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Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein
Accepted Manuscript
Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schrodinger equations in an alpha helical protein ¨ Zhong Du, Bo Tian, Han-Peng Chai, Yu-Qiang Yuan PII: DOI: Reference:
S1007-5704(18)30190-4 10.1016/j.cnsns.2018.06.014 CNSNS 4556
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
13 December 2017 18 April 2018 11 June 2018
Please cite this article as: Zhong Du, Bo Tian, Han-Peng Chai, Yu-Qiang Yuan, Vector multirogue waves for the three-coupled fourth-order nonlinear Schrodinger equations in an alpha ¨ helical protein, Communications in Nonlinear Science and Numerical Simulation (2018), doi: 10.1016/j.cnsns.2018.06.014
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Highlights • The vector multi-rogue-wave solutions are derived via the Darboux-dressing transformation.
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• The baseband modulation instability theory are verified.
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• The single vector rogue wave, vector rogue wave pair and triple vector rogue wave are presented.
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Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schr¨odinger equations in an alpha helical protein Zhong Du, Bo Tian∗, Han-Peng Chai, Yu-Qiang Yuan
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State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
Abstract
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In this paper, we investigate the vector multi-rogue waves for the three-coupled fourth-order nonlinear Schr¨ odinger equations, which describe the dynamics of an alpha helical protein with the nearest and next-nearest neighbour interactions and interspine coupling. Via the Darboux-dressing transformation, the vector multi-rogue-wave solutions are derived. Based on such solutions, we present the single vector rogue wave, vector rogue wave pair and triple vector rogue wave graphically. We show that a fourpetaled rogue wave with two humps and two valleys appears in two components, while the other component has an eye-shaped rogue wave. Existing time of the rogue wave decreases with the strength of higher-order linear and nonlinear effects. We also obtain the separated and interacting vector rogue wave pairs, as well as the triple vector rogue waves. Moreover, we verify the baseband modulation instability through the linear stability analysis.
Keywords: Alpha helical protein; Three-coupled fourth-order nonlinear Schr¨odinger equations; Vector multi-rogue waves; Darboux-dressing transformation; Modulation instability
∗
Corresponding author, with e-mail address as tian− [email protected]
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1. Introduction
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The scalar nonlinear Schr¨odinger (NLS)-type equations have been used to model the nonlinear dynamics of such fields as molecular biology, nonlinear optics, oceanography, BoseEinstein condensation and plasmas physics [1–7]. Rogue-wave solutions for the scalar NLS equation have been expressed in the rational form [8]. It has been found that the rogue waves occur and disappear unpredictably [9], and can be generated by the modulation instability (MI) for the NLS equation [10]. The scalar NLS equation has been shown to admit the eye-shaped rogue wave, which has one hump and two valleys [11]. For the resonant interaction processes in molecular biology, nonlinear optics, etc., multiple waves have been investigated [12, 13]. Coupled NLS equations, which have the vector rogue waves including the anti-eye-shaped rogue waves and four-petaled rogue waves, have been proposed [12–15]. In the field of molecular biology, it has been found that the bio-energy through the protein molecules is released by the hydrolysis of adenosine triphosphate (ATP), and the energy from the ATP can be stored in the C=O vibration (amide-I) of a peptide group (H-N-C=O) of a protein and transported in an alpha helical protein [16–19]. Bio-energy has been thought to be disorganized rapidly due to the dispersion effect caused by the resonant interaction of intra peptide dipole vibrations (amide-I) [20]. Nonlinear effect, generated by the interaction of those vibrations with the local displacements of equilibrium positions of the peptide groups (H-N-C=O), has been proved to provide a potential well [21–23]. Solitons in the protein molecules, arisen as a result of the balance between the dispersion and nonlinear effects, have been found to appear as the stable carriers capable of transferring the bio-energy without any losses [18, 23]. Soliton dynamics of a single chain with one exciton band and one phonon mode corresponding to one atom per peptide group (H-N-C=O) of the protein chain has been investigated [24–27]. In reality, an alpha helical protein structure has been thought to consist of the peptide groups (H-N-C=O), which is connected by three quasilinear strands of hydrogen bonds [18, 28, 29]. For the molecular dynamics along the hydrogen bonding spine in the alpha helical protein, the coupled Hirota equations containing the third-order dispersion, self-steepening and decayed nonlinear response have been claimed to be more precise than the coupled NLS equations [30, 31]. To describe the bio-energy transfer along the hydrogen bonding spine in the alpha helical protein with their right and left nearest-neighbor interactions, researchers have considered the three coupled Hirota equations [19, 22, 23, 32]. Studying the dynamics of an alpha helical protein with the nearest and next-nearest neighbour interactions and interspine coupling, people have investigated the following three-coupled fourth-order NLS
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equations [18, 32, 33]:
ρ=1
+2
3 X
"
|qρ |2 qα + γ qα,xxxx + 2
∗ qα,x + 6 qρ qρ,x
3 X
qρ∗ qρ,x qα,x + 4
ρ=1
ρ=1
+2
3 X
3 X
∗ qα + 6 qρ qρ,xx
ρ=1
ρ=1
|qρ |2
!2
3 X ρ=1
3 X ρ=1
|qρ,x |2 qα
|qρ |2 qα,xx + 4
3 X
qρ∗ qρ,xx qα
ρ=1
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iqα,t + qα,xx + 2
3 X
qα = 0
(α = 1, 2, 3),
(1)
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where qα (x, t) denotes the amplitude of molecular excitation in the α-th spine, γ represents the strength of higher-order linear and nonlinear effects, the subscripts x and t represent the partial derivatives of the scaled space and retarded time variables, respectively, “ * ” denotes the complex conjugate. For Eqs. (1), multi-soliton solutions have been derived via the Darboux transformation (DT) [18]; bilinear forms and multi-soliton solutions have been obtained via the binary Bell-polynomial approach [32]; semirational rogue waves have been reported via the generalized DT [33]. However, to our knowledge, vector multi-rogue waves for Eqs. (1) have not been presented. In Section 2, vector multi-rogue-wave solutions for Eqs. (1) will be constructed via the Darboux-dressing transformation (DDT). In Section 3, three types of the vector multirogue waves will be discussed. Through the linear stability analysis, the M I for Eqs. (1) will be investigated in Section 4. Section 5 will be our conclusions. 2. Vector multi-rogue-wave solutions for Eqs. (1)
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with
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Based on the Ablowitz-Kaup-Newell-Segur system [34], Lax pair for Eqs. (1) can be given as [33]
U =
−iλ q1∗ q2∗ q3∗ −q1 iλ 0 0 , −q2 0 iλ 0 −q3 0 0 iλ
4
V11 = −8iγλ + 4i 1 + γ − iγ
"
3 X ρ=1
Ψx = U Ψ,
(|qρ |2xx
3 X ρ=1 2
2
|qρ |
Ψt = V Ψ,
!
V = 2
λ + 2γ
V11 V21 V31 V41
3 X
V12 V22 V32 V42
qρ∗ qρ,x
ρ=1
2
2
2
2
(2)
V13 V23 V33 V43 −
V14 V24 V34 V44
∗ qρ qρ,x
2
,
λ−i 2
3 X ρ=1
− 3|qρ,x | ) + 6(|q1 | |q2 | + |q2 | |q3 | + |q1 | |q3 | ) + 3 4
|qρ |2
3 X ρ=1
4
|qρ |
#
,
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∗ qj−1,xxx +3
− iγ
3 X ρ=1
∗ |qρ |2 qj−1,x +3
3 X
ρ=1
ρ=1
qj−1,xxx + 3 4
Vj,j = 8iγλ − + iγ
|qρ |2 qs−1,x + 3
ρ=1 4iγ|qj−1 |2 λ2
|qj−1 |2xx −
3 X
!
3 X ρ=1
qρ∗ qρ,x qs−1
(j = 2, 3, 4), !
|qρ |2 qs−1 + qs−1 λ − iqs−1,x
!
(s = 2, 3, 4),
ρ=1 ∗ ∗ + 2γ(qj−1 qj−1,x − qj−1,x qj−1 )λ + i|qj−1 |2 ! 3 X 2 3|qj−1,x | + 3 |qρ |2 |qj−1 |2 (j = 2, 3, 4) ρ=1 ∗ 2γ(qs−1 qj−1,x
∗ ∗ − qs−1,x qj−1 )λ + iqs−1 qj−1 " # 3 X ∗ ∗ 2 ∗ +iγ (qs−1 qj−1 )xx − 3qs−1,x qj−1,x + 3 |qρ | qs−1 qj−1 (j, s = 2, 3, 4, s 6= j),
Vs,j =
∗ −4iγqs−1 qj−1 λ2
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− iγ
3 X
∗ ∗ ∗ |qρ |2 qj−1 + qj−1 λ − iqj−1,x
∗ ∗ qρ qρ,x qj−1
Vs,1 = −8γqs−1 λ3 + 4iγqs−1,x λ2 + 2 γqs−1,xx + 2γ
!
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∗ ∗ ∗ V1,j = 8γqj−1 λ3 + 4iγqj−1,x λ2 − 2 γqj−1,xx + 2γ
3 X
+
ρ=1
M
where Ψ is a 4×4 matrix solution of Lax Pair (2), and λ is the spectral parameter. According to the compatibility condition Ut − Vx + U V − V U = 0, Eqs. (1) can be reproduced. By virtue of the similar method in Ref. [35], the DDT for Eqs. (1) can be presented as 2i(λ0 − λ∗0 )ψ1∗ ψ2 , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 2i(λ0 − λ∗0 )ψ1∗ ψ3 q2 = q2 [0] − , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 2i(λ0 − λ∗0 )ψ1∗ ψ4 q3 = q3 [0] − , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2
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q1 = q1 [0] −
(3a) (3b) (3c)
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where λ0 is a given value of the spectral parameter, q1 [0], q2 [0] and q3 [0] denote the seed solutions for Eqs. (1), (ψ1 , ψ2 , ψ3 , ψ4 )T = Ψ0 Z0 , ψ1 , ψ2 , ψ3 and ψ4 are all the functions with respect to x and t, the superscript T denotes the transpose of a vector/matrix, Ψ0 is the corresponding 4 × 4 matrix solution for Lax Pair (2) at λ = λ0 , Z0 = (ζ1 , ζ2 , ζ3 , ζ4 )T is a nonzero complex constant vector, ζ1 ,ζ2 , ζ3 and ζ4 are all complex constants. To construct the DDT solutions for Eqs. (1) via DDT (3), we may start with the seed
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solutions for Eqs. (1) as follows: qα [0] = aα ei(µα x+να t) , 3 3 3 X X X να = −4γµ2α a2ρ − 4γµα (a2ρ µρ ) − 4γ (a2ρ µ2ρ ) + 6γ
3 X ρ=1
a2ρ
!2
ρ=1
+2
3 X ρ=1
ρ=1
a2ρ + γµ4α − µ2α ,
(4)
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ρ=1
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where aα ’s (α = 1, 2, 3) and µα ’s are all the real constants. Then, substituting the transformation 1 0 0 0 0 ei(µ1 x+ν1 t) 0 0 Ψ0 = GΦ, G = (5) , 0 0 ei(µ2 x+ν2 t) 0 0 0 0 ei(µ3 x+ν3 t) into Lax Pair (2), we can derive that
Φx = G−1 U G + (G−1 )x G Φ = U0 Φ, Φt = G−1 V G + (G−1 )t G Φ = V0 Φ, U0 =
−iλ a1 a2 a3 −a1 i(λ − µ1 ) 0 0 , −a2 0 i(λ − µ2 ) 0 −a3 0 0 i(λ − µ3 ) ∆1 −a1 A1 −a2 A2 −a3 A3 a1 A 1 ∆2 ia1 a2 A12 ia1 a3 A13 , a2 A2 ia1 a2 A12 ∆3 ia2 a3 A23 a3 A3 ia1 a3 A13 ia2 a3 A23 ∆4
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(6b)
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where
(6a)
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V0 =
AC
with
∆1 = i 4γλ2 −
3 X ρ=1
∆j = i
"
3 X
a2ρ
ρ=1
!
3 X
+ 4γλ
ρ=1
#
a2ρ µρ
!
+ 3γ
3 X ρ=1
a2ρ µ2ρ − 3γ
3 X ρ=1
a2ρ
!2
a2ρ − 8γλ4 + 4λ2 ,
a2j−1
3γ
3 X ρ=1
a2ρ
2
− 4γλ − 4γλµj−1 − 6
3γµ2j−1
+1
!
4
#
+ 8γλ − νj−1 , (j = 2, 3, 4),
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ρ=1
3
a2ρ − 2γµ2α + 2
− 8γλ − 4γλ µα − γµ3α , A12 = 3a21 γ + 3a22 γ + 3a23 γ − A13 = 3a21 γ + 3a22 γ + 3a23 γ − A23 = 3a21 γ + 3a22 γ + 3a23 γ −
!
+ 3γ
3 X
a2ρ µρ + µα
ρ=1
2
3γ
3 X
a2ρ + 1
ρ=1
!
4γλ2 − 2γλµ1 − 2γλµ2 − γµ21 − γµ1 µ2 − γµ22 + 1,
4γλ2 − 2γλµ1 − 2γλµ3 − γµ21 − γµ1 µ3 − γµ23 + 1, 4γλ2 − 2γλµ2 − 2γλµ3 − γµ22 − γµ2 µ3 − γµ23 + 1,
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Aα = λ 4γ
3 X
where Φ is a 4 × 4 matrix function with respect to x and t, U0 and V0 are both the 4 × 4 constant matrices, G is a 4 × 4 nonsingular matrix, and the superscript “−1” denotes the matrix inverse. Thus, the solution Ψ0 for Lax Pair (2) can be written as Ψ0 = GeU0 x+V0 t .
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Thus, we have
(ψ1 , ψ2 , ψ3 , ψ4 )T = GeU0 x+V0 t Z0 .
(7)
(8)
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Substituting Expression (8) into DDT (3), we can obtain the DDT solutions for Eqs. (1). Similar to the analysis in Ref. [15], the vector multi-rogue-wave solutions can be derived when aα ’s and µα ’s satisfy √ √ 2 2 a1 , µ2 = 0, a3 = a1 , a2 = a1 . (9) µ1 = −µ3 = − 2 2 √ With Conditions (9), when λ0 = i 2a1 , we can work out the vector rogue wave solutions √ via DDT (3). Taking a1 = 2, we can achieve the vector multi-rogue-wave solutions for Eqs. (1), √
8ψ1∗ ψ2 , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 8ψ1∗ ψ3 , q2 = e2i(67γ+5)t + |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 √ 8ψ1∗ ψ4 q3 = 2ei[(115γ+9)t+x] + , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 2e−i[x−(115γ+9)t] +
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q1 =
(10a) (10b) (10c)
where ψ1 , ψ2 , ψ3 and ψ4 are presented in Appendix A. 3. Discussions on the vector multi-rogue waves Based on the vector multi-rogue-wave solutions, i.e., Solutions (10), in this section, we will discuss three types of the vector multi-rogue waves: single vector rogue waves, vector rogue wave pairs and triple vector rogue waves. 7
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3.1 Single vector rogue waves
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√ √ When ζ3 = −iζ1 − i 2ζ2 and ζ4 = − 2ζ1 − ζ2 , the dominators and numerators of Solutions (10) are both the polynomials of the second degree with respect to x and t, which generate the single vector rogue wave, as shown in Figs. 1. In the q1 and q3 components, a four-petaled rogue wave with two humps and two valleys appears, as shown in Figs. 1(a) and 1(c), while in the q2 component, an eye-shaped rogue wave with one hump and two valleys can be observed in Fig. 1(b). According to Solutions (10), when the maximum or minimum value of q1 appears, the minimum or maximum value of q3 occurs, which can be seen from the comparison between Figs. 1(a) and 1(c), i.e., the humps or valleys in the q1 component correspond to the valleys or humps in the q3 component. When we decrease the value of the strength of higher-order linear and nonlinear effects, γ, range of the rogue wave along the t axis increases, which means that the existing time of the rogue wave increases, as seen in Figs. 2. 3.0
2.0
2.0
2.5
1.5
1.5
2.0
1.5
1.0
1.0
1.0
0
(b)
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(a)
Figs. 1 Single vector rogue wave via Solutions (10) with γ =
0.5
0.5
M
0.5
(c) 1 20 ,
√ ζ1 = 1, ζ2 = 0, ζ3 = −i, ζ4 = − 2. 3.0
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2.0
2.0
2.5
1.5
1.5 2.0
1.5
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1.0
1.0
1.0 0.5
0.5 0.5
(a)
Figs. 2 The same as Figs. 1 except that γ =
(b)
(c)
1 2000 .
3.2 Vector rogue wave pairs √ (1+i)ζ √ 1 − iζ2 − √ 3 , the dominators and numerators When ζ3 6= −iζ1 − i 2ζ2 and ζ4 = − (1+i)ζ 2 2 of Solutions (10) are both the polynomials of the fourth degree with respect to x and t. In this 8
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case, there are two vector rogue waves in each component, as shown in Figs. 3 − 5. In Figs. 3, we show the separated vector rogue wave pair, where the two vector rogue waves of the vector rogue wave pair are separated and appear at almost the same time. As seen in Figs. 3(a) and 3(c), the q1 and q3 components have the similar structures: An eye-shaped rogue wave coexists with a four-petaled rogue wave, and the humps or valleys of the four-petaled rogue wave in the q1 component correspond to the valleys or humps of the four-petaled rogue wave in the q3 component. Fig. 3(b) displays the two eye-shaped rogue waves with different amplitudes in the q2 component. When ζ3 = −1, we can observe the interacting vector rogue wave pair, the two vector rogue waves of which interact with each other and merge together, as seen in Figs. 4. When |ζ3 | 0, we can obtain two vector rogue waves with three different structures, as seen in Figs. 5: Fig. 5(a) exhibits a four-petaled rogue wave and an anti-eye-shaped rogue wave with two humps and a valley in the q1 component; Shown in Fig. 5(b) is that the q2 component has two eye-shaped rogue waves; Fig. 5(c) depicts an eye-shaped rogue wave and a four-petaled rogue wave. Two vector rogue waves in Figs. 5 emerge at the different values of t and are separated. Here, the vector rogue wave pair can also be thought as the so-called separated vector rogue wave pair. 2.5
2.5
2.5
2.0
2.0
2.0
1.5
1.0
M
1.5
1.5
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0.5
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0.5
0.5
0
Figs. 3 Vector rogue wave pair via Solutions (10) with γ =
AC
1.0
(b)
PT
(a)
1.0
(c)
1 20 ,
ζ1 = ζ2 = 1, ζ3 = −i, ζ4 = −i −
√
2.
2.5 2.5
3 2.0
2.0 2
1.5 1.5
1.0
1.0
1 0.5
0
0.5
0
(a)
(b)
Figs. 4 The same as Figs. 3 except that ζ3 = −1, ζ4 = −i.
9
0
(c)
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3 2.5 1.5 2.0 2 1.5
1.0
1.0 1 0.5 0.5
0
0
0
(b)
Figs. 5 The same as Figs. 3 except that ζ3 = 100, ζ4 = −i −
(c)
1+i √ 2
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(a)
√ − (50 + 50i) 2.
3.3 Triple vector rogue waves
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(1+i)ζ3 √ 1 − iζ2 − √ When ζ4 6= − (1+i)ζ , the dominators and numerators of Solutions (10) are 2 2 both the polynomials of the sixth degree with respect to x and t. In such case, we can obtain the triple vector rogue wave, as shown in Figs. 6 and 7. Figs. 6 depict the three vector rogue waves with ζ3 = −i: In the q1 component, a four-petaled rogue wave emerges first, then an eye-shaped rogue wave and a four-petaled rogue wave appear at almost the same time, as seen in Fig. 6(a); In the q2 component, as shown in Fig. 6(b), a four-petaled rogue wave takes place first, and then the two eye-shaped rogue waves with the different amplitudes occur; As seen in Fig. 6(c), in the q3 component, an eye-shaped rogue wave occurs first, then a four-petaled and an anti-eye-shaped rogue wave appear. Triple vector rogue wave in Figs. 6 consists of three separated vector rogue waves, which can be considered as the separated triple vector rogue wave. Figs. 7 display the interacting triple vector rogue wave, the three vector rogue waves of which interact with each other and merge together. In the q1 and q3 components, two four-petaled rogue waves coexist with an eye-shaped rogue wave, as shown in Figs. 7(a) and 7(c). Triple vector rogue wave looks like a type of the second-order rogue wave with three single rogue wave [36, 37]. However, the distribution structures of the triple vector rogue wave obtained in this paper are different from the second-order rogue wave with three single rogue waves, because the three single rogue waves of the the triple vector rogue wave have different shapes. 3
2.5 3 2.0
2 1.5
2
1.0 1
1 0.5
0
0
10
0
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(a)
(b)
(c)
√ Figs. 6 Triple vector rogue wave with the same parameters as Figs. 3 except that ζ4 = − 2. 3.5
3.5
3.0
3.0
2.0
2.5
2.5 1.5 2.0
1.5
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2.0
1.0
1.0 0.5 0.5 0
(a)
(b)
1.5
1.0
0.5
(c)
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Figs. 7 The same as Figs. 6 except that ζ3 = 0.
4. The MI for Eqs. (1)
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In this section, according to Seed Solutions (4), we will study the MI for Eqs. (1) through the linear stability analysis. Incorporating the perturbation term into Seed Solutions (4), we obtain √
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1 3 q1 = [a1 + u1 ]e 4 ia1 (115a1 γt+18a1 t−2 2x) , 1 2 a1 2 q2 = √ + u2 e 2 ia1 t(67a1 γ+10) , 2 √ 1 3 q = [a + u ]e 4 ia1 (115a1 γt+18a1 t+2 2x) ,
1
3
(11b) (11c)
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3
(11a)
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where is a small parameter, and uα ’s (α = 1, 2, 3) are the functions with respect to x and t. Substituting Solutions (11) into Eqs. (1), we can acquire the linearized disturbance equations of uα ’s at O(), expressed as
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√ √ √ iu1,t − ia31 γ(u∗2,x + 2 2u∗3,x + 15 2u1,x + 3u2,x ) + a21 (2γu∗1,xx + 2γu∗2,xx + 2γu∗3,xx √ √ √ +11γu1,xx + 2 2γu2,xx + 4γu3,xx + 2u∗2 + 2u∗3 + 2u1 + 2u2 + 2u3 + 2u∗1 ) √ √ −i 2a1 (u1,x + 2γu1,xxx ) + u1,xx + γu1,xxxx + 2a41 γ(12u∗1 + 7 2u∗2 √ +14u∗3 + 12u1 + 7 2u2 + 14u3 ) = 0, (12a) √ 2 ∗ √ 2 ∗ 3 ∗ 3 ∗ 2 ∗ 3 iu2,t + ia1 γu1,x − ia1 γu3,x + 2a1 γu1,xx + a1 γu2,xx + 2a1 γu3,xx − 3ia1 γu1,x √ √ +3ia31 γu3,x + 2 2a21 γu1,xx + 12a21 γu2,xx + 2 2a21 γu3,xx + u2,xx + γu2,xxxx √ √ √ √ +14 2a41 γu∗1 + 15a41 γu∗2 + 14 2a41 γu∗3 + 2a21 u∗1 + a21 u∗2 + 2a21 u∗3 √ √ √ +14 2a41 γu3 + 2a21 u1 (14a21 γ + 1) + u2 (15a41 γ + a21 ) + 2a21 u3 = 0, (12b) 11
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√ √ √ iu3,t + ia31 (2 2γu∗1,x + 3γu2,x + 15 2γu3,x + γu∗2,x ) + a21 (2γu∗1,xx + 2γu∗2,xx + 2γu∗3,xx √ √ √ +4γu1,xx + 2 2γu2,xx + 11γu3,xx + 2u∗1 + 2u∗2 + 2u∗3 + 2u1 + 2u2 + 2u3 ) √ √ +i 2a1 (u3,x + 2γu3,xxx ) + u3,xx + γu3,xxxx + 2a41 γ(14u∗1 + 7 2u∗2 + 14u1 √ +7 2u2 + 12u3 + 12u∗3 ) = 0, (12c) the solutions of which can be supposed as (13)
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uα = uα,1 (t)eikx + uα,2 (t)e−ikx ,
where uα ’s are the t-periodic functions, k is the frequency of perturbation, uα,1 ’s and uα,2 ’s are the coefficients of linear combination. Substituting Solutions (13) into Eqs. (12), we derive that
the coefficient matrix M can be expressed as M13 M14 M15 M16 M23 M24 M25 M26 M33 M34 M35 M36 , M43 M44 M45 M46 M53 M54 M55 M56 M63 M64 M65 M66
M
where Ω = (u11 , u∗12 , u21 , u∗22 , u31 , u∗32 )T , and M11 M12 M21 M22 M31 M32 M = M 41 M42 M51 M52 M61 M62
(14)
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Ωt = M Ω,
AC
CE
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with Mm,n ’s (m, n = 1, 2, 3, 4, 5, 6) given in Appendix B. Solving the characteristic equation of the coefficient matrix M , we can achieve six roots as follows: q Γ1 = k 2a21 − k 2 γk 2 − 11a21 γ − 1 , (15a) q Γ2 = −k 2a21 − k 2 γk 2 − 11a21 γ − 1 , (15b) q √ (15c) Γ3 = Γ21,2 + 2a21 γk 4 (46a21 γ − 5γk 2 + 4) − 2 2Λ, q √ Γ4 = − Γ21,2 + 2a21 γk 4 (46a21 γ − 5γk 2 + 4) − 2 2Λ, (15d) q √ Γ5 = Γ21,2 + 2a21 γk 4 (46a21 γ − 5γk 2 + 4) + 2 2Λ, (15e) q √ Γ6 = − Γ21,2 + 2a21 γk 4 (46a21 γ − 5γk 2 + 4) + 2 2Λ, (15f) with
Λ = a1 k 3 143a41 γ 2 + 3a21 γ 8 − 11γk 2 + 2γ 2 k 4 − 3γk 2 + 1 .
The MI with respect to Seed Solutions (4) for Eqs. (1) occurs when the characteristic equation of M has a negative imaginary root. Based on Expressions (15), we can conclude that: If 12
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√ √ 2a21 −k 2 > 0 and Γ21,2 +2a21 γk 4 (46a21 γ − 5γk 2 + 4) > max{2 2Λ, −2 2Λ}, the characteristic equation of M has six real roots and no MI occurs; Otherwise, the characteristic equation of M has at least one pair of complex conjugate roots, and the baseband MI for Eqs. (1) takes place. It can be seen that if the MI for Eqs. (1) occurs, it is of baseband type. 5. Conclusions
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In conclusion, investigation has been made on the three-coupled fourth-order nonlinear Schr¨odinger equations, i.e., Eqs. (1), which describe the dynamics of an alpha helical protein with nearest and next-nearest neighbour interactions and interspine coupling. Via DDT (3), the vector multi-rogue-wave solutions for Eqs. (1). i.e., Solutions (10), have been derived. Based on Solutions (10), single vector rogue wave, vector rogue wave pair and triple vector rogue wave have been presented graphically: Figs. 1 have shown the single vector rogue wave: In the q1 and q3 components, a fourpetaled rogue wave with two humps and two valleys appears, while in the q2 component, an eye-shaped rogue wave with one hump and two valleys can be observed. Decreasing the value of the strength of higher-order linear and nonlinear effects, γ, we have observed that the range of the rogue wave along the t axis increases, which means that the existing time of the rogue wave increases, as shown in Figs. 2. In Figs. 3, we have shown the separated vector rogue wave pair, the two vector rogue waves of which are separated and appear at almost the same time. We have observed the interacting vector rogue wave pair, the two vector rogue waves of which interact with each other and merge together, as seen in Figs. 4. In Figs. 5, we have observed the separated vector rogue wave pair, where the two vector rogue waves emerge at the different values of t. Figs. 6 have depicted the separated triple vector rogue wave which consists of three separated vector rogue waves. Figs. 7 have displayed the interacting triple vector rogue wave, the three vector rogue waves of which interact with each other and merge together. Appendix A
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The expressions of ψ1 , ψ2 , ψ3 and ψ4 can be obtained as 1 ψ1 = (ψ11 ζ1 + ψ12 ζ2 + ψ13 ζ3 + ψ14 ζ4 )e−x−i(27γ+11)t , 6 ψ11 = 2{−12(22γ + 1)t2 (70γ + 22γx + x + 3) + 8i(22γt + t)3 ψ12
−6it[122γ + (22γ + 1)x2 + (114γ + 5)x + 5] + x3 + 6x2 + 9x + 3}, √ = 2{−12(22γ + 1)t2 [(70 + 48i)γ + (1 + i)(22γ + 1)x + (3 + 2i)] −(8 − 8i)(22γt + t)3 − 6it{(100 + 30i)γ + (1 + i)(22γ + 1)x2
+[(114 + 70i)γ + (5 + 3i)]x + (4 + i)} + x[(1 + i)x2 + (6 + 3i)x + 6]}, 13
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ψ13 = 2[−12(22γ + 1)t2 (48γ + 22γx + x + 2) + 8i(22γt + t)3 ψ14
−6it(x + 2)(26γ + 22γx + x + 1) + x(x2 + 3x + 3)], √ = 2{12i(22γ + 1)t2 [(48 + 70i)γ + (1 + i)(22γ + 1)x + (2 + 3i)] +(8 + 8i)(22γt + t)3 − 6t{(30 + 100i)γ + (1 + i)(22γ + 1)x2
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+[(70 + 114i)γ + (3 + 5i)]x + (1 + 4i)} + x((1 − i)x2 + (6 − 3i)x + 6)}, 1 ψ2 = (ψ21 ζ1 + ψ22 ζ2 + ψ23 ζ3 + ψ24 ζ4 )e2i(44γ−1)t+(−1−i)x , 6√ ψ21 = 2{12(22γ + 1)t2 [(70 + 48i)γ + (1 + i)(22γ + 1)x + (3 + 2i)] +(8 − 8i)(22γt + t)3 + 6it{(100 + 30i)γ + (1 + i)(22γ + 1)x2
+[(114 + 70i)γ + (5 + 3i)]x + (4 + i)} − x[(1 + i)x2 + (6 + 3i)x + 6]},
ψ22 = 2{12i(22γ + 1)t2 [(48 − 22i)γ + 22γx + x + (2 − i)] + 8(22γt + t)3 −6t{(8 − 48i)γ + (22γ + 1)x2 + [(70 − 44i)γ + (3 − 2i)]x − 2i}
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ψ23
−ix3 − (3 + 3i)x2 − (3 − 3i)x + 3}, √ = 2{12(22γ + 1)t2 [(48 + 26i)γ + (1 + i)(22γ + 1)x + (2 + i)] +(8 − 8i)(22γt + t)3 + 6it{(30 + 4i)γ + (1 + i)(22γ + 1)x2
+[(70 + 26i)γ + (3 + i)]x + 1} − x2 [3 + (1 + i)x]},
ψ24 = 2{12(22γ + 1)t2 (48γ + 22γx + x + 2) − 8i(22γt + t)3
+6it(x + 2)(26γ + 22γx + x + 1) − x(x2 + 3x + 3)], √ = 2{12(22γ + 1)t2 [(48 + 26i)γ + (1 + i)(22γ + 1)x + (2 + i)]
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ψ32
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+6it[30γ + (22γ + 1)x2 + (70γ + 3)x + 1] − x2 (x + 3)}, 1 ψ3 = (ψ31 ζ1 + ψ32 ζ2 + ψ33 ζ3 + ψ34 ζ4 )e−x+i(107γ−1)t , 6 ψ31 = 2[12(22γ + 1)t2 (48γ + 22γx + x + 2) − 8i(22γt + t)3
+(8 − 8i)(22γt + t)3 + 6it{(30 + 4i)γ + (1 + i)(22γ + 1)x2
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+[(70 + 26i)γ + (3 + i)]x + 1} − x2 [3 + (1 + i)x]},
ψ33 = 2{12(22γ + 1)t2 (26γ + 22γx + x + 1) − 8i(22γt + t)3
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ψ34
+6it[26γ + (22γ + 1)x2 + 26γx + x + 1] − x3 − 3x + 3}, √ = 2{−12i(22γ + 1)t2 [(26 + 48i)γ + (1 + i)(22γ + 1)x + (1 + 2i)] −(8 + 8i)(22γt + t)3 + 6t{(4 + 30i)γ + (1 + i)(22γ + 1)x2
+[(26 + 70i)γ + (1 + 3i)]x + i} + x2 [−3 − (1 − i)x]}, 1 ψ4 = (ψ41 ζ1 + ψ42 ζ2 + ψ43 ζ3 + ψ44 ζ4 )e2i(44γ−1)t+(−1+i)x , 6√ ψ41 = 2{−12i(22γ + 1)t2 [(48 + 70i)γ + (1 + i)(22γ + 1)x + (2 + 3i)] −(8 + 8i)(22γt + t)3 + 6t{(30 + 100i)γ + (1 + i)(22γ + 1)x2
+[(70 + 114i)γ + (3 + 5i)]x + (1 + 4i)} + x[−(1 − i)x2 − (6 − 3i)x − 6]}, 14
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ψ42 = 2{12(22γ + 1)t2 (48γ + 22γx + x + 2) − 8i(22γt + t)3 ψ43
+6it[30γ + (22γ + 1)x2 + (70γ + 3)x + 1] − x2 (x + 3)}, √ = 2{−12i(22γ + 1)t2 [(26 + 48i)γ + (1 + i)(22γ + 1)x + (1 + 2i)] −(8 + 8i)(22γt + t)3 + 6t{(4 + 30i)γ + (1 + i)(22γ + 1)x2
+[(26 + 70i)γ + (1 + 3i)]x + i} + x2 (−3 − (1 − i)x)},
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ψ44 = 2{−12i(22γ + 1)t2 [(48 + 22i)γ + 22γx + x + (2 + i)] − 8(22γt + t)3 +6t{(8 + 48i)γ + (22γ + 1)x2 + [(70 + 44i)γ + (3 + 2i)]x + 2i} +ix3 − (3 − 3i)x2 − (3 + 3i)x + 3}. Appendix B
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The expressions of Mm,n ’s (m, n = 1, 2, 3, 4, 5, 6) can be written as h √ √ i M11 = i 24a41 γ + 15 2a31 γk + a21 2 − 11γk 2 + 2a1 k 1 − 2γk 2 + k 2 γk 2 − 1 , M12 = −M21 = 2ia21 12a21 γ − γk 2 + 1 , h √ √ i M13 = M31 = ia21 14 2a21 γ + 3a1 γk + 2 1 − 2γk 2 , h √ √ i M14 = −M41 = ia21 14 2a21 γ + a1 γk + 2 1 − γk 2 , M15 = M51 = 2ia21 14a21 γ − 2γk 2 + 1 , √ 2 2 2 M16 = −M61 = 2ia1 14a1 γ + 2a1 γk − γk + 1 , h √ √ i M22 = −i 24a41 γ − 15 2a31 γk + a21 2 − 11γk 2 + 2a1 k 2γk 2 − 1 + k 2 γk 2 − 1 , h √ √ i M23 = −M32 = −ia21 14 2a21 γ − a1 γk + 2 1 − γk 2 , h √ √ i M24 = M42 = −ia21 14 2a21 γ − 3a1 γk + 2 1 − 2γk 2 , √ M25 = −M52 = −2ia21 14a21 γ − 2a1 γk − γk 2 + 1 , M26 = M62 = −2ia21 14a21 γ − 2γk 2 + 1 , M33 = i 15a41 γ − 12a21 γk 2 + a21 + γk 4 − k 2 , M34 = −M43 = ia21 15a21 γ − γk 2 + 1 , h √ √ i M35 = M53 = ia21 14 2a21 γ − 3a1 γk + 2 1 − 2γk 2 , h √ √ i M36 = −M63 = ia21 14 2a21 γ + a1 γk + 2 1 − γk 2 , M44 = −i 15a41 γ − 12a21 γk 2 + a21 + γk 4 − k 2 , h √ √ i 2 2 2 M45 = −M54 = −ia1 14 2a1 γ − a1 γk + 2 1 − γk , h √ i √ M46 = M64 = −ia21 14 2a21 γ + 3a1 γk + 2 1 − 2γk 2 , 15
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h √ √ i M55 = i 24a41 γ − 15 2a31 γk + a21 2 − 11γk 2 + 2a1 k 2γk 2 − 1 + k 2 γk 2 − 1 , M56 = −M65 = 2ia21 12a21 γ − γk 2 + 1 , h √ 3 i √ 2 2 4 2 2 2 M66 = −i 24a1 γ + 15 2a1 γk + a1 2 − 11γk + 2a1 k 1 − 2γk + k γk − 1 .
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Acknowledgments
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This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023 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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[34] M. J. Ablowitz, Nonlinear Dispersive Waves (Cambridge: Cambridge Univ. Press, 2011). [35] W. R. Sun, D. Y. Liu and X. Y. Xie, Chaos, 27, 043114 (2017). [36] B. L. Guo, L. M. Ling and Q. P. Liu, Phys. Rev. E 85, 026607 (2012). [37] Y. Ohta and J. K. Yang, Proc. R. Soc. A 468, 1716 (2012).
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Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schrodinger equations in an alpha helical protein ¨ Zhong Du, Bo Tian, Han-Peng Chai, Yu-Qiang Yuan PII: DOI: Reference:
S1007-5704(18)30190-4 10.1016/j.cnsns.2018.06.014 CNSNS 4556
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
13 December 2017 18 April 2018 11 June 2018
Please cite this article as: Zhong Du, Bo Tian, Han-Peng Chai, Yu-Qiang Yuan, Vector multirogue waves for the three-coupled fourth-order nonlinear Schrodinger equations in an alpha ¨ helical protein, Communications in Nonlinear Science and Numerical Simulation (2018), doi: 10.1016/j.cnsns.2018.06.014
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Highlights • The vector multi-rogue-wave solutions are derived via the Darboux-dressing transformation.
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• The baseband modulation instability theory are verified.
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• The single vector rogue wave, vector rogue wave pair and triple vector rogue wave are presented.
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Vector multi-rogue waves for the three-coupled fourth-order nonlinear Schr¨odinger equations in an alpha helical protein Zhong Du, Bo Tian∗, Han-Peng Chai, Yu-Qiang Yuan
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State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
Abstract
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In this paper, we investigate the vector multi-rogue waves for the three-coupled fourth-order nonlinear Schr¨ odinger equations, which describe the dynamics of an alpha helical protein with the nearest and next-nearest neighbour interactions and interspine coupling. Via the Darboux-dressing transformation, the vector multi-rogue-wave solutions are derived. Based on such solutions, we present the single vector rogue wave, vector rogue wave pair and triple vector rogue wave graphically. We show that a fourpetaled rogue wave with two humps and two valleys appears in two components, while the other component has an eye-shaped rogue wave. Existing time of the rogue wave decreases with the strength of higher-order linear and nonlinear effects. We also obtain the separated and interacting vector rogue wave pairs, as well as the triple vector rogue waves. Moreover, we verify the baseband modulation instability through the linear stability analysis.
Keywords: Alpha helical protein; Three-coupled fourth-order nonlinear Schr¨odinger equations; Vector multi-rogue waves; Darboux-dressing transformation; Modulation instability
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Corresponding author, with e-mail address as tian− [email protected]
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1. Introduction
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The scalar nonlinear Schr¨odinger (NLS)-type equations have been used to model the nonlinear dynamics of such fields as molecular biology, nonlinear optics, oceanography, BoseEinstein condensation and plasmas physics [1–7]. Rogue-wave solutions for the scalar NLS equation have been expressed in the rational form [8]. It has been found that the rogue waves occur and disappear unpredictably [9], and can be generated by the modulation instability (MI) for the NLS equation [10]. The scalar NLS equation has been shown to admit the eye-shaped rogue wave, which has one hump and two valleys [11]. For the resonant interaction processes in molecular biology, nonlinear optics, etc., multiple waves have been investigated [12, 13]. Coupled NLS equations, which have the vector rogue waves including the anti-eye-shaped rogue waves and four-petaled rogue waves, have been proposed [12–15]. In the field of molecular biology, it has been found that the bio-energy through the protein molecules is released by the hydrolysis of adenosine triphosphate (ATP), and the energy from the ATP can be stored in the C=O vibration (amide-I) of a peptide group (H-N-C=O) of a protein and transported in an alpha helical protein [16–19]. Bio-energy has been thought to be disorganized rapidly due to the dispersion effect caused by the resonant interaction of intra peptide dipole vibrations (amide-I) [20]. Nonlinear effect, generated by the interaction of those vibrations with the local displacements of equilibrium positions of the peptide groups (H-N-C=O), has been proved to provide a potential well [21–23]. Solitons in the protein molecules, arisen as a result of the balance between the dispersion and nonlinear effects, have been found to appear as the stable carriers capable of transferring the bio-energy without any losses [18, 23]. Soliton dynamics of a single chain with one exciton band and one phonon mode corresponding to one atom per peptide group (H-N-C=O) of the protein chain has been investigated [24–27]. In reality, an alpha helical protein structure has been thought to consist of the peptide groups (H-N-C=O), which is connected by three quasilinear strands of hydrogen bonds [18, 28, 29]. For the molecular dynamics along the hydrogen bonding spine in the alpha helical protein, the coupled Hirota equations containing the third-order dispersion, self-steepening and decayed nonlinear response have been claimed to be more precise than the coupled NLS equations [30, 31]. To describe the bio-energy transfer along the hydrogen bonding spine in the alpha helical protein with their right and left nearest-neighbor interactions, researchers have considered the three coupled Hirota equations [19, 22, 23, 32]. Studying the dynamics of an alpha helical protein with the nearest and next-nearest neighbour interactions and interspine coupling, people have investigated the following three-coupled fourth-order NLS
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equations [18, 32, 33]:
ρ=1
+2
3 X
"
|qρ |2 qα + γ qα,xxxx + 2
∗ qα,x + 6 qρ qρ,x
3 X
qρ∗ qρ,x qα,x + 4
ρ=1
ρ=1
+2
3 X
3 X
∗ qα + 6 qρ qρ,xx
ρ=1
ρ=1
|qρ |2
!2
3 X ρ=1
3 X ρ=1
|qρ,x |2 qα
|qρ |2 qα,xx + 4
3 X
qρ∗ qρ,xx qα
ρ=1
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iqα,t + qα,xx + 2
3 X
qα = 0
(α = 1, 2, 3),
(1)
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where qα (x, t) denotes the amplitude of molecular excitation in the α-th spine, γ represents the strength of higher-order linear and nonlinear effects, the subscripts x and t represent the partial derivatives of the scaled space and retarded time variables, respectively, “ * ” denotes the complex conjugate. For Eqs. (1), multi-soliton solutions have been derived via the Darboux transformation (DT) [18]; bilinear forms and multi-soliton solutions have been obtained via the binary Bell-polynomial approach [32]; semirational rogue waves have been reported via the generalized DT [33]. However, to our knowledge, vector multi-rogue waves for Eqs. (1) have not been presented. In Section 2, vector multi-rogue-wave solutions for Eqs. (1) will be constructed via the Darboux-dressing transformation (DDT). In Section 3, three types of the vector multirogue waves will be discussed. Through the linear stability analysis, the M I for Eqs. (1) will be investigated in Section 4. Section 5 will be our conclusions. 2. Vector multi-rogue-wave solutions for Eqs. (1)
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with
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Based on the Ablowitz-Kaup-Newell-Segur system [34], Lax pair for Eqs. (1) can be given as [33]
U =
−iλ q1∗ q2∗ q3∗ −q1 iλ 0 0 , −q2 0 iλ 0 −q3 0 0 iλ
4
V11 = −8iγλ + 4i 1 + γ − iγ
"
3 X ρ=1
Ψx = U Ψ,
(|qρ |2xx
3 X ρ=1 2
2
|qρ |
Ψt = V Ψ,
!
V = 2
λ + 2γ
V11 V21 V31 V41
3 X
V12 V22 V32 V42
qρ∗ qρ,x
ρ=1
2
2
2
2
(2)
V13 V23 V33 V43 −
V14 V24 V34 V44
∗ qρ qρ,x
2
,
λ−i 2
3 X ρ=1
− 3|qρ,x | ) + 6(|q1 | |q2 | + |q2 | |q3 | + |q1 | |q3 | ) + 3 4
|qρ |2
3 X ρ=1
4
|qρ |
#
,
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∗ qj−1,xxx +3
− iγ
3 X ρ=1
∗ |qρ |2 qj−1,x +3
3 X
ρ=1
ρ=1
qj−1,xxx + 3 4
Vj,j = 8iγλ − + iγ
|qρ |2 qs−1,x + 3
ρ=1 4iγ|qj−1 |2 λ2
|qj−1 |2xx −
3 X
!
3 X ρ=1
qρ∗ qρ,x qs−1
(j = 2, 3, 4), !
|qρ |2 qs−1 + qs−1 λ − iqs−1,x
!
(s = 2, 3, 4),
ρ=1 ∗ ∗ + 2γ(qj−1 qj−1,x − qj−1,x qj−1 )λ + i|qj−1 |2 ! 3 X 2 3|qj−1,x | + 3 |qρ |2 |qj−1 |2 (j = 2, 3, 4) ρ=1 ∗ 2γ(qs−1 qj−1,x
∗ ∗ − qs−1,x qj−1 )λ + iqs−1 qj−1 " # 3 X ∗ ∗ 2 ∗ +iγ (qs−1 qj−1 )xx − 3qs−1,x qj−1,x + 3 |qρ | qs−1 qj−1 (j, s = 2, 3, 4, s 6= j),
Vs,j =
∗ −4iγqs−1 qj−1 λ2
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− iγ
3 X
∗ ∗ ∗ |qρ |2 qj−1 + qj−1 λ − iqj−1,x
∗ ∗ qρ qρ,x qj−1
Vs,1 = −8γqs−1 λ3 + 4iγqs−1,x λ2 + 2 γqs−1,xx + 2γ
!
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∗ ∗ ∗ V1,j = 8γqj−1 λ3 + 4iγqj−1,x λ2 − 2 γqj−1,xx + 2γ
3 X
+
ρ=1
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where Ψ is a 4×4 matrix solution of Lax Pair (2), and λ is the spectral parameter. According to the compatibility condition Ut − Vx + U V − V U = 0, Eqs. (1) can be reproduced. By virtue of the similar method in Ref. [35], the DDT for Eqs. (1) can be presented as 2i(λ0 − λ∗0 )ψ1∗ ψ2 , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 2i(λ0 − λ∗0 )ψ1∗ ψ3 q2 = q2 [0] − , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 2i(λ0 − λ∗0 )ψ1∗ ψ4 q3 = q3 [0] − , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2
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q1 = q1 [0] −
(3a) (3b) (3c)
AC
CE
where λ0 is a given value of the spectral parameter, q1 [0], q2 [0] and q3 [0] denote the seed solutions for Eqs. (1), (ψ1 , ψ2 , ψ3 , ψ4 )T = Ψ0 Z0 , ψ1 , ψ2 , ψ3 and ψ4 are all the functions with respect to x and t, the superscript T denotes the transpose of a vector/matrix, Ψ0 is the corresponding 4 × 4 matrix solution for Lax Pair (2) at λ = λ0 , Z0 = (ζ1 , ζ2 , ζ3 , ζ4 )T is a nonzero complex constant vector, ζ1 ,ζ2 , ζ3 and ζ4 are all complex constants. To construct the DDT solutions for Eqs. (1) via DDT (3), we may start with the seed
5
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solutions for Eqs. (1) as follows: qα [0] = aα ei(µα x+να t) , 3 3 3 X X X να = −4γµ2α a2ρ − 4γµα (a2ρ µρ ) − 4γ (a2ρ µ2ρ ) + 6γ
3 X ρ=1
a2ρ
!2
ρ=1
+2
3 X ρ=1
ρ=1
a2ρ + γµ4α − µ2α ,
(4)
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ρ=1
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where aα ’s (α = 1, 2, 3) and µα ’s are all the real constants. Then, substituting the transformation 1 0 0 0 0 ei(µ1 x+ν1 t) 0 0 Ψ0 = GΦ, G = (5) , 0 0 ei(µ2 x+ν2 t) 0 0 0 0 ei(µ3 x+ν3 t) into Lax Pair (2), we can derive that
Φx = G−1 U G + (G−1 )x G Φ = U0 Φ, Φt = G−1 V G + (G−1 )t G Φ = V0 Φ, U0 =
−iλ a1 a2 a3 −a1 i(λ − µ1 ) 0 0 , −a2 0 i(λ − µ2 ) 0 −a3 0 0 i(λ − µ3 ) ∆1 −a1 A1 −a2 A2 −a3 A3 a1 A 1 ∆2 ia1 a2 A12 ia1 a3 A13 , a2 A2 ia1 a2 A12 ∆3 ia2 a3 A23 a3 A3 ia1 a3 A13 ia2 a3 A23 ∆4
PT
M
(6b)
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where
(6a)
CE
V0 =
AC
with
∆1 = i 4γλ2 −
3 X ρ=1
∆j = i
"
3 X
a2ρ
ρ=1
!
3 X
+ 4γλ
ρ=1
#
a2ρ µρ
!
+ 3γ
3 X ρ=1
a2ρ µ2ρ − 3γ
3 X ρ=1
a2ρ
!2
a2ρ − 8γλ4 + 4λ2 ,
a2j−1
3γ
3 X ρ=1
a2ρ
2
− 4γλ − 4γλµj−1 − 6
3γµ2j−1
+1
!
4
#
+ 8γλ − νj−1 , (j = 2, 3, 4),
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ρ=1
3
a2ρ − 2γµ2α + 2
− 8γλ − 4γλ µα − γµ3α , A12 = 3a21 γ + 3a22 γ + 3a23 γ − A13 = 3a21 γ + 3a22 γ + 3a23 γ − A23 = 3a21 γ + 3a22 γ + 3a23 γ −
!
+ 3γ
3 X
a2ρ µρ + µα
ρ=1
2
3γ
3 X
a2ρ + 1
ρ=1
!
4γλ2 − 2γλµ1 − 2γλµ2 − γµ21 − γµ1 µ2 − γµ22 + 1,
4γλ2 − 2γλµ1 − 2γλµ3 − γµ21 − γµ1 µ3 − γµ23 + 1, 4γλ2 − 2γλµ2 − 2γλµ3 − γµ22 − γµ2 µ3 − γµ23 + 1,
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Aα = λ 4γ
3 X
where Φ is a 4 × 4 matrix function with respect to x and t, U0 and V0 are both the 4 × 4 constant matrices, G is a 4 × 4 nonsingular matrix, and the superscript “−1” denotes the matrix inverse. Thus, the solution Ψ0 for Lax Pair (2) can be written as Ψ0 = GeU0 x+V0 t .
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Thus, we have
(ψ1 , ψ2 , ψ3 , ψ4 )T = GeU0 x+V0 t Z0 .
(7)
(8)
PT
ED
M
Substituting Expression (8) into DDT (3), we can obtain the DDT solutions for Eqs. (1). Similar to the analysis in Ref. [15], the vector multi-rogue-wave solutions can be derived when aα ’s and µα ’s satisfy √ √ 2 2 a1 , µ2 = 0, a3 = a1 , a2 = a1 . (9) µ1 = −µ3 = − 2 2 √ With Conditions (9), when λ0 = i 2a1 , we can work out the vector rogue wave solutions √ via DDT (3). Taking a1 = 2, we can achieve the vector multi-rogue-wave solutions for Eqs. (1), √
8ψ1∗ ψ2 , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 8ψ1∗ ψ3 , q2 = e2i(67γ+5)t + |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 √ 8ψ1∗ ψ4 q3 = 2ei[(115γ+9)t+x] + , |ψ1 |2 + |ψ2 |2 + |ψ3 |2 + |ψ4 |2 2e−i[x−(115γ+9)t] +
AC
CE
q1 =
(10a) (10b) (10c)
where ψ1 , ψ2 , ψ3 and ψ4 are presented in Appendix A. 3. Discussions on the vector multi-rogue waves Based on the vector multi-rogue-wave solutions, i.e., Solutions (10), in this section, we will discuss three types of the vector multi-rogue waves: single vector rogue waves, vector rogue wave pairs and triple vector rogue waves. 7
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3.1 Single vector rogue waves
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√ √ When ζ3 = −iζ1 − i 2ζ2 and ζ4 = − 2ζ1 − ζ2 , the dominators and numerators of Solutions (10) are both the polynomials of the second degree with respect to x and t, which generate the single vector rogue wave, as shown in Figs. 1. In the q1 and q3 components, a four-petaled rogue wave with two humps and two valleys appears, as shown in Figs. 1(a) and 1(c), while in the q2 component, an eye-shaped rogue wave with one hump and two valleys can be observed in Fig. 1(b). According to Solutions (10), when the maximum or minimum value of q1 appears, the minimum or maximum value of q3 occurs, which can be seen from the comparison between Figs. 1(a) and 1(c), i.e., the humps or valleys in the q1 component correspond to the valleys or humps in the q3 component. When we decrease the value of the strength of higher-order linear and nonlinear effects, γ, range of the rogue wave along the t axis increases, which means that the existing time of the rogue wave increases, as seen in Figs. 2. 3.0
2.0
2.0
2.5
1.5
1.5
2.0
1.5
1.0
1.0
1.0
0
(b)
ED
(a)
Figs. 1 Single vector rogue wave via Solutions (10) with γ =
0.5
0.5
M
0.5
(c) 1 20 ,
√ ζ1 = 1, ζ2 = 0, ζ3 = −i, ζ4 = − 2. 3.0
PT
2.0
2.0
2.5
1.5
1.5 2.0
1.5
AC
CE
1.0
1.0
1.0 0.5
0.5 0.5
(a)
Figs. 2 The same as Figs. 1 except that γ =
(b)
(c)
1 2000 .
3.2 Vector rogue wave pairs √ (1+i)ζ √ 1 − iζ2 − √ 3 , the dominators and numerators When ζ3 6= −iζ1 − i 2ζ2 and ζ4 = − (1+i)ζ 2 2 of Solutions (10) are both the polynomials of the fourth degree with respect to x and t. In this 8
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case, there are two vector rogue waves in each component, as shown in Figs. 3 − 5. In Figs. 3, we show the separated vector rogue wave pair, where the two vector rogue waves of the vector rogue wave pair are separated and appear at almost the same time. As seen in Figs. 3(a) and 3(c), the q1 and q3 components have the similar structures: An eye-shaped rogue wave coexists with a four-petaled rogue wave, and the humps or valleys of the four-petaled rogue wave in the q1 component correspond to the valleys or humps of the four-petaled rogue wave in the q3 component. Fig. 3(b) displays the two eye-shaped rogue waves with different amplitudes in the q2 component. When ζ3 = −1, we can observe the interacting vector rogue wave pair, the two vector rogue waves of which interact with each other and merge together, as seen in Figs. 4. When |ζ3 | 0, we can obtain two vector rogue waves with three different structures, as seen in Figs. 5: Fig. 5(a) exhibits a four-petaled rogue wave and an anti-eye-shaped rogue wave with two humps and a valley in the q1 component; Shown in Fig. 5(b) is that the q2 component has two eye-shaped rogue waves; Fig. 5(c) depicts an eye-shaped rogue wave and a four-petaled rogue wave. Two vector rogue waves in Figs. 5 emerge at the different values of t and are separated. Here, the vector rogue wave pair can also be thought as the so-called separated vector rogue wave pair. 2.5
2.5
2.5
2.0
2.0
2.0
1.5
1.0
M
1.5
1.5
ED
0.5
CE
0.5
0.5
0
Figs. 3 Vector rogue wave pair via Solutions (10) with γ =
AC
1.0
(b)
PT
(a)
1.0
(c)
1 20 ,
ζ1 = ζ2 = 1, ζ3 = −i, ζ4 = −i −
√
2.
2.5 2.5
3 2.0
2.0 2
1.5 1.5
1.0
1.0
1 0.5
0
0.5
0
(a)
(b)
Figs. 4 The same as Figs. 3 except that ζ3 = −1, ζ4 = −i.
9
0
(c)
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3 2.5 1.5 2.0 2 1.5
1.0
1.0 1 0.5 0.5
0
0
0
(b)
Figs. 5 The same as Figs. 3 except that ζ3 = 100, ζ4 = −i −
(c)
1+i √ 2
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(a)
√ − (50 + 50i) 2.
3.3 Triple vector rogue waves
AC
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PT
ED
M
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(1+i)ζ3 √ 1 − iζ2 − √ When ζ4 6= − (1+i)ζ , the dominators and numerators of Solutions (10) are 2 2 both the polynomials of the sixth degree with respect to x and t. In such case, we can obtain the triple vector rogue wave, as shown in Figs. 6 and 7. Figs. 6 depict the three vector rogue waves with ζ3 = −i: In the q1 component, a four-petaled rogue wave emerges first, then an eye-shaped rogue wave and a four-petaled rogue wave appear at almost the same time, as seen in Fig. 6(a); In the q2 component, as shown in Fig. 6(b), a four-petaled rogue wave takes place first, and then the two eye-shaped rogue waves with the different amplitudes occur; As seen in Fig. 6(c), in the q3 component, an eye-shaped rogue wave occurs first, then a four-petaled and an anti-eye-shaped rogue wave appear. Triple vector rogue wave in Figs. 6 consists of three separated vector rogue waves, which can be considered as the separated triple vector rogue wave. Figs. 7 display the interacting triple vector rogue wave, the three vector rogue waves of which interact with each other and merge together. In the q1 and q3 components, two four-petaled rogue waves coexist with an eye-shaped rogue wave, as shown in Figs. 7(a) and 7(c). Triple vector rogue wave looks like a type of the second-order rogue wave with three single rogue wave [36, 37]. However, the distribution structures of the triple vector rogue wave obtained in this paper are different from the second-order rogue wave with three single rogue waves, because the three single rogue waves of the the triple vector rogue wave have different shapes. 3
2.5 3 2.0
2 1.5
2
1.0 1
1 0.5
0
0
10
0
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(a)
(b)
(c)
√ Figs. 6 Triple vector rogue wave with the same parameters as Figs. 3 except that ζ4 = − 2. 3.5
3.5
3.0
3.0
2.0
2.5
2.5 1.5 2.0
1.5
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2.0
1.0
1.0 0.5 0.5 0
(a)
(b)
1.5
1.0
0.5
(c)
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Figs. 7 The same as Figs. 6 except that ζ3 = 0.
4. The MI for Eqs. (1)
M
In this section, according to Seed Solutions (4), we will study the MI for Eqs. (1) through the linear stability analysis. Incorporating the perturbation term into Seed Solutions (4), we obtain √
ED
1 3 q1 = [a1 + u1 ]e 4 ia1 (115a1 γt+18a1 t−2 2x) , 1 2 a1 2 q2 = √ + u2 e 2 ia1 t(67a1 γ+10) , 2 √ 1 3 q = [a + u ]e 4 ia1 (115a1 γt+18a1 t+2 2x) ,
1
3
(11b) (11c)
PT
3
(11a)
CE
where is a small parameter, and uα ’s (α = 1, 2, 3) are the functions with respect to x and t. Substituting Solutions (11) into Eqs. (1), we can acquire the linearized disturbance equations of uα ’s at O(), expressed as
AC
√ √ √ iu1,t − ia31 γ(u∗2,x + 2 2u∗3,x + 15 2u1,x + 3u2,x ) + a21 (2γu∗1,xx + 2γu∗2,xx + 2γu∗3,xx √ √ √ +11γu1,xx + 2 2γu2,xx + 4γu3,xx + 2u∗2 + 2u∗3 + 2u1 + 2u2 + 2u3 + 2u∗1 ) √ √ −i 2a1 (u1,x + 2γu1,xxx ) + u1,xx + γu1,xxxx + 2a41 γ(12u∗1 + 7 2u∗2 √ +14u∗3 + 12u1 + 7 2u2 + 14u3 ) = 0, (12a) √ 2 ∗ √ 2 ∗ 3 ∗ 3 ∗ 2 ∗ 3 iu2,t + ia1 γu1,x − ia1 γu3,x + 2a1 γu1,xx + a1 γu2,xx + 2a1 γu3,xx − 3ia1 γu1,x √ √ +3ia31 γu3,x + 2 2a21 γu1,xx + 12a21 γu2,xx + 2 2a21 γu3,xx + u2,xx + γu2,xxxx √ √ √ √ +14 2a41 γu∗1 + 15a41 γu∗2 + 14 2a41 γu∗3 + 2a21 u∗1 + a21 u∗2 + 2a21 u∗3 √ √ √ +14 2a41 γu3 + 2a21 u1 (14a21 γ + 1) + u2 (15a41 γ + a21 ) + 2a21 u3 = 0, (12b) 11
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√ √ √ iu3,t + ia31 (2 2γu∗1,x + 3γu2,x + 15 2γu3,x + γu∗2,x ) + a21 (2γu∗1,xx + 2γu∗2,xx + 2γu∗3,xx √ √ √ +4γu1,xx + 2 2γu2,xx + 11γu3,xx + 2u∗1 + 2u∗2 + 2u∗3 + 2u1 + 2u2 + 2u3 ) √ √ +i 2a1 (u3,x + 2γu3,xxx ) + u3,xx + γu3,xxxx + 2a41 γ(14u∗1 + 7 2u∗2 + 14u1 √ +7 2u2 + 12u3 + 12u∗3 ) = 0, (12c) the solutions of which can be supposed as (13)
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uα = uα,1 (t)eikx + uα,2 (t)e−ikx ,
where uα ’s are the t-periodic functions, k is the frequency of perturbation, uα,1 ’s and uα,2 ’s are the coefficients of linear combination. Substituting Solutions (13) into Eqs. (12), we derive that
the coefficient matrix M can be expressed as M13 M14 M15 M16 M23 M24 M25 M26 M33 M34 M35 M36 , M43 M44 M45 M46 M53 M54 M55 M56 M63 M64 M65 M66
M
where Ω = (u11 , u∗12 , u21 , u∗22 , u31 , u∗32 )T , and M11 M12 M21 M22 M31 M32 M = M 41 M42 M51 M52 M61 M62
(14)
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Ωt = M Ω,
AC
CE
PT
ED
with Mm,n ’s (m, n = 1, 2, 3, 4, 5, 6) given in Appendix B. Solving the characteristic equation of the coefficient matrix M , we can achieve six roots as follows: q Γ1 = k 2a21 − k 2 γk 2 − 11a21 γ − 1 , (15a) q Γ2 = −k 2a21 − k 2 γk 2 − 11a21 γ − 1 , (15b) q √ (15c) Γ3 = Γ21,2 + 2a21 γk 4 (46a21 γ − 5γk 2 + 4) − 2 2Λ, q √ Γ4 = − Γ21,2 + 2a21 γk 4 (46a21 γ − 5γk 2 + 4) − 2 2Λ, (15d) q √ Γ5 = Γ21,2 + 2a21 γk 4 (46a21 γ − 5γk 2 + 4) + 2 2Λ, (15e) q √ Γ6 = − Γ21,2 + 2a21 γk 4 (46a21 γ − 5γk 2 + 4) + 2 2Λ, (15f) with
Λ = a1 k 3 143a41 γ 2 + 3a21 γ 8 − 11γk 2 + 2γ 2 k 4 − 3γk 2 + 1 .
The MI with respect to Seed Solutions (4) for Eqs. (1) occurs when the characteristic equation of M has a negative imaginary root. Based on Expressions (15), we can conclude that: If 12
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√ √ 2a21 −k 2 > 0 and Γ21,2 +2a21 γk 4 (46a21 γ − 5γk 2 + 4) > max{2 2Λ, −2 2Λ}, the characteristic equation of M has six real roots and no MI occurs; Otherwise, the characteristic equation of M has at least one pair of complex conjugate roots, and the baseband MI for Eqs. (1) takes place. It can be seen that if the MI for Eqs. (1) occurs, it is of baseband type. 5. Conclusions
CE
PT
ED
M
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In conclusion, investigation has been made on the three-coupled fourth-order nonlinear Schr¨odinger equations, i.e., Eqs. (1), which describe the dynamics of an alpha helical protein with nearest and next-nearest neighbour interactions and interspine coupling. Via DDT (3), the vector multi-rogue-wave solutions for Eqs. (1). i.e., Solutions (10), have been derived. Based on Solutions (10), single vector rogue wave, vector rogue wave pair and triple vector rogue wave have been presented graphically: Figs. 1 have shown the single vector rogue wave: In the q1 and q3 components, a fourpetaled rogue wave with two humps and two valleys appears, while in the q2 component, an eye-shaped rogue wave with one hump and two valleys can be observed. Decreasing the value of the strength of higher-order linear and nonlinear effects, γ, we have observed that the range of the rogue wave along the t axis increases, which means that the existing time of the rogue wave increases, as shown in Figs. 2. In Figs. 3, we have shown the separated vector rogue wave pair, the two vector rogue waves of which are separated and appear at almost the same time. We have observed the interacting vector rogue wave pair, the two vector rogue waves of which interact with each other and merge together, as seen in Figs. 4. In Figs. 5, we have observed the separated vector rogue wave pair, where the two vector rogue waves emerge at the different values of t. Figs. 6 have depicted the separated triple vector rogue wave which consists of three separated vector rogue waves. Figs. 7 have displayed the interacting triple vector rogue wave, the three vector rogue waves of which interact with each other and merge together. Appendix A
AC
The expressions of ψ1 , ψ2 , ψ3 and ψ4 can be obtained as 1 ψ1 = (ψ11 ζ1 + ψ12 ζ2 + ψ13 ζ3 + ψ14 ζ4 )e−x−i(27γ+11)t , 6 ψ11 = 2{−12(22γ + 1)t2 (70γ + 22γx + x + 3) + 8i(22γt + t)3 ψ12
−6it[122γ + (22γ + 1)x2 + (114γ + 5)x + 5] + x3 + 6x2 + 9x + 3}, √ = 2{−12(22γ + 1)t2 [(70 + 48i)γ + (1 + i)(22γ + 1)x + (3 + 2i)] −(8 − 8i)(22γt + t)3 − 6it{(100 + 30i)γ + (1 + i)(22γ + 1)x2
+[(114 + 70i)γ + (5 + 3i)]x + (4 + i)} + x[(1 + i)x2 + (6 + 3i)x + 6]}, 13
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ψ13 = 2[−12(22γ + 1)t2 (48γ + 22γx + x + 2) + 8i(22γt + t)3 ψ14
−6it(x + 2)(26γ + 22γx + x + 1) + x(x2 + 3x + 3)], √ = 2{12i(22γ + 1)t2 [(48 + 70i)γ + (1 + i)(22γ + 1)x + (2 + 3i)] +(8 + 8i)(22γt + t)3 − 6t{(30 + 100i)γ + (1 + i)(22γ + 1)x2
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+[(70 + 114i)γ + (3 + 5i)]x + (1 + 4i)} + x((1 − i)x2 + (6 − 3i)x + 6)}, 1 ψ2 = (ψ21 ζ1 + ψ22 ζ2 + ψ23 ζ3 + ψ24 ζ4 )e2i(44γ−1)t+(−1−i)x , 6√ ψ21 = 2{12(22γ + 1)t2 [(70 + 48i)γ + (1 + i)(22γ + 1)x + (3 + 2i)] +(8 − 8i)(22γt + t)3 + 6it{(100 + 30i)γ + (1 + i)(22γ + 1)x2
+[(114 + 70i)γ + (5 + 3i)]x + (4 + i)} − x[(1 + i)x2 + (6 + 3i)x + 6]},
ψ22 = 2{12i(22γ + 1)t2 [(48 − 22i)γ + 22γx + x + (2 − i)] + 8(22γt + t)3 −6t{(8 − 48i)γ + (22γ + 1)x2 + [(70 − 44i)γ + (3 − 2i)]x − 2i}
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ψ23
−ix3 − (3 + 3i)x2 − (3 − 3i)x + 3}, √ = 2{12(22γ + 1)t2 [(48 + 26i)γ + (1 + i)(22γ + 1)x + (2 + i)] +(8 − 8i)(22γt + t)3 + 6it{(30 + 4i)γ + (1 + i)(22γ + 1)x2
+[(70 + 26i)γ + (3 + i)]x + 1} − x2 [3 + (1 + i)x]},
ψ24 = 2{12(22γ + 1)t2 (48γ + 22γx + x + 2) − 8i(22γt + t)3
+6it(x + 2)(26γ + 22γx + x + 1) − x(x2 + 3x + 3)], √ = 2{12(22γ + 1)t2 [(48 + 26i)γ + (1 + i)(22γ + 1)x + (2 + i)]
PT
ψ32
ED
M
+6it[30γ + (22γ + 1)x2 + (70γ + 3)x + 1] − x2 (x + 3)}, 1 ψ3 = (ψ31 ζ1 + ψ32 ζ2 + ψ33 ζ3 + ψ34 ζ4 )e−x+i(107γ−1)t , 6 ψ31 = 2[12(22γ + 1)t2 (48γ + 22γx + x + 2) − 8i(22γt + t)3
+(8 − 8i)(22γt + t)3 + 6it{(30 + 4i)γ + (1 + i)(22γ + 1)x2
CE
+[(70 + 26i)γ + (3 + i)]x + 1} − x2 [3 + (1 + i)x]},
ψ33 = 2{12(22γ + 1)t2 (26γ + 22γx + x + 1) − 8i(22γt + t)3
AC
ψ34
+6it[26γ + (22γ + 1)x2 + 26γx + x + 1] − x3 − 3x + 3}, √ = 2{−12i(22γ + 1)t2 [(26 + 48i)γ + (1 + i)(22γ + 1)x + (1 + 2i)] −(8 + 8i)(22γt + t)3 + 6t{(4 + 30i)γ + (1 + i)(22γ + 1)x2
+[(26 + 70i)γ + (1 + 3i)]x + i} + x2 [−3 − (1 − i)x]}, 1 ψ4 = (ψ41 ζ1 + ψ42 ζ2 + ψ43 ζ3 + ψ44 ζ4 )e2i(44γ−1)t+(−1+i)x , 6√ ψ41 = 2{−12i(22γ + 1)t2 [(48 + 70i)γ + (1 + i)(22γ + 1)x + (2 + 3i)] −(8 + 8i)(22γt + t)3 + 6t{(30 + 100i)γ + (1 + i)(22γ + 1)x2
+[(70 + 114i)γ + (3 + 5i)]x + (1 + 4i)} + x[−(1 − i)x2 − (6 − 3i)x − 6]}, 14
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ψ42 = 2{12(22γ + 1)t2 (48γ + 22γx + x + 2) − 8i(22γt + t)3 ψ43
+6it[30γ + (22γ + 1)x2 + (70γ + 3)x + 1] − x2 (x + 3)}, √ = 2{−12i(22γ + 1)t2 [(26 + 48i)γ + (1 + i)(22γ + 1)x + (1 + 2i)] −(8 + 8i)(22γt + t)3 + 6t{(4 + 30i)γ + (1 + i)(22γ + 1)x2
+[(26 + 70i)γ + (1 + 3i)]x + i} + x2 (−3 − (1 − i)x)},
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ψ44 = 2{−12i(22γ + 1)t2 [(48 + 22i)γ + 22γx + x + (2 + i)] − 8(22γt + t)3 +6t{(8 + 48i)γ + (22γ + 1)x2 + [(70 + 44i)γ + (3 + 2i)]x + 2i} +ix3 − (3 − 3i)x2 − (3 + 3i)x + 3}. Appendix B
AC
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PT
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The expressions of Mm,n ’s (m, n = 1, 2, 3, 4, 5, 6) can be written as h √ √ i M11 = i 24a41 γ + 15 2a31 γk + a21 2 − 11γk 2 + 2a1 k 1 − 2γk 2 + k 2 γk 2 − 1 , M12 = −M21 = 2ia21 12a21 γ − γk 2 + 1 , h √ √ i M13 = M31 = ia21 14 2a21 γ + 3a1 γk + 2 1 − 2γk 2 , h √ √ i M14 = −M41 = ia21 14 2a21 γ + a1 γk + 2 1 − γk 2 , M15 = M51 = 2ia21 14a21 γ − 2γk 2 + 1 , √ 2 2 2 M16 = −M61 = 2ia1 14a1 γ + 2a1 γk − γk + 1 , h √ √ i M22 = −i 24a41 γ − 15 2a31 γk + a21 2 − 11γk 2 + 2a1 k 2γk 2 − 1 + k 2 γk 2 − 1 , h √ √ i M23 = −M32 = −ia21 14 2a21 γ − a1 γk + 2 1 − γk 2 , h √ √ i M24 = M42 = −ia21 14 2a21 γ − 3a1 γk + 2 1 − 2γk 2 , √ M25 = −M52 = −2ia21 14a21 γ − 2a1 γk − γk 2 + 1 , M26 = M62 = −2ia21 14a21 γ − 2γk 2 + 1 , M33 = i 15a41 γ − 12a21 γk 2 + a21 + γk 4 − k 2 , M34 = −M43 = ia21 15a21 γ − γk 2 + 1 , h √ √ i M35 = M53 = ia21 14 2a21 γ − 3a1 γk + 2 1 − 2γk 2 , h √ √ i M36 = −M63 = ia21 14 2a21 γ + a1 γk + 2 1 − γk 2 , M44 = −i 15a41 γ − 12a21 γk 2 + a21 + γk 4 − k 2 , h √ √ i 2 2 2 M45 = −M54 = −ia1 14 2a1 γ − a1 γk + 2 1 − γk , h √ i √ M46 = M64 = −ia21 14 2a21 γ + 3a1 γk + 2 1 − 2γk 2 , 15
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h √ √ i M55 = i 24a41 γ − 15 2a31 γk + a21 2 − 11γk 2 + 2a1 k 2γk 2 − 1 + k 2 γk 2 − 1 , M56 = −M65 = 2ia21 12a21 γ − γk 2 + 1 , h √ 3 i √ 2 2 4 2 2 2 M66 = −i 24a1 γ + 15 2a1 γk + a1 2 − 11γk + 2a1 k 1 − 2γk + k γk − 1 .
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Acknowledgments
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This work has been supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023 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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