Breather-to-soliton transition for a sixth-order nonlinear Schrödinger equation in an optical fiber

Accepted Manuscript Breather-to-soliton transition for a sixth-order nonlinear Schr¨odinger equation in an optical fiber

Qian-Min Huang, Yi-Tian Gao, Lei Hu

PII: DOI: Reference:

S0893-9659(17)30218-5 http://dx.doi.org/10.1016/j.aml.2017.06.015 AML 5291

To appear in:

Applied Mathematics Letters

Received date : 11 April 2017 Revised date : 27 June 2017 Accepted date : 28 June 2017 Please cite this article as: Q. Huang, Y. Gao, L. Hu, Breather-to-soliton transition for a sixth-order nonlinear Schr¨odinger equation in an optical fiber, Appl. Math. Lett. (2017), http://dx.doi.org/10.1016/j.aml.2017.06.015 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.

Breather-to-soliton transition for a sixth-order nonlinear Schr¨odinger equation in an optical fiber Qian-Min Huang, Yi-Tian Gao∗, Lei Hu Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

Abstract In this letter, breather-to-soliton transition is studied for an integrable sixth-order nonlinear Schr¨odinger equation in an optical fiber. Constraint for the breather-to-soliton transition is given. Breathers could be transformed into the different types of solitons, which are determined by the values of the real and imaginary parts of the eigenvalues in the Darboux transformation. Interactions of the breathers and breathers, of the breathers and solitons, as well as of the solitons and solitons, are discussed graphically.

PACS numbers: 05. 45. Yv; 47. 35. Fg; 02. 30. Jr Keywords: Optical fiber; Sixth-order nonlinear Schr¨odinger equation; Darboux transformation; Breather-to-soliton transition

∗

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

1

1. Introduction Solitons and breathers are two types of the analytic solutions for the nonlinear Schr¨odinger (NLS) equations [1–8]. It has been known that when the group-velocity dispersion in an optical fiber is anomalous or when the nonlinearity of a planar waveguide is self-focusing, a constantamplitude continuous wave has been seen to be unstable due to the modulational instability, and to break down into a sequence of localized pulses or beams for the spatial domain, and those pulses have been named the bright solitons [11, 12]. Correspondingly, in the case of normal group-velocity dispersion in the fibers or self-defocusing nonlineariy in the waveguides, initial pulses or spatially localized beams undergo the enhanced dispersion- or diffraction-induced broadening and chirping, and in that case a constant amplitude wave is modulationally stable and localized pulses can appear only as the “holes” on a continuous wave background which have been named the dark solitons [11, 12]. Breathers have been classified as the Akhmediev breathers (ABs), Kuznetsov-Ma (KM) solitons and Peregrine solitons [14, 15]. The ABs have exhibited the spatial periodic properties, while the KM solitons have exhibited the temporal periodic properties [16, 19, 20]. Both of the ABs and KM solitons could be transformed into the Peregrine solitons under certain conditions [16]. Relation between the solitons and breathers in the relevant optical fibers has been conducted via an integrable NLS hierarchy [15–18], i.e., iqx + α2 (qtt + 2q|q|2 ) − iα3 (qttt + 6|q|2 qt ) + α4 (qtttt + 6q ∗ qt2 + 4q|qt |2 + 8|q|2 qtt + 2q 2 qtt∗

+ 6|q|4 q) − iα5 (qttttt + 10|q|2 qttt + 30|q|4 qt + 10qqt qtt∗ + 10qqt∗ qtt + 20q ∗ qt qtt + 10qt2 qt∗ )  ∗ ∗ + α6 qtttttt + q 2 [60|qt |2 q ∗ + 50qtt (q ∗ )2 + 2qtttt ] + q[12qtttt q ∗ + 8qt qttt + 22|qtt |2 + 18qttt qt∗ (1) + 70qt2 (q ∗ )2 ] + 20qt2 qtt∗ + 10qt (5qtt qt∗ + 3qttt q ∗ ) + 20qtt2 q ∗ + 10q 3 [(qt∗ )2 + 2q ∗ qtt∗ ] + 20q|q|6 + · · · = 0,

where q = q(x, t) represents a normalized complex amplitude of the optical pulse envelope, the subscripts x and t respectively denote the propagation variable and transverse variable (time in a moving frame) [17, 18], αl ’s (l = 2, 3, 4, · · · ) are the real constant parameters, and ∗ represents the complex conjugation. Breather-to-soliton transitions have been conducted for the third-, fourth- and fifth-order NLS equations respectively [16–18], and for a special case of the sixth-order NLS equation, i.e., α3 = α4 = α5 = 0 [15]. Core idea of the breather-to-soliton transition is to restrict the real and imaginary parts of the eigenvalues of the breathers. Under certain constraints, the breathers would not exhibit periodicity, thus we call the breathers as the solitons. It is worth to note that the solitons are essentially still belong to the breathers, because they are located on the non-zero planes. To our knowledge, breather-to-soliton transition has not been conducted on the sixth-order case of Hierarchy (1), with αm ’s (m = 7, 8, 9, · · · ) equal to zero. In Section 2, the firstand second-order breathers will be constructed via the Darboux transformation (DT), and the condition for the breather-to-soliton transition will be given. In Section 3, conclusions will be given. 2

2. Breather solutions and breather-to-soliton transition The N -th order DT for Hierarchy (1) with αm ’s (m = 7, 8, 9, · · · ) can be expressed as [15–18] ∗

q [N ] = q [0] − 2 with [0]

φ1,1 [0] φ2,1 T [ι] =

!

[ι−1] N X (λι − λ∗ι )φ1,ι φ2,ι [ι−1] ι=1

[ι−1] [ι−1]

φ1,ι φ1,ι

∗

[ι−1] [ι−1]

+ φ2,ι φ2,ι

∗

(2)

,

! ! ! [ι−1] φ1,1 φ1,ι φ 1,ι = , = (T [ι−1] · · · T [1] )|λ=λι , [ι−1] φ2,1 φ2,ι φ2,ι ! ! ! ! [ι−1] [ι−1] ∗ [ι−1] [ι−1] ∗ −1 λ 0 φ1,ι −φ2,ι λι 0 φ1,ι −φ2,ι − , [ι−1] [ι−1] ∗ [ι−1] [ι−1] ∗ 0 λ 0 λ∗ι φ2,ι φ1,ι φ2,ι φ1,ι

(3)

[ι−1]

[ι−1]

where the superscripts [ι]’s (ι=1, 2,· · · ,N ) represent the ι-th order DT, (φ1,ι , φ2,ι ) and (φ1,ι , φ2,ι ) are the eigenvectors, which are the complex functions of x and t, λ and λl are the eigenvalues [ι−1] [ι−1] of the eigenvectors (φ1,ι , φ2,ι ) and (φ1,ι , φ2,ι ), and “−1” means the inverse operation. To construct the breathers for Hierarchy (1) with αm ’s (m = 7, 8, 9, · · · ) equal to zero, the seed solution is selected as q [0] = ei2(α2 +3α4 +10α6 )x . The eigenfunctions (φ1,1 , φ2,1 ) and (φ1,2 , φ2,2 ) at λ1 = l1 + ih1 and λ2 = l2 + ih2 are respectively i h√ √ √ λ21 +1F1 (x)+ λ21 +1t+ 12 arccos λ21 +1 − iπ − 2i m(x) i 4 φ1,1 = e h√ i √ √ −i λ21 +1F1 (x)+ λ21 +1t+ 12 arccos λ21 +1 + iπ − 2i m(x) 4 −e , i h√ √ √ λ21 +1F1 (x)+ λ21 +1t− 12 arccos λ21 +1 + iπ + 2i m(x) −i 4 φ2,1 = e i h√ √ √ λ21 +1F1 (x)+ λ21 +1t− 21 arccos λ21 +1 − iπ i + 2i m(x) 4 +e , h√ i (4) √ √ i i λ22 +1F2 (x)+ λ22 +1t+ 12 arccos λ22 +1 − iπ − m(x) 4 2 φ1,2 = e h√ i √ √ −i λ22 +1F2 (x)+ λ22 +1t+ 12 arccos λ22 +1 + iπ − 2i m(x) 4 −e , i h√ √ √ λ22 +1F2 (x)+ λ22 +1t− 12 arccos λ22 +1 + iπ −i + 2i m(x) 4 φ2,2 = e h√ √ 2 √ 2 i iπ i 1 2 λ2 +1F2 (x)+ λ2 +1t− 2 arccos λ2 +1 − 4 + 2 m(x) i +e with

m(x) = 2(α2 + 3α4 + 10α6 )x, F1 (x) = −2[−λ1 α2 + (1 − 2λ21 )α3 − 2λ1 α4 + 4λ31 α4 + 3α5 − 4λ21 α5 + 8λ41 α5 − 6λ1 α6 + 8λ31 α6 − 16λ51 α6 ]x,

F2 (x) = −2[−λ2 α2 + (1 −

2λ22 )α3

− 2λ2 α4 +

4λ32 α4

+ 3α5

− 4λ22 α5 + 8λ42 α5 − 6λ2 α6 + 8λ32 α6 − 16λ52 α6 ]x, 3

(5)

where l1 , h1 , l2 and h2 are the real parameters. According to Eq. (2), the first-order breathers can be expressed as q [1] = q [0] −

4ih1 φ∗1,1 φ2,1 . φ1,1 φ∗1,1 + φ2,1 φ∗2,1

(6)

Similarly, the second-order breathers can be expressed as q [2] = q [0] − 2

[l−1] ∗ [l−1] φ2,l ∗ [l−1] [l−1] [l−1] [l−1] ∗ + φ2,l φ2,l l=1 φ1,l φ1,l

2 X

(λl − λ∗l )φ1,l

= q [1] −

[1] ∗ [1]

4ih2 φ1,2 φ2,2 [1]

[1] ∗

[1]

[1] ∗

φ1,2 φ1,2 + φ2,2 φ2,2

,

(7)

where [1]

φ1,2 [1] φ2,2

!

[1]

= (T )|λ2 =l2 +ih2

! φ1,2 . φ2,2

(8)

To proceed with the breather-to-soliton transition, we rewrite The First Order Breathers (6) as q [1] = ei2(α2 +3α4 +10α6 )x (1 + 2h1

S1 + iN1 ), M1

(9)

where S1 = cos(Γr t + Vr x)cosh(2χi ) − cosh(Γi t + Vi x)sin(2χr ), N1 = sinh(Γi t + Vi x)cos(2χr ) + sin(Γr t + Vr x)sinh(2χi ),

(10)

M1 = cosh(Γi t + Vi x)cosh(2χi ) − cosh(Γr t + Vr x)sin(2χr ), with q q 1 Γ = Γr + iΓi = 2 1 + λ21 , χ = χr + iχi = cos−1 ( 1 + λ21 ), 2 2 Vr = −{i(h1 + il1 )α2 + [1 + 2(h1 + il1 ) ]α3 + i[2(h1 + il1 ) + 4(h1 + il1 )3 ]α4 + [3 + 4(h1 + il1 )2 + 8(h1 + il1 )4 ]α5 + i[6(h1 + il1 ) + 16(h1 + il1 )5

+ 8(h1 + il1 )3 ]α6 }(−iΓi + Γr ) − {−(ih1 + l1 )α2 + [1 + 2(h1 − il1 )2 ]α3 + [−2(ih1 + l1 ) + 4(ih1 + l1 )3 ]α4 + [3 + 4(h1 − il1 )2 + 8(ih1 + l1 )4 ]α5

(11)

+ [−6(ih1 + l1 ) + 8(ih1 + l1 )3 − 16(ih1 + l1 )5 ]α6 }(iΓi + Γr ),

Vi = 2{l1 α2 + (−1 − 2h21 + 2l12 )α3 + 2(l1 + 6h21 l1 − 2l13 )α4

− (3 + 4h21 + 8h41 − l12 − 12h21 l12 + 2l14 )α5 }Γi + 2h1{α2 + 4l1 α3 + 2(1 + 2h21 − 6l12 )α4 + 8l1 (1 + 4h21 − 4l12 )α5 + 2[3 + 4h21 + 8h41 − 4(3 + 20h21 l12 + 40l14 )]α6 }Γr .

According to Refs. [15, 16], the constraint that a breather can be transformed into a continuous soliton on a constant background is Vr Vi = . Γr Γi

(12) 4

Based on (12), we have α2 + 2[2l1 α3 + (1 + 2h22 − 6l12 )α4 + (4l1 + 16h21 l1 − 16l13 )α5 + (3 + 4h21 + 8h41 − 12l12 − 80h21 l12 + 40l14 )α6 ] = 0.

(a)

(13)

(b)

Figs. 1. The first-order breathers via Solutions (6), with α2 = 0.5, α3 = α4 = α5 = α6 = 0.06, (a): λ1 = −0.6 − 0.469i; (b): λ1 = −0.6 + 0.469i.

Eq. (13) indicates that the higher-order terms, i.e., the third-, fourth-, fifth- and sixth-order terms, are necessary for the breather-to-soliton transition. Eq. (13) admits the following solutions: q √ 1 h1 = ± √ −α4 − 8l1 α5 + (−2 + 40l12 )α6 + Θ, 2 2α6 or

1 h1 = ± √ 2 2α6

q √ α4 + 8l1 α5 + (2 − 40l12 )α6 + Θ,

(14)

with Θ = (α4 + 8l1 α5 )2 − [4α2 + 4α4 + 16l1 (α3 + 2l1 α4 + 32l12 α5 )]α6 + 4(−5 − 16l12 + 320l14 )α62 .

(15)

Therefore, the breather could be transformed into a continuous soliton on a constant background when the real part and imaginary part of the eigenvalue λ1 accord with Relations (14).

5

(a)

(b)

Figs. 2. The second-order breathers via Solutions (7), with α2 = 0.5, α3 = α4 = α5 = α6 = 0.06, (a); λ1 = −0.6 + 0.469i, λ2 = −0.95 + 0.645i; (b); λ1 = −0.6 − 0.469i, λ2 = −0.4 + 0.757i.

(a)

(b)

Figs. 3. The second-order breathers via Solutions (7), with α2 = 0.5, α3 = α4 = α5 = α6 = 0.06, (a); λ1 = −0.6 + 0.469i, λ2 = 0.5 + 0.8i; (b): λ1 = −0.8 + 0.5i and λ2 = 0.5 + 0.9i.

Figs. 1 present two types of the breather-to-soliton transitions. In Figs. 1 (a), the breather is transformed into an oscillatory M-shape soliton on a constant background. While in Figs. 1 (b), the breather is transformed into an oscillatory W-shape soliton on a constant background. According to Ref. [15], the oscillatory M-shape soliton corresponds to the dark soliton, while the oscillatory W-shape soliton corresponds to the bright soliton. Breather-to-soliton transitions for the second-order breathers are similar to those of the firstorder breathers, i.e., when the real and imaginary parts of λ1 and λ2 accord with Relations (14), there will be an interaction between two solitons, as shown in Figs. 2 (a) and (b), which are the interactions between the oscillatory M-shape and oscillatory W-shape solitons, respectively; when the real and imaginary parts of λ1 or λ2 accord with Relations (14), there will be an 6

interaction between a breather and a soliton, as shown in Figs. 3 (a); when both of the real and imaginary parts of λ1 and λ2 do not accord with Relations (14), there will be an interaction between the two breathers, as shown in Figs. 3 (b).

3. Conclusions In this letter, breather-to-soliton transition has been studied for the integrable sixth-order NLS equation in an optical fiber, i.e., Hierarchy (1) with αm ’s (m = 7, 8, 9, · · · ) equal to zero. Constraint for the breather-to-soliton transition has been given in Relations (14). Breathers have been shown to be transformed into the different types of solitons, which are determined by the values of the real and imaginary parts of the eigenvalues λ1 and λ2 . Interactions of the breathers and breathers, of the breathers and solitons, as well as of the solitons and solitons, have been shown in Figs. 2 and 3.

Acknowledgements We express our sincere thanks to all the members of our discussion group for their valuable comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 11272023, and by the Fundamental Research Funds for the Central Universities under Grant No. 50100002016105010.

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