- Email: [email protected]
Jacobi elliptic function solutions to (1 + 1) dimensional nonlinear Schrödinger equation in management system
Optik 122 (2011) 1289–1292
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Jacobi elliptic function solutions to (1 + 1) dimensional nonlinear Schrödinger equation in management system Pingyuan Liang a,∗ , Jianchu Liang b a b
College of Physics Science and Information Engineering, University of Jishou, Hunan 416000, China Department of Electronic Science, Huizhou University, Guangdong 516000, China
a r t i c l e
i n f o
Article history: Received 4 February 2010 Accepted 10 August 2010
Keywords: Analytical solution Optical spatial soliton
a b s t r a c t A new kind of solutions namely elliptic function solutions to the (1 + 1) dimensional nonlinear Schrödinger equation (NLSE) in dispersion, nonlinearity and gain management system are obtained. The solutions are the generalization of analytical solutions to one-dimensional NLSE. The properties of the amplitudes and the phases of the solutions are investigated. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction
2. Analytical solutions to the 1D GNLSE
The analytical solution to one-dimensional (1D) normalized nonlinear Schrödinger equation [1–4] which governs the propagation of light pulses in optical fiber:
The normalized GNLSE in management system that governs the propagation of a light pulse in optical fiber can be written as: i
1 ∂2 u ∂u + + |u2 |u = 0, i 2 ∂t 2 ∂z has been obtained by various kinds of method, such as inverse scattering transformation [5], Darboux–Bäcklund transformation [6], self-similarity technique [7], and variational method [8] as well as moment method [9]. The analytical soliton solution is in the form of travelling wave hyperbolic secant function u = sec h(z + 2 2 vt + 0 )ei(−kz+1/2( −v )t+ϕ0 ) , where and v represent the amplitude and the velocity of the soliton. 1D generalized nonlinear Schrödinger equation (GNLSE) in management system [10] is investigated recently with self-similar method. In management system, all the parameters are variable along with propagation distance z. Kruglov et al. [7] obtained sech and tanh self-similar solitary wave solutions to GNLSE; Chen et al. [11] obtained self-similar parabolic, Hermite–Gaussian, and hybrid solutions to the GNLSE. In this paper we investigate the GNLSE in management system with another method namely elliptic function method, and a novel kind of exact solution will be obtained.
∂u ∂2 u 1 + ˇ(z) 2 + (z)|u|2 u − i(z)u = 0, 2 ∂z ∂t
where ˇ, and are dispersive coefficient nonlinear coefficient and gain (loss) coefficient, respectively. Eq. (1) governs the propagation of a light pulse in optical fiber. Our strategy to solve partial equation (1) is as follows: Considering that 1D NLSE possesses traveling wave hyperbolic secant function solution, and the hyperbolic secant function is a special case of Jacobi elliptic function, we search for the traveling wave Jacobi elliptic function solution to the 1D GNLSE. According to the strategy, we look for the solution to the 1D GNLSE (1) in the form: u(z, t) = A(z, t)eiB(z,t) ,
0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.08.018
(2)
where A = f0 (z) + f1 (z)F() + f−1 (z)F−1 (), B = a(z)t2 + b(z)t + c(z), F() is a Jacobi elliptic function which is the solution of the ordinary differential equation (F )2 = q0 + q2 F 2 + q4 F 4 , (z, t) = k(z)t + w(z)z is the traveling wave variable, and by incorporating z into w(z), is rewritten as (z, t) = k(z)t + w(z). a is the curvature of the wavefront, the spatial frequency shift b is relative to the parallel motion of the wave-front, and c is homogeneous phase shift. All the coefficients are distance-dependent. Substituting Eq. (2) into (1), we get the coupling equations:
∗ Corresponding author. E-mail address: [email protected] (P. Liang).
(1)
∂A ∂A ∂B A ∂2 B +ˇ + 2 ∂t 2 ∂z ∂t ∂t
− A = 0,
(3)
1290
P. Liang, J. Liang / Optik 122 (2011) 1289–1292
Table 1 Relations between values of (q0 , q2 , q4 ) and corresponding F() in ode F 2 = q0 + q2 F 2 + q4 F 4 . q0
q2
q4
F 2 = q0 + q2 F 2 + q4 F 4
F
1
−(1 + m2 )
m2
F 2 = (1 − F 2 )(1 − m2 F 2 )
sn, cd =
1 − m2
2m2 − 1
−m2
F 2 = (1 − F 2 )(m2 F 2 + 1 − m2 )
cn
2
m −1
2−m
−1
F 2 = (1 − F 2 )(F 2 + m2 − 1)
dn
2
−(1 + m )
m
2 2
1
−m2
2m2 − 1
1 − m2
−1
2−m
m −1
2
2
= (1 − F )(m − F )
F
2
= (1 − F 2 )[(m2 − 1)F 2 − m2 ]
nc = (cn)−1
F
2
= (1 − F )[(1 − m )F − 1]
nd = (dn)−1
2
2
2
ns = (sn)
2
2
2
2
2 − m2
1 − m2
F
= (1 + F 2 )[(1 − m2 )F 2 + 1]
sc =
sn cn
1
2m2 − 1
−m2 (1 − m2 )
F 2 = (1 + m2 F 2 )[(1 + (m2 − 1)F 2 ]
sd =
sn dn
2 − m2
1
F 2 = (1 + F 2 )(F 2 + 1 − m2 )
cs =
cn sn
ds =
dn sn
−m (1 − m ) 2
2m − 1
2
∂B ˇ −A + 2 ∂z
2
∂2 A −A ∂t 2
∂B ∂t
1
F
2 + A3 = 0.
(4)
Further substituting of the expressions of A, B, into (3) and (4), utilizing the properties of elliptic function: (F )2 = q0 + q2 F 2 + q4 F 4 and F = q2 F + 2q4 F3 , and imposing the coefficients of tq Fn and tq Fn F
−1
F
2
1
1 − m2
cn dn
2
= (F + m )(F + m − 1) 2
2
2
2
, dc =
dn cn
(q = 0, 1, 2; n = −3, −2, −1, 0, 1, 2, 3) be separately equal to zero, we obtain a system of algebraic or first-order ordinary differential equations for fi , k, ω, a, b, and c, from which we obtain the results about the coefficients:
z f0 = 0;
fi = fi0 e
0
(−aˇ)dz
(i = 1, −1),
(5a)
Fig. 1. 3D intensity |u(t, x)|2 plots of Jacobian SN (rows 1 and 2), Jacobian CN function (row 3) and Jacobian DN function (row 4) solutions. The distributed parameters are set as ˇ = 1 + 0.1 cos z, and a0 = 0.05 in row 1; ˇ = 1 + 0.1 cos z, a0 = 0 in rows 2–4; = 0.1 cos t in columns 1 and 2, = 0 in columns 3 and 4. The four columns corresponding to m = −0.5, 0, 0, and 1, respectively.
P. Liang, J. Liang / Optik 122 (2011) 1289–1292
z
w=− 0
k = k0 e c= 0
b = b0 e a=
z
−2
z
ˇbkdz + w0 ,
z
ˇadz
0
k2 f12
(5e)
t 0
ˇdt
=εa0 ,
f−1 = ±
t
where ε= 1+2a0
−1 ˇdt
,
(5f)
0
ˇq4 (when f1 = / 0) or = −
(5d)
,
a0
= −
(5c)
ˇ 2 ˇ k q2 − b2 + 3f1 f−1 dz + c0 , 2 2
−2
1+2a0
(5b)
ˇadz
0
1291
k2 2 f−1
ˇq0 (when f−1 = / 0)
(5g)
q0 f1 (when f1 = / 0 and f−1 = / 0). q4
(5h)
What is amazing and delectable is that all the formulas are self-consistent and self-contained, which means that our solving procedure is successful. The only necessary and sufficient condition for the existence of Jacobi elliptic function solution to the 1D GNLSE is that the nonlinearity and the dispersion should fulfill a certain relation expressed by Eq. (5g). This condition is actually a generalized constraint for generalized NLSE to have elliptic function solutions including the hypersecant and hypertangent soliton solutions. As for 1D NLSE with varying parameters, solitons exist only under certain conditions [10,11]. From Eq. (5) of reference [10], the necessary condition for 1D GNLSE to have soliton solutions is = cˇ, where c is a constant, and Eq. (1) in Ref. [10] can be normalized so as that = ± ˇ. In our paper, constraint (5 g) results in = − q4 ˇ assuming that k0 = 1, f10 = 1, = 0, and a = 0. As for elliptic functions, cn → sec h when m = 1, and then q4 = − 1 (see Table 1) and = ˇ is obtained in this case; sn → tan h when m = 1 and then q4 = 1 and = − ˇ. This is consistent with the above special case in Ref. [11] as well as the 1D normalized equation in the introduction with constant coefficients. So the condition for the existence of the soliton solutions of 1D NLSE is just a special case of the constraint in our paper. This constraint can be realized in some situations, especially in the case of constant coefficients. This constraint applies to the physically relevant systems apart from the cases when = a = 0. Substituting the resulting expressions (5a)–(5h) into Eq. (2), we get the following analytical solutions (6) and (7) to the 1D GNLSE. Solutions (6) and (7) are the single elliptic function and hybrid elliptic function solution, respectively:
z
u(z, t) = f10 e or
0
z
u(z, t) = f10 e
0
(−aˇ)dz
(−aˇ)dz
F()ei[at
2 +bt+c]
F() ±
(f−1 = 0),
(6)
q0 −1 2 F () ei[at +bt+c] q4
Fig. 2. 3D intensity |u(t, x)|2 of Hybrid Jacobi elliptic function solutions:
cn(, m) +
(f1 = / 0 and f−1 = / 0), where −
z 0
a
is
ˇb0 k0 e
−4
determined
z 0
ˇadz
by
dz + w0 ,
c = [(q2 /2)(k02 + l02 + p20 ) − (b20 /2)] c0
in
√ 3 q0 q4 k02 ]
Eq.
z 0
ˇe
(6) −4
z 0
or ˇadz
Eq.
t 0
b = b0 e ˇe
= k0 t e
(5f),
−4
t 0
−2
t
ˇadt
0
ˇadt
,
−2
z 0
(7) ˇadz
m2 − 1/m2 nc(, m)
m2 − 1/m2 nc(, m)
(b),
1 − m2 nd(, m) (c) and dn(, m) − 1 − m2 nd(, m) (d), where the dn(, m) + parameters are set to be = 0.5 cos t, ˇ = cos 0.5t, and a0 = 0. m is set to be 2 in (a) and (b), and 0.5 in (c) and (d).
and
dt +
If the initial phase chirp is presented, i.e., a0 = / 0, the solutions are respectively
c = [(q2 /2)(k02 + l02 + p20 ) − (b20 /2) ∓
dt + c0 in Eq. (7).
cn(, m) +
(a),
t u = f10 ε3/2 e
0
dt
F()ei[at
2 +bt+c]
,
(6a)
1292
P. Liang, J. Liang / Optik 122 (2011) 1289–1292
and
t
u = f1,0 ε3/2 e
0
dt
F() ±
q0 −1 2 F () ei[at +bt+c] , q4
(7a)
where = εk0 t + (b0 /2a0 )k0 (ε − 1) + w0 , b = εb0 , and c = (1 − ε/4a0 )(k02 q2 − b20 ) + c0 in Eq. (6a) or c = (1 − ε/4a0 )(k02 q2 − b20 ∓ √ 6 q0 q4 k02 ) + c0 in (7a). When a0 = 0, the solutions are
z
u(z, t) = f10 e or
z
u(z, t) = f10 e where
0
z
0
dz
dz
b = b0 ,
F()ei(bt+c) ,
F() ±
(6b)
q0 −1 F () ei(bt+c) , q4
(7b)
ˇdz + w0 ,
c = [(q2 /2)k02 −
= k0 t − b0 k0
z 0
(b20 /2)] 0 ˇdz + c0 in Eq. (6b) z √ 3 q0 q4 k02 ] 0 ˇdz + c0 in Eq. (7b).
or
c = [(q2 /2)k02 − (b20 /2) ∓
3. Analyses and discussion In order to manifest the solution, we substitute Eq. (6) into Eq. (1), where we set the following periodically distributed parameters: = 0.1 cos z, ˇ = 1 + 0.1 cos z, The nonlinear coefficient is = 2 (1 + 2a (z + 0.1 sin z)))e−0.2 sin z according the constraint −(k02 /f10 0 (5g). We find that Eq. (1) is satisfied, which proves that Eq. (6) is just the solution to Eq. (1). It should be pointed out that the functions , ˇ, and s are chosen ad hoc in order to get analytical expressions for the parameters a, b, and , and to simplify the calculation. There are 12 different Jacobi elliptic functions, each of which differentiated by different order m (see Table 1). Jacobi elliptic functions degenerate into triangular functions when m = 0, and hyperbolic functions when m = 1. The amplitude of the light wave or matter wave should be finite, thus the elliptic functions that ending with “s” (ns, cs, ds) cannot be selected to be the amplitude function; all that ending with “n” (sn, cn, dn) can absolutely be amplitude function in any order; and all that ending with “d” or “c” can be amplitude function in some orders. As for hybrid solution, only F() = cn(, m) ± m2 − 1/m2 nc(, m), (m > 1) and
F() = dn(, m) ± 1 − m2 nd(, m) (−1 < m < 1) can be wave functions. Depicted in Fig. 1 are the wave intensities of Jacobi sn (rows 1 and 2), Jacobi cn (row 3) and Jacobi dn solution (row 4). Without losing generality, the initial conditions are set to be: b0 = k0 = 1, w0 = c0 = 0, and f10 = 1. The varying parameters are still set as ˇ = 1 + 0.1 cos z, and a0 = 0.05 in row 1; ˇ = 1 + 0.1 cos z, and a = 0 in rows 2–4. We set = 0.1 cos t in columns 1 and 2, = 0 in columns 3 and 4. The four columns correspond to m = −0.5, 0, 0, and 1, respectively. The conventional one-dimensional hyperbolic secant (hyperbolic tangent) soliton [12,13] solution is the special case of our results (see Fig. 1(2d) and (3d)). In these cases, and ˇ must be of the same sign for hypersecant (bright) soliton, and of the contrary sign for hypertangent (black) soliton solution which is derived from the necessary condition (5g). There are sine function solutions (Fig. 1(2c)) and cosine function solutions (Fig. 1(3c)) to the NLSE. The periodical gain (loss) periodically amplifies (deduces) the
amplitude of the intensity, and net gain (loss) amplifies (deduces) the amplitude. In the case in which the chirp is absent (i.e., a = 0), the energy conserves provided that there is no gain (loss) (see the lower two lines in Fig. 1); however in the case in which an initial chirp is present (i.e., a = / 0), the energy fluctuate along with distance even if there is no gain (loss), as shown in the upper row of Fig. 1. This phenomenon has never been found before. We think that the unconservation of the energy is mathematically due to the presence of the wave-front curvature. When an initial phase chirp / 0) is input, the curvature of the phase wave-front evolves in (a0 = t the form of a = a0 /(1 + 2a0 0 ˇdt ), but the curvature center fixes at ( − b0 /2a0 , − b0 /2a0 , − b0 /2a0 ). If the initial wave is unchirped (a0 = 0), it remains unchirped. Fig. 2 depicts the hybrid Jacobi elliptic function solutions cn(, m) + m2 − 1/m2 nc(, m)), cn(, m) + m2 − 1/m2 nc(, m) (b),
dn(, m) +
1 − m2 nd(, m)
(c)
and
dn(, m) −
1 − m2 nd(, m)
(d), where the parameters are set to be = 0.5 cos t, ˇ = cos 0.5t, and a0 = 0. m is set to be 2 in (a) and (b), and 0.5 in (c) and (d). These solutions have never been obtained before, and they are absolutely different with the single elliptic function solutions. 4. Conclusions We obtain a new kind of analytical solutions to 1D GNLSE with distributed coefficients. The solutions are in the form of travelling wave elliptic function, which are the generalization of the conventional hyperbolic function soliton solutions to 1D NLSE. The method that we use to solve the NLSE will pave the way to new methods for solving partial differential equations. References [1] J.E. Bjorkholm, A. Ashkin, cw Self-focusing and self-trapping of light in sodium vapor, Phys. Rev. Lett. 32 (1974) 129–132. [2] J.S. Aitchison, A.M. Weiner, Y. Silberberg, M.K. Oliver, J.L. Jackel, D.E. Leaird, E.M. Vogel, P.W.E. Smith, Observation of spatial optical solitons in a nonlinear glass waveguide, Opt. Lett. 15 (1990) 471–473. [3] R. de la Fuente, O. Varela, H. Michinel, Fourier analysis of non-paraxial selffocusing, Opt. Commun. 173 (2000) 403–411. [4] L. Yan, X. Mi, Y. Wu, J. Liang, Incoherently coupled anti-dark soliton pairs, Optik 121 (2010) 54–56. [5] M.J. Ablowitz, Nonlinear Schrödinger Equation and Inverse Scattering, Cambridge University Press, New York, 1991. [6] V.B. Matveev, M.A. Salle, Dardoux Transformations and Solitons (Springer Series in Non-linear Dynamics), Springer, Berlin, 1991. [7] V.I. Kruglov, A.C. Peacock, J.D. Harvey, Exact Self-Similar Solutions of the Generalized Nonlinear Schrödinger Equation with Distributed Coefficients, Phys. Rev. Lett. 90 (1-4) (2003) 113902. [8] V.S. Filho, F.K. Abdullaev, A. Gammal, L. Tomip, Theory of a mode-locked atom laser with toroidal geometry, Phys. Rev. A 63 (1–4) (2001) 053603. [9] V.M. Pérez-García, P. Torres, J.J. García-Ripoll, H. Michinel, Moment analysis of paraxial propagation in a nonlinear graded index fibre, J. Opt. B: Quant. Semiclass. Opt. 2 (2000) 353–358. [10] V.N. Serkin, A. Hasegawa, Novel soliton solutions of the nonlinear Schrödinger equation model, Phys. Rev. Lett. 85 (2000) 4502–4505. [11] S. Chen, L. Yi, D.S. Guo, P. Lu, Self-similar evolutions of parabolic, Hermite–Gaussian, and hybrid optical pulses: universality and diversity, Phys. Rev. E 72 (1–5) (2005) 016622. [12] N.J. Zabusky, M.D. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965) 240–243. [13] J. Liang, Z. Cai, Y. Sun, L. Yi, Various kinds of spatial solitons associated with photoisomerization, J. Opt. Soc. Am. B 26 (2009) 36–41.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Jacobi elliptic function solutions to (1 + 1) dimensional nonlinear Schrödinger equation in management system Pingyuan Liang a,∗ , Jianchu Liang b a b
College of Physics Science and Information Engineering, University of Jishou, Hunan 416000, China Department of Electronic Science, Huizhou University, Guangdong 516000, China
a r t i c l e
i n f o
Article history: Received 4 February 2010 Accepted 10 August 2010
Keywords: Analytical solution Optical spatial soliton
a b s t r a c t A new kind of solutions namely elliptic function solutions to the (1 + 1) dimensional nonlinear Schrödinger equation (NLSE) in dispersion, nonlinearity and gain management system are obtained. The solutions are the generalization of analytical solutions to one-dimensional NLSE. The properties of the amplitudes and the phases of the solutions are investigated. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction
2. Analytical solutions to the 1D GNLSE
The analytical solution to one-dimensional (1D) normalized nonlinear Schrödinger equation [1–4] which governs the propagation of light pulses in optical fiber:
The normalized GNLSE in management system that governs the propagation of a light pulse in optical fiber can be written as: i
1 ∂2 u ∂u + + |u2 |u = 0, i 2 ∂t 2 ∂z has been obtained by various kinds of method, such as inverse scattering transformation [5], Darboux–Bäcklund transformation [6], self-similarity technique [7], and variational method [8] as well as moment method [9]. The analytical soliton solution is in the form of travelling wave hyperbolic secant function u = sec h(z + 2 2 vt + 0 )ei(−kz+1/2( −v )t+ϕ0 ) , where and v represent the amplitude and the velocity of the soliton. 1D generalized nonlinear Schrödinger equation (GNLSE) in management system [10] is investigated recently with self-similar method. In management system, all the parameters are variable along with propagation distance z. Kruglov et al. [7] obtained sech and tanh self-similar solitary wave solutions to GNLSE; Chen et al. [11] obtained self-similar parabolic, Hermite–Gaussian, and hybrid solutions to the GNLSE. In this paper we investigate the GNLSE in management system with another method namely elliptic function method, and a novel kind of exact solution will be obtained.
∂u ∂2 u 1 + ˇ(z) 2 + (z)|u|2 u − i(z)u = 0, 2 ∂z ∂t
where ˇ, and are dispersive coefficient nonlinear coefficient and gain (loss) coefficient, respectively. Eq. (1) governs the propagation of a light pulse in optical fiber. Our strategy to solve partial equation (1) is as follows: Considering that 1D NLSE possesses traveling wave hyperbolic secant function solution, and the hyperbolic secant function is a special case of Jacobi elliptic function, we search for the traveling wave Jacobi elliptic function solution to the 1D GNLSE. According to the strategy, we look for the solution to the 1D GNLSE (1) in the form: u(z, t) = A(z, t)eiB(z,t) ,
0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.08.018
(2)
where A = f0 (z) + f1 (z)F() + f−1 (z)F−1 (), B = a(z)t2 + b(z)t + c(z), F() is a Jacobi elliptic function which is the solution of the ordinary differential equation (F )2 = q0 + q2 F 2 + q4 F 4 , (z, t) = k(z)t + w(z)z is the traveling wave variable, and by incorporating z into w(z), is rewritten as (z, t) = k(z)t + w(z). a is the curvature of the wavefront, the spatial frequency shift b is relative to the parallel motion of the wave-front, and c is homogeneous phase shift. All the coefficients are distance-dependent. Substituting Eq. (2) into (1), we get the coupling equations:
∗ Corresponding author. E-mail address: [email protected] (P. Liang).
(1)
∂A ∂A ∂B A ∂2 B +ˇ + 2 ∂t 2 ∂z ∂t ∂t
− A = 0,
(3)
1290
P. Liang, J. Liang / Optik 122 (2011) 1289–1292
Table 1 Relations between values of (q0 , q2 , q4 ) and corresponding F() in ode F 2 = q0 + q2 F 2 + q4 F 4 . q0
q2
q4
F 2 = q0 + q2 F 2 + q4 F 4
F
1
−(1 + m2 )
m2
F 2 = (1 − F 2 )(1 − m2 F 2 )
sn, cd =
1 − m2
2m2 − 1
−m2
F 2 = (1 − F 2 )(m2 F 2 + 1 − m2 )
cn
2
m −1
2−m
−1
F 2 = (1 − F 2 )(F 2 + m2 − 1)
dn
2
−(1 + m )
m
2 2
1
−m2
2m2 − 1
1 − m2
−1
2−m
m −1
2
2
= (1 − F )(m − F )
F
2
= (1 − F 2 )[(m2 − 1)F 2 − m2 ]
nc = (cn)−1
F
2
= (1 − F )[(1 − m )F − 1]
nd = (dn)−1
2
2
2
ns = (sn)
2
2
2
2
2 − m2
1 − m2
F
= (1 + F 2 )[(1 − m2 )F 2 + 1]
sc =
sn cn
1
2m2 − 1
−m2 (1 − m2 )
F 2 = (1 + m2 F 2 )[(1 + (m2 − 1)F 2 ]
sd =
sn dn
2 − m2
1
F 2 = (1 + F 2 )(F 2 + 1 − m2 )
cs =
cn sn
ds =
dn sn
−m (1 − m ) 2
2m − 1
2
∂B ˇ −A + 2 ∂z
2
∂2 A −A ∂t 2
∂B ∂t
1
F
2 + A3 = 0.
(4)
Further substituting of the expressions of A, B, into (3) and (4), utilizing the properties of elliptic function: (F )2 = q0 + q2 F 2 + q4 F 4 and F = q2 F + 2q4 F3 , and imposing the coefficients of tq Fn and tq Fn F
−1
F
2
1
1 − m2
cn dn
2
= (F + m )(F + m − 1) 2
2
2
2
, dc =
dn cn
(q = 0, 1, 2; n = −3, −2, −1, 0, 1, 2, 3) be separately equal to zero, we obtain a system of algebraic or first-order ordinary differential equations for fi , k, ω, a, b, and c, from which we obtain the results about the coefficients:
z f0 = 0;
fi = fi0 e
0
(−aˇ)dz
(i = 1, −1),
(5a)
Fig. 1. 3D intensity |u(t, x)|2 plots of Jacobian SN (rows 1 and 2), Jacobian CN function (row 3) and Jacobian DN function (row 4) solutions. The distributed parameters are set as ˇ = 1 + 0.1 cos z, and a0 = 0.05 in row 1; ˇ = 1 + 0.1 cos z, a0 = 0 in rows 2–4; = 0.1 cos t in columns 1 and 2, = 0 in columns 3 and 4. The four columns corresponding to m = −0.5, 0, 0, and 1, respectively.
P. Liang, J. Liang / Optik 122 (2011) 1289–1292
z
w=− 0
k = k0 e c= 0
b = b0 e a=
z
−2
z
ˇbkdz + w0 ,
z
ˇadz
0
k2 f12
(5e)
t 0
ˇdt
=εa0 ,
f−1 = ±
t
where ε= 1+2a0
−1 ˇdt
,
(5f)
0
ˇq4 (when f1 = / 0) or = −
(5d)
,
a0
= −
(5c)
ˇ 2 ˇ k q2 − b2 + 3f1 f−1 dz + c0 , 2 2
−2
1+2a0
(5b)
ˇadz
0
1291
k2 2 f−1
ˇq0 (when f−1 = / 0)
(5g)
q0 f1 (when f1 = / 0 and f−1 = / 0). q4
(5h)
What is amazing and delectable is that all the formulas are self-consistent and self-contained, which means that our solving procedure is successful. The only necessary and sufficient condition for the existence of Jacobi elliptic function solution to the 1D GNLSE is that the nonlinearity and the dispersion should fulfill a certain relation expressed by Eq. (5g). This condition is actually a generalized constraint for generalized NLSE to have elliptic function solutions including the hypersecant and hypertangent soliton solutions. As for 1D NLSE with varying parameters, solitons exist only under certain conditions [10,11]. From Eq. (5) of reference [10], the necessary condition for 1D GNLSE to have soliton solutions is = cˇ, where c is a constant, and Eq. (1) in Ref. [10] can be normalized so as that = ± ˇ. In our paper, constraint (5 g) results in = − q4 ˇ assuming that k0 = 1, f10 = 1, = 0, and a = 0. As for elliptic functions, cn → sec h when m = 1, and then q4 = − 1 (see Table 1) and = ˇ is obtained in this case; sn → tan h when m = 1 and then q4 = 1 and = − ˇ. This is consistent with the above special case in Ref. [11] as well as the 1D normalized equation in the introduction with constant coefficients. So the condition for the existence of the soliton solutions of 1D NLSE is just a special case of the constraint in our paper. This constraint can be realized in some situations, especially in the case of constant coefficients. This constraint applies to the physically relevant systems apart from the cases when = a = 0. Substituting the resulting expressions (5a)–(5h) into Eq. (2), we get the following analytical solutions (6) and (7) to the 1D GNLSE. Solutions (6) and (7) are the single elliptic function and hybrid elliptic function solution, respectively:
z
u(z, t) = f10 e or
0
z
u(z, t) = f10 e
0
(−aˇ)dz
(−aˇ)dz
F()ei[at
2 +bt+c]
F() ±
(f−1 = 0),
(6)
q0 −1 2 F () ei[at +bt+c] q4
Fig. 2. 3D intensity |u(t, x)|2 of Hybrid Jacobi elliptic function solutions:
cn(, m) +
(f1 = / 0 and f−1 = / 0), where −
z 0
a
is
ˇb0 k0 e
−4
determined
z 0
ˇadz
by
dz + w0 ,
c = [(q2 /2)(k02 + l02 + p20 ) − (b20 /2)] c0
in
√ 3 q0 q4 k02 ]
Eq.
z 0
ˇe
(6) −4
z 0
or ˇadz
Eq.
t 0
b = b0 e ˇe
= k0 t e
(5f),
−4
t 0
−2
t
ˇadt
0
ˇadt
,
−2
z 0
(7) ˇadz
m2 − 1/m2 nc(, m)
m2 − 1/m2 nc(, m)
(b),
1 − m2 nd(, m) (c) and dn(, m) − 1 − m2 nd(, m) (d), where the dn(, m) + parameters are set to be = 0.5 cos t, ˇ = cos 0.5t, and a0 = 0. m is set to be 2 in (a) and (b), and 0.5 in (c) and (d).
and
dt +
If the initial phase chirp is presented, i.e., a0 = / 0, the solutions are respectively
c = [(q2 /2)(k02 + l02 + p20 ) − (b20 /2) ∓
dt + c0 in Eq. (7).
cn(, m) +
(a),
t u = f10 ε3/2 e
0
dt
F()ei[at
2 +bt+c]
,
(6a)
1292
P. Liang, J. Liang / Optik 122 (2011) 1289–1292
and
t
u = f1,0 ε3/2 e
0
dt
F() ±
q0 −1 2 F () ei[at +bt+c] , q4
(7a)
where = εk0 t + (b0 /2a0 )k0 (ε − 1) + w0 , b = εb0 , and c = (1 − ε/4a0 )(k02 q2 − b20 ) + c0 in Eq. (6a) or c = (1 − ε/4a0 )(k02 q2 − b20 ∓ √ 6 q0 q4 k02 ) + c0 in (7a). When a0 = 0, the solutions are
z
u(z, t) = f10 e or
z
u(z, t) = f10 e where
0
z
0
dz
dz
b = b0 ,
F()ei(bt+c) ,
F() ±
(6b)
q0 −1 F () ei(bt+c) , q4
(7b)
ˇdz + w0 ,
c = [(q2 /2)k02 −
= k0 t − b0 k0
z 0
(b20 /2)] 0 ˇdz + c0 in Eq. (6b) z √ 3 q0 q4 k02 ] 0 ˇdz + c0 in Eq. (7b).
or
c = [(q2 /2)k02 − (b20 /2) ∓
3. Analyses and discussion In order to manifest the solution, we substitute Eq. (6) into Eq. (1), where we set the following periodically distributed parameters: = 0.1 cos z, ˇ = 1 + 0.1 cos z, The nonlinear coefficient is = 2 (1 + 2a (z + 0.1 sin z)))e−0.2 sin z according the constraint −(k02 /f10 0 (5g). We find that Eq. (1) is satisfied, which proves that Eq. (6) is just the solution to Eq. (1). It should be pointed out that the functions , ˇ, and s are chosen ad hoc in order to get analytical expressions for the parameters a, b, and , and to simplify the calculation. There are 12 different Jacobi elliptic functions, each of which differentiated by different order m (see Table 1). Jacobi elliptic functions degenerate into triangular functions when m = 0, and hyperbolic functions when m = 1. The amplitude of the light wave or matter wave should be finite, thus the elliptic functions that ending with “s” (ns, cs, ds) cannot be selected to be the amplitude function; all that ending with “n” (sn, cn, dn) can absolutely be amplitude function in any order; and all that ending with “d” or “c” can be amplitude function in some orders. As for hybrid solution, only F() = cn(, m) ± m2 − 1/m2 nc(, m), (m > 1) and
F() = dn(, m) ± 1 − m2 nd(, m) (−1 < m < 1) can be wave functions. Depicted in Fig. 1 are the wave intensities of Jacobi sn (rows 1 and 2), Jacobi cn (row 3) and Jacobi dn solution (row 4). Without losing generality, the initial conditions are set to be: b0 = k0 = 1, w0 = c0 = 0, and f10 = 1. The varying parameters are still set as ˇ = 1 + 0.1 cos z, and a0 = 0.05 in row 1; ˇ = 1 + 0.1 cos z, and a = 0 in rows 2–4. We set = 0.1 cos t in columns 1 and 2, = 0 in columns 3 and 4. The four columns correspond to m = −0.5, 0, 0, and 1, respectively. The conventional one-dimensional hyperbolic secant (hyperbolic tangent) soliton [12,13] solution is the special case of our results (see Fig. 1(2d) and (3d)). In these cases, and ˇ must be of the same sign for hypersecant (bright) soliton, and of the contrary sign for hypertangent (black) soliton solution which is derived from the necessary condition (5g). There are sine function solutions (Fig. 1(2c)) and cosine function solutions (Fig. 1(3c)) to the NLSE. The periodical gain (loss) periodically amplifies (deduces) the
amplitude of the intensity, and net gain (loss) amplifies (deduces) the amplitude. In the case in which the chirp is absent (i.e., a = 0), the energy conserves provided that there is no gain (loss) (see the lower two lines in Fig. 1); however in the case in which an initial chirp is present (i.e., a = / 0), the energy fluctuate along with distance even if there is no gain (loss), as shown in the upper row of Fig. 1. This phenomenon has never been found before. We think that the unconservation of the energy is mathematically due to the presence of the wave-front curvature. When an initial phase chirp / 0) is input, the curvature of the phase wave-front evolves in (a0 = t the form of a = a0 /(1 + 2a0 0 ˇdt ), but the curvature center fixes at ( − b0 /2a0 , − b0 /2a0 , − b0 /2a0 ). If the initial wave is unchirped (a0 = 0), it remains unchirped. Fig. 2 depicts the hybrid Jacobi elliptic function solutions cn(, m) + m2 − 1/m2 nc(, m)), cn(, m) + m2 − 1/m2 nc(, m) (b),
dn(, m) +
1 − m2 nd(, m)
(c)
and
dn(, m) −
1 − m2 nd(, m)
(d), where the parameters are set to be = 0.5 cos t, ˇ = cos 0.5t, and a0 = 0. m is set to be 2 in (a) and (b), and 0.5 in (c) and (d). These solutions have never been obtained before, and they are absolutely different with the single elliptic function solutions. 4. Conclusions We obtain a new kind of analytical solutions to 1D GNLSE with distributed coefficients. The solutions are in the form of travelling wave elliptic function, which are the generalization of the conventional hyperbolic function soliton solutions to 1D NLSE. The method that we use to solve the NLSE will pave the way to new methods for solving partial differential equations. References [1] J.E. Bjorkholm, A. Ashkin, cw Self-focusing and self-trapping of light in sodium vapor, Phys. Rev. Lett. 32 (1974) 129–132. [2] J.S. Aitchison, A.M. Weiner, Y. Silberberg, M.K. Oliver, J.L. Jackel, D.E. Leaird, E.M. Vogel, P.W.E. Smith, Observation of spatial optical solitons in a nonlinear glass waveguide, Opt. Lett. 15 (1990) 471–473. [3] R. de la Fuente, O. Varela, H. Michinel, Fourier analysis of non-paraxial selffocusing, Opt. Commun. 173 (2000) 403–411. [4] L. Yan, X. Mi, Y. Wu, J. Liang, Incoherently coupled anti-dark soliton pairs, Optik 121 (2010) 54–56. [5] M.J. Ablowitz, Nonlinear Schrödinger Equation and Inverse Scattering, Cambridge University Press, New York, 1991. [6] V.B. Matveev, M.A. Salle, Dardoux Transformations and Solitons (Springer Series in Non-linear Dynamics), Springer, Berlin, 1991. [7] V.I. Kruglov, A.C. Peacock, J.D. Harvey, Exact Self-Similar Solutions of the Generalized Nonlinear Schrödinger Equation with Distributed Coefficients, Phys. Rev. Lett. 90 (1-4) (2003) 113902. [8] V.S. Filho, F.K. Abdullaev, A. Gammal, L. Tomip, Theory of a mode-locked atom laser with toroidal geometry, Phys. Rev. A 63 (1–4) (2001) 053603. [9] V.M. Pérez-García, P. Torres, J.J. García-Ripoll, H. Michinel, Moment analysis of paraxial propagation in a nonlinear graded index fibre, J. Opt. B: Quant. Semiclass. Opt. 2 (2000) 353–358. [10] V.N. Serkin, A. Hasegawa, Novel soliton solutions of the nonlinear Schrödinger equation model, Phys. Rev. Lett. 85 (2000) 4502–4505. [11] S. Chen, L. Yi, D.S. Guo, P. Lu, Self-similar evolutions of parabolic, Hermite–Gaussian, and hybrid optical pulses: universality and diversity, Phys. Rev. E 72 (1–5) (2005) 016622. [12] N.J. Zabusky, M.D. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965) 240–243. [13] J. Liang, Z. Cai, Y. Sun, L. Yi, Various kinds of spatial solitons associated with photoisomerization, J. Opt. Soc. Am. B 26 (2009) 36–41.