- Email: [email protected]
Interaction of the nonautonomous soliton in the optical fiber
Optik 127 (2016) 11282–11287
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Interaction of the nonautonomous soliton in the optical fiber Da-Wei Zuo a,∗ , Hui-Xian Jia b a b
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Department of Basic, Shijiazhuang Post and Telecommunication Technical College, Shijiazhuang 050021, China
a r t i c l e
i n f o
Article history: Received 9 May 2016 Received in revised form 3 September 2016 Accepted 5 September 2016 PACS: 05.45.Yv 52.35.Mw 52.35.Sb
a b s t r a c t Investigated in this paper is a generalized nonautonomous nonlinear Schrödinger equation with both external potential and phase modulation, which can be used to described the nonautonomous soliton in the optical fiber. With our results as follows: Lax pair, Daboux transformation, condition of the modulation instability and N−soliton solutions have been obtained; Propagation and interaction of the solitons have been analysed: (1) the group velocity and linear potential affect the direction and trajectory of one/two solitons propagation, respectively; (2) two solitons have the same average velocity at some fixed time or a period. © 2016 Elsevier GmbH. All rights reserved.
Keywords: Nonautonomous soliton Darboux transformation Optical fiber
1. Introduction Optical solitons1 have been playing an important role in the fundamental research of the nonlinear science and in the field of the next generation optical communication systems with long distance, high speed and large capacity, which caused by the balance of the self-phase modulation and group-velocity dispersion [4,5]. Propagation of the picosecond optical solitons in a nonlinear, dispersive medium can be governed by the nonlinear Schrödinger (NLS) equation, iQ Z + QTT + 2Q 2 Q ∗ = 0,
(1)
which also appears in the fluid mechanics, plasma physics, turbulence, quantum field theory, condensed matter, quantum electronics, phase transitions, biophysics and star formation, where i2 =−1, * is the complex conjugate and Q is a function of the scaled temporal coordinate T and spatial coordinate Z [1,6–9]. Besides, nonautonomous solitons which can be achieved by virtue of the inhomogeneous distribution of the nonlinear medium have also been potently useful for the various applications in the optical solitons communication system of their special features [10]. For example, existing solitons in nonautonomous physical systems need not only the varying in time nonlinearity and dispersion dependent on each other but also they satisfy some integrability conditions; this law is called as Serkin–Hasegawa theorem [11]. This theorem is applied to a nonautonomous system and soliton management are discussed in Ref. [11]. Nonautonomous solitons can be described by the the generalized nonautonomous NLS equations with variable coefficients.
∗ Corresponding author. E-mail address: [email protected] (D.-W. Zuo). 1 A soliton is a solitary wave which preserves its velocity and shape after the interaction [1], i.e., the soliton can be considered as a quasi-particle [2,3]. http://dx.doi.org/10.1016/j.ijleo.2016.09.022 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
D.-W. Zuo, H.-X. Jia / Optik 127 (2016) 11282–11287
11283
Variable coefficients NLS equation has been widely investigated, e.g., law of the soliton adaptation to an external potential has been studied in the nonautonomous solitons [12], and nonautonomous solitons in parity time symmetric potentials for both exponential diffraction decreasing and periodically modulated waveguide have been reported [13]. However, we find that there has not much discussion about the propagation and interaction of the soliton with the time/space dependent phase modulation and external potential. An important problem is that how the time/space dependent phase modulation and external potentials are affecting the propagation and interaction of the nonautonomous optical soliton? How one can control the dynamical behavior of the nonautonomous optical soliton through the time/space dependent phase modulation and external potential? Being motivated by the above aspects and to attain the answers of those questions, this paper will discuss the nonautonomous optical soliton in nonlinear optical fibers with the time/space dependent phase modulation and external potential. Based on the above discussions, we will work on a generalized nonautonomous NLS equation with both external potential and phase modulation, which can be written as [14] qz + a(z)qt − id(z)qtt − ir(z)q2 q∗ − d(z)(z)q + b1 (z)q + ib2 (z, t)q = 0 , where b1 (z) =
r (z)
1 2
r(z)
b22 (z, t) =
1 2
−
d (z) , d(z)
b2 (z, t) = b21 (z, t) + b22 (z, t),
(2)
b21 (z, t) = [a(z)(z) − v1 (z)]t,
(z) − d(z)(z)2 t 2 ,
where q is the solitary envelope of the optical pulse and depend on the scaled temporal coordinate t and spatial coordinate z for the optical soliton in the fiber. Real functions d(z), r(z), b1 (z) and (z) denote the dispersion, nonlinearity, loss or gain and increment of the phase modulation respectively. b2 (z, t) represents the external potential and it is assumed to be a function of the propagation distance and time. Especially, v1 (z) and b22 (z, t) are represents the linear and harmonic potential respectively. a(z) results from the group velocity. Physically speaking, Eq. (2) is not only used to describe the propagation of optical pulse in the inhomogeneous optical fiber, but also character the core of a dispersion-managed soliton. Besides, the combined external potentials in Eq. (2) are efficient in supporting the matter-wave solitons [14]. However, to our knowledge, lax pair, N-soliton solutions, Darboux transformation (DT) and linear stability analysis of Eq. (2) have not been obtained. With the aid of symbolic computation [15–17], Section 2, lax pair, N-soliton solutions and DT for Eq. (2) will be obtained. Section 3, Linear stability analysis, propagation and interaction of solitons will be discussed. Section 4 will be the conclusions. 2. Lax pair, DT and N−Soliton solutions of Eq. (2) 2.1. Lax pair of Eq. (2) Nonlinear evolution equations (NLEEs) with soliton solutions have some special characters, the basic property of which is that those can be equivalent to a pair of linear system, i.e. the Lax pair [18]. People call those equations as the Lax integrable. By virtue of the Ablowitz–Kaup–Newell–Segur technology [19], Lax pair of Eq. (2) are attained as t = U,
(3a)
z = V,
(3b) )T ,
where = (1 , 2 while 1 and 2 are both the functions of z and t, the superscript T denotes the matrix/vector transpose and matrices U = (Uij ) and V = (Vij ) have the forms
U11 = −U22 = −(z), V11 = −V22 =
V12 =
V21 =
∗ U12 = −U21 =
r(z) it 2 (z) e 2 q, 2d(z)
1 i[qq∗ r(z) + t v1 (z)] + [a(z) − 2td(z)(z)](z) + 2id(z)(z)2 , 2
r[z] it 2 (z) e 2 −a(z)q + d(z)(z)tq − 2i(z)d(z)q + id(z)qt , 2d(z) r[z] −it 2 (z) e 2 [a[z]q∗ − td(z)(z)q∗ + 2id(z)(z)q∗ + id(z)q∗t ]. 2d(z)
Eq. (2) can be obtained via the compatibility condition Uz − Vt + UV − VU = 0, if and only if the spectrum parameter satisfies the following equation (z)z = 2d(z)(z)(z) −
iv1 (z) , 2
(4)
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D.-W. Zuo, H.-X. Jia / Optik 127 (2016) 11282–11287
i.e. the spectrum parameter (z) = e
2d(z)(z)dz
[(0) +
z 0
e
−
2d(z)(z)dz
v1 (z)dz], where (0) is a integration constant.
2.2. DT of Eq. (2) It is well known that DT is an efficient way to construct the exact solutions of the NLEEs, especially to the Lax integral system. Based on Lax Pair (3), we will attain the exact solutions of Eq. (2) by using of the DT. We introduce a transformation of Lax Pair (3) as [1] = M,
(5)
where the sign [k] (k = 1, 2, 3, . . ., N) with N as a positive integer means that the matrix/function is engendered from the k-th DT, M is a matrix function depending on the spectrum parameter (z), z and t. Expression (5) can transform Lax Pair (3) into [1]
t
[1]
= U [1] [1] ,
z
= V [1] [1] ,
where the matrices U[1] and V[1] have the same forms as those of U and V. It can be found that M should satisfy the following conditions via Expressions (3) and (5), Mt + MU = U [1] M,
Mz + MV = V [1] M.
Thus, we can attain the DT M is
M=
(z)
0
0
(z)
−
11 [1 (z)]
∗ [ (z)] 12 1
12 [1 (z)]
∗ [ (z)] −11 1
1 (z)
0
0
−∗1 (z)
11 [1 (z)]
∗ [ (z)] 12 1
12 [1 (z)]
∗ [ (z)] −11 1
−1 ,
(6)
where [11 [1 (z)], 12 [1 (z)]]T is a solution of Lax Pair (3) at (z) = 1 (z) and q = q[0] , while q[0] is the seed solution of Eq. (2), 11 and 12 are both the functions of z and t, 1 (z) is a complex function and dependent on the variable z, while “−1 ” represents the inverse matrix. By virtue of DT (6), we can attain the analytic solutions of Eq. (2). Therefore, the first-order solutions of Eq. (2) can be given as [1]
q
[0]
=q
+e
−
√ ∗ [ (z)] 2 2[1 (z) + ∗1 z(z)]12 [1 (z)]11 d(z) 1
i(z)t 2 2
∗ [ (z)] + [ (z)]∗ [ (z)]] [11 [1 (z)]11 1 12 1 12 1
r(z)
.
(7)
When the seed solution q[0] = 0, the One-soliton solutions of Eq. (2) can be attained via Expression (7) with
z
[a(z)1 (z) + 2id(z)1 (z)2 ]dz − 1 (z)t},
11 [1 (z)] = exp{ 0
12 [1 (z)] = exp{−
z
[a(z)1 (z) + 2id(z)1 (z)2 ]dz + 1 (z)t},
0
by virtue of Lax pair (3), where exp means the e−exponential function. Iterating the DT N times, we can give the Nth-iterated potential transformation. N-order solutions of Eq. (2) can be attained as
q[N]
√ − = q[0] + 2 2e
i(z)t 2 N 2 k=1
∗ [ (z)] [k (z) + ∗k z(z)]k2 [k (z)]k1 k
d(z)
∗ [ (z)] + [ (z)]∗ [ (z)]] [k1 [k (z)]k1 k k2 k k2 k
r(z)
(8)
,
where k (z) s is the different complex spectrum parameters and m+1j [m+1 (z)] = [m+1 (z) + ∗m (z)]mj [m+1 (z)] −
Am [m (z) + ∗m (z)]mj [m (z)]
∗ [ (z)] + ∗ m1 [m (z)]m1 m m2 [m (z)]m2 [m (z)]
∗ ∗ Am = m1 [m+1 (z)]m1 [m (z)] + m2 [m+1 (z)]m2 [m (z)],
,
j = 1, 2, m = 1, 2, . . ., N − 1,
while [k1 [k (z)], k2 [k (z)]]T is a solution of Lax Pair (3) at (z) = k (z) and q = q[k−1] while k1 [k (z)] and k2 [k (z)] are both the functions of z and t.
D.-W. Zuo, H.-X. Jia / Optik 127 (2016) 11282–11287
11285
3. Propagation and interaction of the solitons and linear stability analysis 3.1. Linear stability analysis We will study the modulation instability (MI) of the plane-wave solutions for Eq. (2). The plane-wave solution of Eq. (2) is
q=
v21 − 2a v1 4dr
exp
i
−2a + v1 t + t2 2 2d
,
(9)
we rewrite v1 (z), a(z), (z), d(z) and r(z) as v1 , a, , d and r correspondingly, for getting the plane-wave solutions. Perturbation term is added to Expression (9) as
q=
v21 − 2a v1 4dr
+ u
exp
i
−2a + v1 t + t2 2 2d
,
(10)
where is a formal parameter and u is a function of z and t. Substituting Expression (10) into Eq. (2), we attain the linearized perturbation equation without the high-order terms of u, 2ia v1 u − iv21 u + 2ia v1 u∗ − iv21 u∗ − 4ad 2 ut + 4d v1 ut − 4id2 2 utt + 4d 2 uz = 0.
(11)
Solutions of Eq. (11) can be assumed as u = ı1 exp[i(kz − wt)] + ı2 exp[−i(kz − wt)] ,
(12)
where ı1 and ı2 are both the complex constants, while k and w are both the real wave number and perturbation frequency. Substitution Expression (12) to (11), we have 4idk 2 ı1 + 4iadw 2 ı1 + 4id2 w2 2 ı1 + 2ia v1 ı1 − 4idw v1 ı1 − iv21 ı1 + 2ia v1 ı2 − iv21 ı2 = 0,
(13a)
2ia v1 ı1 − iv21 ı1
(13b)
2
2
2
2 2
− 4idk ı2 − 4iadw ı2 + 4id w
ı2 + 2ia v1 ı2 + 4idw v1 ı2 − iv21 ı2
= 0.
Making the determinant of the matrix of coefficients for Eq. (13) to be zero, we find that the dispersion relation between k and w as
wv1 ± k = −aw +
w2 2 2d2 w2 2 + 2a v1 − v21 , √ 2 2
(14)
From Expression (14), it can be found that if w2 2 (2d2 w2 2 + 2a v1 − v21 ) ≥ 0, soliton solutions of Eq. (2) will be stable because the wave number is always real. Under the condition w2 2 (2d2 w2 2 + 2a v1 − v21 ) < 0, k will have the imaginary part, which means that for the perturbations w grow exponentially. Thus, the MI of Eq. (2) will occur. 3.2. Propagation and interaction of the solitons In the following, we discuss the propagation and interaction of the solitons, based on the linear stability analysis of Eq. (2). For investigating the effect of periodically varying potential, we can take the generalized external potential function v1 (z) as 1 is the period. Direction of the solitons is decided by the sign k sin(ωz), where k is the amplitude of solitons propagation, ω of the group velocity, a(z). The increment of the phase modulation, (z), equal to zero, when the amplitude modulation are considered. As for dispersion d(z), nonlinearity r(z), we will let those as the constants without loss of generality. While all of those properties still allowing for the analytical treatment of the equation.
Fig. 1. One solitons of Expression (7) with the parameters (z) = 0, d(z) = 2.3, r(z) = 1.5 and 1 (0) = 1.3 + i, and (a) of v1 (z) = 3 and a(z) = 2.2, (b) of v1 (z) = 12 sin(6z) and a(z) = 2.2, (c) of v1 (z) = 12 sin(6z) and a(z) =−2.2.
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Fig. 2. Two solitons of Expression (8) with the parameters (z) = 0, d(z) = 2.3, r(z) = 1.5, 1 (0) = 1.3 + i and 2 (0) = 1.7 + i, and (a) of v1 (z) = 3 and a(z) = 2.2, (b) of v1 (z) = 12 sin(6z) and a(z) = 2.2, (c) of v1 (z) = 12 sin(6z) and a(z) =−2.2.
Fig. 1 show the propagation of the one solitons. Fig. 1(a) exhibits that the trajectory of the soliton propagation is the two branches of a parabola from the excited time. Fig. 1(b) display that the propagation of one-soliton is the sin curve, which caused by the external potential function v1 (z). Fig. 1(b) and (c) show that the direction of one solitons propagation can be changed with the sign of the group velocity, a(z). As described above, we can say that: the external potential function affect the track of the optical soliton propagation, while the group velocity affect the direction of the optical soliton propagation. But, all of those phenomena depend on the analytical property of the control equation. Fig. 2 exhibit the interaction of the two solitons. Fig. 2(a) shows that: the two solitons are both propagate along the two breaches of a parabola; they are not separated; the two solitons have the same average velocity at some fixed time. Fig. 2(b) displays that: the two solitons are both propagate along the sin curve; the two solitons meet each other after a period. Fig. 2(b) and (c) exhibit that the direction of the two solitons interaction can also be varied with the sign of a(z). As mentioned above, the interaction of the two optical solitons with the time/space dependent phase modulation and external potential still interwoven with each other, i.e., the interaction of the two optical solitons display the bounded state. 4. Conclusions In this paper, a generalized nonautonomous NLS equation with both external potential and phase modulation, Eq. (2), has been studied, which can be used to described the nonautonomous solitons in the optical fiber. Main results have been attained as follows. 1) Lax Pair (3), DT (5) and N−soliton solutions (8) of Eq. (2) have been obtained. 2) Condition (14) of the MI for Eq. (2) have been attained. 3) Propagation and interaction of the solitons have been analysed: Fig. 1(a) has exhibited that the trajectory of the soliton propagation is the two branches of a parabola from the excited time. Fig. 1(b) has displayed that the propagation of onesoliton is a sin curve, which caused by the potential function v1 (z); Fig. 1(b) and (c) have shown that the direction of one solitons propagation can be changed with the sign of a(z). Fig. 2(a) has shown that: the two solitons are both propagate along the two breaches of a parabola; they are not separated; the two solitons have the same average velocity at some fixed time. Fig. 2(b) has displayed that: the two solitons are both propagate along the sin curve; the two solitons meet each other after a period. Fig. 2(b) and (c) have exhibited that the direction of the two solitons interaction can also be varied with the sign of a(z). Acknowledgements This work has been supported by the Foundation of Hebei Education Department of China under Grant No. QN2015051. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991. N. Benes, A. Kasman, K. Young, J. Nonlinear Sci. 16 (2006) 179. I. Christov, C.I. Christov, Phys. Lett. A 372 (2008) 841. A. Hasegawa, Solitons in Optical Fibers Optical Solitons in Fibers, Springer, Berlin, 1989. J.R. Taylor, Optical Solitons: Theory and Experiment, Cambridge Univ. Press, Cambridge, 1992. M.I. Weinstein, Commun. Math. Phys. 87 (1983) 567. C.H. Dai, Y.Y. Wang, Nonlinear Dyn. 83 (2016) 2453. C.H. Dai, Y. Wang, J. Liu, Nonlinear Dyn. 84 (2016) 1157. C.H. Dai, Y.J. Xu, Appl. Math. Model. 39 (2015) 7420. V.N. Serkin, A. Hasegawa, T.L. Belyaeva, Phys. Rev. Lett. 98 (2007) 074102.
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V.N. Serkin, A. Hasegawa, Phys. Rev. Lett. 85 (2000) 4502. V. Bagnato, D.E. Pritchard, D. Kleppner, Phys. Rev. A 35 (1987) 4354. C.Q. Dai, Y.Y. Wang, Opt. Commun. 315 (2014) 303. A. Mahalingama, M.S. Mani Rajan, Opt. Fiber Technol. 25 (2015) 44. B. Tian, Y.T. Gao, Phys. Lett. A 342 (2005) 228. Y.T. Gao, B. Tian, Phys. Lett. A 361 (2007) 523. B. Tian, Y.T. Gao, Phys. Plasmas 12 (2005) 070703. R. Hirota, The Direct Method in Soliton Theory, Cambridge Univ. Press, Cambridge, 2004. M.J. Ablowitz, D.J. Kaup, A.C. Newell, H. Segur, Phys. Rev. Lett. 31 (1973) 125.
11287
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Original research article
Interaction of the nonautonomous soliton in the optical fiber Da-Wei Zuo a,∗ , Hui-Xian Jia b a b
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Department of Basic, Shijiazhuang Post and Telecommunication Technical College, Shijiazhuang 050021, China
a r t i c l e
i n f o
Article history: Received 9 May 2016 Received in revised form 3 September 2016 Accepted 5 September 2016 PACS: 05.45.Yv 52.35.Mw 52.35.Sb
a b s t r a c t Investigated in this paper is a generalized nonautonomous nonlinear Schrödinger equation with both external potential and phase modulation, which can be used to described the nonautonomous soliton in the optical fiber. With our results as follows: Lax pair, Daboux transformation, condition of the modulation instability and N−soliton solutions have been obtained; Propagation and interaction of the solitons have been analysed: (1) the group velocity and linear potential affect the direction and trajectory of one/two solitons propagation, respectively; (2) two solitons have the same average velocity at some fixed time or a period. © 2016 Elsevier GmbH. All rights reserved.
Keywords: Nonautonomous soliton Darboux transformation Optical fiber
1. Introduction Optical solitons1 have been playing an important role in the fundamental research of the nonlinear science and in the field of the next generation optical communication systems with long distance, high speed and large capacity, which caused by the balance of the self-phase modulation and group-velocity dispersion [4,5]. Propagation of the picosecond optical solitons in a nonlinear, dispersive medium can be governed by the nonlinear Schrödinger (NLS) equation, iQ Z + QTT + 2Q 2 Q ∗ = 0,
(1)
which also appears in the fluid mechanics, plasma physics, turbulence, quantum field theory, condensed matter, quantum electronics, phase transitions, biophysics and star formation, where i2 =−1, * is the complex conjugate and Q is a function of the scaled temporal coordinate T and spatial coordinate Z [1,6–9]. Besides, nonautonomous solitons which can be achieved by virtue of the inhomogeneous distribution of the nonlinear medium have also been potently useful for the various applications in the optical solitons communication system of their special features [10]. For example, existing solitons in nonautonomous physical systems need not only the varying in time nonlinearity and dispersion dependent on each other but also they satisfy some integrability conditions; this law is called as Serkin–Hasegawa theorem [11]. This theorem is applied to a nonautonomous system and soliton management are discussed in Ref. [11]. Nonautonomous solitons can be described by the the generalized nonautonomous NLS equations with variable coefficients.
∗ Corresponding author. E-mail address: [email protected] (D.-W. Zuo). 1 A soliton is a solitary wave which preserves its velocity and shape after the interaction [1], i.e., the soliton can be considered as a quasi-particle [2,3]. http://dx.doi.org/10.1016/j.ijleo.2016.09.022 0030-4026/© 2016 Elsevier GmbH. All rights reserved.
D.-W. Zuo, H.-X. Jia / Optik 127 (2016) 11282–11287
11283
Variable coefficients NLS equation has been widely investigated, e.g., law of the soliton adaptation to an external potential has been studied in the nonautonomous solitons [12], and nonautonomous solitons in parity time symmetric potentials for both exponential diffraction decreasing and periodically modulated waveguide have been reported [13]. However, we find that there has not much discussion about the propagation and interaction of the soliton with the time/space dependent phase modulation and external potential. An important problem is that how the time/space dependent phase modulation and external potentials are affecting the propagation and interaction of the nonautonomous optical soliton? How one can control the dynamical behavior of the nonautonomous optical soliton through the time/space dependent phase modulation and external potential? Being motivated by the above aspects and to attain the answers of those questions, this paper will discuss the nonautonomous optical soliton in nonlinear optical fibers with the time/space dependent phase modulation and external potential. Based on the above discussions, we will work on a generalized nonautonomous NLS equation with both external potential and phase modulation, which can be written as [14] qz + a(z)qt − id(z)qtt − ir(z)q2 q∗ − d(z)(z)q + b1 (z)q + ib2 (z, t)q = 0 , where b1 (z) =
r (z)
1 2
r(z)
b22 (z, t) =
1 2
−
d (z) , d(z)
b2 (z, t) = b21 (z, t) + b22 (z, t),
(2)
b21 (z, t) = [a(z)(z) − v1 (z)]t,
(z) − d(z)(z)2 t 2 ,
where q is the solitary envelope of the optical pulse and depend on the scaled temporal coordinate t and spatial coordinate z for the optical soliton in the fiber. Real functions d(z), r(z), b1 (z) and (z) denote the dispersion, nonlinearity, loss or gain and increment of the phase modulation respectively. b2 (z, t) represents the external potential and it is assumed to be a function of the propagation distance and time. Especially, v1 (z) and b22 (z, t) are represents the linear and harmonic potential respectively. a(z) results from the group velocity. Physically speaking, Eq. (2) is not only used to describe the propagation of optical pulse in the inhomogeneous optical fiber, but also character the core of a dispersion-managed soliton. Besides, the combined external potentials in Eq. (2) are efficient in supporting the matter-wave solitons [14]. However, to our knowledge, lax pair, N-soliton solutions, Darboux transformation (DT) and linear stability analysis of Eq. (2) have not been obtained. With the aid of symbolic computation [15–17], Section 2, lax pair, N-soliton solutions and DT for Eq. (2) will be obtained. Section 3, Linear stability analysis, propagation and interaction of solitons will be discussed. Section 4 will be the conclusions. 2. Lax pair, DT and N−Soliton solutions of Eq. (2) 2.1. Lax pair of Eq. (2) Nonlinear evolution equations (NLEEs) with soliton solutions have some special characters, the basic property of which is that those can be equivalent to a pair of linear system, i.e. the Lax pair [18]. People call those equations as the Lax integrable. By virtue of the Ablowitz–Kaup–Newell–Segur technology [19], Lax pair of Eq. (2) are attained as t = U,
(3a)
z = V,
(3b) )T ,
where = (1 , 2 while 1 and 2 are both the functions of z and t, the superscript T denotes the matrix/vector transpose and matrices U = (Uij ) and V = (Vij ) have the forms
U11 = −U22 = −(z), V11 = −V22 =
V12 =
V21 =
∗ U12 = −U21 =
r(z) it 2 (z) e 2 q, 2d(z)
1 i[qq∗ r(z) + t v1 (z)] + [a(z) − 2td(z)(z)](z) + 2id(z)(z)2 , 2
r[z] it 2 (z) e 2 −a(z)q + d(z)(z)tq − 2i(z)d(z)q + id(z)qt , 2d(z) r[z] −it 2 (z) e 2 [a[z]q∗ − td(z)(z)q∗ + 2id(z)(z)q∗ + id(z)q∗t ]. 2d(z)
Eq. (2) can be obtained via the compatibility condition Uz − Vt + UV − VU = 0, if and only if the spectrum parameter satisfies the following equation (z)z = 2d(z)(z)(z) −
iv1 (z) , 2
(4)
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D.-W. Zuo, H.-X. Jia / Optik 127 (2016) 11282–11287
i.e. the spectrum parameter (z) = e
2d(z)(z)dz
[(0) +
z 0
e
−
2d(z)(z)dz
v1 (z)dz], where (0) is a integration constant.
2.2. DT of Eq. (2) It is well known that DT is an efficient way to construct the exact solutions of the NLEEs, especially to the Lax integral system. Based on Lax Pair (3), we will attain the exact solutions of Eq. (2) by using of the DT. We introduce a transformation of Lax Pair (3) as [1] = M,
(5)
where the sign [k] (k = 1, 2, 3, . . ., N) with N as a positive integer means that the matrix/function is engendered from the k-th DT, M is a matrix function depending on the spectrum parameter (z), z and t. Expression (5) can transform Lax Pair (3) into [1]
t
[1]
= U [1] [1] ,
z
= V [1] [1] ,
where the matrices U[1] and V[1] have the same forms as those of U and V. It can be found that M should satisfy the following conditions via Expressions (3) and (5), Mt + MU = U [1] M,
Mz + MV = V [1] M.
Thus, we can attain the DT M is
M=
(z)
0
0
(z)
−
11 [1 (z)]
∗ [ (z)] 12 1
12 [1 (z)]
∗ [ (z)] −11 1
1 (z)
0
0
−∗1 (z)
11 [1 (z)]
∗ [ (z)] 12 1
12 [1 (z)]
∗ [ (z)] −11 1
−1 ,
(6)
where [11 [1 (z)], 12 [1 (z)]]T is a solution of Lax Pair (3) at (z) = 1 (z) and q = q[0] , while q[0] is the seed solution of Eq. (2), 11 and 12 are both the functions of z and t, 1 (z) is a complex function and dependent on the variable z, while “−1 ” represents the inverse matrix. By virtue of DT (6), we can attain the analytic solutions of Eq. (2). Therefore, the first-order solutions of Eq. (2) can be given as [1]
q
[0]
=q
+e
−
√ ∗ [ (z)] 2 2[1 (z) + ∗1 z(z)]12 [1 (z)]11 d(z) 1
i(z)t 2 2
∗ [ (z)] + [ (z)]∗ [ (z)]] [11 [1 (z)]11 1 12 1 12 1
r(z)
.
(7)
When the seed solution q[0] = 0, the One-soliton solutions of Eq. (2) can be attained via Expression (7) with
z
[a(z)1 (z) + 2id(z)1 (z)2 ]dz − 1 (z)t},
11 [1 (z)] = exp{ 0
12 [1 (z)] = exp{−
z
[a(z)1 (z) + 2id(z)1 (z)2 ]dz + 1 (z)t},
0
by virtue of Lax pair (3), where exp means the e−exponential function. Iterating the DT N times, we can give the Nth-iterated potential transformation. N-order solutions of Eq. (2) can be attained as
q[N]
√ − = q[0] + 2 2e
i(z)t 2 N 2 k=1
∗ [ (z)] [k (z) + ∗k z(z)]k2 [k (z)]k1 k
d(z)
∗ [ (z)] + [ (z)]∗ [ (z)]] [k1 [k (z)]k1 k k2 k k2 k
r(z)
(8)
,
where k (z) s is the different complex spectrum parameters and m+1j [m+1 (z)] = [m+1 (z) + ∗m (z)]mj [m+1 (z)] −
Am [m (z) + ∗m (z)]mj [m (z)]
∗ [ (z)] + ∗ m1 [m (z)]m1 m m2 [m (z)]m2 [m (z)]
∗ ∗ Am = m1 [m+1 (z)]m1 [m (z)] + m2 [m+1 (z)]m2 [m (z)],
,
j = 1, 2, m = 1, 2, . . ., N − 1,
while [k1 [k (z)], k2 [k (z)]]T is a solution of Lax Pair (3) at (z) = k (z) and q = q[k−1] while k1 [k (z)] and k2 [k (z)] are both the functions of z and t.
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3. Propagation and interaction of the solitons and linear stability analysis 3.1. Linear stability analysis We will study the modulation instability (MI) of the plane-wave solutions for Eq. (2). The plane-wave solution of Eq. (2) is
q=
v21 − 2a v1 4dr
exp
i
−2a + v1 t + t2 2 2d
,
(9)
we rewrite v1 (z), a(z), (z), d(z) and r(z) as v1 , a, , d and r correspondingly, for getting the plane-wave solutions. Perturbation term is added to Expression (9) as
q=
v21 − 2a v1 4dr
+ u
exp
i
−2a + v1 t + t2 2 2d
,
(10)
where is a formal parameter and u is a function of z and t. Substituting Expression (10) into Eq. (2), we attain the linearized perturbation equation without the high-order terms of u, 2ia v1 u − iv21 u + 2ia v1 u∗ − iv21 u∗ − 4ad 2 ut + 4d v1 ut − 4id2 2 utt + 4d 2 uz = 0.
(11)
Solutions of Eq. (11) can be assumed as u = ı1 exp[i(kz − wt)] + ı2 exp[−i(kz − wt)] ,
(12)
where ı1 and ı2 are both the complex constants, while k and w are both the real wave number and perturbation frequency. Substitution Expression (12) to (11), we have 4idk 2 ı1 + 4iadw 2 ı1 + 4id2 w2 2 ı1 + 2ia v1 ı1 − 4idw v1 ı1 − iv21 ı1 + 2ia v1 ı2 − iv21 ı2 = 0,
(13a)
2ia v1 ı1 − iv21 ı1
(13b)
2
2
2
2 2
− 4idk ı2 − 4iadw ı2 + 4id w
ı2 + 2ia v1 ı2 + 4idw v1 ı2 − iv21 ı2
= 0.
Making the determinant of the matrix of coefficients for Eq. (13) to be zero, we find that the dispersion relation between k and w as
wv1 ± k = −aw +
w2 2 2d2 w2 2 + 2a v1 − v21 , √ 2 2
(14)
From Expression (14), it can be found that if w2 2 (2d2 w2 2 + 2a v1 − v21 ) ≥ 0, soliton solutions of Eq. (2) will be stable because the wave number is always real. Under the condition w2 2 (2d2 w2 2 + 2a v1 − v21 ) < 0, k will have the imaginary part, which means that for the perturbations w grow exponentially. Thus, the MI of Eq. (2) will occur. 3.2. Propagation and interaction of the solitons In the following, we discuss the propagation and interaction of the solitons, based on the linear stability analysis of Eq. (2). For investigating the effect of periodically varying potential, we can take the generalized external potential function v1 (z) as 1 is the period. Direction of the solitons is decided by the sign k sin(ωz), where k is the amplitude of solitons propagation, ω of the group velocity, a(z). The increment of the phase modulation, (z), equal to zero, when the amplitude modulation are considered. As for dispersion d(z), nonlinearity r(z), we will let those as the constants without loss of generality. While all of those properties still allowing for the analytical treatment of the equation.
Fig. 1. One solitons of Expression (7) with the parameters (z) = 0, d(z) = 2.3, r(z) = 1.5 and 1 (0) = 1.3 + i, and (a) of v1 (z) = 3 and a(z) = 2.2, (b) of v1 (z) = 12 sin(6z) and a(z) = 2.2, (c) of v1 (z) = 12 sin(6z) and a(z) =−2.2.
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Fig. 2. Two solitons of Expression (8) with the parameters (z) = 0, d(z) = 2.3, r(z) = 1.5, 1 (0) = 1.3 + i and 2 (0) = 1.7 + i, and (a) of v1 (z) = 3 and a(z) = 2.2, (b) of v1 (z) = 12 sin(6z) and a(z) = 2.2, (c) of v1 (z) = 12 sin(6z) and a(z) =−2.2.
Fig. 1 show the propagation of the one solitons. Fig. 1(a) exhibits that the trajectory of the soliton propagation is the two branches of a parabola from the excited time. Fig. 1(b) display that the propagation of one-soliton is the sin curve, which caused by the external potential function v1 (z). Fig. 1(b) and (c) show that the direction of one solitons propagation can be changed with the sign of the group velocity, a(z). As described above, we can say that: the external potential function affect the track of the optical soliton propagation, while the group velocity affect the direction of the optical soliton propagation. But, all of those phenomena depend on the analytical property of the control equation. Fig. 2 exhibit the interaction of the two solitons. Fig. 2(a) shows that: the two solitons are both propagate along the two breaches of a parabola; they are not separated; the two solitons have the same average velocity at some fixed time. Fig. 2(b) displays that: the two solitons are both propagate along the sin curve; the two solitons meet each other after a period. Fig. 2(b) and (c) exhibit that the direction of the two solitons interaction can also be varied with the sign of a(z). As mentioned above, the interaction of the two optical solitons with the time/space dependent phase modulation and external potential still interwoven with each other, i.e., the interaction of the two optical solitons display the bounded state. 4. Conclusions In this paper, a generalized nonautonomous NLS equation with both external potential and phase modulation, Eq. (2), has been studied, which can be used to described the nonautonomous solitons in the optical fiber. Main results have been attained as follows. 1) Lax Pair (3), DT (5) and N−soliton solutions (8) of Eq. (2) have been obtained. 2) Condition (14) of the MI for Eq. (2) have been attained. 3) Propagation and interaction of the solitons have been analysed: Fig. 1(a) has exhibited that the trajectory of the soliton propagation is the two branches of a parabola from the excited time. Fig. 1(b) has displayed that the propagation of onesoliton is a sin curve, which caused by the potential function v1 (z); Fig. 1(b) and (c) have shown that the direction of one solitons propagation can be changed with the sign of a(z). Fig. 2(a) has shown that: the two solitons are both propagate along the two breaches of a parabola; they are not separated; the two solitons have the same average velocity at some fixed time. Fig. 2(b) has displayed that: the two solitons are both propagate along the sin curve; the two solitons meet each other after a period. Fig. 2(b) and (c) have exhibited that the direction of the two solitons interaction can also be varied with the sign of a(z). Acknowledgements This work has been supported by the Foundation of Hebei Education Department of China under Grant No. QN2015051. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]
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