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Dromion-like structures in the variable coefficient nonlinear Schrödinger equation
Applied Mathematics Letters 30 (2014) 28–32
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Dromion-like structures in the variable coefficient nonlinear Schrödinger equation Wen-Jun Liu ∗ , Bo Tian, Ming Lei School of Science, P. O. Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China
article
info
Article history: Received 27 October 2013 Received in revised form 9 December 2013 Accepted 9 December 2013 Keywords: Dromion-like structures Variable coefficient nonlinear Schrödinger equation Soliton management
abstract Dromion-like structures, which have generally been investigated in the (2 + 1) or higher dimension partial differential equations, are reported in the (1+1) dimension variable coefficient nonlinear Schrödinger equation for the first time. With Hirota’s method, the analytic solutions for this equation are obtained. The concept of soliton management is introduced when the variable group-velocity dispersion and Kerr nonlinearity functions are suggested. Results show that the single and two dromion-like structures can be derived, and the single dromion-like structures can evolve into two dromion-like structures via different choices of the variable group-velocity dispersion and Kerr nonlinearity functions. The results of this paper will be valuable to the study of the Bose–Einstein condensate and nonlinear optical systems. © 2013 Elsevier Ltd. 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 [1–5]. As is known, nonlinear dynamical problems in physics and other natural fields are usually characterized by nonlinear evolution of partial differential equations (PDEs) [6]. Much work has been done on the subject of obtaining the analytic solutions to the PDEs [7–16], and various methods of obtaining exact solutions of nonlinear system have been proposed. On the other hand, solitons, chaos, and fractals are three most important aspects in nonlinear dynamical problems. Thus, searching for the analytic soliton solutions to the PDEs has long been an important and interesting topic in nonlinear science [6–13]. Soliton theory supplies ample applications to such varied subjects as nonlinear optics, plasmas, condensed matter physics, solids, fluid mechanics, particle physics, astrophysics, and so on [1–5]. In particular, the introduction of the concept of one type of coherent soliton solutions, called dromions, has triggered renewed interest with suggestions for the applications in nonlinear optics, hydrodynamics, plasma physics, and dusty plasma physics [17,18]. Dromions are first introduced in [19], and generally decay exponentially in all spatial directions [20]. Dromion solutions are analytic solutions of a class of two-dimensional PDEs, and usually driven by two or more nonparallel straight-line ghost solitons [21]. Recently, the study of dromions has been attractive and active, and great progress has been made [20–30]. For the Korteweg–de Vries (KdV) equation, dromion structures have been revealed for the (2 + 1) and (3 + 1) dimensions [17,22], respectively. Some special types of dromion solutions have been discussed by selecting arbitrary functions appropriately with the Bäcklund transformation and variable separation approach [23]. For the Davey–Stewartson (DS) equations, the single dromion solutions have been obtained explicitly [24,25], and have been analyzed numerically [26]. Via
∗
Corresponding author. Tel.: +86 13426223665. E-mail address: [email protected] (W.-J. Liu).
0893-9659/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.aml.2013.12.004
W.-J. Liu et al. / Applied Mathematics Letters 30 (2014) 28–32
29
Darboux and binary Darboux transformations, dromion solutions of noncommutative DS equations have been obtained [27]. Applying Hirota’s binary operator approach to the (2 + 1) dimension nonlinear Schrödinger (NLS) equation with radially variable diffraction and nonlinearity coefficients, dromions have been obtained [20]. By selecting arbitrary functions appropriately, two types of multi-dromion excitations have been investigated [28,29]. Using the standard truncated Painlevé expansion, multi-dromion solutions have been obtained [30]. Moreover, the interactions between two dromions have been studied [24]. According to the above analysis, we notice that dromions have been reported in the (2 + 1) or higher dimensions. In this paper, dromionic structures will be obtained in the (1 + 1) dimension variable coefficient NLS equation, which to the best of our knowledge has not been reported before. When the variable group-velocity dispersion (GVD) and Kerr nonlinearity coefficients are selected as Gaussian and Hyperbolic functions respectively, the obtained analytic solutions of the variable coefficient NLS equation can lead to the dromion-like structures. The physical effects affecting dromion-like structures’ properties will be discussed. Such dromion-like structures might be useful in explaining some phenomena in both Bose–Einstein condensates and nonlinear optical systems. The structure of the present paper will be as follows. In Section 2, the analytic solution for the variable coefficient NLS equation will be derived with Hirota’s method. In Section 3, the concept of soliton management will be introduced, the dromion-like structures will be formed, and discussion on the dromion-like structures will be performed. Finally, our conclusions will be given in Section 4. 2. Analytic solutions for the variable coefficient NLS equation Under investigation in this paper is the following variable coefficient NLS equation [3,31,32]: D(z ) ∂ 2 A ∂A −i + iρ(z )|A|2 A = g (z )A, (1) ∂z 2 ∂τ 2 where A(z , τ ) is the slowly varying envelope. z is the longitudinal coordinate, and τ is the time in the moving coordinate system. D(z ) and ρ(z ) represent the GVD and Kerr nonlinearity coefficients respectively. g (z ) is related to the loss or gain coefficient. At first, we use the dependent variable transformation [33] A(z , τ ) =
h( z , τ ) f (z , τ )
,
(2)
where h(z , τ ) is a complex differentiable function, and f (z , τ ) is a real one, and submit expression (2) into Eq. (1), the bilinear forms for Eq. (1) are obtained as Dz h · f −
i 2
D(z )D2τ h · f − g (z )h · f = 0,
(3)
D(z )D2τ f · f + 2ρ(z )hh∗ = 0.
(4)
Dz and Dτ [34] are the Hirota’s bilinear operators, which can be defined by n Dm z Dτ (a · b) =
∂ ∂ − ′ ∂z ∂z
m
∂ ∂ − ′ ∂τ ∂τ
n
a(z , τ )b(z ′ , τ ′ )
z ′ =z , τ ′ =τ
.
(5)
Bilinear forms (3)–(4) can be solved by the following power series expansions for h(z , τ ) and f (z , τ ): h(z , τ ) = ε h1 (z , τ ) + ε 3 h3 (z , τ ) + ε 5 h5 (z , τ ) + · · · ,
(6)
f (z , τ ) = 1 + ε f2 (z , τ ) + ε f4 (z , τ ) + ε f6 (z , τ ) + · · · ,
(7)
2
4
6
where ε is a formal expansion parameter. Substituting expressions (6)–(7) into bilinear forms (3)–(4) and equating coefficients of the same powers of ε to zero yield the recursion relations for hn (z , τ )’s and fn (z , τ )’s. Then, analytic solutions for Eq. (1) can be obtained. In order to derive the analytic solutions for Eq. (1), we assume that
h1 (z , τ ) = exp [a11 (z ) + ia12 (z )] z + (b11 + ib12 )τ + k11 + ik12 ,
(8)
where a1j (z )’s are differentiable functions to be determined, b1j ’s and k1j ’s (j = 1, 2) are the real constants. Substituting h1 (z , τ ) into the resulting set of linear partial differential equations, and after some calculations, we get the constraints on the parameters: a11 (z ) =
1 z
−b11 b12 D(z ) + g (z ) dz ,
a12 (z ) =
1 2z
b211 − b212 D(z )dz ,
30
W.-J. Liu et al. / Applied Mathematics Letters 30 (2014) 28–32
a
b
Fig. 1. (a) Evolution of the dromion-like structures. Parameters are: k11 = 1, k12 = 2, b11 = 1, b12 = −0.05, c1 = −10, c2 = 2, c3 = 5, and c4 = 0.7; (b) the contour plot of (a).
and
ρ(z ) [−b11 b12 D(z ) + g (z )] dz + 2b11 τ + 2k11 , f2 (z , τ ) = − 2 exp 2 4b11 D(z ) ρ(z )Dz (z ) − D(z )ρz (z ) , 2D(z )ρ(z ) hn (z , τ ) = 0 (n = 3, 5, 7, . . .),
(9)
g (z ) =
fn (z , τ ) = 0 (n = 4, 6, 8, . . .).
Without loss of generality, we set ε = 1, and the analytic solution can be expressed as A(z , τ ) =
h(z , τ ) f (z , τ )
=
h1 ( z , τ ) 1 + f2 (z , τ )
=
4b211 D(z ) exp [a11 (z ) + ia12 (z )] z + (b11 + ib12 )τ + k11 + ik12 4b211 D(z ) − ρ(z ) exp 2
[−b11 b12 D(z ) + g (z )] dz + 2b11 τ + 2k11
−ρ(z ) 1 = exp ik12 + ib12 τ + ia12 (z )z − ln 2 4b211 D(z ) −ρ(z ) 1 . × sech k11 + b11 τ + a11 (z )z + ln 2 4b211 D(z )
(10)
3. Discussion With appropriate parameters in solution (10), the dromion-like structures can be formed as shown in Fig. 1. The variable GVD and Kerr nonlinearity coefficients D(z ) and ρ(z ) can be expressed as follows, D(z ) = c1 exp −c2 z 2 ,
ρ(z ) =
c3 1 + c4 z
,
where cl (l = 1, 2, 3, 4) are the arbitrary nonzero constants. Assumptions for D(z ) and ρ(z ) are mainly based on the concept of soliton management in optical communications. D(z ) and ρ(z ) are related to the dispersion profile functions of dispersiondecreasing fibers [35]. The amplitude of the dromion-like structures decay exponentially in all directions. Fig. 1(b) is the contour plot of Fig. 1(a). In order to obtain the dromion-like structures, the sign of c1 should be negative, and the sign of c2 , c3 and c4 should be positive. Changing the values of c1 and c2 , we can adjust the amplitude of the dromion-like structures. Decreasing the value of c1 and increasing the value of c3 can both increase the amplitude of the dromion-like structures. Changing the value of c2 can control the amplitude decay rate of the dromion-like structures. As shown in Fig. 2, the amplitude decay rate of the dromion-like structures become faster than that in Fig. 1 when we increase the value of c2 . In the above analysis, changing the values of c1 , c2 and c3 can just adjust the amplitude of the dromion-like structures. Changing the value of c4 , the amplitude of the dromion-like structures can also be adjusted. Besides, the evolution forms of the dromion-like structures are changed as shown in Fig. 3. The single dromion-like structures evolve into two dromion-like structures. The bigger the value of c4 is, the closer the amplitude of two dromion-like structures will be in Fig. 4.
W.-J. Liu et al. / Applied Mathematics Letters 30 (2014) 28–32
a
31
b
Fig. 2. (a) Evolution of the dromion-like structures with the same parameters as those given in Fig. 1, but with c2 = 10; (b) the contour plot of (a).
a
b
Fig. 3. (a) Evolution of the dromion-like structures with the same parameters as those given in Fig. 1, but with c4 = 3; (b) the contour plot of (a).
a
b
Fig. 4. (a) Evolution of the dromion-like structures with the same parameters as those given in Fig. 1, but with c3 = 35, and c4 = 30; (b) the contour plot of (a).
4. Conclusions In conclusion, the dromion-like structures have been obtained in the (1 + 1) dimension variable coefficient nonlinear Schrödinger equation, i.e., Eq. (1). The analytic solutions (10) for Eq. (1) have been derived with Hirota’s method. Results have revealed that we can achieve the purpose of soliton management by controlling the variable GVD coefficient D(z ) and Kerr nonlinearity coefficient ρ(z ). As seen in Fig. 1, the amplitude of the dromion-like structures can be affected through changing the values of c1 and c3 . As seen in Fig. 2, changing the value of c2 can control the amplitude decay rate of the dromion-like structures. Moreover, two dromion-like structures can be formed when we change the value of c4 , which has been described in Figs. 3 and 4. Such localized structures might be of value in explaining some phenomena in both Bose–Einstein condensates and nonlinear optical systems.
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W.-J. Liu et al. / Applied Mathematics Letters 30 (2014) 28–32
Acknowledgments We express our sincere thanks to the editors and referees for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 60772023, by the Fundamental Research Funds for the Central Universities of China (Grant Nos. 2012RC0706 and 2011BUPTYB02), and by the National Basic Research Program of China (Grant No. 2010CB923200). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
L. Mollenauer, J.P. Gordon, Solitons in Optical Fibers, Academic Press, Burlington, 2006. 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. B. Eliasson, P.K. Shukla, Phys. Rep. 422 (2006) 225. Z.G. Chen, M. Segev, D.N. Christodoulides, Rep. Progr. Phys. 75 (2012) 086401. S.H. Ma, J.P. Fang, C.L. Zheng, Chin. Phys. B 17 (2008) 2676. L. Kavitha, B. Srividya, D. Gopi, Comput. Math. Appl. 62 (2011) 4691. W.X. Ma, Phys. Lett. A 319 (2003) 325. W.X. Ma, T.W. Huang, Y. Zhang, Phys. Scr. 82 (2010) 065003. W.X. Ma, J.H. Lee, Chaos Solitons Fractals 42 (2009) 1356. W.X. Ma, Z.N. Zhu, Appl. Math. Comput. 218 (2012) 11871. W.X. Ma, Y. Zhang, Y.N. Tang, J.Y. Tu, Appl. Math. Comput. 218 (2012) 7174. W.X. Ma, Sci. China Math. 55 (2012) 1769. X.F. Wu, G.S. Hua, Z.Y. Ma, Commun. Nonlinear Sci. Numer. Simul. 18 (2013) 3325. C.Q. Dai, H.P. Zhu, J. Opt. Soc. Amer. B 30 (2013) 3291. H.P. Zhu, Nonlinear Dynam. 72 (2013) 873. K. Annou, R. Annou, Phys. Plasmas 19 (2012) 043705. R. Radha, M. Lakshmanan, Phys. Lett. A 197 (1995) 7. M. Boiti, J.J.P. Leon, L.M. Martina, F. Pempinelli, Phys. Lett. A 132 (1988) 432. W.P. Zhong, M.R. Belić, Y.Z. Xia, Phys. Rev. E 83 (2011) 036603. H.Y. Ruan, S.Y. Lou, J. Math. Phys. 38 (1997) 3123. S.Y. Lou, J. Phys. A 29 (1996) 5989. S.Y. Lou, H.Y. Ruan, J. Phys. A 34 (2001) 305. H.Y. Ruan, Y.X. Chen, Phys. Rev. E 62 (2000) 5738. K. Annou, S. Bahamida, R. Annou, AIP Conf. Proc. 1041 (2008) 155. N. Yoshida, K. Nishinari, J. Satsuma, K. Abe, J. Phys. A 31 (1998) 3325. C.R. Gilson, S.R. Macfarlane, J. Phys. A 42 (2009) 235202. C.Q. Dai, Y.Y. Wang, Nonlinear Dynam. 70 (2012) 189. S.Y. Lou, Phys. Scr. 65 (2002) 7. S.Y. Lou, J. Lin, X.Y. Tang, Eur. Phys. J. B 22 (2001) 473. H.J. Zheng, C.Q. Wu, Z. Wang, H.S. Yu, S.L. Liu, X. Li, Optik 123 (2012) 818. I.O. Zolotovskii, S.G. Novikov, O.G. Okhotnikov, D.I. Sementsov, I.O. Yavtushenko, M.S. Yavtushenko, Opt. Spectrosc. 112 (2012) 893. W.J. Liu, B. Tian, H.L. Zhen, Y. Jiang, Europhys. Lett. 100 (2012) 64003. R. Hirota, Phys. Rev. Lett. 27 (1971) 1192. M.D. Pelusi, H.F. Liu, IEEE J. Quantum Electron. 33 (1997) 1430.
Contents lists available at ScienceDirect
Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml
Dromion-like structures in the variable coefficient nonlinear Schrödinger equation Wen-Jun Liu ∗ , Bo Tian, Ming Lei School of Science, P. O. Box 122, Beijing University of Posts and Telecommunications, Beijing 100876, China
article
info
Article history: Received 27 October 2013 Received in revised form 9 December 2013 Accepted 9 December 2013 Keywords: Dromion-like structures Variable coefficient nonlinear Schrödinger equation Soliton management
abstract Dromion-like structures, which have generally been investigated in the (2 + 1) or higher dimension partial differential equations, are reported in the (1+1) dimension variable coefficient nonlinear Schrödinger equation for the first time. With Hirota’s method, the analytic solutions for this equation are obtained. The concept of soliton management is introduced when the variable group-velocity dispersion and Kerr nonlinearity functions are suggested. Results show that the single and two dromion-like structures can be derived, and the single dromion-like structures can evolve into two dromion-like structures via different choices of the variable group-velocity dispersion and Kerr nonlinearity functions. The results of this paper will be valuable to the study of the Bose–Einstein condensate and nonlinear optical systems. © 2013 Elsevier Ltd. 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 [1–5]. As is known, nonlinear dynamical problems in physics and other natural fields are usually characterized by nonlinear evolution of partial differential equations (PDEs) [6]. Much work has been done on the subject of obtaining the analytic solutions to the PDEs [7–16], and various methods of obtaining exact solutions of nonlinear system have been proposed. On the other hand, solitons, chaos, and fractals are three most important aspects in nonlinear dynamical problems. Thus, searching for the analytic soliton solutions to the PDEs has long been an important and interesting topic in nonlinear science [6–13]. Soliton theory supplies ample applications to such varied subjects as nonlinear optics, plasmas, condensed matter physics, solids, fluid mechanics, particle physics, astrophysics, and so on [1–5]. In particular, the introduction of the concept of one type of coherent soliton solutions, called dromions, has triggered renewed interest with suggestions for the applications in nonlinear optics, hydrodynamics, plasma physics, and dusty plasma physics [17,18]. Dromions are first introduced in [19], and generally decay exponentially in all spatial directions [20]. Dromion solutions are analytic solutions of a class of two-dimensional PDEs, and usually driven by two or more nonparallel straight-line ghost solitons [21]. Recently, the study of dromions has been attractive and active, and great progress has been made [20–30]. For the Korteweg–de Vries (KdV) equation, dromion structures have been revealed for the (2 + 1) and (3 + 1) dimensions [17,22], respectively. Some special types of dromion solutions have been discussed by selecting arbitrary functions appropriately with the Bäcklund transformation and variable separation approach [23]. For the Davey–Stewartson (DS) equations, the single dromion solutions have been obtained explicitly [24,25], and have been analyzed numerically [26]. Via
∗
Corresponding author. Tel.: +86 13426223665. E-mail address: [email protected] (W.-J. Liu).
0893-9659/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.aml.2013.12.004
W.-J. Liu et al. / Applied Mathematics Letters 30 (2014) 28–32
29
Darboux and binary Darboux transformations, dromion solutions of noncommutative DS equations have been obtained [27]. Applying Hirota’s binary operator approach to the (2 + 1) dimension nonlinear Schrödinger (NLS) equation with radially variable diffraction and nonlinearity coefficients, dromions have been obtained [20]. By selecting arbitrary functions appropriately, two types of multi-dromion excitations have been investigated [28,29]. Using the standard truncated Painlevé expansion, multi-dromion solutions have been obtained [30]. Moreover, the interactions between two dromions have been studied [24]. According to the above analysis, we notice that dromions have been reported in the (2 + 1) or higher dimensions. In this paper, dromionic structures will be obtained in the (1 + 1) dimension variable coefficient NLS equation, which to the best of our knowledge has not been reported before. When the variable group-velocity dispersion (GVD) and Kerr nonlinearity coefficients are selected as Gaussian and Hyperbolic functions respectively, the obtained analytic solutions of the variable coefficient NLS equation can lead to the dromion-like structures. The physical effects affecting dromion-like structures’ properties will be discussed. Such dromion-like structures might be useful in explaining some phenomena in both Bose–Einstein condensates and nonlinear optical systems. The structure of the present paper will be as follows. In Section 2, the analytic solution for the variable coefficient NLS equation will be derived with Hirota’s method. In Section 3, the concept of soliton management will be introduced, the dromion-like structures will be formed, and discussion on the dromion-like structures will be performed. Finally, our conclusions will be given in Section 4. 2. Analytic solutions for the variable coefficient NLS equation Under investigation in this paper is the following variable coefficient NLS equation [3,31,32]: D(z ) ∂ 2 A ∂A −i + iρ(z )|A|2 A = g (z )A, (1) ∂z 2 ∂τ 2 where A(z , τ ) is the slowly varying envelope. z is the longitudinal coordinate, and τ is the time in the moving coordinate system. D(z ) and ρ(z ) represent the GVD and Kerr nonlinearity coefficients respectively. g (z ) is related to the loss or gain coefficient. At first, we use the dependent variable transformation [33] A(z , τ ) =
h( z , τ ) f (z , τ )
,
(2)
where h(z , τ ) is a complex differentiable function, and f (z , τ ) is a real one, and submit expression (2) into Eq. (1), the bilinear forms for Eq. (1) are obtained as Dz h · f −
i 2
D(z )D2τ h · f − g (z )h · f = 0,
(3)
D(z )D2τ f · f + 2ρ(z )hh∗ = 0.
(4)
Dz and Dτ [34] are the Hirota’s bilinear operators, which can be defined by n Dm z Dτ (a · b) =
∂ ∂ − ′ ∂z ∂z
m
∂ ∂ − ′ ∂τ ∂τ
n
a(z , τ )b(z ′ , τ ′ )
z ′ =z , τ ′ =τ
.
(5)
Bilinear forms (3)–(4) can be solved by the following power series expansions for h(z , τ ) and f (z , τ ): h(z , τ ) = ε h1 (z , τ ) + ε 3 h3 (z , τ ) + ε 5 h5 (z , τ ) + · · · ,
(6)
f (z , τ ) = 1 + ε f2 (z , τ ) + ε f4 (z , τ ) + ε f6 (z , τ ) + · · · ,
(7)
2
4
6
where ε is a formal expansion parameter. Substituting expressions (6)–(7) into bilinear forms (3)–(4) and equating coefficients of the same powers of ε to zero yield the recursion relations for hn (z , τ )’s and fn (z , τ )’s. Then, analytic solutions for Eq. (1) can be obtained. In order to derive the analytic solutions for Eq. (1), we assume that
h1 (z , τ ) = exp [a11 (z ) + ia12 (z )] z + (b11 + ib12 )τ + k11 + ik12 ,
(8)
where a1j (z )’s are differentiable functions to be determined, b1j ’s and k1j ’s (j = 1, 2) are the real constants. Substituting h1 (z , τ ) into the resulting set of linear partial differential equations, and after some calculations, we get the constraints on the parameters: a11 (z ) =
1 z
−b11 b12 D(z ) + g (z ) dz ,
a12 (z ) =
1 2z
b211 − b212 D(z )dz ,
30
W.-J. Liu et al. / Applied Mathematics Letters 30 (2014) 28–32
a
b
Fig. 1. (a) Evolution of the dromion-like structures. Parameters are: k11 = 1, k12 = 2, b11 = 1, b12 = −0.05, c1 = −10, c2 = 2, c3 = 5, and c4 = 0.7; (b) the contour plot of (a).
and
ρ(z ) [−b11 b12 D(z ) + g (z )] dz + 2b11 τ + 2k11 , f2 (z , τ ) = − 2 exp 2 4b11 D(z ) ρ(z )Dz (z ) − D(z )ρz (z ) , 2D(z )ρ(z ) hn (z , τ ) = 0 (n = 3, 5, 7, . . .),
(9)
g (z ) =
fn (z , τ ) = 0 (n = 4, 6, 8, . . .).
Without loss of generality, we set ε = 1, and the analytic solution can be expressed as A(z , τ ) =
h(z , τ ) f (z , τ )
=
h1 ( z , τ ) 1 + f2 (z , τ )
=
4b211 D(z ) exp [a11 (z ) + ia12 (z )] z + (b11 + ib12 )τ + k11 + ik12 4b211 D(z ) − ρ(z ) exp 2
[−b11 b12 D(z ) + g (z )] dz + 2b11 τ + 2k11
−ρ(z ) 1 = exp ik12 + ib12 τ + ia12 (z )z − ln 2 4b211 D(z ) −ρ(z ) 1 . × sech k11 + b11 τ + a11 (z )z + ln 2 4b211 D(z )
(10)
3. Discussion With appropriate parameters in solution (10), the dromion-like structures can be formed as shown in Fig. 1. The variable GVD and Kerr nonlinearity coefficients D(z ) and ρ(z ) can be expressed as follows, D(z ) = c1 exp −c2 z 2 ,
ρ(z ) =
c3 1 + c4 z
,
where cl (l = 1, 2, 3, 4) are the arbitrary nonzero constants. Assumptions for D(z ) and ρ(z ) are mainly based on the concept of soliton management in optical communications. D(z ) and ρ(z ) are related to the dispersion profile functions of dispersiondecreasing fibers [35]. The amplitude of the dromion-like structures decay exponentially in all directions. Fig. 1(b) is the contour plot of Fig. 1(a). In order to obtain the dromion-like structures, the sign of c1 should be negative, and the sign of c2 , c3 and c4 should be positive. Changing the values of c1 and c2 , we can adjust the amplitude of the dromion-like structures. Decreasing the value of c1 and increasing the value of c3 can both increase the amplitude of the dromion-like structures. Changing the value of c2 can control the amplitude decay rate of the dromion-like structures. As shown in Fig. 2, the amplitude decay rate of the dromion-like structures become faster than that in Fig. 1 when we increase the value of c2 . In the above analysis, changing the values of c1 , c2 and c3 can just adjust the amplitude of the dromion-like structures. Changing the value of c4 , the amplitude of the dromion-like structures can also be adjusted. Besides, the evolution forms of the dromion-like structures are changed as shown in Fig. 3. The single dromion-like structures evolve into two dromion-like structures. The bigger the value of c4 is, the closer the amplitude of two dromion-like structures will be in Fig. 4.
W.-J. Liu et al. / Applied Mathematics Letters 30 (2014) 28–32
a
31
b
Fig. 2. (a) Evolution of the dromion-like structures with the same parameters as those given in Fig. 1, but with c2 = 10; (b) the contour plot of (a).
a
b
Fig. 3. (a) Evolution of the dromion-like structures with the same parameters as those given in Fig. 1, but with c4 = 3; (b) the contour plot of (a).
a
b
Fig. 4. (a) Evolution of the dromion-like structures with the same parameters as those given in Fig. 1, but with c3 = 35, and c4 = 30; (b) the contour plot of (a).
4. Conclusions In conclusion, the dromion-like structures have been obtained in the (1 + 1) dimension variable coefficient nonlinear Schrödinger equation, i.e., Eq. (1). The analytic solutions (10) for Eq. (1) have been derived with Hirota’s method. Results have revealed that we can achieve the purpose of soliton management by controlling the variable GVD coefficient D(z ) and Kerr nonlinearity coefficient ρ(z ). As seen in Fig. 1, the amplitude of the dromion-like structures can be affected through changing the values of c1 and c3 . As seen in Fig. 2, changing the value of c2 can control the amplitude decay rate of the dromion-like structures. Moreover, two dromion-like structures can be formed when we change the value of c4 , which has been described in Figs. 3 and 4. Such localized structures might be of value in explaining some phenomena in both Bose–Einstein condensates and nonlinear optical systems.
32
W.-J. Liu et al. / Applied Mathematics Letters 30 (2014) 28–32
Acknowledgments We express our sincere thanks to the editors and referees for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 60772023, by the Fundamental Research Funds for the Central Universities of China (Grant Nos. 2012RC0706 and 2011BUPTYB02), and by the National Basic Research Program of China (Grant No. 2010CB923200). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
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