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Vector rational and semi-rational rogue waves for the coupled cubic-quintic nonlinear Schrödinger system in a non-Kerr medium
Journal Pre-proof Vector rational and semi-rational rogue waves for the coupled cubic-quintic nonlinear Schrödinger system in a non-Kerr medium
Zhong Du, Bo Tian, Qi-Xing Qu, Xiao-Yu Wu, Xue-Hui Zhao
PII:
S0168-9274(20)30037-4
DOI:
https://doi.org/10.1016/j.apnum.2020.02.002
Reference:
APNUM 3767
To appear in:
Applied Numerical Mathematics
Received date:
16 July 2019
Revised date:
2 November 2019
Accepted date:
5 February 2020
Please cite this article as: Z. Du et al., Vector rational and semi-rational rogue waves for the coupled cubic-quintic nonlinear Schrödinger system in a non-Kerr medium, Appl. Numer. Math. (2020), doi: https://doi.org/10.1016/j.apnum.2020.02.002.
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Vector rational and semi-rational rogue waves for the coupled cubic-quintic nonlinear Schr¨odinger system in a non-Kerr medium Zhong Du1 , Bo Tian1∗, Qi-Xing Qu2 , Xiao-Yu Wu1 , Xue-Hui Zhao1 1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China 2. School of Information, University of International Business and Economics, Beijing 100029, China
Abstract Non-Kerr media possess certain applications in photonic lattices and optical fibers. Studied in this paper are the vector rational and semi-rational rogue waves in a nonKerr medium, through the coupled cubic-quintic nonlinear Schr¨odinger system, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium. Applying the gauge transformations, we derive the N th-order Darboux transformtion and N th-order vector rational and semi-rational rogue wave solutions, where N is a positive integer. With such solutions, we present three types of the second-order rogue waves with the triangle structure: the one with each component containing three four-petalled rogue waves, the one with each component containing three eye-shaped rogue waves, and the other with one component containing three anti-eye-shaped rogue waves and the other component containing three eye-shaped rogue waves. We exhibit the thirdorder vector rogue waves with the merged, triangle and pentagon structures in each component. Moreover, we show the first- and second-order vector semi-rational rogue waves which display the coexistence of the rogue waves and the breathers.
Keywords: Non-Kerr medium; Rational rogue waves; Semi-rational rogue waves; Coupled cubicquintic nonlinear Schr¨odinger system; Darboux transformation ∗
Corresponding author, with e-mail address as tian− [email protected]
1
1. Introduction Non-Kerr media have been reported to possess certain applications in photonic lattices and optical fibers [1–3]. For instance, Refs. [4–6] have investigated the propagation of the Gauss beams, laser beams and pulses in the non-Kerr media. Scalar nonlinear Schr¨odinger (NLS)-type equations have been called to play a certain role in describing such fields as nonlinear optics, oceanography, plasma physics, hydrodynamics, molecular biology, Bose-Einstein condensation and finance [7–13]. NLS-type equations have been regarded as the models to describe the localized waves which include the solitons, breathers and rogue waves [14–17]. Instabilities in the optics have been found to be described by the optical rogue waves which can be modelled by the NLS equations [18]. To model more dynamic patterns of the localized wave solutions, people have studied the coupled NLS-type systems admitting the solitons, breathers, eye-shaped rogue waves, anti-eye-shaped rogue waves, four-petalled rogue waves and semi-rational rogue waves1 [19–22]. For the high-bit-rate optical fiber communication, people have increased the intensity of the incident power to produce the shorter pulses, and correspondingly investigated the NLStype equations with certain higher-order nonlinear terms [23, 24]. Ref. [25] has reported the transitions between symmetric and asymmetric solitons for a coupled NLS equations with cubicquintic nonlinearity. For the effects of quintic nonlinearity on the ultrashort optical pulse (less than 100-fs optical pulse) propagation in a non-Kerr medium, people have investigated the following coupled cubic-quintic NLS system [26–30]: 2 iq1z + q1tt + 2 |q1 |2 + |q2 |2 q1 + ρ1 |q1 |2 + ρ2 |q2 |2 q1 −2i ρ1 |q1 |2 + ρ2 |q2 |2 q1 t + 2i (ρ1 q1∗ q1t + ρ2 q2∗ q2t ) q1 = 0, (1a) 2 2 iq2z + q2tt + 2 |q1 | + |q2 |2 q2 + ρ1 |q1 |2 + ρ2 |q2 |2 q2 −2i ρ1 |q1 |2 + ρ2 |q2 |2 q2 t + 2i (ρ1 q1∗ q1t + ρ2 q2∗ q2t ) q2 = 0, (1b) where the two components q1 = q1 (z, t) and q2 = q2 (z, t) represent that two electromagnetic fields, z denotes the scaled distance along the direction of the propagation, t is the local time, “ * ” denotes the complex conjugation, and ρ1 and ρ2 are both the real constants. Bilinear forms, soliton solutions, inelastic and head-on interactions of the solitons for System (1) have been shown [26]. Darboux transformation (DT) and multi-soliton solutions for System (1) have been discussed [27]. Generalized DT and rogue wave solutions for System (1) have been derived [28]. DT and rogue waves for the multi-component forms of System (1) have been obtained [29]. Localized waves on the plane wave background for System (1) have been constructed by virtue of the generalized DT [30]. Different from the DTs in Refs. [28, 30], in this paper, we will construct another DT for System (1) through the loop method [31], and correspondingly derive the N th-order vector 1
The semi-rational rogue waves can describe the coexistence of the rogue waves and solitons/breathers
2
rational and semi-rational rogue wave solutions for System (1). N th-order DT for System (1) will be constructed in Section 2. The N th-order vector rational and semi-rational rogue wave solutions for System (1) will be deduced in Section 3. In Section 4, vector rational and semirational rogue waves will be discussed. Our conclusions will be presented in Section 5. 2. N th-order DT associated with System (1) According to the standard procedure of Darboux matrix method, Ref. [28] has constructed a generalized DT for System (1), which could only derive the eye-shaped rogue wave solutions. In this section, through the loop method [31], we will construct the DT with more free parameters for System (1), which can generate the eye-shaped, anti-eye-shaped and four-petalled rogue wave solutions, and the semi-rational solutions consisting of rational and exponential expression. Eqs. (1) have been found to possess a Lax pair [28]: Ψt = M Ψ,
Ψz = LΨ,
(2)
where Ψ is a 3 × 1 vector function with respect to z, t and the complex eigenvalue λ, and M and L are the matrices, expressed as ⎛ ⎞ 0 q 1 q2 1 ⎜ ⎟ M = iλ − i ρ1 |q1 |2 + ρ2 |q2 |2 σ + ⎝ −q1∗ 0 0 ⎠ , (3a) 2 −q2∗ 0 0 ⎛ ⎞ ω + i (|q1 |2 + |q2 |2 ) s1 s2 ⎜ ∗ ⎟ L = ⎝ iq1t (3b) − 2λq1∗ − q1∗ (ρ1 |q1 |2 + ρ2 |q2 |2 ) −ω − i|q1 |2 −iq2 q1∗ ⎠ , ∗ ∗ ∗ 2 2 ∗ 2 iq2t − 2λq2 − q2 (ρ1 |q1 | + ρ2 |q2 | ) −iq1 q2 −ω − i|q2 | with σ = diag(−1, 1, 1), s1 = q1 ρ1 |q1 |2 + ρ2 |q2 |2 + 2λq1 + iq1t , s2 = q2 ρ1 |q1 |2 + ρ2 |q2 |2 + 2λq2 + iq2t , 2 1 1 ∗ ∗ ω = i ρ1 |q1 |2 + ρ2 |q2 |2 + ρ1 (q1 q1t − q1∗ q1t ) + ρ2 (q2 q2t − q2∗ q2t ) − 2iλ2 . 2 2 Our calculation indicates that System (1) can be generated through the compatibility condition Mz − Lt + M L − LM = 0. Observing Lax Pair (2), we see that Lax Pair (2) is not an AblowitzKaup-Newell-Segur (AKNS) system [32]. In what follows, we will convert Lax Pair (2) into an AKNS system. We choose the following gauge transformations: ⎛ 1 ⎞ 2 2 e 2 i (ρ1 |q1 | +ρ2 |q2 | ) dt 0 0 1 2 2 ⎜ ⎟ Ψ=⎝ (4a) 0 e− 2 i (ρ1 |q1 | +ρ2 |q2 | ) dt 0 ⎠ Φ, − 12 i (ρ1 |q1 |2 +ρ2 |q2 |2 ) dt 0 0 e q1 = u1 ei
(ρ1 |u1 |2 +ρ2 |u2 |2 ) dt
,
q2 = u2 ei
(ρ1 |u1 |2 +ρ2 |u2 |2 ) dt
3
,
(4b)
such that we can transform Lax Pair (2) into the following Lax pair: Φt = U Φ, with
Φz = V Φ,
(5)
⎛
⎞ 0 u1 u2 ⎜ ⎟ U = iλσ + Qσ, Q = ⎝ u∗1 0 0 ⎠ , u∗2 0 0 V = 2iσλ2 − 2σQλ + iQt − iQ2 + i |u1 |2 + |u2 |2 (I − σ) 1 ∗ ∗ ∗ ∗ 2 2 + (ρ1 u1 u1t + ρ2 u2 u2t ) − (ρ1 u1 u1t + ρ2 u2 u2t ) + i ρ1 |u1 | + ρ2 |u2 | z dt σ, 2
where Φ is a 3 × 1 vector function with respect to z, t and λ, I is the identity matrix, and u1 = u1 (z, t) and u2 = u2 (z, t) meet the following coupled system: iu1z + u1tt + 2 |u1 |2 + |u2 |2 u1 − i (ρ1 u1 u∗1t + ρ2 u2 u∗2t ) u1 ∗ ∗ ρ1 |u1 |2 + ρ2 |u2 |2 z dt = 0, +i (ρ1 u1 u1t + ρ2 u2 u2t ) u1 − u1 (6a) iu2z + u2tt + 2 |u1 |2 + |u2 |2 u2 − i (ρ1 u1 u∗1t + ρ2 u2 u∗2t ) u2 ∗ ∗ ρ1 |u1 |2 + ρ2 |u2 |2 z dt = 0. +i (ρ1 u1 u1t + ρ2 u2 u2t ) u2 − u2 (6b) From Transformations (4b), we find that System (6) are equivalent to System (1), and the module of uα (α = 1, 2) is equal to that of qα . After calculating, we obtain that the reduction condition of Lax Pair (5) is U † (λ) = −U (λ∗ ) and V † (λ) = −V (λ∗ ), where the sign “ † ” represents the Hermitian adjoint. Similar to the procedure mentioned in Refs. [33–35], via the loop group method [31], the N th-order DT matrix associated with Lax Pair (5) can be written as D[N ] = I − YM−1 (λI − S)−1 Y † ,
(7)
where [N ] (N = 1, 2, 3, · · · ) stands for the N th-iteration, Y = (y1 , y2 , · · · , yN ), M = (Mmj )N ×N † (m, j = 1, 2, · · · , N ), Mmj = ym yj /(λj − λ∗m ), yj ≡ y(z, t, λj ), y(z, t, λ) = v(z, t, λ)Φ(z, t, λ), v(z, t, λ) is a nonzero complex function, Φ(z, t, λj )’s are N distinct vector solutions of Lax Pair (5), S = diag(λ∗1 , λ∗2 , · · · , λ∗N ), λj ’s are N values of λ, and Yk (k = 1, 2, 3) is the k-th row of Y, and the superscript “ − 1” denotes the inverse of a matrix. Moreover, the N th-order DT associated with Lax Pair (5) can be deduced as M Y2† det Y1 0 u1 [N ] = u1 [0] + 2i , (8a) det M M Y3† det Y1 0 u2 [N ] = u2 [0] + 2i , (8b) det M 4
where u1 [0] and u2 [0] are the seed solutions for System (6). 3. Vector rational and semi-rational rogue wave solutions for System (1) Based on N th-Order DT (8), the vector rational and semi-rational rogue wave solutions for System (1) will be constructed. In what follows, we set the seed solutions for System (6) in the forms of u1 [0] = a1 ei(b1 t+c1 z) ,
u2 [0] = a2 ei(b2 t+c2 z) ,
(9)
where cα = 2(a21 + a22 ) − b2α − 2(a21 b1 ρ1 + a22 b2 ρ2 ), aα ’s and bα ’s are the constants. According to Seed Solutions (9) and Lax Pair (5), the solutions of Lax Pair (5) are derived as Φ(z, t, λ) = GHKLe−iβz ,
β = a21 b1 ρ1 + a22 b2 ρ2 − 2a21 − 2a22 − λ2 ,
(10)
with K = diag(e−if (x1 −λ) , e−if (x2 −λ) , e−if (x3 −λ) ), ⎛ ⎞ 1 0 0 ⎜ ⎟ G = ⎝0 a1 ei[b1 t+c1 z] 0 ⎠, i[b2 t+c2 z] 0 0 a2 e
L = (l1 , l2 , l3 )T , ⎛ ⎞ 1 1 1 ⎜ i i i ⎟ H = ⎝− x1 +b − x2 +b − x3 +b ⎠, 1 1 1 i i i − x1 +b2 − x2 +b2 − x3 +b2
where f (ξ) = ξ[t + z(2λ + ξ)], the sign T denotes the transpose for a vector/matrix, l1 , l2 and l3 are three nonzero constants, and x1 , x2 and x3 are three different roots of the cubic equation for x as follows: [(x − 2λ)(x + b1 ) − a21 ](x + b2 ) − a22 (x + b1 ) = 0.
(11)
Choosing v(z, t, λ) = eiβz , we have y(z, t, λ) = GHKL. 3.1 Vector rational rogue wave solutions for System (1) To construct the vector rational rogue wave solutions for System (1) based on N th-Order DT (8), the perturbation expansion method [36] and Transformation (4b), we presume that Cubic Eq. (11) admits a pair of double roots x0 when λ = λ1 . We set λ = λ1 (1 + r2 ), where r is a small parameter. Inserting λ = λ1 (1 + r2 ) into Cubic Eq. (11), we derive that Cubic Eq. (11) possesses three roots x1 , x2 and x3 , which meet that lim x1 = lim x2 = x0 . Taking r→0 r→0 ⎛ ⎞ 1 1 1 1 1 0 ⎜ i i ⎟ , H = ⎝− x1 +b W= , R= − x2 +b ⎠, 1 1 1 1 −1 0 i i r − x1 +b − x2 +b 2 2 J = diag((x1 + b1 )(x1 + b2 ), (x2 + b1 )(x2 + b2 )), 5
K = diag(e−if (x1 −λ) , e−if (x2 −λ) ),
and L = (l1 , l2 )T with lα = form of
N
j=1 lαj r
2(j−1)
, we see that HJ KWRL can be expanded in the
HJ KWRL = 1 ∂ j (HJ KWRL) with gj = j! ∂λj
∞
gj (z, t)r2j ,
j=0
λ=λ1
(j = 0, 1, 2, · · · ). Thus, we obtain the N th-order vector rational
rogue wave solutions for System (6) as det(Γ − 2iΞ†2 Ξ1 ) u1 [N ] = u1 [0], det(Γ)
(12a)
det(Γ − 2iΞ†3 Ξ1 ) u2 [0], det(Γ)
(12b)
u2 [N ] =
where Ξ(z, t) = (g0 , g1 , · · · , gN −1 ), Ξτ is the τ -th row of Ξ(z, t) (τ = 1, 2, 3) and Γ = (Γmj )N ×N with ⎛ ⎞ 1 0 0 min(j−1,k) m+j−2 k+1 1 ⎜ ⎟ † Ckl λ∗k−l (−λ1 )l gm−1−k+l i − Γmj = ⎝0 a21 0 ⎠ gj−1−l , 1 2Im(λ1 ) k=0 l=max(0,k−m+1) 0 0 a22 and Im(∗) being the image part of the complex number. Based on Transformations (4b), the N th-order vector rational rogue wave solutions for System (1) can be derived as q1 [N ] = u1 [N ]ei q2 [N ] = u2 [N ]ei
(ρ1 |u1 [N ]|2 +ρ2 |u2 [N ]|2 ) dt (ρ1 |u1
[N ]|2 +ρ
2 |u2
[N ]|2 ) dt
,
(13a)
.
(13b)
3.2 Vector semi-rational rogue wave solutions for System (1) In Section 3.1, we only employ a pair of double roots of Cubic Eq. (11), and do not consider the third root of Cubic Eq. (11) when we deduce the vector rational rogue wave solutions, i.e., Solutions (13). In what follows, we will employ the third root x3 to construct the vector semi-rational rogue wave solutions for Eqs. (1). We suppose that Cubic Eq. (11) possesses a pair of double roots when λ = λ1 , which is represented by x0 . Inserting λ = λ1 (1 + r2 ) into Cubic Eq. (11), we find that Cubic Eq. (11) yields three roots x1 , x2 and x3 , which meet that x1 → x0 and x2 → x0 as r → 0. We choose ⎞ ⎞ ⎛ ⎛ 1 1 0 1 0 0 ⎟ ⎜ ⎟ ⎜ W = ⎝ 1 −1 0 ⎠ , R = ⎝ 0 1r 0 ⎠ , 0 0 1 0 0 1 ⎛ ⎞ 0 0 (x1 + b1 )(x1 + b2 ) ⎜ ⎟ J=⎝ 0 (x2 + b1 )(x2 + b2 ) 0 ⎠, 0 0 (x3 + b1 )(x3 + b2 ) 6
and L = (l1 , l2 , l3 )T with lτ =
N
j=1 lτ j r
2(j−1)
such that we can rewrite HJKWRL as ∞
HJKWRL =
hj (z, t)r2j ,
j=0
1 ∂ j (HJKWRL) with hj = j! ∂λj
λ=λ1
(j = 0, 1, 2, · · · ). Then, the N th-order vector semi-rational rogue
wave solutions for Eqs. (6) are represented as det(Λ − 2iΘ†2 Θ1 ) u1 [N ] = u1 [0], det(Λ)
(14a)
det(Λ − 2iΘ†3 Θ1 ) u2 [0], det(Λ)
(14b)
u2 [N ] =
where Θ(z, t) = (h0 , h1 , · · · , hN −1 ), Θτ denotes the τ -th row of Λ(z, t) and ⎛ 1 min(j−1,k) m+j−2 k+1 1 ⎜ l ∗k−l l † i − Ck λ1 (−λ1 ) hm−1−k+l ⎝0 Λmj = 2Im(λ1 ) k=0 l=max(0,k−m+1) 0
Λ = (Λmj )N ×N with ⎞ 0 0 ⎟ a21 0 ⎠ hj−1−l . 0 a22
Based on Transformations (4b), we can deduce the N th-order vector semi-rational rogue wave solutions for Eqs. (1) in the forms of q1 [N ] = u1 [N ]ei q2 [N ] = u2 [N ]ei
(ρ1 |u1 [N ]|2 +ρ2 |u2 [N ]|2 ) dt (ρ1 |u1
[N ]|2 +ρ
2 |u2
[N ]|2 ) dt
,
(15a)
.
(15b)
When l31 = 0, Solutions (15) are the rational expressions which can only generate vector rogue waves; When l31 = 0, Solutions (15) are the semi-rational expressions which can describe the coexistence of the rogue waves and breathers. 4. Vector rational and semi-rational rogue waves for System (1) 4.1 Higher-order vector rational rogue waves for System (1) Based on Solutions (13) with N = 2, we will graphically discuss the characteristics of the second-order vector rational rogue waves with the real constants ρ1 and ρ2 through Figs. 1. As seen in Figs. 1(a1 -a2 ), the second-order rational rogue waves with the merged structure take place in the q1 and q2 components. We only increase the values of ρ1 and ρ2 respectively, the second-order rogue waves in the q1 and q2 components keep unchanged, which means that ρ1 and ρ2 have no effects on the second-order vector rational rogue wave, as shown in Figs. 1(b1 -b2 ) and 1(c1 -c2 ). 7
(a1 )
(b1 )
(c1 )
(a2 )
(b2 )
(c2 )
Figs. 1 The second-order vector rational rogue waves via Solutions (13) with N = 2, a1 = 1, a2 = 1, b1 = 25 , b2 = − 25 , λ1 = ρ1 =
1 20 ,
√ 3 6 5 i, l11
= 0, l12 = 0, l21 = 2, l22 = 0, (a1 -a2 ) ρ1 =
1 20 ,
ρ2 =
1 70 ;
(b1 -b2 ) ρ1 = 1, ρ2 =
1 70 ;
(c1 -c2 )
ρ2 = 1. √
Taking N = 2, a1 = 1, a2 = 1, b1 = 25 , b2 = − 25 , λ1 = 3 5 6 i and l12 = 101 in Solutions (13), we observe that the merged structure splits up into three four-petalled rogue waves which form a triangle structure in each component, as seen in Figs. 2(a1 -a2 ). When b1 = b2 = 0, √ λ1 = 2i and l12 = 1001, three eye-shaped rogue waves appear in each component and form the triangle structure, can be seen in Figs. 2(b1 -b2 ). When b1 = 1, b2 = −1, λ1 = √ which √ 1 6 3 − 9 − i 6 3 + 9 and l12 = 1001, the triangle structure consisting of the three 4 anti-eye-shaped rogue waves occur in the q1 component, while the triangle structure consisting of the three eye-shaped rogue waves appear in the q2 component, as shown in Figs. 2(c1 -c2 ).
(a1 )
(b1 )
8
(c1 )
(a2 )
(b2 )
(c2 )
Figs. 2 The second-order vector rational rogue waves via Solutions (13) with N = 2, ρ1 = √
b2 = − 25 , λ1 = 3 5 6 i, l12 = 101; (b1 -b2 ) b1 = 0, √ √ λ1 = 14 6 3 − 9 − i 6 3 + 9 , l12 = 1001.
2 5,
a2 = 1, l11 = 0, l21 = 2, l22 = 0, (a1 -a2 ) b1 = √ λ1 = 2i, l12 = 1001; (c1 -c2 ) b1 = 1, b2 = −1,
1 20 ,
ρ2 =
1 70 ,
a1 = 1,
b2 = 0,
√
When N = 3, a1 = 1, a2 = 1, b1 = 25 , b2 = − 25 and λ1 = 3 5 6 i, Figs. 3 present the third-order vector rogue waves with each component possessing six four-petaled rogue waves. Choosing l12 = 0 and l13 = 0, we observe that the six four-petaled rogue waves form the merged structure in each component, as shown in Figs. 3(a1 -a2 ). When l12 = 101 and l13 = 0, the merged structure turns into the triangle structure in each component, as seen in Figs. 3(b1 -b2 ): Three four-petaled rogue waves form the triangle structure, while the other three four-petaled rogue waves are located at the three sides of the triangle structure. Choosing l12 = 0 and l13 = 30001, we observe that five four-petaled rogue waves constitute the pentagon structure surrounding the other one four-petaled rogue wave in each component, as seen in Figs. 3(c1 -c2 ).
(a1 )
(b1 )
(c1 )
(a2 )
(b2 )
(c2 )
9
Figs. 3 The third-order vector rational rogue waves via Solutions (13) with N = 3, ρ1 = a2 = 1, b1 =
2 5,
b2 =
− 25 ,
√
λ1 =
3 6 5 i, l11
1 20 ,
ρ2 =
1 70 ,
a1 = 1,
= 0, l21 = 2, l22 = 0, l23 = 0, (a1 -a2 ) l12 = 0, l13 = 0; (b1 -b2 )
l12 = 101, l13 = 0; (c1 -c2 ) l12 = 0, l13 = 30001.
4.2 The first- and second-order vector semi-rational rogue waves In what follows, we will graphically analyze the characteristics of the first- and second-order semi-rational rogue waves on the basis of Solutions (15) via Figs. 4-5. √ 3 6 2 2 Choosing a1 = 1, a2 = 1, b1 = 5 , b2 = − 5 and λ1 = 5 i in Solutions (15), we can acquire the N th-order vector semi-rational rogue waves which describe that the N th-order four-petalled rogue waves coexist with the N eye-shaped breathers in each component, as shown in Figs. 4. Figs. 4(a1 -a2 ) present the first-order vector semi-rational rogue wave which demonstrates that one four-petalled rogue wave coexists with one eye-shaped breather in each component. Figs. 4(b1 -b2 ) display the second-order vector semi-rational rogue wave which describes the second-order rogue wave with the merged structure coexists with two eye-shaped breathers in each component. When l12 = 101, Figs. 4(c1 -c2 ) show that the triangle structure consisting of three four-petalled rogue waves coexists with two eye-shaped breathers in each component. Moreover, increasing or decreasing the value of l31 , we find that the rogue waves and the breathers merge together or split up in in each component, the corresponding figures of which are omitted.
(a1 )
(b1 )
(c1 )
(a2 )
(b2 )
(c2 )
10
Figs. 4 Vector semi-rational rogue waves via Solutions (15) with ρ1 = b2 = − 25 , λ1 =
√
3 6 5 i, −10
l22 = 0, l31 = 10
1 20 ,
ρ2 =
1 70 ,
a1 = 1, a2 = 1, b1 = 25 ,
(a1 -a2 ) N = 1, l11 = 0, l21 = 2, l31 = 10−5 ; (b1 -b2 ) N = 2, l11 = 0, l12 = 0, l21 = 2,
, l32 = 0; (c1 -c2 ) N = 2, l11 = 0, l12 = 101, l21 = 2, l22 = 0, l31 = 10−10 , l32 = 0.
√ √ When a1 = 1, a2 = 1, b1 = 1, b2 = −1 and λ1 = 6 3 − 9 − i 6 3 + 9 , Solutions (15) can generate the N th-order vector semi-rational rogue waves: The N th-order antieye-shaped rogue wave coexists with N eye-shaped breathers in the q1 component, while the N th-order eye-shaped rogue wave coexists with N anti-eye-shaped breathers in the q2 component, as shown in Figs. 5. For instance, Figs. 5(a1 -a2 ) describe that one anti-eye-shaped rogue wave coexists with one eye-shaped breather in the q1 component, while one eye-shaped rogue wave coexists with one anti-eye-shaped breather in the q2 component. As seen in Figs. 5(b1 b2 ), the coexistence of the second-order anti-eye-shaped rogue wave with the merged structure and two parallel eye-shaped breathers takes place in the q1 component, while the second-order eye-shaped rogue wave with the merged structure coexists with two parallel anti-eye-shaped breathers in the q2 component. Increasing the value of l12 , we observe that the merged structures in Figs. 5(b1 -b2 ) turn into the triangle structures in Figs. 5(c1 -c2 ). 1 4
(a1 )
(b1 )
(c1 )
(a2 )
(b2 )
(c2 )
1 1 , ρ2 = 70 , a1 = 1, a2 = 1, b1 = 1, Figs. 5 Vector semi-rational rogue waves via Solutions (15) with ρ1 = 20 √ √ 6 3 − 9 − i 6 3 + 9 , (a1 -a2 ) N = 1, l11 = 0, l21 = 2, l31 = 10−5 ; (b1 -b2 ) N = 2, b2 = −1, λ1 = 14
l11 = 0, l12 = 0, l21 = 2, l22 = 0, l31 = 10−10 , l32 = 0; (c1 -c2 ) N = 2, l11 = 0, l12 = 101, l21 = 2, l22 = 0, l31 = 10−10 , l32 = 0.
11
5. Conclusions Non-Kerr media have been reported to possess certain applications in photonic lattices and optical fibers. In this paper, we have studied the vector multi-rational and semi-rational rogue waves in a non-Kerr medium, through the coupled cubic-quintic nonlinear Schr¨odinger system, i.e., System (1), which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium. Employing Transformations (4), we have converted System (1) into System (6) with Lax Pair (5). For q1 and q2 , the electromagnetic fields in the two cores of an optical waveguide, we have derived N th-Order DT (8) associated with Lax Pair (5), the N th-order vector rational and semi-rational rogue wave solutions, i.e., Solutions (13) and Solutions (15). According to Solutions (13), Figs. 1 have implied that ρ1 and ρ2 , the constant coefficients in System (1), have no influence on the second-order vector rational rogue waves; Figs. 2 have presented three types of the second-order rogue waves with the triangle structure which consists of three eye-shaped rogue waves, or three four-petalled rogue waves, or three anti-eye-shaped rogue waves; Figs. 3 have exhibited the third-order vector rogue waves with the merged, triangle and pentagon structures consisting of six four-petalled rogue waves in each component. Based on Solutions (15), Figs. 4 have shown the first-order/second-order vector semi-rational rogue waves which describe the coexistence of first-order/second-order four-petalled rogue waves and the one/two eye-shaped breathers in the q1 and q2 components. Figs. 5 have displayed the first-order/second-order vector semi-rational rogue waves: The first-order/second-order antieye-shaped rogue wave coexists with the one/two eye-shaped breathers in the q1 component, while the first-order/second-order eye-shaped rogue wave coexists with the one/two anti-eyeshaped breathers in the q2 component. Acknowledgments This work has been supported by the BUPT Excellent Ph.D. Students Foundation under No. CX2019201, by the National Natural Science Foundation of China under Grant Nos. 11772017, 11805020 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.
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14
Zhong Du, Bo Tian, Qi-Xing Qu, Xiao-Yu Wu, Xue-Hui Zhao
PII:
S0168-9274(20)30037-4
DOI:
https://doi.org/10.1016/j.apnum.2020.02.002
Reference:
APNUM 3767
To appear in:
Applied Numerical Mathematics
Received date:
16 July 2019
Revised date:
2 November 2019
Accepted date:
5 February 2020
Please cite this article as: Z. Du et al., Vector rational and semi-rational rogue waves for the coupled cubic-quintic nonlinear Schrödinger system in a non-Kerr medium, Appl. Numer. Math. (2020), doi: https://doi.org/10.1016/j.apnum.2020.02.002.
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Vector rational and semi-rational rogue waves for the coupled cubic-quintic nonlinear Schr¨odinger system in a non-Kerr medium Zhong Du1 , Bo Tian1∗, Qi-Xing Qu2 , Xiao-Yu Wu1 , Xue-Hui Zhao1 1. State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China 2. School of Information, University of International Business and Economics, Beijing 100029, China
Abstract Non-Kerr media possess certain applications in photonic lattices and optical fibers. Studied in this paper are the vector rational and semi-rational rogue waves in a nonKerr medium, through the coupled cubic-quintic nonlinear Schr¨odinger system, which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium. Applying the gauge transformations, we derive the N th-order Darboux transformtion and N th-order vector rational and semi-rational rogue wave solutions, where N is a positive integer. With such solutions, we present three types of the second-order rogue waves with the triangle structure: the one with each component containing three four-petalled rogue waves, the one with each component containing three eye-shaped rogue waves, and the other with one component containing three anti-eye-shaped rogue waves and the other component containing three eye-shaped rogue waves. We exhibit the thirdorder vector rogue waves with the merged, triangle and pentagon structures in each component. Moreover, we show the first- and second-order vector semi-rational rogue waves which display the coexistence of the rogue waves and the breathers.
Keywords: Non-Kerr medium; Rational rogue waves; Semi-rational rogue waves; Coupled cubicquintic nonlinear Schr¨odinger system; Darboux transformation ∗
Corresponding author, with e-mail address as tian− [email protected]
1
1. Introduction Non-Kerr media have been reported to possess certain applications in photonic lattices and optical fibers [1–3]. For instance, Refs. [4–6] have investigated the propagation of the Gauss beams, laser beams and pulses in the non-Kerr media. Scalar nonlinear Schr¨odinger (NLS)-type equations have been called to play a certain role in describing such fields as nonlinear optics, oceanography, plasma physics, hydrodynamics, molecular biology, Bose-Einstein condensation and finance [7–13]. NLS-type equations have been regarded as the models to describe the localized waves which include the solitons, breathers and rogue waves [14–17]. Instabilities in the optics have been found to be described by the optical rogue waves which can be modelled by the NLS equations [18]. To model more dynamic patterns of the localized wave solutions, people have studied the coupled NLS-type systems admitting the solitons, breathers, eye-shaped rogue waves, anti-eye-shaped rogue waves, four-petalled rogue waves and semi-rational rogue waves1 [19–22]. For the high-bit-rate optical fiber communication, people have increased the intensity of the incident power to produce the shorter pulses, and correspondingly investigated the NLStype equations with certain higher-order nonlinear terms [23, 24]. Ref. [25] has reported the transitions between symmetric and asymmetric solitons for a coupled NLS equations with cubicquintic nonlinearity. For the effects of quintic nonlinearity on the ultrashort optical pulse (less than 100-fs optical pulse) propagation in a non-Kerr medium, people have investigated the following coupled cubic-quintic NLS system [26–30]: 2 iq1z + q1tt + 2 |q1 |2 + |q2 |2 q1 + ρ1 |q1 |2 + ρ2 |q2 |2 q1 −2i ρ1 |q1 |2 + ρ2 |q2 |2 q1 t + 2i (ρ1 q1∗ q1t + ρ2 q2∗ q2t ) q1 = 0, (1a) 2 2 iq2z + q2tt + 2 |q1 | + |q2 |2 q2 + ρ1 |q1 |2 + ρ2 |q2 |2 q2 −2i ρ1 |q1 |2 + ρ2 |q2 |2 q2 t + 2i (ρ1 q1∗ q1t + ρ2 q2∗ q2t ) q2 = 0, (1b) where the two components q1 = q1 (z, t) and q2 = q2 (z, t) represent that two electromagnetic fields, z denotes the scaled distance along the direction of the propagation, t is the local time, “ * ” denotes the complex conjugation, and ρ1 and ρ2 are both the real constants. Bilinear forms, soliton solutions, inelastic and head-on interactions of the solitons for System (1) have been shown [26]. Darboux transformation (DT) and multi-soliton solutions for System (1) have been discussed [27]. Generalized DT and rogue wave solutions for System (1) have been derived [28]. DT and rogue waves for the multi-component forms of System (1) have been obtained [29]. Localized waves on the plane wave background for System (1) have been constructed by virtue of the generalized DT [30]. Different from the DTs in Refs. [28, 30], in this paper, we will construct another DT for System (1) through the loop method [31], and correspondingly derive the N th-order vector 1
The semi-rational rogue waves can describe the coexistence of the rogue waves and solitons/breathers
2
rational and semi-rational rogue wave solutions for System (1). N th-order DT for System (1) will be constructed in Section 2. The N th-order vector rational and semi-rational rogue wave solutions for System (1) will be deduced in Section 3. In Section 4, vector rational and semirational rogue waves will be discussed. Our conclusions will be presented in Section 5. 2. N th-order DT associated with System (1) According to the standard procedure of Darboux matrix method, Ref. [28] has constructed a generalized DT for System (1), which could only derive the eye-shaped rogue wave solutions. In this section, through the loop method [31], we will construct the DT with more free parameters for System (1), which can generate the eye-shaped, anti-eye-shaped and four-petalled rogue wave solutions, and the semi-rational solutions consisting of rational and exponential expression. Eqs. (1) have been found to possess a Lax pair [28]: Ψt = M Ψ,
Ψz = LΨ,
(2)
where Ψ is a 3 × 1 vector function with respect to z, t and the complex eigenvalue λ, and M and L are the matrices, expressed as ⎛ ⎞ 0 q 1 q2 1 ⎜ ⎟ M = iλ − i ρ1 |q1 |2 + ρ2 |q2 |2 σ + ⎝ −q1∗ 0 0 ⎠ , (3a) 2 −q2∗ 0 0 ⎛ ⎞ ω + i (|q1 |2 + |q2 |2 ) s1 s2 ⎜ ∗ ⎟ L = ⎝ iq1t (3b) − 2λq1∗ − q1∗ (ρ1 |q1 |2 + ρ2 |q2 |2 ) −ω − i|q1 |2 −iq2 q1∗ ⎠ , ∗ ∗ ∗ 2 2 ∗ 2 iq2t − 2λq2 − q2 (ρ1 |q1 | + ρ2 |q2 | ) −iq1 q2 −ω − i|q2 | with σ = diag(−1, 1, 1), s1 = q1 ρ1 |q1 |2 + ρ2 |q2 |2 + 2λq1 + iq1t , s2 = q2 ρ1 |q1 |2 + ρ2 |q2 |2 + 2λq2 + iq2t , 2 1 1 ∗ ∗ ω = i ρ1 |q1 |2 + ρ2 |q2 |2 + ρ1 (q1 q1t − q1∗ q1t ) + ρ2 (q2 q2t − q2∗ q2t ) − 2iλ2 . 2 2 Our calculation indicates that System (1) can be generated through the compatibility condition Mz − Lt + M L − LM = 0. Observing Lax Pair (2), we see that Lax Pair (2) is not an AblowitzKaup-Newell-Segur (AKNS) system [32]. In what follows, we will convert Lax Pair (2) into an AKNS system. We choose the following gauge transformations: ⎛ 1 ⎞ 2 2 e 2 i (ρ1 |q1 | +ρ2 |q2 | ) dt 0 0 1 2 2 ⎜ ⎟ Ψ=⎝ (4a) 0 e− 2 i (ρ1 |q1 | +ρ2 |q2 | ) dt 0 ⎠ Φ, − 12 i (ρ1 |q1 |2 +ρ2 |q2 |2 ) dt 0 0 e q1 = u1 ei
(ρ1 |u1 |2 +ρ2 |u2 |2 ) dt
,
q2 = u2 ei
(ρ1 |u1 |2 +ρ2 |u2 |2 ) dt
3
,
(4b)
such that we can transform Lax Pair (2) into the following Lax pair: Φt = U Φ, with
Φz = V Φ,
(5)
⎛
⎞ 0 u1 u2 ⎜ ⎟ U = iλσ + Qσ, Q = ⎝ u∗1 0 0 ⎠ , u∗2 0 0 V = 2iσλ2 − 2σQλ + iQt − iQ2 + i |u1 |2 + |u2 |2 (I − σ) 1 ∗ ∗ ∗ ∗ 2 2 + (ρ1 u1 u1t + ρ2 u2 u2t ) − (ρ1 u1 u1t + ρ2 u2 u2t ) + i ρ1 |u1 | + ρ2 |u2 | z dt σ, 2
where Φ is a 3 × 1 vector function with respect to z, t and λ, I is the identity matrix, and u1 = u1 (z, t) and u2 = u2 (z, t) meet the following coupled system: iu1z + u1tt + 2 |u1 |2 + |u2 |2 u1 − i (ρ1 u1 u∗1t + ρ2 u2 u∗2t ) u1 ∗ ∗ ρ1 |u1 |2 + ρ2 |u2 |2 z dt = 0, +i (ρ1 u1 u1t + ρ2 u2 u2t ) u1 − u1 (6a) iu2z + u2tt + 2 |u1 |2 + |u2 |2 u2 − i (ρ1 u1 u∗1t + ρ2 u2 u∗2t ) u2 ∗ ∗ ρ1 |u1 |2 + ρ2 |u2 |2 z dt = 0. +i (ρ1 u1 u1t + ρ2 u2 u2t ) u2 − u2 (6b) From Transformations (4b), we find that System (6) are equivalent to System (1), and the module of uα (α = 1, 2) is equal to that of qα . After calculating, we obtain that the reduction condition of Lax Pair (5) is U † (λ) = −U (λ∗ ) and V † (λ) = −V (λ∗ ), where the sign “ † ” represents the Hermitian adjoint. Similar to the procedure mentioned in Refs. [33–35], via the loop group method [31], the N th-order DT matrix associated with Lax Pair (5) can be written as D[N ] = I − YM−1 (λI − S)−1 Y † ,
(7)
where [N ] (N = 1, 2, 3, · · · ) stands for the N th-iteration, Y = (y1 , y2 , · · · , yN ), M = (Mmj )N ×N † (m, j = 1, 2, · · · , N ), Mmj = ym yj /(λj − λ∗m ), yj ≡ y(z, t, λj ), y(z, t, λ) = v(z, t, λ)Φ(z, t, λ), v(z, t, λ) is a nonzero complex function, Φ(z, t, λj )’s are N distinct vector solutions of Lax Pair (5), S = diag(λ∗1 , λ∗2 , · · · , λ∗N ), λj ’s are N values of λ, and Yk (k = 1, 2, 3) is the k-th row of Y, and the superscript “ − 1” denotes the inverse of a matrix. Moreover, the N th-order DT associated with Lax Pair (5) can be deduced as M Y2† det Y1 0 u1 [N ] = u1 [0] + 2i , (8a) det M M Y3† det Y1 0 u2 [N ] = u2 [0] + 2i , (8b) det M 4
where u1 [0] and u2 [0] are the seed solutions for System (6). 3. Vector rational and semi-rational rogue wave solutions for System (1) Based on N th-Order DT (8), the vector rational and semi-rational rogue wave solutions for System (1) will be constructed. In what follows, we set the seed solutions for System (6) in the forms of u1 [0] = a1 ei(b1 t+c1 z) ,
u2 [0] = a2 ei(b2 t+c2 z) ,
(9)
where cα = 2(a21 + a22 ) − b2α − 2(a21 b1 ρ1 + a22 b2 ρ2 ), aα ’s and bα ’s are the constants. According to Seed Solutions (9) and Lax Pair (5), the solutions of Lax Pair (5) are derived as Φ(z, t, λ) = GHKLe−iβz ,
β = a21 b1 ρ1 + a22 b2 ρ2 − 2a21 − 2a22 − λ2 ,
(10)
with K = diag(e−if (x1 −λ) , e−if (x2 −λ) , e−if (x3 −λ) ), ⎛ ⎞ 1 0 0 ⎜ ⎟ G = ⎝0 a1 ei[b1 t+c1 z] 0 ⎠, i[b2 t+c2 z] 0 0 a2 e
L = (l1 , l2 , l3 )T , ⎛ ⎞ 1 1 1 ⎜ i i i ⎟ H = ⎝− x1 +b − x2 +b − x3 +b ⎠, 1 1 1 i i i − x1 +b2 − x2 +b2 − x3 +b2
where f (ξ) = ξ[t + z(2λ + ξ)], the sign T denotes the transpose for a vector/matrix, l1 , l2 and l3 are three nonzero constants, and x1 , x2 and x3 are three different roots of the cubic equation for x as follows: [(x − 2λ)(x + b1 ) − a21 ](x + b2 ) − a22 (x + b1 ) = 0.
(11)
Choosing v(z, t, λ) = eiβz , we have y(z, t, λ) = GHKL. 3.1 Vector rational rogue wave solutions for System (1) To construct the vector rational rogue wave solutions for System (1) based on N th-Order DT (8), the perturbation expansion method [36] and Transformation (4b), we presume that Cubic Eq. (11) admits a pair of double roots x0 when λ = λ1 . We set λ = λ1 (1 + r2 ), where r is a small parameter. Inserting λ = λ1 (1 + r2 ) into Cubic Eq. (11), we derive that Cubic Eq. (11) possesses three roots x1 , x2 and x3 , which meet that lim x1 = lim x2 = x0 . Taking r→0 r→0 ⎛ ⎞ 1 1 1 1 1 0 ⎜ i i ⎟ , H = ⎝− x1 +b W= , R= − x2 +b ⎠, 1 1 1 1 −1 0 i i r − x1 +b − x2 +b 2 2 J = diag((x1 + b1 )(x1 + b2 ), (x2 + b1 )(x2 + b2 )), 5
K = diag(e−if (x1 −λ) , e−if (x2 −λ) ),
and L = (l1 , l2 )T with lα = form of
N
j=1 lαj r
2(j−1)
, we see that HJ KWRL can be expanded in the
HJ KWRL = 1 ∂ j (HJ KWRL) with gj = j! ∂λj
∞
gj (z, t)r2j ,
j=0
λ=λ1
(j = 0, 1, 2, · · · ). Thus, we obtain the N th-order vector rational
rogue wave solutions for System (6) as det(Γ − 2iΞ†2 Ξ1 ) u1 [N ] = u1 [0], det(Γ)
(12a)
det(Γ − 2iΞ†3 Ξ1 ) u2 [0], det(Γ)
(12b)
u2 [N ] =
where Ξ(z, t) = (g0 , g1 , · · · , gN −1 ), Ξτ is the τ -th row of Ξ(z, t) (τ = 1, 2, 3) and Γ = (Γmj )N ×N with ⎛ ⎞ 1 0 0 min(j−1,k) m+j−2 k+1 1 ⎜ ⎟ † Ckl λ∗k−l (−λ1 )l gm−1−k+l i − Γmj = ⎝0 a21 0 ⎠ gj−1−l , 1 2Im(λ1 ) k=0 l=max(0,k−m+1) 0 0 a22 and Im(∗) being the image part of the complex number. Based on Transformations (4b), the N th-order vector rational rogue wave solutions for System (1) can be derived as q1 [N ] = u1 [N ]ei q2 [N ] = u2 [N ]ei
(ρ1 |u1 [N ]|2 +ρ2 |u2 [N ]|2 ) dt (ρ1 |u1
[N ]|2 +ρ
2 |u2
[N ]|2 ) dt
,
(13a)
.
(13b)
3.2 Vector semi-rational rogue wave solutions for System (1) In Section 3.1, we only employ a pair of double roots of Cubic Eq. (11), and do not consider the third root of Cubic Eq. (11) when we deduce the vector rational rogue wave solutions, i.e., Solutions (13). In what follows, we will employ the third root x3 to construct the vector semi-rational rogue wave solutions for Eqs. (1). We suppose that Cubic Eq. (11) possesses a pair of double roots when λ = λ1 , which is represented by x0 . Inserting λ = λ1 (1 + r2 ) into Cubic Eq. (11), we find that Cubic Eq. (11) yields three roots x1 , x2 and x3 , which meet that x1 → x0 and x2 → x0 as r → 0. We choose ⎞ ⎞ ⎛ ⎛ 1 1 0 1 0 0 ⎟ ⎜ ⎟ ⎜ W = ⎝ 1 −1 0 ⎠ , R = ⎝ 0 1r 0 ⎠ , 0 0 1 0 0 1 ⎛ ⎞ 0 0 (x1 + b1 )(x1 + b2 ) ⎜ ⎟ J=⎝ 0 (x2 + b1 )(x2 + b2 ) 0 ⎠, 0 0 (x3 + b1 )(x3 + b2 ) 6
and L = (l1 , l2 , l3 )T with lτ =
N
j=1 lτ j r
2(j−1)
such that we can rewrite HJKWRL as ∞
HJKWRL =
hj (z, t)r2j ,
j=0
1 ∂ j (HJKWRL) with hj = j! ∂λj
λ=λ1
(j = 0, 1, 2, · · · ). Then, the N th-order vector semi-rational rogue
wave solutions for Eqs. (6) are represented as det(Λ − 2iΘ†2 Θ1 ) u1 [N ] = u1 [0], det(Λ)
(14a)
det(Λ − 2iΘ†3 Θ1 ) u2 [0], det(Λ)
(14b)
u2 [N ] =
where Θ(z, t) = (h0 , h1 , · · · , hN −1 ), Θτ denotes the τ -th row of Λ(z, t) and ⎛ 1 min(j−1,k) m+j−2 k+1 1 ⎜ l ∗k−l l † i − Ck λ1 (−λ1 ) hm−1−k+l ⎝0 Λmj = 2Im(λ1 ) k=0 l=max(0,k−m+1) 0
Λ = (Λmj )N ×N with ⎞ 0 0 ⎟ a21 0 ⎠ hj−1−l . 0 a22
Based on Transformations (4b), we can deduce the N th-order vector semi-rational rogue wave solutions for Eqs. (1) in the forms of q1 [N ] = u1 [N ]ei q2 [N ] = u2 [N ]ei
(ρ1 |u1 [N ]|2 +ρ2 |u2 [N ]|2 ) dt (ρ1 |u1
[N ]|2 +ρ
2 |u2
[N ]|2 ) dt
,
(15a)
.
(15b)
When l31 = 0, Solutions (15) are the rational expressions which can only generate vector rogue waves; When l31 = 0, Solutions (15) are the semi-rational expressions which can describe the coexistence of the rogue waves and breathers. 4. Vector rational and semi-rational rogue waves for System (1) 4.1 Higher-order vector rational rogue waves for System (1) Based on Solutions (13) with N = 2, we will graphically discuss the characteristics of the second-order vector rational rogue waves with the real constants ρ1 and ρ2 through Figs. 1. As seen in Figs. 1(a1 -a2 ), the second-order rational rogue waves with the merged structure take place in the q1 and q2 components. We only increase the values of ρ1 and ρ2 respectively, the second-order rogue waves in the q1 and q2 components keep unchanged, which means that ρ1 and ρ2 have no effects on the second-order vector rational rogue wave, as shown in Figs. 1(b1 -b2 ) and 1(c1 -c2 ). 7
(a1 )
(b1 )
(c1 )
(a2 )
(b2 )
(c2 )
Figs. 1 The second-order vector rational rogue waves via Solutions (13) with N = 2, a1 = 1, a2 = 1, b1 = 25 , b2 = − 25 , λ1 = ρ1 =
1 20 ,
√ 3 6 5 i, l11
= 0, l12 = 0, l21 = 2, l22 = 0, (a1 -a2 ) ρ1 =
1 20 ,
ρ2 =
1 70 ;
(b1 -b2 ) ρ1 = 1, ρ2 =
1 70 ;
(c1 -c2 )
ρ2 = 1. √
Taking N = 2, a1 = 1, a2 = 1, b1 = 25 , b2 = − 25 , λ1 = 3 5 6 i and l12 = 101 in Solutions (13), we observe that the merged structure splits up into three four-petalled rogue waves which form a triangle structure in each component, as seen in Figs. 2(a1 -a2 ). When b1 = b2 = 0, √ λ1 = 2i and l12 = 1001, three eye-shaped rogue waves appear in each component and form the triangle structure, can be seen in Figs. 2(b1 -b2 ). When b1 = 1, b2 = −1, λ1 = √ which √ 1 6 3 − 9 − i 6 3 + 9 and l12 = 1001, the triangle structure consisting of the three 4 anti-eye-shaped rogue waves occur in the q1 component, while the triangle structure consisting of the three eye-shaped rogue waves appear in the q2 component, as shown in Figs. 2(c1 -c2 ).
(a1 )
(b1 )
8
(c1 )
(a2 )
(b2 )
(c2 )
Figs. 2 The second-order vector rational rogue waves via Solutions (13) with N = 2, ρ1 = √
b2 = − 25 , λ1 = 3 5 6 i, l12 = 101; (b1 -b2 ) b1 = 0, √ √ λ1 = 14 6 3 − 9 − i 6 3 + 9 , l12 = 1001.
2 5,
a2 = 1, l11 = 0, l21 = 2, l22 = 0, (a1 -a2 ) b1 = √ λ1 = 2i, l12 = 1001; (c1 -c2 ) b1 = 1, b2 = −1,
1 20 ,
ρ2 =
1 70 ,
a1 = 1,
b2 = 0,
√
When N = 3, a1 = 1, a2 = 1, b1 = 25 , b2 = − 25 and λ1 = 3 5 6 i, Figs. 3 present the third-order vector rogue waves with each component possessing six four-petaled rogue waves. Choosing l12 = 0 and l13 = 0, we observe that the six four-petaled rogue waves form the merged structure in each component, as shown in Figs. 3(a1 -a2 ). When l12 = 101 and l13 = 0, the merged structure turns into the triangle structure in each component, as seen in Figs. 3(b1 -b2 ): Three four-petaled rogue waves form the triangle structure, while the other three four-petaled rogue waves are located at the three sides of the triangle structure. Choosing l12 = 0 and l13 = 30001, we observe that five four-petaled rogue waves constitute the pentagon structure surrounding the other one four-petaled rogue wave in each component, as seen in Figs. 3(c1 -c2 ).
(a1 )
(b1 )
(c1 )
(a2 )
(b2 )
(c2 )
9
Figs. 3 The third-order vector rational rogue waves via Solutions (13) with N = 3, ρ1 = a2 = 1, b1 =
2 5,
b2 =
− 25 ,
√
λ1 =
3 6 5 i, l11
1 20 ,
ρ2 =
1 70 ,
a1 = 1,
= 0, l21 = 2, l22 = 0, l23 = 0, (a1 -a2 ) l12 = 0, l13 = 0; (b1 -b2 )
l12 = 101, l13 = 0; (c1 -c2 ) l12 = 0, l13 = 30001.
4.2 The first- and second-order vector semi-rational rogue waves In what follows, we will graphically analyze the characteristics of the first- and second-order semi-rational rogue waves on the basis of Solutions (15) via Figs. 4-5. √ 3 6 2 2 Choosing a1 = 1, a2 = 1, b1 = 5 , b2 = − 5 and λ1 = 5 i in Solutions (15), we can acquire the N th-order vector semi-rational rogue waves which describe that the N th-order four-petalled rogue waves coexist with the N eye-shaped breathers in each component, as shown in Figs. 4. Figs. 4(a1 -a2 ) present the first-order vector semi-rational rogue wave which demonstrates that one four-petalled rogue wave coexists with one eye-shaped breather in each component. Figs. 4(b1 -b2 ) display the second-order vector semi-rational rogue wave which describes the second-order rogue wave with the merged structure coexists with two eye-shaped breathers in each component. When l12 = 101, Figs. 4(c1 -c2 ) show that the triangle structure consisting of three four-petalled rogue waves coexists with two eye-shaped breathers in each component. Moreover, increasing or decreasing the value of l31 , we find that the rogue waves and the breathers merge together or split up in in each component, the corresponding figures of which are omitted.
(a1 )
(b1 )
(c1 )
(a2 )
(b2 )
(c2 )
10
Figs. 4 Vector semi-rational rogue waves via Solutions (15) with ρ1 = b2 = − 25 , λ1 =
√
3 6 5 i, −10
l22 = 0, l31 = 10
1 20 ,
ρ2 =
1 70 ,
a1 = 1, a2 = 1, b1 = 25 ,
(a1 -a2 ) N = 1, l11 = 0, l21 = 2, l31 = 10−5 ; (b1 -b2 ) N = 2, l11 = 0, l12 = 0, l21 = 2,
, l32 = 0; (c1 -c2 ) N = 2, l11 = 0, l12 = 101, l21 = 2, l22 = 0, l31 = 10−10 , l32 = 0.
√ √ When a1 = 1, a2 = 1, b1 = 1, b2 = −1 and λ1 = 6 3 − 9 − i 6 3 + 9 , Solutions (15) can generate the N th-order vector semi-rational rogue waves: The N th-order antieye-shaped rogue wave coexists with N eye-shaped breathers in the q1 component, while the N th-order eye-shaped rogue wave coexists with N anti-eye-shaped breathers in the q2 component, as shown in Figs. 5. For instance, Figs. 5(a1 -a2 ) describe that one anti-eye-shaped rogue wave coexists with one eye-shaped breather in the q1 component, while one eye-shaped rogue wave coexists with one anti-eye-shaped breather in the q2 component. As seen in Figs. 5(b1 b2 ), the coexistence of the second-order anti-eye-shaped rogue wave with the merged structure and two parallel eye-shaped breathers takes place in the q1 component, while the second-order eye-shaped rogue wave with the merged structure coexists with two parallel anti-eye-shaped breathers in the q2 component. Increasing the value of l12 , we observe that the merged structures in Figs. 5(b1 -b2 ) turn into the triangle structures in Figs. 5(c1 -c2 ). 1 4
(a1 )
(b1 )
(c1 )
(a2 )
(b2 )
(c2 )
1 1 , ρ2 = 70 , a1 = 1, a2 = 1, b1 = 1, Figs. 5 Vector semi-rational rogue waves via Solutions (15) with ρ1 = 20 √ √ 6 3 − 9 − i 6 3 + 9 , (a1 -a2 ) N = 1, l11 = 0, l21 = 2, l31 = 10−5 ; (b1 -b2 ) N = 2, b2 = −1, λ1 = 14
l11 = 0, l12 = 0, l21 = 2, l22 = 0, l31 = 10−10 , l32 = 0; (c1 -c2 ) N = 2, l11 = 0, l12 = 101, l21 = 2, l22 = 0, l31 = 10−10 , l32 = 0.
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5. Conclusions Non-Kerr media have been reported to possess certain applications in photonic lattices and optical fibers. In this paper, we have studied the vector multi-rational and semi-rational rogue waves in a non-Kerr medium, through the coupled cubic-quintic nonlinear Schr¨odinger system, i.e., System (1), which describes the effects of quintic nonlinearity on the ultrashort optical pulse propagation in the medium. Employing Transformations (4), we have converted System (1) into System (6) with Lax Pair (5). For q1 and q2 , the electromagnetic fields in the two cores of an optical waveguide, we have derived N th-Order DT (8) associated with Lax Pair (5), the N th-order vector rational and semi-rational rogue wave solutions, i.e., Solutions (13) and Solutions (15). According to Solutions (13), Figs. 1 have implied that ρ1 and ρ2 , the constant coefficients in System (1), have no influence on the second-order vector rational rogue waves; Figs. 2 have presented three types of the second-order rogue waves with the triangle structure which consists of three eye-shaped rogue waves, or three four-petalled rogue waves, or three anti-eye-shaped rogue waves; Figs. 3 have exhibited the third-order vector rogue waves with the merged, triangle and pentagon structures consisting of six four-petalled rogue waves in each component. Based on Solutions (15), Figs. 4 have shown the first-order/second-order vector semi-rational rogue waves which describe the coexistence of first-order/second-order four-petalled rogue waves and the one/two eye-shaped breathers in the q1 and q2 components. Figs. 5 have displayed the first-order/second-order vector semi-rational rogue waves: The first-order/second-order antieye-shaped rogue wave coexists with the one/two eye-shaped breathers in the q1 component, while the first-order/second-order eye-shaped rogue wave coexists with the one/two anti-eyeshaped breathers in the q2 component. Acknowledgments This work has been supported by the BUPT Excellent Ph.D. Students Foundation under No. CX2019201, by the National Natural Science Foundation of China under Grant Nos. 11772017, 11805020 and 11471050, by the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05), and by the Fundamental Research Funds for the Central Universities of China under Grant No. 2011BUPTYB02.
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