balance between the group-velocitydispersion and self-phase modulation [1], solitons in optical fibers have drawn considerable attentionbecause of their robust nature and potential application in all-optical, long-distance communica-tions [2]. In the picosecond regime, the model governing the propagation of optical solitons in a1ingle-mode fibre is the celebrated nonlinear Schr¨odinger equation (NLSE) [1]. However, one hasto take into account some higher-order linear and nonlinear effects for the ultrashort pulses prop-agating in high-bit-rate transmission systems [2]. The governing model for the femtosecond pulsepropagation is the following higher-order NLSE [3]:i u z + σ u tt + | u | u = − i ε (cid:2) σ u ttt + σ ( | u | u ) t + σ u ( | u | ) t (cid:3) , (1)where σ = ± ε is a real small parameter, σ , σ and σ represent the third-order dispersion (TOD),self-steepening (SS, also known as Kerr dispersion) and stimulated Raman scattering (SRS) effects,respectively [2, 3].With σ = 0, Eq. (1) has two important integrable versions: (i) the Hirota equation (HE) [4], σ : σ : σ + σ = 1 : 6 σ : 0; (ii) the Sasa-Satuma equation (SSE) [5], σ : σ : σ + σ = 1 : 6 σ : 3 σ .The SSE usually takes the form [5]i u z + σ u tt + | u | u + i ε (cid:2) u ttt + 6 σ ( | u | u ) t − σu ( | u | ) t (cid:3) = 0 . (2)Although there is a fixed relation among the higher-order terms, the SSE is thought to be morefundamental than the HE for applications in optical fibers because the former contains the SRSterm [2, 3]. Up to now, many integrable properties of Eq. (2) have been detailed, like the inversescattering transform scheme [5, 6], bilinear representation [7], Painlev´e property [8], conservationlaws [9], nonlocal symmetries [10], squared eigenfunctions [11], B¨acklund transformation [12] andDarboux transformation (DT) [13, 14].The presence of the SRS term enriches the solitonic behavior in Eq. (2) [5–7, 14–18]. Underthe vanishing boundary condition (VBC), Eq. (2) with σ = 1 possesses the common single-humpsoliton [5, 15], the double-hump soliton behaving like two in-phase solitons with a fixed separation [5,6], and the multi-hump breather with the periodically-oscillating structure [6, 7]. Under the non-vanishing boundary condition, Eq. (2) with σ = − σ = 1 has the bright-like soliton which is linearly combined of a dark one and abright one [18]. On the other hand, the soliton interaction behavior underlying in the SSE is farmore abundant and complicated than that in the NLSE. Even with the VBC, the shape-changinginteractions between soliton and breather have recently been found in Eq. (2) with σ = 1 [14].2n this letter, we are trying to reveal some novel solitonic phenomena on a continuous wave(CW) background for Eq. (2) with σ = 1. Via the DT technique [14], we obtain three families ofsingle-soliton solutions which can display two completely-different profiles. The first type is theanti-dark (AD) soliton having the form of a bright soliton on a CW background, i.e., it looks like adark soliton with reverse sign amplitude [19]. The second type takes the Mexican-hat (MH) shape,that is, one high hump carries two small dips which have a symmetrical distribution with respectto the hump, hence such new type of soliton is called the MH soliton. More importantly, we findthat the femtosecond AD and MH solitons admit the resonant interaction, elastic interaction, aswell as various partially/completely inelastic interactions which consist of the fundamental resonantand elastic interaction structures. To our knowledge, it is the first time that the coexistence ofelastic and resonant soliton interactions has been found in the NLSE-type models. Physically,the resonant interaction of optical waves excited from a CW background can be used to realize asecond-harmonic generation in the centro-symmetry optical fiber [20]. N-th iterated soliton solutions via the Darboux transformation. — With the simple CW solution u = ρ e i( t ε − z ε + φ ) ( ρ > φ are both real constants) as a seed, we employ the DT-iteratedalgorithm presented in Ref. [14] to obtain the N-th iterated solution in the form u N = e i( t ε − z ε ) (cid:18) ρ e i φ − τ N +1 ,N − ,N τ N,N,N (cid:19) , (3)with τ J,K,L = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F N × J − G N × K − H N × L F ∗ N × J − H ∗ N × K − G ∗ N × L G ∗ N × J F ∗ N × K N × J F N × K ∗ N × J ∗ N × L G N × J N × L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (4)where J + K + L = 6 N , the block matrices F N × J = (cid:0) λ m − k f k (cid:1) ≤ k ≤ N, ≤ m ≤ J , G N × K = (cid:2) ( − λ k ) m − g k (cid:3) ≤ k ≤ N, ≤ m ≤ K ,3 N × L = (cid:2) ( − λ k ) m − h k (cid:3) ≤ k ≤ N, ≤ m ≤ L , and the functions f k , g k , h k (1 ≤ k ≤ N ) are given as f k = e i φ (cid:16) α k e θ k ( t,z ) + β k e − θ k ( t,z ) (cid:17) ,g k = − e − i φ (cid:18) ρα k χ + k e θ k ( t,z ) + ρβ k χ − k e − θ k ( t,z ) − γ k e − ω k ( t,z ) (cid:19) ,h k = − e φ (cid:18) ρα k χ + k e θ k ( t,z ) + ρβ k χ − k e − θ k ( t,z ) + γ k e − ω k ( t,z ) (cid:19) , (5)with θ k ( t, z ) = χ k (cid:2) t − z ε − (cid:0) λ k + ρ (cid:1) ε z (cid:3) , ω k ( t, z ) = λ k (cid:0) t − z ε − λ k ε z (cid:1) , χ k = q λ k − ρ , χ ± k = λ k ± q λ k − ρ , α k , β k and γ k being nonzero complex constants.In order to obtain the solitonic structure from solution (3), we require all χ k ’s (1 ≤ k ≤ N ) bereal numbers, that is, Im( λ k ) = 0 and | λ k | > √ ρ . For convenience of our analysis, we introducethe notations µ (1) k = α ∗ k β k − α k β ∗ k , µ (2) k = α ∗ k γ k + α k γ ∗ k and µ (3) k = β ∗ k γ k + β k γ ∗ k ( k = 1 , n, m )” to represent the soliton interaction with n asymptotic solitons as z → −∞ and m ones as z → ∞ . (a) (b) - - t È u È (c) Figure 1: Evolution of the (a) AD soliton and (b) MH soliton plotted via solution (6), where | α | = | β | = 1, λ = 1 . ρ = 1, φ = 0, ε = 0 . φ (1)1 = 0 for (a) and φ (1)1 = π for (b). (c)Transverse plots of AD (blue dotted line) and MH (red solid line) solitons at z = 0. Anti-dark and Mexican-hat solitons. — For solution (3) with N = 1, we can obtain threefamilies of single-soliton solutions under the reducible cases µ ( i )1 = 0 (1 ≤ i ≤ µ (1)1 = 0 (i.e., β α ∗ − α β ∗ = 0), the solution can be written as u (1)1 = ρ e i( t ε − z ε + φ + π ) + √ χ e i( t ε − z ε + φ + π ) √ ρ + | λ | e i φ (1)1 cosh (cid:0) Θ (1)1 + δ (1)1 (cid:1) , (6)4here Θ (1)1 = 2 θ ( t, z ), δ (1)1 = ln (cid:16) | α | χ − | β | χ +1 (cid:17) , φ (1)1 = Arg( α ) − Arg( β ) = 0 or π . In this solution,the first part ρ e i( t ε − z ε + φ + π ) is a CW solution of Eq. (2), while the second part describes asoliton embedded in the CW background (Note that the denominator has no singularity because | λ | > √ ρ ).The parameter φ (1)1 = 0 implies that the embedded solution has the same phase as that of theCW solution. In this case, u (1)1 represents an AD soliton which displays the bright soliton profile onthe CW pedestal [see Figs. 1(a) and 1(c)]. The soliton velocity and width are, respectively, givenby v = 4 ε (cid:0) ρ + λ (cid:1) + ε and w = χ , and | u (1)1 | reaches the maximum | u (1)1 | max = ρ | λ | + √ ( λ − ρ )( | λ | + √ ρ )when Θ (1)1 = 0. If φ (1)1 = π , the embedded solution and CW solution have the same phases inthe inner region √ ( ρ + χ ) − χ √ λ − ρ ρ | λ | ≤ Θ (1)1 + δ (1)1 ≤ √ ( ρ + χ ) + χ √ λ − ρ ρ | λ | , but their phases areopposite in the outer region Θ (1)1 + δ (1)1 < √ ( ρ + χ ) − χ √ λ − ρ ρ | λ | or Θ (1)1 + δ (1)1 > √ ( ρ + χ ) + χ √ λ − ρ ρ | λ | .Hence, the modulus of u (1)1 exhibits that one high hump is symmetrically accompanied with twosmall dips beneath the CW background, which looks like the MH shape [see Figs. 1(b) and 1(c)].The velocity and width of the MH soliton are the same as those of the AD one, but its maximumamplitude drastically increases to | u (1)1 | max = √ ( ρ + χ ) − ρ | λ || λ |−√ ρ at the center of the hump, and dropsto zero at the centers of two dips. The generation of the MH soliton could be explained as that thephase oppositeness makes some energy be transferred from the CW background to the embeddedsolution, and further leads to rising of one hump and sinking of two dips.For the reducible cases µ (2)1 = 0 and µ (3)1 = 0, we can obtain the other two families of single-soliton solutions as follows: u (2)1 =e i( t ε − z ε + φ ) h ρ tanh (cid:0) Θ (2)1 + δ (2)1 (cid:1) + e i φ (2)1 q λ χ +1 sech (cid:0) Θ (2)1 + δ (2)1 (cid:1)i , (7) u (3)1 =e i( t ε − z ε + φ + π ) h ρ tanh (cid:0) Θ (3)1 + δ (3)1 (cid:1) + e i φ (3)1 q λ χ − sech (cid:0) Θ (3)1 + δ (3)1 (cid:1)i , (8)where Θ (2)1 = θ ( t, z )+ ω ( t, z ), Θ (3)1 = θ ( t, z ) − ω ( t, z ), δ (2)1 = ln (cid:16) | α | χ | γ | λ χ +1 (cid:17) , δ (3)1 = ln (cid:16) | γ | λ χ − | β | χ (cid:17) , φ (2)1 = Arg( α ) − Arg( γ ) = ± π and φ (3)1 = Arg( β ) − Arg( γ ) = ± π . Because | λ | > √ ρ , either u (2)1 or u (3)1 displays only the AD soliton profile, which is similar to the case φ (1)1 = 0 in solution (6).Solutions (7) and (8) are also called the combined solitary wave solutions [18]. The combineddark and bright solitons have a constant phase difference π or − π . Such phase difference causes a5onlinear phase shift, for example, the nonlinear phase shift in solution (7) can be given as φ (2)NL ( t, z ) = arctan (cid:20) sin φ (2)1 · q λ χ +1 sech (cid:0) Θ (2)1 + δ (2)1 (cid:1) ρ tanh (cid:0) Θ (2)1 + δ (2)1 (cid:1) (cid:21) . (9) −60 −30 0 30 60 −30 −15 0 15 300.40.81.2 zt |u| (a) −60 −30 0 30 60 −30 −15 0 15 300.40.81.2 zt |u| (b) Figure 2: Numerical evolution of the AD soliton under the perturbation of a white noise with themaximal value 0 .
08. The initial pulse corresponds to solution (6) at z = 0 with the parameters as α = β = λ = 1, ρ = 0 . ε = 0 .
13, (a) σ : σ : σ + σ = 1 : 6 : 3, (b) σ : σ : σ + σ = 1 : 5 . . σ : σ : σ + σ = 1 : 5 . . z . Thus, the balance between the energy input and output alsoplays an important role in maintaining a long-lived optical soliton [22]. (a) (b) Figure 3: (a) Resonant (2 , λ = 1 . γ = 1 + 2 i. (b)Resonant (1 , λ = − .
42 and γ = 1 − α = 1, β = 1 + i, ρ = 1, φ = 0 and ε = 0 . Resonant and elastic interactions. — If µ ( i )1 = 0 (1 ≤ i ≤
3) in solution (3) with N = 1, thephase difference (which is neither π nor ± π ) between the embedded solution and the CW solutionresults in that there are three asymptotic solitons appearing on top of the same CW background.The asymptotic expressions of the three solitons as z → ±∞ have the same form in Eqs. (6)–(8)except that φ (1)1 = π (cid:2) − sgn (cid:0) µ (2)1 µ (3)1 (cid:1)(cid:3) , δ (1)1 = 12 ln | µ (2)1 | χ − | µ (3)1 | χ +1 ! , (10) φ (2)1 = π (cid:0) i µ (1)1 µ (3)1 (cid:1) , δ (2)1 = 12 ln | µ (1)1 | χ | µ (3)1 | λ χ +1 ! , (11) φ (3)1 = − π (cid:0) i µ (1)1 µ (2)1 (cid:1) , δ (3)1 = 12 ln | µ (2)1 | λ χ − | µ (1)1 | χ ! . (12)Their wave numbers and frequencies can be respectively given as follows: ( K (1)1 , Ω (1)1 ) = (cid:2) − ε (cid:0) ρ + λ (cid:1) χ − χ ε , χ (cid:3) , ( K (2)1 , Ω (2)1 ) = (cid:2) − χ +1 ε − ε ( χ +1 ) (cid:0) λ + χ − (cid:1) , λ + χ (cid:3) , ( K (3)1 , Ω (3)1 ) = (cid:2) χ − ε + 2 ε ( χ − ) (cid:0) λ + χ +1 (cid:1) , χ − λ (cid:3) , which exactly satisfy the three-soliton resonant conditions K (1)1 = K (2)1 + K (3)1 and Ω (1)1 = Ω (2)1 + Ω (3)1 . Associated with λ > λ <
0, the solu-7ion can, respectively, exhibit the (2 , , (cid:0) µ (2)1 µ (3)1 (cid:1) = 1, the three resonant solitons all belongs to the AD case; whilefor sgn (cid:0) µ (2)1 µ (3)1 (cid:1) = −
1, two are still the AD solitons but the other one is of the MH shape. Thatmeans that the CW background exchanges its energy with one interacting soliton, and causes suchsoliton changes its shape after resonant interaction, as shown in Fig. 3(b). (a) (b)
Figure 4: (a) Shape-preserving elastic (2 , α = 1 + i. (b) Shape-changingelastic (2 , α = 1. The other parameters are chosen as α = 1, β = i, β = − γ = 1, γ = 1 − i, ρ = 0 . λ = − λ = 1 . φ = 0 and ε = 0 . , z → ±∞ in solution (3)with N = 2 under the condition µ ( i )1 = µ ( j )2 = 0 (1 ≤ i, j ≤ φ ( i ) k and δ ( i ) k ( k = 1 ,
2; 1 ≤ i ≤
3) (details are omitted for saving the space). Each interacting solitoncould be either the AD or MH one, depending on the concrete parametric choice. For example,Fig. 4(a) illustrates that the AD solitons display the standard elastic interaction, that is, they cancompletely recover their individual intensities and velocities after an interaction except for the phaseshift in their envelops. Note that the phase shift, which corresponds to the instantaneous frequencyat pulse peak being nonzero, will result in the relative motion of interacting solitons [23]. Also, the8nergy exchange may take place between some interacting soliton and the CW background, andresult in the shape change of such soliton after interaction, as seen in Fig. 4(b). However, this kindof soliton interactions are still considered to be elastic in the sense that there is no energy exchangebetween two different solitons.
Partially and completely inelastic interactions. — For other cases in solution (3) with N = 2,one can obtain five different types of inelastic soliton interactions. If there is only one µ ( i ) k equalto 0, the solution can exhibit the (3 , , λ k − > λ k − < z → ±∞ are not equal, but one z → −∞ soliton and one z → ∞ soliton [which are marked by the red arrows in Figs. 5(a) and 5(b)] have the same velocities andintensities and differ only by the phases of their envelops. Accordingly, the (3 , , partially inelastic type . If none of µ ( i ) k ’s (1 ≤ i ≤
3, 1 ≤ k ≤
2) is equal to0, the solution can display the (3 , , , λ λ < λ , λ > λ , λ < completely inelastic type in the sense that the asymptotic solitons as z → −∞ totallydiffer from those as z → ∞ in the velocities and intensities. ❄ ❄ (a) ❄❄ (b) Figure 5: (a) Inelastic (3 , β = 1 + i, β = 2 + i, γ = 0 . λ = 1 . λ = − .
73. (b) Inelastic (2 , β = i, β = 1 + i, γ = 1, λ = 1 .
73 and λ = − .
5. The other parameters are chosen as α = α = γ = ρ = 1, φ = 0 and ε = 0 . a) (b) (c) Figure 6: (a) Inelastic (3 , α = 4, β = −
16 i, β = 1 − γ = 2 − i, γ = 0 . λ = 1 . λ = − .
73. (b) Inelastic (4 , α = 1 − i, β = 5 i, β = 10 i, γ = 1 + 10 i, γ = − i, λ = 1 . λ = 1 .
73. (c) Inelastic (2 , α = 1 − . β = 1 −
10 i, β = i, γ = 1, γ = i, λ = − . λ = − .
73. The otherparameters are chosen as α = 1, ρ = 1, φ = 0 and ε = 0 . z → ±∞ are in general not the same. Conclusion. — In this letter, via the DT method we have constructed new analytic solitonsolutions for Eq. (2) which governs the propagation of femtosecond pulses in a monomode fiberwith the TOD, SS and SRS effects. We have revealed that two new types of femtosecond soli-tons (i.e., the AD and MH solitons) occur in Eq. (2) with σ = 1 on a CW background. Thenumerical experiments have indicated that the AD soliton can propagate stably for a long distancewith presence of a small initial perturbation or slight violation of the fixed ratio of parametersin Eq. (2). More importantly, we have obtained that the AD and MH solitons can exhibit boththe resonant and elastic interactions. Such two fundamental interactions can generate variouscomplicated structures, in which the numbers, velocities and intensities of interacting solitons are10sually not the same before and after interaction. In addition, we have found that some interactingsoliton may exchange its energy with the background in the interaction, which results in one ADsoliton changing into an MH one, or one MH soliton into an AD one. It should be noted thatchanging the propagation direction of optical solitons is an important concept for realizing opticalswitching [24]. Therefore, as a self-induced Y-junction waveguide, the soliton resonant interactionmight bring about some applications in all-optical information processing and routing of opticalsignals [2, 24]. In mathematics, our results will enrich the knowledge of soliton interactions in a(1+1)-dimensional integrable equation with the single field. It is also worthy of being studied tomake a finer classification of soliton interactions in Eq. (2) with σ = 1. Acknowledgements. — This work has been supported by the Science Foundations of China Uni-versity of Petroleum, Beijing (Grant No. BJ-2011-04), by the National Natural Science Foundationsof China under Grant Nos. 11247267, 11371371, 11426105, 61475198, and by the Fundamental Re-search Funds of the Central Universities (Project No. 2014QN30).