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Multiple dromion excitations in sixth order NLS equation with variable coefficients
Optik 158 (2018) 1179–1185
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Multiple dromion excitations in sixth order NLS equation with variable coefficients Natarajan Prathap a , Sekar Arunprakash a , Murugan Senthil Mani Rajan b,∗ , Kumarappa Subramanian c a b c
Department of EEE, University College of Engineering, Ramanathapuram 623513, India Department of Physics, University College of Engineering, Ramanathapuram 623513, India Department of Physics, Jerusalem College of Engineering, Chennai, India
a r t i c l e
i n f o
Article history: Received 6 December 2017 Accepted 26 December 2017 Keywords: Attosecond soliton Dromions Darboux transformation Lax pair Optical fiber
a b s t r a c t We investigated the solitonsolution for sixth order inhomogeneous NLS equation with variable coefficients. This equation describes the attosecond pulse propagation in an inhomogeneous fiber. Lax pair of this system is obtained via the Ablowitz–Kaup–Newell–Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton multi soliton solutions. Using two soliton solutions with variable coeffcients, two and four dromion structures has been obtained. Through proper choice of control parameters in variable coeffcients, dromions are controlled. Especially, two dromions evolved into four dromions has been observed for certain value of control parameter. The obtained results may have promising applications in the study of attosecond soliton propagation in fiber system and dromion structures in nonlinear systems. © 2018 Elsevier GmbH. All rights reserved.
1. Introduction Nonlinear dynamical problems are of interest to engineers, physicists and mathematicians because most physical systems are inherently nonlinear in nature [1,2]. As is known, nonlinear dynamical problems in physics and other natural fields are usually considered by nonlinear evolution of partial differential equations (PDEs) [3]. Much work has been done on the subject of obtaining the analytic solutions to the nonlinear PDEs [4–6], and various methods of obtaining exact solutions of nonlinear system have been proposed. Thus, searching for the analytic soliton solutions to the nonlinear Schrödinger equation has long been an important and interesting topic in nonlinear science [7–9]. It is still an interesting subject to investigate various of exact analytical solutions, in particular solitons, of nonlinear physical equations. With the advent of symbolic computation algorithms, sophisticated methods of algebraic manipulation have become feasible for equations of the NLS type [10,11]. Dromion exponentially decayed in all spatial directions, and was firstly proposed by Boiti [12]. Dromions are generated for generalized (2 + 1) Korteweg-de Vries equations [13]. Localized coherent structures such as dromion, peakon and compacton through modified tanh-function method have been obtained recently in [14]. Ref. [15] has put forward another mechanism to generate dromions and dromion lattice, breathers and instantons for (2 + 1) dimensional integrable equation. Although some dromion like structures have been exposed [16,17], the construction of localized excitations in various NLS equations with
∗ Corresponding author. E-mail address: [email protected] ( Murugan Senthil Mani Rajan). https://doi.org/10.1016/j.ijleo.2017.12.140 0030-4026/© 2018 Elsevier GmbH. All rights reserved.
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Natarajan Prathap et al. / Optik 158 (2018) 1179–1185
variable coefficients is still a challenging and rewarding problem. Very recently, dromion like structure have been obtained for higher order NLS equation with variable coefficients in [18]. Dromion structures have been obtained for (2 + 1) dimensional nonlinear Schrödinger equation with a parity-time-symmetric potential [19]. Stability of dromion like structure have been analyzed for complex Ginzburg–Landau equation with variable coefficients [20]. Give some introduction for attosecond soliton with reference. Attosecond soliton have been obtained for fifth order NLS equation with constant coefficients [21]. Three-soliton solutions for fifth order inhomogeneous NLS equation via the Hirota bilinear method and symbolic computation have been obtained [22]. Very recently, breathers and rogue waves for sixth order NLS variable coefficients equation with constant GVD and nonlinearity parameters have been investigated [23]. Breather to soliton transition have been explained for sixth order NLS equation [24]. In this paper, dromion structures will be obtained for the sixth order variable coefficient NLS equation, which to the best of our knowledge has not been reported before. Moreover, through the Darboux transformation and symbolic computation, dromion structure for Eq. (1) with loss/gain, which has not been discussed before, will be investigated. Based on the above considerations, in this paper, we will devote ourselves to the following variable-coefficient higher-order NLS equation with gain or loss term as follows. i
∂q(z, t) 1 + F1 + i (z) F2 + ˇ(z) F3 − i (z) F4 + ı(z) F5 + i G(z) = 0 2 ∂z
(1)
F1 = D(z) qtt + 2 R(z) |q|2 q F2 = qttt + 6|q|2 qt F3 = qtttt + 8|q|2 qtt + 6|q|4 q + 4q|qt |2 + 6q2t q∗ + 2 q2 q∗tt F 4 = qttttt + 10|q|2 qttt + 10(|q|2 q)t + 20 q∗ qt qtt + 6q2t q∗ + 2 q2 q∗tt F5 = qtttttt + [60q∗ |qt |2 + 50(q∗ )2 qtt + 2 q∗tttt ]q2 + q[12 q∗ qtttt + 8 qt q∗ttt + 22 |qtt |2 ] + q[18 qttt q∗t + 70 (q∗ )2 (qt )2 ] + 20 (qt )2 q∗tt + 10 q3 [(q∗t )2 + 2 q∗ q∗tt ] + 20 |q|6 q Where G(z) =
1 W [R(z), D(z)] 2 R(z), D(z)
W [R(z), D(z)] = R(z)
dD(z) dR(z) − D(z) dz dz
Eq. (1) is not only applied to the attosecond soliton propagation in an inhomogeneous optical fiber system, but also to analyze the various dispersion management soliton through proper choice of variable coefficients. Where q(z, t) is the complex envelope of the optical field, z is the propagation distance and t is the retarded time. D(z), R(z), G(z) are denotes the group velocity dispersion, nonlinearity and amplification or absorption coefficient respectively. (z) is signifies the coefficient of third order dispersion and self-steepening. ˇ(z), (z) and ı(z) are represents the higher order dispersion and nonlinearities. Second term in the above Eq. (1) is represents group velocity. In practical applications, Eq. (1) can be used to describe not only the amplification or absorption of soliton propagation in some inhomogeneous nonlinear fiber systems, but also the stable transmission of controllable soliton. 2. Lax pair In this section, based on the AKNS procedure [25], we will construct a Lax pair of Eq. (1). From the Lax pair, one can generate multi-soliton solutions expediently via Darboux transformation. The linear eigenvalue problem for Eq. (1) can be expressed as follows: t
= U
U=i
V=
6 j=0
,
z
= V
q∗
q
−
j Vj
(2)
Natarajan Prathap et al. / Optik 158 (2018) 1179–1185
Where
Vj = i
Aj
Bj∗
Bj
− Aj
1181
A6 = 32 ı(z), B6 = 0, A5 = 16 (z), B5 = 32 ı(z) q A4 = −8 (z) − 16 ı(z)|q|2 , B4 = 16 (z) q + 16 i ı(z) qt A3 = −4 (z) − 8 (z) |q|2 − 8 i ı(z) |q|2t , B3 = −8ˇ(z) q + 8 i (z) qt − 8 qtt − 16ı(z)|q|2 q A2 = 1 + 4 ˇ(z)|q|2 + 4 i (z) (q∗t q − qt q∗ ) + 12 ı(z)|q|4 − 8ı(z)|qt |2 + 4ı(z) (q∗t q − qt q∗ )t B2 = −4 (z) q − 8 (z) |q|2 q − 24 i ı(z)|q|2 qt − 4 i ˇ(z) qt − 4 i (z)qtt − 4 i ı(z)qttt A1 = 2 (z) |q|2 + 6 (z) |q|4 − 2 i ˇ(z) (q∗t q − qt q∗ ) + 12 i ı(z) |q|2 (qt q∗ − q∗t q) − 2 (z) |qt |2 + 2 (z) (q∗tt q + qtt q∗ ) + 2 i ı(z)(qt q∗tt − q∗t qtt + q∗ qttt − q∗ttt q) B1 = q + 4 ˇ(z) |q|2 q − 2 i (z) qt − 12 i (z) |q|2 qt + 12 ı(z) q∗ q2t + 16 ı(z)|q|2 qtt + 4 ı(z) q2 q∗tt − 2 i (z) qttt + 2 ˇ(z) qtt + 2 ı(z) qtttt + 12ı(z) |q|4 q + 8 ı(z) |qqt |2 Because the solutions include two distributed functions D(Z) and R(Z), thus by choosing the form most approximate to the real conditions, one can explain the various soliton control systems or dispersion management systems. 3. Darboux transformation and multi-soliton solutions In order to derive the soliton solutions for nonlinear evolution equations, the Darboux transformation [26] method can be adopted, which plays a vital role in soliton theory. As application of the Darboux transformation technique, many analytical solutions for some nonlinear evolution equations have been constructed [27,28]. In this section, we have symbolically constructed the Darboux transformation for Eq. (1) based on the Lax pair which is given in the previous section.
qn = q + 2
D R
∗ ( ) (m + ∗m )1,m (m )2,m m
Am
Where k,m+1 (m+1 ) = (m+1 + ∗m )k,m (m+1 ) −
2
Bm (m + ∗m )k,m (m ) Am
(3)
2
Am = 1,m (m ) + 2,m (m )
∗ ∗ Bm = 1,m (m+1 )1,m (m ) + 2,m (m+1 )2,m (m )
In high-bit-rate and long-distance optical communication systems, there are always multiple pulses. Therefore, it is necessary to study multi-soliton transmissions. For two-soliton solutions (n = 2 in Eq. (3)), we can control their dynamic behaviors by choosing suitable parameters.
q2 = 2
D A R B
A = J0 Exp(2 i (1 + 1 t)) + J1 Exp(2 i (2 + 2 t)) + J2 Exp(2 i (2∗ + ∗2 t))+ J3 Exp(2 i (1∗ + ∗1 t)) B = J4 Exp(2 i (1 + 1∗ ) + 2 i (1 + ∗1 )t) + J5 Exp(2 i (2 + 1∗ ) + 2 i (2 + ∗1 )t)+ J6 Exp(2 i (1 + 2∗ ) + 2 i (1 + ∗2 )t) + J7 Exp(2 i (2 + 2∗ ) + 2 i (2 + ∗2 )t)+ J8 Exp(2 i (1 + 2 ) + 2 i (1 + 2 )t) + J9 Exp(2 i (1∗ + 2∗ ) + 2 i (∗1 + ∗2 )t)
(4)
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1 = 21
(D(z) − 4 1 ˛(z) − 8 21 ˇ(z) + 16 31 (z) + 32 41 ı(z)) dz
2 = 22
(D(z) − 4 2 ˛(z) − 8 22 ˇ(z) + 16 32 (z) + 32 42 ı(z)) dz
1 = 1 + i 1 2 = 2 + i 2 From the soliton solution, one can clearly see that, if one manages to control the parameters j and j , it is possible to form a separate evolution and the interaction of solitons. Although the fundamental soliton propagation cannot be obtained in standard fibers, pulse propagation over relatively long distances can still be obtained through an appropriate combination of dispersion and nonlinearity management. 4. Dromion structures via variable coeffcients When the variable group-velocity dispersion (GVD) and Kerr nonlinearity coefficients are selected as exponential functions, the obtained analytic solutions of the variable coefficient sixth order NLS equation can lead to the dromion-like structures. With proper choice of parameters in solution (10), the dromion-like structures can be formed as shown in below figures. For this special case, the distributed coefficients D(z) and R(z) can be expressed as follows [29], 2
2
D(z) = Exp (− a (z − b) ) + Exp (− a (z + b) )
(5)
R(z) = 1 As shown in Fig. 1, we present the double dromion-like structures with different values of controlled parameters for obtained soliton solutions. The structure with respect to its center exhibits exactly symmetry which means that it is steeper in negative and positive direction are equal, and is more compressed in time direction. From Figs. 1 and 2, we may infer that the double dromion-like structures with maximum amplitude at z = 0 and suppressed width, from which we find that the width of dromions can be influenced by the parameters. As seen in Figs. 1 and 2, the amplitude of the dromion-like structures cannot be affected through changing the value of a while the position of dromion remains unchanged. We further observe that the distance between dromions still close and appear as a pair without merging. In addition, the evolution forms of the dromion-like structures are changed as shown in Fig. 3. The two dromion-like structures change into four dromion-like structures. The lower the value of b is, the closer the four dromion-like structures along the z direction as depicted in Fig. 4. For b = 1, two pair of dromions merged and form a single pair as depicted in Figs. 1 and 2. Hereby, we increase the value of b especially for b > 1, two dromion-like structures are evolved into four dromion-like structures. Through the above analysis, we find that the parameters a and b have a crucial effect on the pulse width and the number of dromions. Changing the
Fig. 1. (a) Evolution of the dromion-like structures. Parameters are1 = −1.1, 2 = 1, 1 = 0.8, 2 = −0.8, a= 0.1 and b = 1 (b) contour plot.
Natarajan Prathap et al. / Optik 158 (2018) 1179–1185
Fig. 2. (a) Evolution of the dromion-like structures. Parameters are1 = −1.1, 2 = 1, 1 = 0.8, 2 = −0.8, a= 0.5 and b = 1 (b) contour plot.
Fig. 3. (a) Evolution of the dromion-like structures. Parameters are1 = −1.1, 2 = 1, 1 = 0.8, 2 = −0.8, a= 0.5 and b = 5 (b) contour plot.
Fig. 4. (a) Evolution of the dromion-like structures. Parameters are1 = −1.1, 2 = 1, 1 = 0.8, 2 = −0.8, a= 0.5 and b = 2 (b) contour plot.
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Natarajan Prathap et al. / Optik 158 (2018) 1179–1185
Fig. 5. (a) Variation of gain and (b) GVD along the propagation distance (z). (For interpretation of the references to colour in text, the reader is referred to the web version of this article.).
values of a and b, we can adjust the distance between dromions and width of dromions. By keeping a as constant with varying (decreasing) b, the pair of dromions comes closer. For b = 1, by decreasing the value of a the width of the dromion can be changed. By comparing the Figs. 2 and 4, one may conclude that the amplitude of dromions will decay when number of dromions increased. 5. Variation of gain and GVD with propagation distance By closely observing the 2-Dimensional plots demonstrated in Fig. 5(a) and (b), we have found that abruptly change in the profiles of gain and GVD for b = 1 (purple) & b = 5 (blue) with a = 0.5. In both profiles, opposite behaviors have been observed for the values of b = 1 & b = 5. Particularly, significant changes have been detected in the gain and GVD profiles for b > 1 which generate the four dromions as depicted in Fig.4(a). 6. Conclusions The dromion-like optical waves have been obtained for sixth order NLS variable coefficient nonlinear Schrödinger equation with gain/loss, i.e., Eq. (1). The analytic solutions (10) for Eq. (1) have been obtained with Darboux transformation technique using Lax pair. As seen in figures, the width and amplitude of the dromion structures can be affected through changing the values of a and b. Besides, double dromions can be transformed into four dromions when we change the value of b from 1 to 2, which is portrayed in Figs. 2 and 4. In all obtained plots, the GVD varies exponentially and the nonlinear coefficient is remains constant. This means that, the GVD coefficient D(z) plays an important role in the excitation of dromions in the considered system. Such dromions might be of value in explaining some phenomena in nonlinear optical systems. Our results show that the dromion-like structure can be generated in the anomalous dispersion medium. Moreover, obtained results have revealed that we can achieve the purpose of soliton control management by controlling the variable coefficients D(z) and R(z). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic Press, San Diego, 2003. G.P. Agrawal, Nonlinear Fiber Optics, fourth ed., Academic Press, San Diego, 2007. S.H. Ma, J.P. Fang, C.L. Zheng, Chin. Phys. B 17 (2008) 2676. Z.Y. Yan, Phys. Lett. A 374 (2010) 672. W.X. Ma, Z.N. Zhu, Appl. Math. Comput. 218 (2012) 11871. M.S. Mani Rajan, Nonlinear Dyn. 85 (2016) 599. C.Q. Dai, Y.Y. Wang, Commun. Nonlinear Sci. Numer. Simul. 19 (2014) 19. W.P. Zhong, M. Belic, G. Assanto, B.A. Malomed, T. Huang, Phys. Rev. A 84 (2011) 043801. A. Mahalingam, M.S. Mani Rajan, Opt. Fiber Technol. 25 (2015) 44. M.S. Mani Rajan, A. Mahalingam, A. Uthayakumar, Ann. Phys. 346 (2014) 1. S. Vijayalekshmi, A. Mahalingam, M.S. Mani Rajan, Optik 145 (2017) 240. M. Boiti, J.J.P. Leon, L. Martina, F. Pempinelli, Phys. Lett. A 132 (1988) 432. J. Hiretarinta, Phys. Lett. A 149 (1990) 113. B. Zhang, X.L. Zhang, C.Q. Dai, Nonlinear Dyn. 87 (2017) 2385. S. Lou, Phys. Scr. 65 (2002) 7. H. Kaur, R. Pal, T.S. Raju, C.N. Kumar, Ann. Phys. 374 (2016) 366. W.J. Liu, B. Tian, M. Lei, Appl. Math. Lett. 30 (2014) 28. J.W. Yang, Y.T. Gao, Y.J. Feng, C.Q. Su, Nonlinear Dyn. 87 (2017) 851. Y.Q. Li, W.J. Liu, P. Wong, L.G. Huang, N. Pan, Appl. Math. Lett. 47 (2015) 8. P. Wong, L.H. Pang, L.G. Huang, Y.Q. Li, M. Lei, W.J. Liu, Ann. Phys. 360 (2015) 341. J. Chai, B. Tian, H.L. Zhen, W.R. Sun, Ann. Phys. 359 (2015) 371. J.W. Yang, Y.T. Gao, C. Qi, D.W. Su, Y.J. Zuo, Feng, Commun. Nonlinear Sci. Numer. Simul. 42 (2017) 477. S.L. Jia, Y.T. Gaoa, C. Zhao, Z.Z. Lan, Y.J. Feng, Eur. Phys. J. Plus 132 (2017) 34.
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Q.M. Huang, Y.T. Gao, L. Hu, Appl. Math. Lett. 75 (2018) 135. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, 125, Phys. Rev. Lett. (1973) 31. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons, Springer, Berlin, 1991. M.S. Mani Rajan, J. Hakkim, A. Mahalingam, A. Uthayakumar, Eur. Phys. J. D 67 (2013) 150. S. Loomba, M.S. Mani Rajan, R. Gupta, A. Mahalingam, Eur. Phys. J. D 68 (2014) 130. W.J. Liu, L. Huang, P. Huang, Y. Li, M. Lei, Appl. Math. Lett. 61 (2016) 80.
1185
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Multiple dromion excitations in sixth order NLS equation with variable coefficients Natarajan Prathap a , Sekar Arunprakash a , Murugan Senthil Mani Rajan b,∗ , Kumarappa Subramanian c a b c
Department of EEE, University College of Engineering, Ramanathapuram 623513, India Department of Physics, University College of Engineering, Ramanathapuram 623513, India Department of Physics, Jerusalem College of Engineering, Chennai, India
a r t i c l e
i n f o
Article history: Received 6 December 2017 Accepted 26 December 2017 Keywords: Attosecond soliton Dromions Darboux transformation Lax pair Optical fiber
a b s t r a c t We investigated the solitonsolution for sixth order inhomogeneous NLS equation with variable coefficients. This equation describes the attosecond pulse propagation in an inhomogeneous fiber. Lax pair of this system is obtained via the Ablowitz–Kaup–Newell–Segur (AKNS) scheme and the corresponding Darboux transformation is constructed to derive the soliton multi soliton solutions. Using two soliton solutions with variable coeffcients, two and four dromion structures has been obtained. Through proper choice of control parameters in variable coeffcients, dromions are controlled. Especially, two dromions evolved into four dromions has been observed for certain value of control parameter. The obtained results may have promising applications in the study of attosecond soliton propagation in fiber system and dromion structures in nonlinear systems. © 2018 Elsevier GmbH. All rights reserved.
1. Introduction Nonlinear dynamical problems are of interest to engineers, physicists and mathematicians because most physical systems are inherently nonlinear in nature [1,2]. As is known, nonlinear dynamical problems in physics and other natural fields are usually considered by nonlinear evolution of partial differential equations (PDEs) [3]. Much work has been done on the subject of obtaining the analytic solutions to the nonlinear PDEs [4–6], and various methods of obtaining exact solutions of nonlinear system have been proposed. Thus, searching for the analytic soliton solutions to the nonlinear Schrödinger equation has long been an important and interesting topic in nonlinear science [7–9]. It is still an interesting subject to investigate various of exact analytical solutions, in particular solitons, of nonlinear physical equations. With the advent of symbolic computation algorithms, sophisticated methods of algebraic manipulation have become feasible for equations of the NLS type [10,11]. Dromion exponentially decayed in all spatial directions, and was firstly proposed by Boiti [12]. Dromions are generated for generalized (2 + 1) Korteweg-de Vries equations [13]. Localized coherent structures such as dromion, peakon and compacton through modified tanh-function method have been obtained recently in [14]. Ref. [15] has put forward another mechanism to generate dromions and dromion lattice, breathers and instantons for (2 + 1) dimensional integrable equation. Although some dromion like structures have been exposed [16,17], the construction of localized excitations in various NLS equations with
∗ Corresponding author. E-mail address: [email protected] ( Murugan Senthil Mani Rajan). https://doi.org/10.1016/j.ijleo.2017.12.140 0030-4026/© 2018 Elsevier GmbH. All rights reserved.
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Natarajan Prathap et al. / Optik 158 (2018) 1179–1185
variable coefficients is still a challenging and rewarding problem. Very recently, dromion like structure have been obtained for higher order NLS equation with variable coefficients in [18]. Dromion structures have been obtained for (2 + 1) dimensional nonlinear Schrödinger equation with a parity-time-symmetric potential [19]. Stability of dromion like structure have been analyzed for complex Ginzburg–Landau equation with variable coefficients [20]. Give some introduction for attosecond soliton with reference. Attosecond soliton have been obtained for fifth order NLS equation with constant coefficients [21]. Three-soliton solutions for fifth order inhomogeneous NLS equation via the Hirota bilinear method and symbolic computation have been obtained [22]. Very recently, breathers and rogue waves for sixth order NLS variable coefficients equation with constant GVD and nonlinearity parameters have been investigated [23]. Breather to soliton transition have been explained for sixth order NLS equation [24]. In this paper, dromion structures will be obtained for the sixth order variable coefficient NLS equation, which to the best of our knowledge has not been reported before. Moreover, through the Darboux transformation and symbolic computation, dromion structure for Eq. (1) with loss/gain, which has not been discussed before, will be investigated. Based on the above considerations, in this paper, we will devote ourselves to the following variable-coefficient higher-order NLS equation with gain or loss term as follows. i
∂q(z, t) 1 + F1 + i (z) F2 + ˇ(z) F3 − i (z) F4 + ı(z) F5 + i G(z) = 0 2 ∂z
(1)
F1 = D(z) qtt + 2 R(z) |q|2 q F2 = qttt + 6|q|2 qt F3 = qtttt + 8|q|2 qtt + 6|q|4 q + 4q|qt |2 + 6q2t q∗ + 2 q2 q∗tt F 4 = qttttt + 10|q|2 qttt + 10(|q|2 q)t + 20 q∗ qt qtt + 6q2t q∗ + 2 q2 q∗tt F5 = qtttttt + [60q∗ |qt |2 + 50(q∗ )2 qtt + 2 q∗tttt ]q2 + q[12 q∗ qtttt + 8 qt q∗ttt + 22 |qtt |2 ] + q[18 qttt q∗t + 70 (q∗ )2 (qt )2 ] + 20 (qt )2 q∗tt + 10 q3 [(q∗t )2 + 2 q∗ q∗tt ] + 20 |q|6 q Where G(z) =
1 W [R(z), D(z)] 2 R(z), D(z)
W [R(z), D(z)] = R(z)
dD(z) dR(z) − D(z) dz dz
Eq. (1) is not only applied to the attosecond soliton propagation in an inhomogeneous optical fiber system, but also to analyze the various dispersion management soliton through proper choice of variable coefficients. Where q(z, t) is the complex envelope of the optical field, z is the propagation distance and t is the retarded time. D(z), R(z), G(z) are denotes the group velocity dispersion, nonlinearity and amplification or absorption coefficient respectively. (z) is signifies the coefficient of third order dispersion and self-steepening. ˇ(z), (z) and ı(z) are represents the higher order dispersion and nonlinearities. Second term in the above Eq. (1) is represents group velocity. In practical applications, Eq. (1) can be used to describe not only the amplification or absorption of soliton propagation in some inhomogeneous nonlinear fiber systems, but also the stable transmission of controllable soliton. 2. Lax pair In this section, based on the AKNS procedure [25], we will construct a Lax pair of Eq. (1). From the Lax pair, one can generate multi-soliton solutions expediently via Darboux transformation. The linear eigenvalue problem for Eq. (1) can be expressed as follows: t
= U
U=i
V=
6 j=0
,
z
= V
q∗
q
−
j Vj
(2)
Natarajan Prathap et al. / Optik 158 (2018) 1179–1185
Where
Vj = i
Aj
Bj∗
Bj
− Aj
1181
A6 = 32 ı(z), B6 = 0, A5 = 16 (z), B5 = 32 ı(z) q A4 = −8 (z) − 16 ı(z)|q|2 , B4 = 16 (z) q + 16 i ı(z) qt A3 = −4 (z) − 8 (z) |q|2 − 8 i ı(z) |q|2t , B3 = −8ˇ(z) q + 8 i (z) qt − 8 qtt − 16ı(z)|q|2 q A2 = 1 + 4 ˇ(z)|q|2 + 4 i (z) (q∗t q − qt q∗ ) + 12 ı(z)|q|4 − 8ı(z)|qt |2 + 4ı(z) (q∗t q − qt q∗ )t B2 = −4 (z) q − 8 (z) |q|2 q − 24 i ı(z)|q|2 qt − 4 i ˇ(z) qt − 4 i (z)qtt − 4 i ı(z)qttt A1 = 2 (z) |q|2 + 6 (z) |q|4 − 2 i ˇ(z) (q∗t q − qt q∗ ) + 12 i ı(z) |q|2 (qt q∗ − q∗t q) − 2 (z) |qt |2 + 2 (z) (q∗tt q + qtt q∗ ) + 2 i ı(z)(qt q∗tt − q∗t qtt + q∗ qttt − q∗ttt q) B1 = q + 4 ˇ(z) |q|2 q − 2 i (z) qt − 12 i (z) |q|2 qt + 12 ı(z) q∗ q2t + 16 ı(z)|q|2 qtt + 4 ı(z) q2 q∗tt − 2 i (z) qttt + 2 ˇ(z) qtt + 2 ı(z) qtttt + 12ı(z) |q|4 q + 8 ı(z) |qqt |2 Because the solutions include two distributed functions D(Z) and R(Z), thus by choosing the form most approximate to the real conditions, one can explain the various soliton control systems or dispersion management systems. 3. Darboux transformation and multi-soliton solutions In order to derive the soliton solutions for nonlinear evolution equations, the Darboux transformation [26] method can be adopted, which plays a vital role in soliton theory. As application of the Darboux transformation technique, many analytical solutions for some nonlinear evolution equations have been constructed [27,28]. In this section, we have symbolically constructed the Darboux transformation for Eq. (1) based on the Lax pair which is given in the previous section.
qn = q + 2
D R
∗ ( ) (m + ∗m )1,m (m )2,m m
Am
Where k,m+1 (m+1 ) = (m+1 + ∗m )k,m (m+1 ) −
2
Bm (m + ∗m )k,m (m ) Am
(3)
2
Am = 1,m (m ) + 2,m (m )
∗ ∗ Bm = 1,m (m+1 )1,m (m ) + 2,m (m+1 )2,m (m )
In high-bit-rate and long-distance optical communication systems, there are always multiple pulses. Therefore, it is necessary to study multi-soliton transmissions. For two-soliton solutions (n = 2 in Eq. (3)), we can control their dynamic behaviors by choosing suitable parameters.
q2 = 2
D A R B
A = J0 Exp(2 i (1 + 1 t)) + J1 Exp(2 i (2 + 2 t)) + J2 Exp(2 i (2∗ + ∗2 t))+ J3 Exp(2 i (1∗ + ∗1 t)) B = J4 Exp(2 i (1 + 1∗ ) + 2 i (1 + ∗1 )t) + J5 Exp(2 i (2 + 1∗ ) + 2 i (2 + ∗1 )t)+ J6 Exp(2 i (1 + 2∗ ) + 2 i (1 + ∗2 )t) + J7 Exp(2 i (2 + 2∗ ) + 2 i (2 + ∗2 )t)+ J8 Exp(2 i (1 + 2 ) + 2 i (1 + 2 )t) + J9 Exp(2 i (1∗ + 2∗ ) + 2 i (∗1 + ∗2 )t)
(4)
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1 = 21
(D(z) − 4 1 ˛(z) − 8 21 ˇ(z) + 16 31 (z) + 32 41 ı(z)) dz
2 = 22
(D(z) − 4 2 ˛(z) − 8 22 ˇ(z) + 16 32 (z) + 32 42 ı(z)) dz
1 = 1 + i 1 2 = 2 + i 2 From the soliton solution, one can clearly see that, if one manages to control the parameters j and j , it is possible to form a separate evolution and the interaction of solitons. Although the fundamental soliton propagation cannot be obtained in standard fibers, pulse propagation over relatively long distances can still be obtained through an appropriate combination of dispersion and nonlinearity management. 4. Dromion structures via variable coeffcients When the variable group-velocity dispersion (GVD) and Kerr nonlinearity coefficients are selected as exponential functions, the obtained analytic solutions of the variable coefficient sixth order NLS equation can lead to the dromion-like structures. With proper choice of parameters in solution (10), the dromion-like structures can be formed as shown in below figures. For this special case, the distributed coefficients D(z) and R(z) can be expressed as follows [29], 2
2
D(z) = Exp (− a (z − b) ) + Exp (− a (z + b) )
(5)
R(z) = 1 As shown in Fig. 1, we present the double dromion-like structures with different values of controlled parameters for obtained soliton solutions. The structure with respect to its center exhibits exactly symmetry which means that it is steeper in negative and positive direction are equal, and is more compressed in time direction. From Figs. 1 and 2, we may infer that the double dromion-like structures with maximum amplitude at z = 0 and suppressed width, from which we find that the width of dromions can be influenced by the parameters. As seen in Figs. 1 and 2, the amplitude of the dromion-like structures cannot be affected through changing the value of a while the position of dromion remains unchanged. We further observe that the distance between dromions still close and appear as a pair without merging. In addition, the evolution forms of the dromion-like structures are changed as shown in Fig. 3. The two dromion-like structures change into four dromion-like structures. The lower the value of b is, the closer the four dromion-like structures along the z direction as depicted in Fig. 4. For b = 1, two pair of dromions merged and form a single pair as depicted in Figs. 1 and 2. Hereby, we increase the value of b especially for b > 1, two dromion-like structures are evolved into four dromion-like structures. Through the above analysis, we find that the parameters a and b have a crucial effect on the pulse width and the number of dromions. Changing the
Fig. 1. (a) Evolution of the dromion-like structures. Parameters are1 = −1.1, 2 = 1, 1 = 0.8, 2 = −0.8, a= 0.1 and b = 1 (b) contour plot.
Natarajan Prathap et al. / Optik 158 (2018) 1179–1185
Fig. 2. (a) Evolution of the dromion-like structures. Parameters are1 = −1.1, 2 = 1, 1 = 0.8, 2 = −0.8, a= 0.5 and b = 1 (b) contour plot.
Fig. 3. (a) Evolution of the dromion-like structures. Parameters are1 = −1.1, 2 = 1, 1 = 0.8, 2 = −0.8, a= 0.5 and b = 5 (b) contour plot.
Fig. 4. (a) Evolution of the dromion-like structures. Parameters are1 = −1.1, 2 = 1, 1 = 0.8, 2 = −0.8, a= 0.5 and b = 2 (b) contour plot.
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Fig. 5. (a) Variation of gain and (b) GVD along the propagation distance (z). (For interpretation of the references to colour in text, the reader is referred to the web version of this article.).
values of a and b, we can adjust the distance between dromions and width of dromions. By keeping a as constant with varying (decreasing) b, the pair of dromions comes closer. For b = 1, by decreasing the value of a the width of the dromion can be changed. By comparing the Figs. 2 and 4, one may conclude that the amplitude of dromions will decay when number of dromions increased. 5. Variation of gain and GVD with propagation distance By closely observing the 2-Dimensional plots demonstrated in Fig. 5(a) and (b), we have found that abruptly change in the profiles of gain and GVD for b = 1 (purple) & b = 5 (blue) with a = 0.5. In both profiles, opposite behaviors have been observed for the values of b = 1 & b = 5. Particularly, significant changes have been detected in the gain and GVD profiles for b > 1 which generate the four dromions as depicted in Fig.4(a). 6. Conclusions The dromion-like optical waves have been obtained for sixth order NLS variable coefficient nonlinear Schrödinger equation with gain/loss, i.e., Eq. (1). The analytic solutions (10) for Eq. (1) have been obtained with Darboux transformation technique using Lax pair. As seen in figures, the width and amplitude of the dromion structures can be affected through changing the values of a and b. Besides, double dromions can be transformed into four dromions when we change the value of b from 1 to 2, which is portrayed in Figs. 2 and 4. In all obtained plots, the GVD varies exponentially and the nonlinear coefficient is remains constant. This means that, the GVD coefficient D(z) plays an important role in the excitation of dromions in the considered system. Such dromions might be of value in explaining some phenomena in nonlinear optical systems. Our results show that the dromion-like structure can be generated in the anomalous dispersion medium. Moreover, obtained results have revealed that we can achieve the purpose of soliton control management by controlling the variable coefficients D(z) and R(z). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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