Conservation laws, bilinear Bäcklund transformations and solitons for a nonautonomous nonlinear Schrödinger equation with external potentials

Commun Nonlinear Sci Numer Simulat 39 (2016) 472–480

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Conservation laws, bilinear Bäcklund transformations and solitons for a nonautonomous nonlinear Schrödinger equation with external potentials Jun Chai, Bo Tian∗, Xi-Yang Xie, Ya Sun State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

a r t i c l e

i n f o

Article history: Received 16 July 2015 Revised 3 December 2015 Accepted 28 February 2016 Available online 10 March 2016 Keywords: Nonautonomous Schrödinger equation with external potentials Conservation laws Bilinear Bäcklund transformations Soliton solutions

a b s t r a c t Under investigation in this paper is a nonautonomous nonlinear Schrödinger equation with external potentials, which can govern the dynamics of nonautonomous solitons in the nonlinear optical medium non-uniformly distributed in both the transverse and longitudinal directions. Based on the Lax pair, we present an infinite sequence of the conservation laws. Bilinear forms, bilinear Bäcklund transformations, one-, two- and N-soliton solutions under a known variable-coefficient constraint are generated via the Hirota method. With G(t ) = 0 and RB((tt )) being a constant, amplitude of the soliton remains unvarying during the propagation, where t is the scaled time, G(t) is the gain/loss coefficient, B(t), the group velocity dispersion coefficient, and R(t), the nonlinearity coefficient. If we set G(t) = 0 or RB((tt )) as a variable, the amplitude becomes varying. Due to the different choices of the linear oscillator potential coefficient α (t), periodic-, parabolic-, S- and V-type solitons are observed. Meanwhile, we find that α (t) has no influence on the soliton amplitude. Interaction between the two amplitude-unvarying solitons and that between the two amplitude-varying ones are displayed, respectively. The velocity of a moving soliton always keeps varying. © 2016 Elsevier B.V. All rights reserved.

1. Introduction With the development of nonlinear science, nonlinear evolution equations (NLEEs) have the attractive applications in such fields as optics, plasmas physics, fluids and condensed matter physics [1–6]. Some NLEEs, in which the independent variable “time” does not appear explicitly, are called the autonomous systems [7,8]. Solitons from those systems are capable of propagating over the long distances without any change of shape and with the negligible attenuation [9]. However, for the so-called nonautonomous systems or NLEEs, “time” appears explicitly [8,10]. Those systems have been seen in plasma physics, Bose–Einstein condensation and nonlinear optics [11,12]. The nonautonomous solitons, obtained from a nonautonomous system with external potentials, move with the varying amplitudes and velocities adapted both to the external potentials and to the varying dispersion, nonlinearity and gain/loss [10,13,14]. For the nonautonomous system with the varying external potentials, two questions arise: In which condition can solitons exist? Do the solitons still maintain their identities after the soliton interaction [7,8]? To solve the above questions, people need to study the dynamics of nonautonomous solitons which, when considered in the nonlinear optical medium ∗

Corresponding author. E-mail address: [email protected] (B. Tian).

http://dx.doi.org/10.1016/j.cnsns.2016.02.024 1007-5704/© 2016 Elsevier B.V. All rights reserved.

J. Chai et al. / Commun Nonlinear Sci Numer Simulat 39 (2016) 472–480

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non-uniformly distributed in both the transverse and longitudinal directions [15], can be governed by the following nonlinear Schrödinger (NLS) equation with varying dispersion, nonlinearity, gain/loss and external potentials [10,15,16]:



iQt +



B(t ) (t ) 2 Qzz + R(t )|Q |2 Q + iG(t )Q − 2α (t )z + z Q = 0, 2 2

(1)

where Q = Q (z, t ) is the complex envelope of the electric field, the subscripts z and t respectively denote the partial derivatives with respect to the scaled distance and time, B(t), R(t), G(t), α (t) and (t) are the group velocity dispersion (GVD), nonlinearity, gain/loss, linear potential and harmonic oscillator potential coefficients, respectively. Eq. (1) has been shown to satisfy the Painlevé integrability under the constraint [10,16]

−

(t ) 2

d d2 d2 1 d ln B(t ) + R(t ) 2 − ln B(t ) ln R(t ). 2 dt dt dt dt R(t )

B(t ) =

(2)

Lax pair and Darboux transformation for Eq. (1) have been given [10]. Rogue wave solutions for Eq. (1) have been obtained [15]. To our knowledge, however, for Eq. (1), an infinite sequence of the conservation laws, bilinear forms, bilinear Bäcklund transformations (BTs)1 and multiple soliton solutions via the Hirota method [18–20] and symbolic computation [21–24] have not been discussed yet. Motivated by the above, an infinite sequence of the conservation laws based on the Lax pair will be got in Section 2. In Section 3, introducing the auxiliary function, we will assume a relation between (t) and B(t) for the purpose of deriving the bilinear forms for Eq. (1), and then, the analytic one-, two- and N-soliton solutions will be presented under Constraint (2). In Section 4, graphical analysis will be conducted to investigate the influence of the variable coefficients on the soliton propagation and interaction. Bilinear BTs for Eq. (1) will be given in Section 5. Section 6 will be our conclusions. 2. Conservation laws Generally, existence of an infinite sequence of the conservation laws has been claimed to assure the complete integrability for the NLEEs, i.e., the NLEEs can be solved through the inverse scattering technique and possess the multiple soliton solutions [25,26]. On the other hand, some completely-integrable NLEEs, e.g., the Lax-integrable ones, admit an infinite sequence of the conservation laws [25,26]. In the following, through the Lax pair, we will construct the conservation laws for Eq. (1). Under Constraint (2), we start from the Lax pair [10]

z = U , t = V ,

(3)

)T

where  = (1 , 2 is the vector eigenfunction (T denotes the transpose of the vector), 1 and 2 are the complex functions of z and t, and U and V are expressible in the forms



U= V =



−iλ(t )

q

−q∗

iλ(t )

2 

iλ (t )

=0

with

 q=



(4)



A

C

C∗

−A

R(t ) Qe B(t )



i z2 2B (t )



λ(t ) = e

,

[ln RB((tt )) ]t −G(t )

,

(5)

[ln RB((tt )) ]t − 2iGB((tt)) z2

dt

 λ (0 ) +





,

e



A0 = C0 =



1 R(t ) B(t )qq∗ − α (t )z, A1 = z G(t ) − ln 2 B(t ) 1 B(t )qt − i 2



ln

R(t ) B(t )







 G(t )−[ln RB((tt )) ] dt t

α (t )dt ,  , A2 = −B(t ), t

− G(t ) zq, C1 = −iB(t )q, C2 = 0, t

and ∗ as the complex conjugate.  Introducing the function (z, t ) = 2 [27], we get the Ricatti-type equation from Lax Pair (3), 1

z = −q + 2iλ(t ) − q . ∗

1

2

BTs provide a means of constructing the new solutions from the known solutions to the different (same) equations [17].

(6)

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J. Chai et al. / Commun Nonlinear Sci Numer Simulat 39 (2016) 472–480

Then, according to Ref. [27], setting

q =

∞  d j (z, t ) , λ j (t )

(7)

j=1

substituting it into Expression (6) and equating the coefficients of the same power of λ(t) to zero, we have the recurrence relations,

d1 = −

i |q|2 , 2

d2 = −

i qz d1,z − d1 , 2 q

(8a)



.. .





dn+1

(8b)

n−1



 i qz =− dn,z − dn − dc dn−c (n = 2, 3, . . . ), 2 q

(8c)

c=1

where dj ’s ( j = 1, 2, . . .) are the functions of z and t to be determined. From the compatibility condition (ln 1 )zt = (ln 1 )tz , we can obtain the following conservation form:

[−iλ(t ) + q ]t = (R1 + R2 )z ,

(9)

where

R1 = iλ2 (t )A2 + iλ(t )A1 + iA0 , R2 = iλ2 (t )C2 + iλ(t )C1 + iC0 . Substituting Expressions (7) and (8) into Expression (9) and collecting the coefficients of the same power of λ(t), we can obtain an infinite sequence of the conservation laws for Eq. (1),

∂ ∂  + J = 0, ∂t j ∂ z j

(10)

with

i |q|2 , 2  

i R(t ) 1 J1 = − ln − G(t ) z|q|2 + B(t )(q∗ qz − qq∗z ), 2 B(t ) 4

1 = −

(11)

t

1 2 = − qq∗z , 4  

R(t ) i 1 i J2 = − B(t )|q|4 − ln − G(t ) zqq∗z − B(t )(qz q∗z − qq∗zz ), 8 4 B(t ) 8 t i i 3 = − |q|4 + qq∗zz , 8 8 J3 = −

i 8



ln

R(t ) B(t )



− G(t ) (z|q|4 − zqq∗zz ) − t

(13)

1 B(t )|q|2 qq∗z 4

1 − B(t )(qz q∗zz − qq∗zzz ), 16 .. .



(12)

(14)

ι−2



 i z ι = − ι−1,z − ι−1 − c ι−1−c , ι = 4, 5, . . . , 2 q

(15)

c=1

Jι =

i (C2 2+ι + C1 1+ι + C0 ι ), q

(16)

where j ’s and Jj ’s are the conserved densities and conserved fluxes, respectively. Thus, we can conclude that, under Constraint (2), Eq. (1) possesses an infinite sequence of the conservation laws, i.e., Conservation Laws (10).

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3. Bilinear forms and soliton solutions for Eq. (1) 3.1. Bilinear forms To get the bilinear forms for Eq. (1), we assume that [28]

(t ) = γtt (t ) − B(t )γt2 (t ),

(17)

and introduce the dependent variable transformation [29]

Q (z, t ) =

g(z, t ) [− i γt (t )z2 + 1 B(t )γt (t )dt− G(t )dt+iξ (t )z+iω (t )] 2 e 2 , f (z, t )

where

ξ (t ) = −2e



B(t )γt (t )dt









1 e− B(t )γt (t )dt α (t ) dt , ω (t ) = − 2

(18)







B(t )ξ 2 (t ) dt,

γ (t) is a real auxiliary function, g(z, t) is the complex differentiable function, and f(z, t) is a real one. Substituting Relation (17) and Transformation (18) into Eq. (1), we can get the bilinear forms for Eq. (1) as



iDt +

B(t ) 2 Dz + iB(t )[ξ (t ) − γt (t )z]Dz g · f = 0, 2



B(t ) 2 Dz f · f = R(t )|g|2 e [ B(t )γt (t )dt−2 G(t )dt] , 2

(19a) (19b)

where the bilinear operators Dz and Dt are defined by [18–20]



m2 1 Dm z Dt (G · F ) =

∂ ∂ − ∂ z ∂ z

m1 

∂ ∂ − ∂t ∂t

m2

G(z, t )F (z , t  )|z =z,t  =t ,

with G(z, t) as a differentiable function of z and t, F(z , t ) as a differentiable function of the formal variables z and t , and m1 and m2 as the non-negative integers. 3.2. Soliton solutions To construct the soliton solutions for Eq. (1), we expand g(z, t) and f(z, t) with respect to a formal expansion parameter

ε as

g(z, t ) = ε g1 (z, t ) + ε 3 g3 (z, t ) + ε 5 g5 (z, t ) + · · · ,

(20a)

f (z, t ) = 1 + ε 2 f2 (z, t ) + ε 4 f4 (z, t ) + ε 6 f6 (z, t ) + · · · ,

(20b)

where gL (z, t)’s (L = 1, 3, 5, . . . ) are the complex functions, and fP (z, t)’s (P = 2, 4, 6, . . . ) are the real ones. 3.2.1. One-soliton solutions Truncating Expressions (20) as g(z, t ) = ε g1 (z, t ) and f (z, t ) = 1 + ε 2 f2 (z, t ), setting ε = 1 and substituting them into Bilinear Forms (19), we obtain the one-soliton solutions for Eq. (1) as

Q (z, t ) =



i g1 (z, t ) 2 1 e [− 2 γt (t )z + 2 B(t )γt (t )dt − G(t )dt +iξ (t )z+iω (t )], 1 + f2 (z, t )

under Constraint (2), where

 ln RB((tt )) − 2G[t] t γ (t ) = dt, g1 (z, t ) = ηeθ (z,t ) , B[t]



ηη∗ R(t ) ∗ f2 (z, t ) = e[−2 G(t )dt − B(t )γt (t )dt ] eθ (z,t )+θ (z,t ) , (ρ + ρ ∗ )2 B(t )  



 i θ (z, t ) = ρ e B(t )γt (t )dt z + ρ e B(t )γt (t )dt B(t ) ρ e B(t )γt (t )dt B(t ) + 2iξ (t ) dt, 2 



with ρ and η as the complex constants.

(21)

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3.2.2. Two-soliton solutions To derive the two-soliton solutions for Eq. (1), we truncate Expressions (20) as g(z, t ) = ε g1 (z, t ) + ε 3 g3 (z, t ) and f (z, t ) = 1 + ε 2 f2 (z, t ) + ε 4 f4 (z, t ), set ε = 1, substitute them into Bilinear Forms (19), and get

Q (z, t ) =

g1 (z, t ) + g3 (z, t ) [− i γt (t )z2 + 1 B(t )γt (t )dt− G(t )dt+iξ (t )z+iω (t )] 2 e 2 , 1 + f2 (z, t ) + f4 (z, t )

(22)

under Constraint (2), where

γ (t ) =





 ln RB((tt )) − 2G[t] t dt, g1 (z, t ) = η1 eθ1 (z,t ) + η2 eθ2 (z,t ) , B[t]

 η η∗ η (ρ − ρ )2 R(t ) ∗ 1 1 2 1 2 eθ1 (z,t )+θ2 (z,t )+θ1 (z,t ) (ρ1 + ρ1∗ )2 (ρ2 + ρ1 )2 B(t ) η1 η2∗ η2 (ρ1 − ρ2 )2 R(t ) θ1 (z,t )+θ2 (z,t )+θ2∗ (z,t )  + e , (ρ1 + ρ2∗ )2 (ρ2 + ρ2 )2 B(t )  η η∗ R(t )



η1 η2∗ R(t ) ∗ ∗ 1 1 f2 (z, t ) = e[−2 G(t )dt − B(t )γt (t )dt ] eθ1 (z,t )+θ1 (z,t ) + eθ1 (z,t )+θ2 (z,t ) ∗ 2 (ρ1 + ρ1 ) B(t ) (ρ1 + ρ2∗ )2 B(t )  η2 η1∗ R(t ) η2 η2∗ R(t ) θ2 (z,t )+θ1∗ (z,t ) + θ2 (z,t )+θ2∗ (z,t ) , + e e (ρ2 + ρ1∗ )2 B(t ) (ρ2 + ρ2∗ )2 B(t )



g3 (z, t ) = e[−2 G(t )dt − B(t )γt (t )dt ]





f4 (z, t ) = e[−4 G(t )dt −2 B(t )γt (t )dt ]

|η1 |2 |η2 |2 (ρ1 − ρ2 )2 (ρ1∗ − ρ2∗ )2 R(t )2 (ρ1 + ρ1∗ )2 (ρ2 + ρ1∗ )2 (ρ1 + ρ2∗ )2 (ρ2 + ρ2∗ )2 B(t )2

∗ ∗ × eθ1 (z,t )+θ2 (z,t )+θ1 (z,t )+θ2 (z,t ) ,  



 i θ1 (z, t ) = ρ1 e B(t )γt (t )dt z + ρ1 e B(t )γt (t )dt B(t ) ρ1 e B(t )γt (t )dt B(t ) + 2iξ (t ) dt, 2  



 i θ2 (z, t ) = ρ2 e B(t )γt (t )dt z + ρ2 e B(t )γt (t )dt B(t ) ρ2 e B(t )γt (t )dt B(t ) + 2iξ (t ) dt, 2

with ρ 1 , ρ 2 , η1 and η2 as the complex constants. 3.2.3. N-soliton solutions The N-soliton solutions for Eq. (1) can be expressed as

Q (z, t ) =

g(z, t ) [− i γt (t )z2 + 1 B(t )γt (t )dt− G(t )dt+iξ (t )z+iω (t )] 2 e 2 , f (z, t )

under Constraint (2), where

f (z, t ) =





exp

ϑ =0,1

g(z, t ) =

2N 

 

exp

 

ϑl [θl (z, t ) + ln ηl ] ,

l=1

2N 

2N 

ψlk (z, t )ϑl ϑk +

 ϑl [θl (z, t ) + ln ηl ] ,

l
l=1

2N 

2N 

 exp



l


ϑ =0,1

g∗ (z, t ) =

ψlk (z, t )ϑl ϑk +

2N 

ϑ =0,1

ψlk (z, t )ϑl ϑk +

l
(23)

 ϑl [θl (z, t ) + ln ηl ] ,

l=1

 



 i ρl e B(t )γt (t )dt B(t ) ρl e B(t )γt (t )dt B(t ) + 2iξ (t ) dt, 2 θl+N (z, t ) = θl∗ (z, t ), ρl+N = ρl∗ , ηl+N = ηl∗ ,   R(t ) ψlk (z, t ) = −2 G(t )dt − B(t )γt (t )dt + ln , for l = 1, . . . , N (ρl + ρk∗ )2 B(t )

θl (z, t ) = ρl e



B(t )γt (t )dt

z+

and k = N + 1, . . . , 2N or l = N + 1, . . . , 2N and k = 1, . . . , N,   (ρ − ρk )2 B(t ) ψlk (z, t ) = 2 G(t )dt + B(t )γt (t )dt + ln l , for l = 1, . . . , N R(t ) and k = 1, . . . , N or l = N + 1, . . . , 2N and k = N + 1, . . . , 2N,    ln R(t ) − 2G[t] B(t ) t γ (t ) = dt, B[t]

J. Chai et al. / Commun Nonlinear Sci Numer Simulat 39 (2016) 472–480

477

Fig. 1. One soliton via Solutions (21) under Constraint (2) with the parameters as η = 1, ρ = 1, α (t ) = 1.2, G(t ) = 0, (a) B(t ) = sin(t ), R(t ) = 0.6 sin(t ); (b) B(t ) = sin(t ), R(t ) = 1.2 sin(t ).

Fig. 2. The same as Fig. 1 except that (a) B(t ) = 0.4e0.4t , R(t ) = 0.4; and that (b) G(t ) =

t 1+t 2

, B(t ) = 2t, R(t ) = 4t.

 N denoting the summation over all possible pairs taken from 2N elements with ηl ’s and ρ l ’s being the complex constants, 2l
N 



ϑl =

l=1

N 

ϑl+N ,

N 

l=1





ϑl = 1 +

l=1

N  l=1

ϑl+N , 1 +

N  l=1





ϑl =

N 

ϑl+N .

l=1

4. Discussions on the solitons Although Eq. (1) contains five variable coefficients, Solutions (21)–(23) are related to four of them, i.e, G(t), R(t), B(t) and α (t), under Constraint (2). In this part, based on Solutions (21) and (22), influence of those variable coefficients on the solitons will be discussed. Solutions (21) can be rewritten as



Q (z, t ) = Re(ρ )

R(t ) [− i γt (t )z2 −2 G(t )dt+iξ (t )z+iω (t )+ 1 θ (z,t )− 1 θ ∗ (z,t )] 2 2 e 2 B(t )

1

× Sech where τ = e[−2



2



[θ (z, t ) + θ ∗ (z, t ) + ln τ ] ,

G (t )dt − B (t )γt (t )dt ]

ηη∗ R(t ) , and “Re” denotes the real part. (ρ +ρ ∗ )2 B(t )

From Solutions (24), we can find that the soliton amplitude , which is expressed as = |Re(ρ ) related to G(t) and

R (t ) . B (t )

(24)



R (t ) −2 G (t )dt e |, B (t )

is

Figs. 1–3 illustrate the effects of G(t), R(t), B(t) and α (t) on the soliton propagation. With G(t ) = 0 and RB((tt )) being a constant, amplitude of the soliton shown in Fig. 1 keeps unvarying during the propagation. Besides, comparison between Figs. 1(a) and (b) shows that the soliton amplitude may increase as the increase of RB((tt )) . If we keep G(t ) = 0 but set RB((tt )) as a variable, the soliton amplitude becomes varying, as seen in Fig. 2(a), where the soliton amplitude increases exponentially. To investigate the effects further, we present the case that G(t) = 0 in Fig. 2(b), and find that the amplitude is varying. The soliton amplitude increases first, but then decreases. In Fig. 3, due to the different values of α (t), a periodic-type soliton is displayed in Fig. 3(a), while a parabolic-type one appears in Fig. 3(b). Actually, the different choices of α (t) lead to the different velocities, which give rise to the different types of solitons. Meanwhile, we notice that the soliton amplitude shown in Fig. 3 is equal to that shown in Fig. 1(a). That is to say, the soliton amplitude cannot be affected by α (t). Further, we can observe that the soliton moves with the varying velocity in each figure.

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J. Chai et al. / Commun Nonlinear Sci Numer Simulat 39 (2016) 472–480

Fig. 3. The same as Fig. 1(a) except that (a) α (t ) = 3 sin(t ); and that (b) α (t ) = 1.8t cos(t ) + 1.3t.

Fig. 4. Interaction between the two solitons via Solutions (22) under Constraint (2) with the parameters as η1 = η2 = 1, ρ1 = 1, ρ2 = 0.7 − 1.3i, G(t ) = 0, B(t ) = R(t ) = 4t, (a) α (t ) = 0.35t; (b) α (t ) = 0.4t 2 .

Fig. 5. The same as Fig. 4 except that B(t ) = 0.3e0.4t , R(t ) = 0.4, (a) α (t ) = 0.65; and that (b) α (t ) = −1.3t − 2.

Interactions between the two solitons are displayed in Figs. 4 and 5. In Fig. 4, the interaction is between two amplitudeunvarying solitons, where G(t ) = 0 and RB((tt )) is a constant. With the different values of α (z), we observe two parabolic-type solitons in Fig. 4(a) but two S-type ones in Fig. 4(b). Meanwhile, we can find that those interactions are elastic. In Fig. 5, as R (t ) is a variable, the interaction between the two amplitude-varying solitons occurs: With the two solitons propagating forB (t ) ward, their amplitudes decrease exponentially. As the different values of α (z), the interaction between the two exponentialtype solitons appears in Fig. 5(a), while that between the two V-type ones is shown in Fig. 5(b). 5. Bilinear BTs The bilinear BTs can be used to construct the new solutions from the known ones [17]. In this section, we aim to obtain the bilinear BTs for Eq. (1). By defining a bilinear operator [18–20]

H = iDt +

B(t ) 2 Dz + iB(t )[ξ (t ) − γt (t )z]Dz , 2

we consider the expression





P ≡ f 2 H (g · f ) − f 2 H g · f  = 0.

(25)

With the exchange formulas of the bilinear operators, P can be rewritten as



P = ( f  f ) − iDt (g · f + f  · g) − iB(t )[ξ (t ) − γt (t )z]Dz (g · f + f  · g) +





B(t ) 2  Dz (g · f − f  · g) + (g f + f  g) iDt f  · f + μ(t )B(t )Dz f  · f 2

J. Chai et al. / Commun Nonlinear Sci Numer Simulat 39 (2016) 472–480



1 R(t )e [ B(t )γt (t )dt−2 G(t )dt] (gg∗ − g g∗ ) 2

+ iB(t )[ξ (t ) − γt (t )z]Dz f  · f +



B(t ) 2  + (g · f − f  · g) − Dz f · f − μ(t )2 B(t )Dt f  · f 2



1 + R(t )e [ B(t )γt (t )dt−2 G(t )dt] (gg∗ + g g∗ ) . 2

479



(26)

Through Expression (26), we can derive the bilinear BTs for Eq. (1) as

{iDt + iB(t )[ξ (t ) − γt (t )z]Dz }(g · f + f  · g) B(t ) 2  − Dz (g · f − f  · g) +  (t )(g · f − f  · g) = 0,

(27a)

2

{iDt + μ(t )B(t )Dz + iB(t )[ξ (t ) − γt (t )z]Dz } f  · f +



1 R(t )e [ B(t )γt (t )dt−2 G(t )dt] (gg∗ − g g∗ ) = 0, 2

 B(t ) 2 −

(27b)



D2z + μ2 (t )B(t ) −  (t ) f  · f



1 R(t )e[ B(t )γt (t )dt−2 G(t )dt] (gg∗ + g g∗ ) = 0, 2

(27c)

Dz (g · f + f  · g) = μ(t )(g f − f  g),

(27d)

where g = g (z, t) is the complex differentiable function, f = f (z, t) is a real one, (g, f) and (g , f ) are two pairs of the solutions for Bilinear Forms (19), and μ(t) and ϖ(t) are the real functions. Hereby, we choose f = 1 and g = 0 that correspond to the seed solution Q = 0, and substitute them into Bilinear BTs (27). Then, the new one-soliton solutions for Eq. (1) can be expressed in the following form:

Q  (z, t ) =

g (z, t ) [− i γt (t )z2 + 1 B(t )γt (t )dt− G(t )dt+iξ (t )z+iω (t )] 2 e 2 , f  (z, t )

(28)

under the constraint

4b1 b2 B(t )e2



with

G(t )dt + B(t )γt (t )dt



g (z, t ) =

βe

f  (z, t ) = b1 e

ie B(t )γt (t )dt z+2









e B(t )γt (t )dt z+ e2



= ββ ∗ ,

e2







B (t )γt (t )dt



B(t ) e− B(t )γt (t )dt α (t )dt dt

 B (t )γt (t )dt

−e B(t )γt (t )dt z− e2

(29)





B(t ) −1+2 e−



,

 B (t )γt (t )dt



α (t )dt dt



B (t )γt (t )dt







B(t ) −1+2 e− B(t )γt (t )dt α (t )dt dt

+ b2 e

B(t ) 2 B(t )γt (t )dt  (t ) = − e , μ(t ) = ie B(t )γt (t )dt , 2



,

where β is the complex constant, b1 and b2 are the real ones. 6. Conclusions In this paper, our main attention has been focused on a nonautonomous NLS equation with external potentials, i.e., Eq. (1), which can govern the dynamics of nonautonomous solitons in the nonlinear optical medium non-uniformly distributed in both the transverse and longitudinal directions. Based on Lax Pair (3), Conservation Laws (10) have been given. Bilinear Forms (19), as well as One-, Two- and N-soliton Solutions (21)–(23), under Constraint (2), have been obtained via the Hirota method. In addition, we have got Bilinear BTs (27), from which the new one-soliton solutions for Eq. (1), i.e., Solutions (28), have been derived. The main results of this paper can be summarized as follows: (1) Although Eq. (1) contains five variable coefficients, Solutions (21)–(23) and (28) are related to four of them, i.e, G(t), R(t), B(t) and α (t), under Constraint (2), where G(t), R(t), B(t) and α (t) are the gain/loss, nonlinearity, GVD and linear potential coefficients, respectively. (2) The soliton amplitude can be affected by G(t), R(t) and B(t): With G(t ) = 0 and RB((tt )) being a constant, the amplitude shown in Fig. 1 remains unvarying. If we set G(t) = 0 or RB((tt )) as a variable, the amplitude becomes varying, as seen in Fig. 2.

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(3) α (t) has no influence on the soliton amplitude, but can affect the soliton type, as seen in Fig. 3: Periodic-type in Fig. 3(a), while parabolic-type in Fig. 3(b). Besides, those solitons have the same amplitude. (4) Interaction between the two amplitude-unvarying solitons and that between the two amplitude-varying ones have been presented in Figs. 4 and 5, respectively. (5) For a moving soliton, its velocity always keeps varying. Acknowledgments We express our science thanks to the Editors and Reviewers for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant nos. 11272023 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), and by the Fundamental Research Funds for the Central Universities of China under Grant no. 2011BUPTYB02. 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]

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