Conservation laws, nonautonomous breathers and rogue waves for a higher-order nonlinear Schrödinger equation in the inhomogeneous optical fiber

Accepted Manuscript Conservation laws, nonautonomous breathers and rogue waves for a higher-order nonlinear Schrzdinger equation in the inhomogeneous optical fiber Chuan-Qi Su, Nan Qin, Jian-Guang Li PII:

S0749-6036(16)30736-4

DOI:

10.1016/j.spmi.2016.09.052

Reference:

YSPMI 4542

To appear in:

Superlattices and Microstructures

Received Date: 20 August 2016 Revised Date:

23 September 2016

Accepted Date: 28 September 2016

Please cite this article as: C.-Q. Su, N. Qin, J.-G. Li, Conservation laws, nonautonomous breathers and rogue waves for a higher-order nonlinear Schrzdinger equation in the inhomogeneous optical fiber, Superlattices and Microstructures (2016), doi: 10.1016/j.spmi.2016.09.052. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

RI PT

Conservation laws, nonautonomous breathers and rogue waves for a higher-order nonlinear Schr¨odinger equation in the inhomogeneous optical fiber

SC

Chuan-Qi Su ∗, Nan Qin, Jian-Guang Li

College of Electromechanical Engineering, Qingdao University of

AC C

EP

TE

D

M AN U

Science and Technology, Qingdao 266061, China

∗

Corresponding author, with e-mail address as [email protected]

1

ACCEPTED MANUSCRIPT Abstract Under investigation in this paper is a higher-order nonlinear Schr¨odinger equation which can describe the propagation of ultrashort pulse in the inhomogeneous optical fiber. Lax pair and conservation laws are constructed from which the integrability of the equation can be verified. Nonautonomous breathers and rogue waves for the equation are derived based on the Darboux transformation (DT) and generalized DT, respectively. Influence of the

RI PT

group-velocity dispersion, third- and forth-order dispersion and gain or loss coefficient on the propagation and interaction of the nonautonomous breathers and rogue waves, is also discussed specifically. There exist two types of nonautonomous breathers. Expressions of the quasi-periods for the two types of nonautonomous breathers are given and the effects of coefficients on the quasi-periods are also discussed. Gain or loss coefficient affects the

SC

background for the nonautonomous breather, while third-order dispersion has an influence on the trajectory of the nonautonomous breather. Group-velocity dispersion affects the range of the nonautonomous rogue wave. In addition, group-velocity dispersion also pro-

M AN U

duces a skew angle and the skew angle rotates in the clockwise direction with the increase of group-velocity dispersion. When forth-order dispersion coefficient is taken as a linear function of distance, the structure of rogue wave pair can be formed. The separation and relative locations for the two constituents of rogue wave pair are affected by the coefficients of the linear function.

D

PACS numbers: 05.45.Yv, 04.30.Nk, 42.81.Dp

AC C

EP

TE

Keywords: Higher-order nonlinear Schr¨odinger equation; nonautonomous breathers; rogue waves; inhomogeneous optical fibers; conservation laws

2

ACCEPTED MANUSCRIPT

1. Introduction

SC

  1 iqz + qtt + |q|2 q − iι qttt + 6|q|2 qt 2   + κ qtttt + 6qt2 q ∗ + 4q|qt |2 + 8qtt |q|2 + 2qtt∗ q 2 + 6q|q|4 = 0 ,

RI PT

In the nonlinear optics, the nonlinear Schr¨odinger (NLS) equation describes the optical pulse propagation in optical fiber when the pulse width is greater than 100 femtosecond [1, 2]. However, over the past years, ultrashort (femtosecond) optical pulses have been studied due to their potential applications in such areas as ultrahigh-bit-rate optical communication systems and optical sampling systems [3, 4]. To produce ultrashort pulses, the intensity of the incident light field increases, which leads that the non-Kerr effects come into play [5–7]. The dynamics of such systems should be described by the NLS equation with high-order terms such as third- and forthorder dispersion, quintic nonlinearity, self-steeping and self-frequency shift [5–7]. In Refs. [8–10], the following higher-order NLS equation have been introduced, (1)

AC C

EP

TE

D

M AN U

where q = q(z, t) represents the slowly varying normalized envelope of the electric field, z and t are the normalized distance and retarded time, respectively, the asterisk represents the complex conjugation, while the real parameters ι and κ are the coefficients of the third- and forth-order dispersion, respectively [8–12]. Solitons, breathers and rogue waves for Eq. (1) have been obtained via the methods of Darboux transformation (DT) and generalized DT [8–11]. Eq. (1) degenerates to the Hirota equation when κ = 0 [13–15]. The Lakshmanan-Porsezian-Daniel (LPD) equation is obtained if ι is set to be zero [16]. Solitons, breathers and rogue waves for this case have been investigated in some papers [17–20]. If ι = κ = 0, Eq. (1) will degenerates to the NLS equation [1, 2]. In practice, however, the inhomogeneity of the optical fiber has to be taken into account. Thus, the variable-coefficient NLS equations have been considered to be more realistic models than their constant-coefficient counterparts [21–24]. The inhomogeneities of optical fibers may arise from the imperfection of manufacture, variation in the lattice parameters, fluctuation of the fiber diameters, fiber loss, and the distance dependence of group-velocity dispersion [4, 25– 27]. Based on the above considerations, in this paper, we will devote ourselves to the following variable-coefficient higher-order NLS equation with gain or loss term [28],    iqz + a(z)qtt + b(z)|q|2 q + i c(z)qttt + d(z)|q|2 qt + f (z) qtttt + γ1 (z)qt2 q ∗ + γ2 (z)q|qt |2 (2)  + γ3 (z)qtt |q|2 + γ4 (z)qtt∗ q 2 + γ5 (z)q|q|4 + ig(z)q = 0 , where a(z) denotes the group-velocity dispersion, b(z) accounts for the self-phase modulation, c(z) and f (z) are the coefficients for the third- and forth-order dispersion, respectively, d(z) is related to the time-delay correction for the cubic nonlinearity term |q|2 q, g(z) is the gain [g(z) < 0] or loss [g(z) > 0] coefficient, and γj (z) (j = 1, 2, 3, 4, 5) are real functions of z [28–31]. The similar transformation between Eq. (2) and its constant-coefficient counterpart has been constructed [28]. 3

ACCEPTED MANUSCRIPT

TE

D

M AN U

SC

RI PT

To our knowledge, Lax pair, conservation laws, nonautonomous breathers and rogue waves for Eq. (2) have not been given in the existing literatures. Investigation on the analytic solutions of the nonlinear evolution equations, e.g., the soliton, breather and rouge-wave solutions, has played an important role in the study of nonlinear phenomena. The concept of solitons has been introduced to characterize the solitary waves which keep their features unchanged, during the propagation and after the interaction, just like the particles [32–37]. The name “breather” reflects that the behavior of the solution is periodic in time or space, and localized in space or time [38–41]. The space-periodic Akhmediev breather and time-periodic Kuznetsov-Ma soliton are two types of known breathers [38–41]. Compared with breathers, rogue waves are localized in both space and time, appear from nowhere and disappear without a trace [42–46]. Roguewave solutions can be obtained via the Taylor expansion of the breather solutions [15, 47]. It should be noted that the concepts of breathers and rogue waves have been developed for the autonomous systems, in which the independent variable “space” or (and) “time” has not appeared explicitly [48, 49]. While, in the practical situations, we may encounter the media with time- or space-dependent nonlinearity, dispersion and external potentials, leading that the systems become nonautonomous [27, 48–53]. Properties of the nonautonomous breathers and rogue waves are different from those in the autonomous systems [27, 48–53]. This paper is organized as follows. In Section 2, Lax pair and conservation laws for Eq. (2) will be constructed from which the integrability of Eq. (2) can be verified. In Section 3, DT and generalized DT for Eq. (2) will be derived. Property of the nonautonomous breathers and rogue waves for Eq. (2) will be investigated in Section 4 and 5, respectively. Influence of the groupvelocity dispersion coefficient a(z), third- and forth-order dispersion coefficients c(z) and f (z), time-delay correction coefficient d(z), and the gain or loss coefficient g(z) will also be discussed analytically and graphically. The last section will be the conclusion.

EP

2. Lax pair and conservation laws

AC C

Employing the Ablowitz-Kaup-Newell-Segur formalism [54], we can derive the Lax pair for Eq. (2) as Φt = U Φ ,

Φz = V Φ ,

(3)

where Φ = (Φ1 , Φ2 )T is a vector eigenfunction with Φ1 and Φ2 both the functions of z and t, the superscript T denotes the transpose of a matrix. The matrices U and V have the form of ! ! iλ α(z)q A(z, t) B(z, t) U= , V = , (4) −α(z)q ∗ −iλ C(z, t) −A(z, t) with 1 A(z, t) = 8if (z)λ4 + 4ic(z)λ3 − i[2a(z) + γ2 (z)f (z)|q|2 ]λ2 + [γ4 (z)f (z)(qqt∗ − qt q ∗ ) − d(z)|q|2 ]λ 3 4

ACCEPTED MANUSCRIPT

d(z) a(z)d(z) , 3γ1 (z) = 2γ2 (z) = 4γ3 (z) = γ4 (z) = , 3c(z) 3c(z) d(z)2 d0 (z) c0 (z) γ5 (z) = , g(z) = − , 6c(z)2 2d(z) 2c(z)

(6)

M AN U

SC

b(z) =

RI PT

i i 1 i + b(z)|q|2 + γ5 (z)f (z)|q|4 + d(z)(qqt∗ − qt q ∗ ) − γ4 (z)f (z)(qtt q ∗ + qqtt∗ − qt qt∗ ) , (5a) 2 2 6 2  3 2 B(z, t) = α(z) 8f (z)qλ + 4[c(z)q − if (z)qt ]λ − 2[a(z)q + γ2 (z)f (z)|q|2 q + ic(z)qt 1 + f (z)qtt ]λ + ia(z)qt + iγ1 (z)f (z)|q|2 qt − c(z)qtt + if (z)qttt − d(z)|q|2 q , (5b) 3  ∗ 3 ∗ ∗ 2 ∗ C(z, t) = −α(z) 8f (z)q λ + 4[c(z)q − if (z)qt ]λ − 2[a(z)q + γ2 (z)f (z)|q|2 q − ic(z)qt∗ 1 ∗ − d(z)|q|2 q ∗ , (5c) + f (z)qtt∗ ]λ − ia(z)qt∗ − iγ1 (z)f (z)|q|2 qt∗ − c(z)qtt∗ − if (z)qttt 3 p where λ is the spectral parameter and α(z) = d(z)/6c(z). It can be verified that the compatibility condition Uz − Vt + U V − V U = 0 gives rise to Eq. (2) under the following conditions,

where 0 represents the differential with respect to z. It can be verified that under the appropriate choices of a(z), c(z), d(z) and f (z), Eq. (2) can be degenerated to Eq. (1). Thus, we have shown that Eq. (2) is Lax integrable by giving its corresponding Lax Pair (3). In the following, we will prove the existence of infinitely-many conservation laws [55] which further verify the integrability 2 , then we obtain the following Riccati-type of Eq. (2). For that, we introduce the function Γ = Φ Φ1 equation from Lax Pair (3),

D

Γt = −α(z)q ∗ − 2iλΓ − α(z)qΓ2 .

(7)

TE

P −n , where Γn 0 s (n = 1, 2, ...) are functions of z and t to be determined, Setting qΓ = ∞ n=1 Γn λ substituting it into Eq. (7) and equating the coefficients of the same power of λ to zero, we have

AC C

EP

i Γ1 = α(z)qq ∗ , 2 i qt 1 Γ2 = (Γ1,t − Γ1 ) = − α(z)qqt∗ , 2 q 4 k−1   X qt i Γk+1 = Γk,t − Γk + α(z) Γj Γk−j , (k = 2, 3, ...) . 2 q j=1

(8a) (8b) (8c)

From the compatibility condition (lnΦ1 )zt = (lnΦ1 )tz [55], the infinitely-many conservation laws for Eq. (2) can be expressed as ∂Un ∂Fn + = 0, ∂z ∂t

(9)

with i U1 = α(z)Γ1 = α(z)2 qq ∗ , 2

(10a) 5

ACCEPTED MANUSCRIPT α(z)  8f (z)qΓ4 + 4[c(z)q − if (z)qt ]Γ3 − 2[a(z)q + γ2 (z)f (z)|q|2 q + ic(z)qt q 1 + f (z)qtt ]Γ1 + [ia(z)qt + iγ1 (z)f (z)|q|2 qt − c(z)qtt + if (z)qttt − d(z)|q|2 q]Γ1 , (10b) 3 Un = α(z)Γn , (10c) α(z)  Fn = − 8f (z)qΓn+3 + 4[c(z)q − if (z)qt ]Γn+2 − 2[a(z)q + γ2 (z)f (z)|q|2 q + ic(z)qt q 1 + f (z)qtt ]Γn+1 + [ia(z)qt + iγ1 (z)f (z)|q|2 qt − c(z)qtt + if (z)qttt − d(z)|q|2 q]Γn , (10d) 3

RI PT

F1 = −

where Un 0 s and Fn 0 s represent the conserved fluxes and conserved densities, respectively.

SC

3. DT and generalized DT

φ11 −φ∗12 φ12 φ∗11

(11)

TE

H [1] =

S [1] = H [1] Λ(H [1] )−1 , ! ! λ1 0 , , Λ= 0 λ∗1

D

M [1] =λI − S [1] ,

M AN U

In this section, DT and generalized DT for Eq. (2) will be constructed. DT is a method to construct soliton and breather solutions, but fails to be iterated at the same spectral parameter which makes it impossible to construct rogue wave solutions [56]. To overcome that problem, the so-called generalized DT has been put forward and applied to different physical models [56–59]. To construct the DT for Lax Pair (3), we assume that q [0] is a seed solution of Eq. (2), and (φ11 , φ12 )T is a solution of Lax Pair (3) with λ = λ1 , q = q [0] . One can verify that (−φ∗12 , φ∗11 )T is also a solution of Lax Pair (3) with λ = λ∗1 , q = q [0] . The DT matrix M [1] can be constructed as

AC C

EP

where I denotes the 2×2 identity matrix, (H [1] )−1 is the inverse of H [1] . The superscripts [n] (n = 1, 2, . . .) are introduced to represent the matrices or functions which are derived from the ithorder DT. With the aid of symbolic computation [60], it can be verified that Lax Pair (3) remains covariant under the operation of matrix M [1] , i.e., the form of Lax Pair (3) keeps unchanged except that q [0] is replaced by q [1] , which is defined by q [1] = q [0] +

2i(λ1 − λ∗1 )φ11 φ∗12 . α(z)(φ11 φ∗11 + φ12 φ∗12 )

(12)

If (ψ11 , ψ12 )T is a solution of Lax Pair (3) with λ = λ2 , q = q [0] , then M [1] |λ=λ2 (ψ11 , ψ12 )T = (φ21 , φ22 )T is a solution of Lax Pair (3) with λ = λ2 , q = q [1] . The second-order solutions for Eq. (2) can be generated by use of (φ21 , φ22 )T , q [2] = q [1] +

2i(λ2 − λ∗2 )φ21 φ∗22 . α(z)(φ21 φ∗21 + φ22 φ∗22 )

(13)

Based on the above DT, we now derive the generalized DT for Eq. (2). Let us start with the assumption that Φ(λ1 + δ) is a solution of Lax Pair (3) with λ = λ1 + δ and q = q [0] , where δ is 6

ACCEPTED MANUSCRIPT a small parameter. By expanding Φ(λ1 + δ) into the Taylor series at λ = λ1 , i.e., δ = 0, we have Φ(λ1 + δ) = Φ0 + Φ1 δ + Φ2 δ 2 + · · · + Φj δ j + · · · , j

M [1] |λ=λ1 +δ Φ(λ1 + δ) = Φ0 + M [1] |λ=λ1 Φ1 = (φ21 , φ22 )T , δ→0 δ lim

RI PT

Φ(λ) T where Φj = j!1 ∂ ∂λ is a solution of Lax j |λ=λ1 (j = 0, 1, 2, . . .). It is true that Φ0 = (φ11 , φ12 ) [0] Pair (3) with λ = λ1 , q = q . The first-order generalized DT can be constructed via Eqs. (11) and (12) by the use of Φ0 = (φ11 , φ12 )T . It can be shown that M [1] |λ=λ1 +δ Φ(λ1 +δ) is a solution of Lax Pair (3) with λ = λ1 +δ, q = q [1] . With the help of the identity M [1] |λ=λ1 Φ0 = 0 [56], the limit process,

(14)

M AN U

SC

provides a solution of Lax Pair (3) with λ = λ1 , q = q [1] . Thus, the second-order solutions for Eq. (2) can be constructed based on Eq. (13). The N th-order DT and generalized DT can be constructed in a similar way. For the sake of brevity, we omit the specific derivations here.

4. Nonautonomous breathers

D

In order to obtain the nonautonomous breathers for Eq. (2), the nonzero continuous background should be taken into consideration. It can be justified that Eq. (2) has the solution of the form

TE

 q [0] = m(z)exp in(z)] ,

(15)

where m(z) and n(z) satisfy the following conditions, m(z) = 1/α(z) , n0 (z) = 2[a(z) + 3f (z)] .

EP

(16)

AC C

If we take the spectral parameter λ as a pure imaginary number, i.e., λ = ih with h as a real constant, and substituting Eq. (15) into Lax Pair (3), we can obtain the solution for Lax Pair (3) as ! i (c1 eA + c2 e−A )e 2 n(z) , (17) Φ= i (c3 eA + c4 e−A )e− 2 n(z) where c1 and c2 are both the constants, and   Z Z Z 2 2 A = µ t − 2ih a(z)dz − 2(1 + 2h ) c(z)dz − 4ih(1 + 2h ) f (z)dz , √ µ = h2 − 1 , c3 = (h + µ)c1 , c4 = (h − µ)c2 .

(18)

As shown in the following discussions, there exist two types of nonautonomous breathers based √ on the sign of h2 − 1. When h > 1, h2 − 1 > 0, µ = h2 − 1 is a real number in this situation. 7

ACCEPTED MANUSCRIPT Substituting Eqs. (15) and (17) into Eq. (12), we can obtain the first-type nonautonomous breathers for Eq. (2) as h M i   1 q [1] = − m(z) exp in(z) , (19) α(z)N1 with

RI PT

h i √ M1 =4c1 c2 e2Re(A) (1 − h2 )cos[2Im(A)] + ih h2 − 1sin[2Im(A)] , √ √ N1 =c21 h( h2 − 1 + h)e4Re(A) + c1 c2 e2Re(A) cos[2Im(A)] + c22 h(h − h2 − 1) ,

(20)

Tz =

M AN U

and the quasi-period along z axis can be expressed as

SC

where Re(A) and Im(A) denote the real and imaginary parts of the spectral parameter A, respectively. It can be seen from Eq. (20) that the quasi-period1 for the first-type nonautonomous breather is related to the imaginary part of A, Z  Z √ 2 2 Im(A) = −2h h − 1 a(z)dz + 2(1 + 2h ) f (z)dz , (21)

πz √ , R R 2 2h h − 1 a(z)dz + 2(1 + 2h2 ) f (z)dz

(22)

AC C

EP

TE

D

which is influenced by the group-velocity dispersion coefficient a(z) and forth-order dispersion coefficient f (z), but independent of the third-order dispersion coefficient c(z) and time-delay correction coefficient d(z). Eq. (22) shows that if a(z) and f (z) are constants, Tz increases with the decrease of a(z) and f (z), which can be seen from the comparison between Figs. 1(a) and 1(b). While, if a(z) or f (z) is z-dependent, then Tz will change along z axis. f (z) is taken as 5z in Figs. 1(c) and in this case, Tz becomes small with the increase of |z|, while enlarges as z → 0.

(a)

(b)

(c)

Figs. 1. Nonautonomous breathers via Solutions (19) with parameters λ = 1 + 2i, c1 = c2 = 1, c(z) = d(z) = 1, (a) a(z) = 1, f (z) = 0.2, (b) a(z) = 5, f (z) = 0.2, (c) a(z) = 2, f (z) = 5z.

1

For the nonautonomous breathers, the “period” becomes z-dependent [48], thus the term “quasi-period” is used in this paper.

8

ACCEPTED MANUSCRIPT

M AN U

SC

RI PT

Figs. 2 show that c(z) and d(z) affect the background for the nonautonomous breather. When c(z) and d(z) are taken as constants, the gain or loss coefficient g(z) = [d0 (z)/d(z) − c0 (z)/c(z)]/2 = 0, background for this case is flat, as shown in Figs. 1. The linear backgrounds are exhibited in Figs. 2(a) and 2(b) where g(z) are constants. Different from Figs. 2(a) and 2(b), g(z) in Figs. 2(c) is a function of z, which is equal to 1/z. In addition, c(z) has also an influence on the trajectory of the nonautonomous breather. Exponential and parabolic trajectories are exhibited in Figs. 2(b) and 2(c) under different choices of c(z).

(a)

(b)

(c)

D

Figs. 2. Nonautonomous breathers via Solutions (19) with parameters λ = 1 + 2i, c1 = c2 = 1, a(z) = 5, f (z) = 0.2, (a) c(z) = 1, d(z) = e4z , (b) c(z) = e6z , d(z) = 5, (c) c(z) = 6z, d(z) = 5.

EP

TE

√ √ When 0 < h < 1, h2 − 1 < 0, µ = h2 − 1 = i 1 − h2 is a pure imaginary number in this situation. By the use of Eqs. (15), (17) and (12), we can obtain the second-type nonautonomous breathers for Eq. (2) as h M i   2 q [1] = − m(z) exp in(z) , (23) α(z)N2 where

AC C

√ √   M2 = 2 c21 (1 − h2 + ih 1 − h2 )e4Re(A) + c22 (1 − h2 − ih 1 − h2 ) , i h √ 2 2 4Re(A) 2Re(A) 2 N2 = c 2 + c 1 e + 2c1 c2 he hcos[2Im(A)] − 1 − h sin[2Im(A)] .

(24)

As shown in Eq. (24), the quasi-period for the second-type nonautonomous breather is also related to the imaginary part of A,   Z √ 2 2 Im(A) = h − 1 t − 2(1 + 2h ) c(z)dz , (25) and the quasi-periods along t and z axes can be expressed respectively as π Tt = √ , 1 − h2

πz R Tz = √ . 2 2 1 − h (1 + 2h2 ) c(z)dz 9

(26)

ACCEPTED MANUSCRIPT

M AN U

SC

RI PT

It can be seen from the above equations that Tt is a constant which is merely related to h; Tz is influenced by c(z), but independent of a(z), d(z) and f (z). c(z) influences the trajectory of the nonautonomous breather in one quasi-period from Figs. 3, which is different from Figs. 2(b) and 2(c). If c(z) is a constant, the linear trajectory in one quasi-period will be exhibited and the slope formed with z axis enlarges with the increase of c(z), as shown in Figs. 3(a) and 3(b). If c(z) is a linear function of z, the parabolic trajectory in one quasi-period can be formed, as shown in Figs. 3(c), and a curved background is exhibited because the gain or loss coefficient g(z) is nonzero for this case.

(a)

(b)

(c)

D

Figs. 3. Nonautonomous breathers via Solutions (23) with parameters λ = 1 + 0.5i, c1 = c2 = 1, a(z) = 2, d(z) = 1, f (z) = 0.2, (a) c(z) = 1, (b) c(z) = 3, (c) c(z) = 2t.

AC C

EP

TE

The comparison between Figs. 3(a) and 4(a) shows that the forth-order dispersion coefficient f (z) influences the range of the nonautonomous breather along z axis. The range decreases when f (z) is changed from 0.2 in Fig. 3(a) to 1 in Fig. 4(a). For Fig. 4(b) where f (z) = z, the nonautonomous breather has two peaks in one quasi-period. While, if we take a(z) = 3z, a platform in one quasi-period is formed, as shown in Fig. 4(c).

10

(a)

(b)

RI PT

ACCEPTED MANUSCRIPT

(c)

SC

Figs. 4. Nonautonomous breathers via Solutions (23) with parameters λ = 1 + 0.5i, c1 = c2 = 1, c(z) = d(z) = 1, (a) a(z) = 2, f (z) = 1, (b) a(z) = 2, f (z) = z, (c) a(z) = 3z, f (z) = 0.2.

AC C

EP

TE

D

M AN U

It can be seen from Figs. 5 that the gain or loss coefficient g(z) influences the background for the nonautonomous breather. The linear background can be formed if g(z) is a constant, as shown in Figs. 5(a) and 5(b). If g(z) > 0 (loss), the background will curve along the negative z axis, while if g(z) < 0 (gain), along the positive z axis. Fig. 5(c) shows the parabolic background under the condition g(z) = [d0 (z)/d(z)−c0 (z)/c(z)]/2 = 5t. Different form that Figs. 2 shows, for the second-type nonautonomous breather, g(z) has no effect on the amplitude of nonautonomous breather.

(a)

(b)

(c)

Figs. 5. Nonautonomous breathers via Solutions (23) with parameters λ = 1 + 0.5i, c1 = c2 = 1, a(z) = 2, c(z) = f (z) = 1, (a) 2

d(z) = et , (b) d(z) = e−t , (c) d(z) = e5t .

5. Nonautonomous rogue waves The h = 1 case has not been discussed in the above section. In fact, when h = 1, the quasi-period becomes infinity from Eqs. (22) and (26). On the other hand, when h = 1, µ and A in Eq. (17) become zero, which leads to the degenerated solutions. In order to obtain the 11

ACCEPTED MANUSCRIPT non-degenerated solutions (nonautonomous rogue waves), the method of generalized DT which has been given in Section 3, will be used. For that, we set c1 and c2 in Eq. (17) as p p √ √ h − h2 − 1 h + h2 − 1 √ c1 = , c2 = − √ , (27) h2 − 1 h2 − 1

Φ = Φ0 + Φ1 δ + Φ2 δ 2 + · · · = Φ0 + Φ1 2 + Φ2 4 + · · · .

RI PT

and h = 1−i2 with  as a small real parameter independent of z and t, then λ = ih = i(1−i2 ) = i + 2 . Thus, λ1 and δ introduced in Section 3 equal i and 2 , respectively. Expanding Φ at δ = 2 = 0, we have (28)

(a)

EP

TE

D

M AN U

SC

Via generalized DT, the nonautonomous rogue waves for Eq. (2) can be derived with the help of Φj (j = 0, 1, 2, . . .). Figs. 6 exhibit the first-order nonautonomous rogue waves under different choices of the group-velocity dispersion coefficient a(z), which shows that a(z) affects the range of the first-order nonautonomous rogue wave along z axis. The range decreases with the increase of a(z) if we take a(z) as a constant. In addition, a(z) also produces a skew angle2 and the skew angle rotates in the clockwise direction with the increase of a(z).

(b)

(c)

AC C

Figs. 6. First-order nonautonomous rogue waves with parameters c(z) = d(z) = 1, f (z) = 0, (a) a(z) = 2, (b) a(z) = 5, (c) a(z) = 8.

The gain or loss coefficient g(z), which is related to c(z) and d(z), influences the background for the first-order nonautonomous rogue wave. The linear, parabolic and periodic backgrounds are exhibited in Fig. 7 under different choices of d(z). 2

The angle formed by the nonautonomous rogue wave along the z axis [61].

12

(a)

(b)

RI PT

ACCEPTED MANUSCRIPT

(c)

2

SC

Figs. 7. First-order nonautonomous rogue waves with parameters a(z) = 2, c(z) = 1, f (z) = 0.2, (a) d(z) = e−5t , (b) d(z) = e25t , (c) d(z) = 2 + cos(50z − 4).

AC C

EP

TE

D

M AN U

A different type of first-order nonautonomous rogue wave is demonstrated in Figs. 8, where the forth-order dispersion coefficient f (z) is taken as linear function of z, i.e., f (z) = k1 z + k2 with k1 and k2 as real constants. Such structure can be viewed as the “composite rogue wave (rogue wave pair)” consisting of two fundamental rogue waves, which has been reported in some vector equations [62–64]. The separation and relative locations for the two fundamental rogue waves are affected by k1 and k2 , as shown in Figs. 8.

(a)

(b)

(c)

Figs. 8. First-order nonautonomous rogue waves with parameters a(z) = 2, c(z) = d(z) = 1, (a) f (z) = z − 0.8, (b) f (z) = z − 1, (c) f (z) = 2z − 1.

Figs. 9 and 10 exhibit some figures for the second-order nonautonomous rogue waves, from which it can be seen that coefficients a(z), c(z), d(z) and f (z) have the similar effects as the firstorder case. If the forth-order dispersion coefficient f (z) is a linear function of z, the structure of rogue wave pair can be formed, as shown in Figs. 10. For the two constituents of such structure, one is second-order rogue wave, while the other splits into three first-order rogue waves. To our 13

ACCEPTED MANUSCRIPT

(b)

SC

(a)

RI PT

knowledge, such phenomenon has not been reported in the existing literatures.

(c)

TE

D

M AN U

Figs. 9. Second-order nonautonomous rogue waves with parameters c(z) = 1, f (z) = 0.5, (a) a(z) = 2, d(z) = 1, (b) a(z) = z, d(z) = 1, (c) a(z) = 2, d(z) = 2 + cos(50z).

(b)

(c)

EP

(a)

Figs. 10. Second-order nonautonomous rogue waves with parameters a(z) = 2, c(z) = d(z) = 1, (a) f (z) = z − 1, (b) f (z) = 2z − 1.5, (c) f (z) = 2z − 1.1.

AC C

6. Conclusion

In this paper, a variable-coefficient higher-order NLS equation which can describe the propagation of ultrashort pulse in the inhomogeneous optical fiber, i.e., Eq. (2), has been investigated. Lax pair (3) and infinitely-many conservation laws (9) have been constructed from which the integrability of Eq. (2) can be verified. Nonautonomous breathes (19), (23) have been derived based on the DT, while nonautonomous rogue waves have been obtained via the generalized DT. Influence of the group-velocity dispersion coefficient a(z), third- and forth-order dispersion coefficients c(z) and f (z), time-delay correction coefficient d(z), and the gain or loss coefficient g(z) on the propagation and interaction of the nonautonomous breathers and rogue waves, has also been discussed analytically and graphically. 14

ACCEPTED MANUSCRIPT

EP

TE

D

M AN U

SC

RI PT

There exist two types of nonautonomous breathers based on the sign of h2 − 1. Nonautonomous breathers for the h2 > 1 case are expressed by Eq. (19), from which we can see that the quasi-period along z axis, Tz , is influenced by a(z) and f (z), but independent of c(z) and d(z). If a(z) and f (z) are constants, then Tz increases with the decrease of a(z) and f (z), which can be seen from the comparison between Figs. 1(a) and 1(b). If a(z) or f (z) is z-dependent, then Tz will change along z axis, as shown in Figs. 1(c). From Figs. 2, it can be seen that c(z) and d(z) affect the background for the nonautonomous breather, while c(z) has also an influence on the trajectory of the nonautonomous breather. Solutions for the second-type nonautonomous breathers (h2 < 1) are expressed by Eq. (23), from which we find that Tt is a constant; while Tz is influenced by c(z), but independent of a(z), d(z) and f (z). If c(z) is a constant, the linear trajectory in one quasi-period is exhibited and the slope formed with z axis enlarges with the increase of c(z), as shown in Figs. 3(a) and 3(b). If c(z) is a linear function of z, the parabolic trajectory in one quasi-period can be formed, as shown in Figs. 3(c). f (z) influences the range of the second-type nonautonomous breather along z axis from the comparison between Figs. 3(a) and 4(a). The linear background can be formed if g(z) is a constant, as shown in Figs. 5(a) and 5(b). If g(z) > 0 (loss), the background will curve along the negative z axis, while if g(z) < 0 (gain), along the positive z axis. Figs. 6 show that a(z) affects the range of the nonautonomous rogue wave along z axis. The range decreases with the increase of a(z) if we take a(z) as a constant. In addition, a(z) also produces a skew angle and the skew angle rotates in the clockwise direction with the increase of a(z). When f (z) is taken as linear function of z, the structure of rogue wave pair can be formed, as shown in Figs. 8 and 10. The separation and relative locations for the two constituents are affected by the coefficients of the linear function. For the two constituents in Figs. 10, one is second-order rogue wave, while the other splits into three first-order rogue waves.

Acknowledgements

AC C

We express our sincere thanks to the editors, reviewers and all the members of our discussion group for their valuable discussions. This work has been supported by the National Natural Science Foundation of China under Grant No. 51374134 and 51674149.

References

[1] B. Kibler, J. Fatome, C. Finot, G. Millot, et al. Nat. Phys., 6, 790 (2010) [2] M. Erkintalo, K. Hammani, B. Kibler, et al. Phys. Rev. Lett., 107, 253901 (2011) [3] Alka, A. Goyal, R. Gupta and C. N. Kumar. Phys. Rev. A, 84, 063830 (2011)

15

ACCEPTED MANUSCRIPT [4] G. P. Agrawal. Applications of Nonlinear Fiber, Academic, San Diego (2001). [5] W. P. Hong. Opt. Commun., 194, 214 (2001) [6] F. H. Qi, H. M. Ju, X. H. Meng and J. Li. Nonl. Dyn., 77, 1331 (2014)

RI PT

[7] P. Wang and B. Tian. Opt. Commun., 285, 3567 (2012) [8] A. Ankiewicz and N. Akhmediev. Phys. Lett. A, 378, 358 (2014)

[9] A. Ankiewicz, Y. Wang, S. Wabnitz and N. Akhmediev. Phys. Rev. E, 89, 012907 (2014)

SC

[10] A. Chowdury, D. J. Kedziora, A. Ankiewicz and N. Akhmediev. Phys. Rev. E, 90, 032922 (2014) [11] W. R. Sun, B. Tian, H. L. Zhen and Y. Sun. Nonl. Dyn., 81, 725 (2015)

M AN U

[12] Q. M. Wang, Y. T. Gao, C. Q. Su and D. W. Zuo. Phys. Scr., 90, 105202 (2015) [13] R. Hirota. J. Math. Phys., 14, 805 (1973)

[14] A. Ankiewicz, J. M. Sota-Crespo and N. Akhmediev. Phys. Rev. E, 81, 046602 (2010) [15] Y. S. Tao and J. S. He. Phys. Rev. E, 85, 026601 (2012)

D

[16] M. Lakshmaman, K. Porsezian and M. Daniel. Phys. Lett. A, 133, 483 (1988)

TE

[17] B. Yang, W. G. Zhang, H. Q. Zhang and S. B. Pei. Phys. Scr., 88, 065004 (2013) [18] H. Q. Zhang, B. Tian, X. H. Meng, X. Lv and W. J. Liu. Eur. Phys. J. B, 72, 233 (2009)

EP

[19] R. Guo and H. Q. Hao. Commun. Nonl. Sci. Numer. Simulat., 18, 2426 (2013) [20] L. H. Wang, K. Porsezian and J. S. He. Phys. Rev. E, 87, 053202 (2013)

AC C

[21] V. I. Kruglov, A. C. Peacock and J. D. Harvey. Phys. Rev. Lett., 90, 113902 (2003) [22] C. Q. Dai, Y. Y. Wang and G. Q. Zhou. Chaos, Solitons Fract., 45, 1291 (2012) [23] Z. Y. Yan. Phys. Lett. A, 374, 672 (2010) [24] C. Q. Dai, X. G. Wang and J. F. Zhang. Ann. Phys., 326, 645 (2011) [25] G. P. Agrawal. Nonlinear Fiber Optics, Elsevier, Sigapore (2004). [26] F. Abdullaev, S. Darmanyan and P. Khabibullaev. Optical Solitons, Springer, Berlin (1991). [27] L. Wang, M. Li, F. H. Qi and C. Geng. Eur. Phys. J. D, 69, 108 (2015)

16

ACCEPTED MANUSCRIPT [28] F. J. Yu. Commun. Nonl. Sci. Numer. Simulat., 34, 142 (2016) [29] J. Li, H. Q. Zhang, T. Xu, Y. X. Zhang and B. Tian. J. Phys. A, 40, 13299 (2007) [30] X. H. Meng, W. J. Liu, H. W. Zhu, C. Y. Zhang and B. Tian. Physica A, 387, 97 (2008)

RI PT

[31] C. Q. Dai and J. F. Zhang. J. Phys. A, 39, 723 (2006) [32] X. Lv, W. X. Ma and C. M. Khalique. Appl. Math. Lett., 50, 37 (2015) [33] X. Lv, F. H. Lin and F. H. Qi. Appl. Math. Model., 39, 3221 (2015) [34] X. Lv. Nonl. Dyn., 81, 239 (2015)

SC

[35] X. Lv, W. X. Ma, J. Yu, F. H. Lin and C. M. Khalique. Nonl. Dyn., 82, 1211 (2015)

M AN U

[36] X. Lv and F. H. Lin. Commun. Nonl. Sci. Numer. Simulat., 32, 241 (2016) [37] X. Lv and F. H. Lin. Commun. Nonl. Sci. Numer. Simulat., 31, 40 (2016) [38] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur. Phys. Rev. Lett., 30, 1262 (1973) [39] E. Kuznetsov. Sov. Phys. Dokl., 22, 507 (1977) [40] Y. C. Ma. Stud. Appl. Math., 60, 43 (1979)

D

[41] N. Akhmediev and V. I. Korneev. Theor. Math. Phys., 69, 1089 (1986)

TE

[42] D. R. Solli, C. Ropers, P. Koonath and B. Jalali. Nature, 450, 1054 (2007) [43] N. Akhmediev, A. Ankiewicz and J. M. Soto-Crespo. Phys. Rev. E, 80, 026601 (2009)

EP

[44] Y. V. Bludov, V. V. Konotop and N. Akhmediev. Phys. Rev. A, 80, 033610 (2009)

AC C

[45] W. M. Moslem, P. K. Shukla and B. Fliasson. Europhys. Lett., 96, 25002 (2011) [46] N. Akhmediev, A. Ankiewicz and M. Taki. Phys. Lett. A, 373, 675 (2009) [47] S. B. Shan, C. Z. Li and J. S. He. Commun. Nonl. Sci. Numer. Simulat., 18, 3337 (2013) [48] V. N. Serkin, A. Hasegawa and T. L. Belyaeva. Phys. Rev. A, 81, 023610 (2010) [49] V. N. Serkin, A. Hasegawa and T. L. Belyaeva. Phys. Rev. Lett., 98, 074102 (2007) [50] L. Wang, Y. J. Zhu, F. H. Qi and R. Guo. Chaos, 25, 063111 (2015) [51] L. Wang, X. Li, F. H. Qi and L. L. Zhang. Ann. Phys., 359, 97 (2015) [52] L. Wang, J. H. Zhang, C. Liu, M. Li and F. H. Qi. Phys. Rev. E, 93, 062217 (2016) 17

ACCEPTED MANUSCRIPT [53] L. Wang, J. H. Zhang, Z. Q. Wang, C. Liu, M. Li, F. H. Qi and R. Guo. Phys. Rev. E, 93, 012214 (2016) [54] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Segur. Phys. Rev. Lett., 31, 125 (1973) [55] X. Lv and M. S. Peng. Commun. Nonl. Sci. Numer. Simulat., 18, 2304 (2013)

RI PT

[56] B. L. Guo, L. M. Ling and Q. P. Liu. Phys. Rev. E, 85, 026607 (2012)

[57] L. J. Li, Z. W. Wu, L. H. Wang and J. S. He. Ann. Phys., 334, 198 (2013) [58] X. Wang, Y. Q. Li and Y. Chen. Wave Motion, 51, 1149 (2014)

SC

[59] X. Wang, Y. Q. Li, F. Huang and Y. Chen. Commun. Nonl. Sci. Numer. Simulat., 20, 434 (2015)

M AN U

[60] B. Tian and Y. T. Gao. Phys. Lett. A, 342, 228 (2005)

[61] X. Wang, B. Yang, Y. Chen and Y. Q. Yang. Phys. Scr., 89, 095210 (2014) [62] L. Wang, L. L. Zhang and F. H. Qi. Z. Naturforsch. A, 70, 251 (2015) [63] F. Baronio, M. Conforti, A. Degasperis and S. Lombardo. Phys. Rev. Lett., 111, 114101 (2013)

AC C

EP

TE

D

[64] L. C. Zhao and J. Liu. J. Opt. Soc. Am. B, 29, 3119 (2012)

18

ACCEPTED MANUSCRIPT

Highlight

AC C

EP

TE D

M AN U

SC

RI PT

1. Lax pair and conservation laws for the equation has been constructed from which the integrability of the equation can be verified. 2. Nonautonomous breathers and rogue waves for the equation have been derived by Darboux transformation (DT) and generalized DT, respectively. 3. Influence of the group-velocity dispersion, third- and forth-order dispersion and gain or loss coefficient on the propagation and interaction of the nonautonomous breathers and rogue waves have been discussed.

1