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Novel rogue waves and dynamics in the integrable pair-transition-coupled nonlinear Schrödinger equation
Applied Mathematics Letters 99 (2020) 105987
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Novel rogue waves and dynamics in the integrable pair-transition-coupled nonlinear Schrödinger equation✩ Xiubin Wang ∗, Bo Han School of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
article
info
Article history: Received 21 June 2019 Received in revised form 21 July 2019 Accepted 24 July 2019 Available online 30 July 2019 Keywords: The integrable pair-transition-coupled nonlinear Schrödinger equation Rogue waves Dynamic analysis
abstract Under investigation in this work is the integrable pair-transition-coupled nonlinear Schrödinger equation, which is an important integrable system. Through the Darboux-dressing transformation, we derive a family of rational solutions describing the extreme events (i.e., rogue waves). This family of solutions contains famous vector rogue waves, bright- and dark-rogue waves, and novel vector unusual freak waves. Moreover, the dynamic behaviors of the rational solutions are discussed with some graphics. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the investigation and solution of rogue waves for the nonlinear integrable systems has become more and more attractive. In 1983, the first rogue wave of the famous nonlinear Schr¨odinger equation was explicitly obtained by Peragrine [1]. After that, a mass of approaches have been proposed for seeking rogue wave solutions [2–19]. Due to the localized wave solutions for the coupled NLS equations are more meaningful and complex than single NLS equation. As a consequence, the foremost purpose of this article is to construct the matter rogue waves for the pair-transition-coupled nonlinear Schr¨odinger (pt-CNLS) equation [20–24], whose form reads ⎧ ( ) 1 ⎪ ⎨ iq1,t + q1,xx + σ |q1 |2 + 2|q2 |2 q1 + σq22 q¯1 = 0, 2 (1.1) ( ) 1 ⎪ ⎩ iq2,t + q2,xx + σ |q2 |2 + 2|q1 |2 q2 + σq 2 q¯2 = 0, 1 2 where q1 = q1 (x, t) and q1 = q2 (x, t) are the complex wave envelopes, σ = ±1 corresponds to attractive (+) or repulsive (−) interactions, respectively, and the overline represents the complex conjugation. In addition, ✩ This work is supported by the National Natural Science Foundation of China under Grant No. 11871180. ∗ Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (B. Han).
https://doi.org/10.1016/j.aml.2019.07.018 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
2
Eq. (1.1) is associated with iq1t = δH/(δ q¯1 ), iq2t = δH/(δ q¯2 ) with the following Hamiltonian [ ( )] ∫ 1 1 +∞ 2 2 4 4 2 2 2 2 2 2 |q1,x | + |q2,x | − 2σ H= |q1 | + |q2 | + 2|q1 | |q2 | + q¯1 q2 + q1 q¯2 dx. 2 −∞ 2
(1.2)
It is well known that Darboux-dressing transformation (DDT) is a powerful and effective approach to seek exact solutions. However, since Eq. (1.1) involves a 4 × 4 matrix spectral problem, the DDT and rogue wave solutions for Eq. (1.1) are rather complicated to find. The research in this work, to the best of the authors’ knowledge, has not been reported so far. The chief purpose of the present work is to seek the rogue-wave solutions of Eq. (1.1) by utilizing the DDT. Besides, the key features of those solutions are graphically discussed. 2. Darboux-dressing transformation The integrable pt-CNLS equation (1.1) with σ = 1 can be found to yield the following Lax pair [21,22,24] Ψx = U Ψ , Ψt = V Ψ , with and
) 1 ( U = iλ (σ4 + Q) , V = λ (iλσ4 + Q) + σ4 Qx − iQ2 , 2 ( ) ( ) † q1 q2 0 O Q= , O= , q2 q1 O 0
(2.1) (2.2) (2.3)
where † denotes the Hermitian conjugation, and Ψ = Ψ (x, t, λ) is a column vector function of the spectral parameter λ, σ4 = diag(1, 1, −1, −1). Based on the loop group method [22,24–26], the following theorem can be established. Theorem 1. The DT reads
( ) ¯ 1 ∇1 ∇† λ1 − λ 1 , T1 = I − ( ) † ¯ λ − λ1 ∇1 ∇1 where ⎛ ⎞ s1 ⎜ s2 ⎟ ⎟ ∇1 = ⎜ ⎝ s3 ⎠ , s4 and the transformations between the q1 , q2 and q1,[1] , q2,[1] read ( ) ( ) ( ) ( ) ¯ 1 s¯1 2 λ1 − λ s3 q1,[1] q1 . = − 2 2 2 2 q2,[1] s4 q2 |s1 | + |s2 | + |s3 | + |s4 |
(2.4)
(2.5)
(2.6)
3. Rogue wave solutions To obtain the rogue wave solutions of the integrable pt-CNLS equation (1.1), we start with the seed solutions of the integrable pt-CNLS equation (1.1) as ( ) 2 2 a1 a2 O= ei(a1 +3a2 )t , (3.1) a2 a1 where a1 and a2 are real numbers. According to the study of [27], we can get the corresponding Q in Lax pair (2.1) as ⎛ ⎞ s1 ⎜ s2 ⎟ ⎟ Ψ =⎜ (3.2) ⎝ s3 ⎠ = AFGZ, s4
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
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Fig. 1. (Color online) Bright–dark rogue wave via solution (3.11) with parameters: µ21 = 1, µ31 = 1, µ41 = 1; (a, b): µ11 = 1; (c, d): µ11 = i.
and
⎛ ⎜ ⎜ A=⎜ ⎝
1 0 0 0
0 1 0 0
0 0 e
2 i(a2 1 +3a2 )t
0
⎞
0 0 0 2
2
ei(a1 +3a2 )t
⎟ ⎟ ⎟, ⎠
(3.3)
with F = eiΘx , G = eiΩt , where Z is a free complex vector. Then it can be checked that the matrices Θ and Ω meet ⎛ ⎞ λ 0 a1 a2 ⎜ 0 λ a2 a1 ⎟ 1 ⎟ Θ =⎜ ⎝ a1 a2 −λ 0 ⎠ , Ω = λΘ − 2 . a2 a1 0 −λ
(3.4)
(3.5)
For simplicity, here we choose a1 = 0, a2 = 1. As mentioned above, F is a 4 × 4 matrix and can be expressed as ⎛ ⎞ Θ1 0 0 Θ2 1 ⎜ 0 Θ1 Θ2 0 ⎟ ⎟, F= ⎜ (3.6) τ ⎝ 0 Θ2 Θ3 0 ⎠ Θ2 0 0 Θ3 where { √ Θ1 = τ cos(τ x) + iλ sin(τ x), τ = λ2 + 1, (3.7) Θ2 = i sin(τ x), Θ3 = τ cos(τ x) − iλ sin(τ x).
4
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
Fig. 2. (Color online) Bright rogue wave via solution (3.11) with parameters:µ11 = 1, µ21 = 1, µ41 = 1. (a, b): µ31 = 2; (c, d): µ31 = 2i.
Following the same method, F is also a 4 × 4 matrix and can be expressed as ⎛
Ω1 1⎜ 0 ⎜ G= ⎝ 0 ξ Ω2
0 Ω1 Ω2 0
0 Ω2 Ω3 0
⎞ Ω2 0 ⎟ ⎟ e−it/2 , 0 ⎠ Ω3
(3.8)
where {
Ω1 = ξ cos(ξt) + iλ2 sin(ξt), Ω2 = i sin(ξt), Ω3 = ξ cos(ξt) + iλ2 sin(ξt), ξ = λτ.
(3.9)
Then by utilizing the seed solution in (3.1), we can get a new breather wave solution of the integrable pt-CNLS equation (1.1). In what follows, using the Taylor expansion formulas
sin(x) =
∞ ∑ n=0
(−1)n
∞ ∑ x2n+1 x2n , cos(x) = (−1)n , (2n + 1)! (2n)! n=0
(3.10)
and choosing λ = i in (3.8) and (3.9), we can construct the first-order rational solution of Eq. (1.1) as follows ( ) ( ) ( ) 2i¯ r1 q1,[1] 0 r3 3it = e − , (3.11) 2 2 2 2 q2,[1] 1 r4 |r1 | + |r2 | + |r3 | + |r4 |
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
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Fig. 3. (Color online) Beak-shaped rogue wave via solution (3.11) with parameters: µ11 = 1, µ21 = 1, µ41 = 1. (a): µ31 = 1.5; (b): µ31 = 5; (c): µ31 = 10.
Fig. 4. (Color online) Beak-shaped rogue wave via solution (3.11) with parameters: µ21 = 100, µ31 = 1000, µ41 = 10. (a, b): µ31 = 10; (c, d): µ31 = −10.
where
⎛ ⎧ ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ F 0 =⎝ ⎪ ⎪ ⎪ ⎪ ⎨ ⎛ ⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎪ Ψ0 = ⎜ ⎝ ⎪ ⎪ ⎩
⎞ ⎛ 1−x 0 0 ix 1 − it ⎜ 0 0 1−x ix 0 ⎟ ⎟ , G0 = ⎜ ⎝ 0 0 ix 1+x 0 ⎠ ix 0 0 1+x −t ⎞ ⎛ ⎞ r1 µ11 ⎜ µ21 ⎟ r2 ⎟ −it/2 ⎟=e ⎟ AF0 G0 Z0 , Z0 = ⎜ ⎝ µ31 ⎠ . r3 ⎠ r4 µ41
0 1 − it −t 0
0 −t 1 + it 0
⎞ −t 0 ⎟ ⎟, 0 ⎠ 1 + it
(3.12)
6
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
This is the first-order rogue wave solution. In order to help readers understand (3.11) better, we examine the first-order rogue waves. Figs. 1–4 show three different kinds of rogue waves, such as bright-rogue waves (Fig. 1(a), (c), bright-rogue wave (Fig. 1(b), (d)), beak-shaped rogue waves (Figs. 3,4)). These results are very helpful for enriching rogue wave phenomena. Acknowledgments We express our sincere thanks to the Editor and Reviewer for their valuable comments. This work is supported by the National Key Research and Development Program of China under Grant No. 2017YFB0202901 and the National Natural Science Foundation of China under Grant No. 11871180. References [1] D.H. Peregrine, Water waves, nonlinear Schr¨ odinger equations and their solutions, J. Aust. Math. Soc. B 25 (1983) 16–43. [2] Y.C. Ma, The perturbed plane-wave solutions of the cubic Schr¨ odinger equation, Stud. Appl. Math. 60 (1979) 43–58. [3] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schr¨ odinger equation, Phys. Rev. E 80 (2009) 026601. [4] B. Guo, L. Ling, Q.P. Liu, Nonlinear Schr¨ odinger equation: generalized Darboux transformation and rogue wave solutions, Phys. Rev. E 85 (2012) 026607. [5] Z. Yan, Vector financial rogue waves, Phys. Lett. A 375 (2011) 4274–4279. [6] Y. Ohta, J. Yang, General high-order rogue waves and their dynamics in the nonlinear Schr¨ odinger equation, Proc. R. Soc. A. 468 (2012) 1716. [7] S. Chen, Z. Yan, The Hirota equation: Darboux transform of the Riemann–Hilbert problem and higher-order rogue waves, Appl. Math. Lett. 95 (2019) 65–71. [8] G. Mu, Z. Qin, R. Grimshaw, Dynamics of rogue waves on a multisoliton background in a vector nonlinear Schr¨ odinger equation, SIAM J. Appl. Math. 75 (2015) 1–20. [9] C.Q. Dai, G.Q. Zhou, J.F. Zhang, Controllable optical rogue waves in the femtosecond regime, Phys. Rev. E 85 (2012) 016603. [10] W.X. Ma, Lump solutions to the Kadomtsev–Petviashvili equation, Phys. Lett. A 379 (2015) 1975–1978. [11] W.X. Ma, Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differ. Equ. 264 (2018) 2633–2659. [12] Z.Z. Lan, Rogue wave solutions for a coupled nonlinear Schr¨ odinger equation in the birefringent optical fiber, Appl. Math. Lett. 98 (2019) 128–134. [13] Z.L. Zhao, L.C. He, Multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa–Fukuyama equation, Appl. Math. Lett. 95 (2019) 114–121. [14] Z.Z. Lan, Multi–soliton solutions for a (2+ 1)–dimensional variable–coefficient nonlinear Schr¨ odinger equation, Appl. Math. Lett. 86 (2018) 243–248; Z. Lan, Solitons, breather and bound waves for a generalized higher-order nonlinear Schr¨ odinger equation in an optical fiber or a planar waveguide, Eur. Phys. J. Plus 132 (2017) 512. [15] Z.Z. Lan, J.J. Su, Solitary and rogue waves with controllable backgrounds for the non-autonomous generalized AB system, Nonlinear Dyn. 96 (2019) 2535–2546; Z.Z. and Hu, W.Q. and Guo, B.L Lan, General propagation lattice boltzmann model for a variable-coefficient compound KdV-Burgers equation, Appl. Math. Model. 73 (2019) 695-714. [16] X.B. Wang, S.F. Tian, T.T. Zhang, Characteristics of the breather and rogue waves in a (2+1)–dimensional nonlinear Schr¨ odinger equation, Proc. Amer. Math. Soc. 146 (2018) 3353–3365. [17] X.B. Wang, S.F. Tian, C.Y. Qin, T.T. Zhang, Dynamics of the breathers, rogue waves and solitary waves in the (2+1)–dimensional Ito equation, Appl. Math. Lett. 68 (2017) 40–47. [18] X.B. Wang. B. Han, The three-component coupled nonlinear Schr¨ odinger equation: Rogue waves on a multi-soliton background and dynamics, Europhys. Lett. 126 (2019) 15001. [19] X.B. Wang, B. Han, Characteristics of rogue waves on a soliton background in a coupled nonlinear Schr¨ odinger equation, Math. Methods Appl. Sci. 42 (2019) 2586–2596. [20] Boris A. Malomed, Bound solitons in coupled nonlinear Schr¨ odinger equations, Phys. Rev. A 45 (1992) R8321. [21] S.F. Tian, Initial–boundary value problems for the general coupled nonlinear Schr¨ odinger equation on the interval via the Fokas method, J. Differ. Equ. 262 (2017) 506–558. [22] G. Zhang, Z. Yan, X.Y. Wen, Modulational instability, beak-shaped rogue waves, multi-dark-dark solitons and dynamics in pair-transition-coupled nonlinear Schr¨ odinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 473 (2017) 201702. [23] S.F. Tian, T.T. Zhang, Long–time asymptotic behavior for the Gerdjikov–Ivanov type of derivative nonlinear Schr¨ odinger equation with time-periodic boundary condition, Proc. Amer. Math. Soc. 146 (2018) 1713–1729. [24] L. Ling, L.C. Zhao, Integrable pair-transition-coupled nonlinear Schr¨ odinger equations, Phys. Rev. E 92 (2015) 022924.
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[25] C.L. Terng, K. Uhlenbeck, B¨ acklund transformations and loop group actions, Comm. Pure Appl. Math. 53 (2000) 1. [26] W.Q. Peng, S.F. Tian, T.T. Zhang, Dynamics of breather waves and higher-order rogue waves in a coupled nonlinear Schr¨ odinger equation, Europhys. Lett. 123 (5) (2018) 50005. [27] G. Gui, Z. Qin, Nth-order rogue waves to nonlinear Schr¨ odinger equation revisited: A variable separation technique, J. Phys. Soc. Japan 83 (2014) 104001.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Novel rogue waves and dynamics in the integrable pair-transition-coupled nonlinear Schrödinger equation✩ Xiubin Wang ∗, Bo Han School of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
article
info
Article history: Received 21 June 2019 Received in revised form 21 July 2019 Accepted 24 July 2019 Available online 30 July 2019 Keywords: The integrable pair-transition-coupled nonlinear Schrödinger equation Rogue waves Dynamic analysis
abstract Under investigation in this work is the integrable pair-transition-coupled nonlinear Schrödinger equation, which is an important integrable system. Through the Darboux-dressing transformation, we derive a family of rational solutions describing the extreme events (i.e., rogue waves). This family of solutions contains famous vector rogue waves, bright- and dark-rogue waves, and novel vector unusual freak waves. Moreover, the dynamic behaviors of the rational solutions are discussed with some graphics. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction In recent years, the investigation and solution of rogue waves for the nonlinear integrable systems has become more and more attractive. In 1983, the first rogue wave of the famous nonlinear Schr¨odinger equation was explicitly obtained by Peragrine [1]. After that, a mass of approaches have been proposed for seeking rogue wave solutions [2–19]. Due to the localized wave solutions for the coupled NLS equations are more meaningful and complex than single NLS equation. As a consequence, the foremost purpose of this article is to construct the matter rogue waves for the pair-transition-coupled nonlinear Schr¨odinger (pt-CNLS) equation [20–24], whose form reads ⎧ ( ) 1 ⎪ ⎨ iq1,t + q1,xx + σ |q1 |2 + 2|q2 |2 q1 + σq22 q¯1 = 0, 2 (1.1) ( ) 1 ⎪ ⎩ iq2,t + q2,xx + σ |q2 |2 + 2|q1 |2 q2 + σq 2 q¯2 = 0, 1 2 where q1 = q1 (x, t) and q1 = q2 (x, t) are the complex wave envelopes, σ = ±1 corresponds to attractive (+) or repulsive (−) interactions, respectively, and the overline represents the complex conjugation. In addition, ✩ This work is supported by the National Natural Science Foundation of China under Grant No. 11871180. ∗ Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (B. Han).
https://doi.org/10.1016/j.aml.2019.07.018 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
2
Eq. (1.1) is associated with iq1t = δH/(δ q¯1 ), iq2t = δH/(δ q¯2 ) with the following Hamiltonian [ ( )] ∫ 1 1 +∞ 2 2 4 4 2 2 2 2 2 2 |q1,x | + |q2,x | − 2σ H= |q1 | + |q2 | + 2|q1 | |q2 | + q¯1 q2 + q1 q¯2 dx. 2 −∞ 2
(1.2)
It is well known that Darboux-dressing transformation (DDT) is a powerful and effective approach to seek exact solutions. However, since Eq. (1.1) involves a 4 × 4 matrix spectral problem, the DDT and rogue wave solutions for Eq. (1.1) are rather complicated to find. The research in this work, to the best of the authors’ knowledge, has not been reported so far. The chief purpose of the present work is to seek the rogue-wave solutions of Eq. (1.1) by utilizing the DDT. Besides, the key features of those solutions are graphically discussed. 2. Darboux-dressing transformation The integrable pt-CNLS equation (1.1) with σ = 1 can be found to yield the following Lax pair [21,22,24] Ψx = U Ψ , Ψt = V Ψ , with and
) 1 ( U = iλ (σ4 + Q) , V = λ (iλσ4 + Q) + σ4 Qx − iQ2 , 2 ( ) ( ) † q1 q2 0 O Q= , O= , q2 q1 O 0
(2.1) (2.2) (2.3)
where † denotes the Hermitian conjugation, and Ψ = Ψ (x, t, λ) is a column vector function of the spectral parameter λ, σ4 = diag(1, 1, −1, −1). Based on the loop group method [22,24–26], the following theorem can be established. Theorem 1. The DT reads
( ) ¯ 1 ∇1 ∇† λ1 − λ 1 , T1 = I − ( ) † ¯ λ − λ1 ∇1 ∇1 where ⎛ ⎞ s1 ⎜ s2 ⎟ ⎟ ∇1 = ⎜ ⎝ s3 ⎠ , s4 and the transformations between the q1 , q2 and q1,[1] , q2,[1] read ( ) ( ) ( ) ( ) ¯ 1 s¯1 2 λ1 − λ s3 q1,[1] q1 . = − 2 2 2 2 q2,[1] s4 q2 |s1 | + |s2 | + |s3 | + |s4 |
(2.4)
(2.5)
(2.6)
3. Rogue wave solutions To obtain the rogue wave solutions of the integrable pt-CNLS equation (1.1), we start with the seed solutions of the integrable pt-CNLS equation (1.1) as ( ) 2 2 a1 a2 O= ei(a1 +3a2 )t , (3.1) a2 a1 where a1 and a2 are real numbers. According to the study of [27], we can get the corresponding Q in Lax pair (2.1) as ⎛ ⎞ s1 ⎜ s2 ⎟ ⎟ Ψ =⎜ (3.2) ⎝ s3 ⎠ = AFGZ, s4
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
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Fig. 1. (Color online) Bright–dark rogue wave via solution (3.11) with parameters: µ21 = 1, µ31 = 1, µ41 = 1; (a, b): µ11 = 1; (c, d): µ11 = i.
and
⎛ ⎜ ⎜ A=⎜ ⎝
1 0 0 0
0 1 0 0
0 0 e
2 i(a2 1 +3a2 )t
0
⎞
0 0 0 2
2
ei(a1 +3a2 )t
⎟ ⎟ ⎟, ⎠
(3.3)
with F = eiΘx , G = eiΩt , where Z is a free complex vector. Then it can be checked that the matrices Θ and Ω meet ⎛ ⎞ λ 0 a1 a2 ⎜ 0 λ a2 a1 ⎟ 1 ⎟ Θ =⎜ ⎝ a1 a2 −λ 0 ⎠ , Ω = λΘ − 2 . a2 a1 0 −λ
(3.4)
(3.5)
For simplicity, here we choose a1 = 0, a2 = 1. As mentioned above, F is a 4 × 4 matrix and can be expressed as ⎛ ⎞ Θ1 0 0 Θ2 1 ⎜ 0 Θ1 Θ2 0 ⎟ ⎟, F= ⎜ (3.6) τ ⎝ 0 Θ2 Θ3 0 ⎠ Θ2 0 0 Θ3 where { √ Θ1 = τ cos(τ x) + iλ sin(τ x), τ = λ2 + 1, (3.7) Θ2 = i sin(τ x), Θ3 = τ cos(τ x) − iλ sin(τ x).
4
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
Fig. 2. (Color online) Bright rogue wave via solution (3.11) with parameters:µ11 = 1, µ21 = 1, µ41 = 1. (a, b): µ31 = 2; (c, d): µ31 = 2i.
Following the same method, F is also a 4 × 4 matrix and can be expressed as ⎛
Ω1 1⎜ 0 ⎜ G= ⎝ 0 ξ Ω2
0 Ω1 Ω2 0
0 Ω2 Ω3 0
⎞ Ω2 0 ⎟ ⎟ e−it/2 , 0 ⎠ Ω3
(3.8)
where {
Ω1 = ξ cos(ξt) + iλ2 sin(ξt), Ω2 = i sin(ξt), Ω3 = ξ cos(ξt) + iλ2 sin(ξt), ξ = λτ.
(3.9)
Then by utilizing the seed solution in (3.1), we can get a new breather wave solution of the integrable pt-CNLS equation (1.1). In what follows, using the Taylor expansion formulas
sin(x) =
∞ ∑ n=0
(−1)n
∞ ∑ x2n+1 x2n , cos(x) = (−1)n , (2n + 1)! (2n)! n=0
(3.10)
and choosing λ = i in (3.8) and (3.9), we can construct the first-order rational solution of Eq. (1.1) as follows ( ) ( ) ( ) 2i¯ r1 q1,[1] 0 r3 3it = e − , (3.11) 2 2 2 2 q2,[1] 1 r4 |r1 | + |r2 | + |r3 | + |r4 |
X. Wang and B. Han / Applied Mathematics Letters 99 (2020) 105987
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Fig. 3. (Color online) Beak-shaped rogue wave via solution (3.11) with parameters: µ11 = 1, µ21 = 1, µ41 = 1. (a): µ31 = 1.5; (b): µ31 = 5; (c): µ31 = 10.
Fig. 4. (Color online) Beak-shaped rogue wave via solution (3.11) with parameters: µ21 = 100, µ31 = 1000, µ41 = 10. (a, b): µ31 = 10; (c, d): µ31 = −10.
where
⎛ ⎧ ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎜ ⎪ F 0 =⎝ ⎪ ⎪ ⎪ ⎪ ⎨ ⎛ ⎪ ⎪ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎪ Ψ0 = ⎜ ⎝ ⎪ ⎪ ⎩
⎞ ⎛ 1−x 0 0 ix 1 − it ⎜ 0 0 1−x ix 0 ⎟ ⎟ , G0 = ⎜ ⎝ 0 0 ix 1+x 0 ⎠ ix 0 0 1+x −t ⎞ ⎛ ⎞ r1 µ11 ⎜ µ21 ⎟ r2 ⎟ −it/2 ⎟=e ⎟ AF0 G0 Z0 , Z0 = ⎜ ⎝ µ31 ⎠ . r3 ⎠ r4 µ41
0 1 − it −t 0
0 −t 1 + it 0
⎞ −t 0 ⎟ ⎟, 0 ⎠ 1 + it
(3.12)
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This is the first-order rogue wave solution. In order to help readers understand (3.11) better, we examine the first-order rogue waves. Figs. 1–4 show three different kinds of rogue waves, such as bright-rogue waves (Fig. 1(a), (c), bright-rogue wave (Fig. 1(b), (d)), beak-shaped rogue waves (Figs. 3,4)). These results are very helpful for enriching rogue wave phenomena. Acknowledgments We express our sincere thanks to the Editor and Reviewer for their valuable comments. This work is supported by the National Key Research and Development Program of China under Grant No. 2017YFB0202901 and the National Natural Science Foundation of China under Grant No. 11871180. References [1] D.H. Peregrine, Water waves, nonlinear Schr¨ odinger equations and their solutions, J. Aust. Math. Soc. B 25 (1983) 16–43. [2] Y.C. Ma, The perturbed plane-wave solutions of the cubic Schr¨ odinger equation, Stud. Appl. Math. 60 (1979) 43–58. [3] N. Akhmediev, A. Ankiewicz, J.M. Soto-Crespo, Rogue waves and rational solutions of the nonlinear Schr¨ odinger equation, Phys. Rev. E 80 (2009) 026601. [4] B. Guo, L. Ling, Q.P. Liu, Nonlinear Schr¨ odinger equation: generalized Darboux transformation and rogue wave solutions, Phys. Rev. E 85 (2012) 026607. [5] Z. Yan, Vector financial rogue waves, Phys. Lett. A 375 (2011) 4274–4279. [6] Y. Ohta, J. Yang, General high-order rogue waves and their dynamics in the nonlinear Schr¨ odinger equation, Proc. R. Soc. A. 468 (2012) 1716. [7] S. Chen, Z. Yan, The Hirota equation: Darboux transform of the Riemann–Hilbert problem and higher-order rogue waves, Appl. Math. Lett. 95 (2019) 65–71. [8] G. Mu, Z. Qin, R. Grimshaw, Dynamics of rogue waves on a multisoliton background in a vector nonlinear Schr¨ odinger equation, SIAM J. Appl. Math. 75 (2015) 1–20. [9] C.Q. Dai, G.Q. Zhou, J.F. Zhang, Controllable optical rogue waves in the femtosecond regime, Phys. Rev. E 85 (2012) 016603. [10] W.X. Ma, Lump solutions to the Kadomtsev–Petviashvili equation, Phys. Lett. A 379 (2015) 1975–1978. [11] W.X. Ma, Y. Zhou, Lump solutions to nonlinear partial differential equations via Hirota bilinear forms, J. Differ. Equ. 264 (2018) 2633–2659. [12] Z.Z. Lan, Rogue wave solutions for a coupled nonlinear Schr¨ odinger equation in the birefringent optical fiber, Appl. Math. Lett. 98 (2019) 128–134. [13] Z.L. Zhao, L.C. He, Multiple lump solutions of the (3+1)-dimensional potential Yu-Toda-Sasa–Fukuyama equation, Appl. Math. Lett. 95 (2019) 114–121. [14] Z.Z. Lan, Multi–soliton solutions for a (2+ 1)–dimensional variable–coefficient nonlinear Schr¨ odinger equation, Appl. Math. Lett. 86 (2018) 243–248; Z. Lan, Solitons, breather and bound waves for a generalized higher-order nonlinear Schr¨ odinger equation in an optical fiber or a planar waveguide, Eur. Phys. J. Plus 132 (2017) 512. [15] Z.Z. Lan, J.J. Su, Solitary and rogue waves with controllable backgrounds for the non-autonomous generalized AB system, Nonlinear Dyn. 96 (2019) 2535–2546; Z.Z. and Hu, W.Q. and Guo, B.L Lan, General propagation lattice boltzmann model for a variable-coefficient compound KdV-Burgers equation, Appl. Math. Model. 73 (2019) 695-714. [16] X.B. Wang, S.F. Tian, T.T. Zhang, Characteristics of the breather and rogue waves in a (2+1)–dimensional nonlinear Schr¨ odinger equation, Proc. Amer. Math. Soc. 146 (2018) 3353–3365. [17] X.B. Wang, S.F. Tian, C.Y. Qin, T.T. Zhang, Dynamics of the breathers, rogue waves and solitary waves in the (2+1)–dimensional Ito equation, Appl. Math. Lett. 68 (2017) 40–47. [18] X.B. Wang. B. Han, The three-component coupled nonlinear Schr¨ odinger equation: Rogue waves on a multi-soliton background and dynamics, Europhys. Lett. 126 (2019) 15001. [19] X.B. Wang, B. Han, Characteristics of rogue waves on a soliton background in a coupled nonlinear Schr¨ odinger equation, Math. Methods Appl. Sci. 42 (2019) 2586–2596. [20] Boris A. Malomed, Bound solitons in coupled nonlinear Schr¨ odinger equations, Phys. Rev. A 45 (1992) R8321. [21] S.F. Tian, Initial–boundary value problems for the general coupled nonlinear Schr¨ odinger equation on the interval via the Fokas method, J. Differ. Equ. 262 (2017) 506–558. [22] G. Zhang, Z. Yan, X.Y. Wen, Modulational instability, beak-shaped rogue waves, multi-dark-dark solitons and dynamics in pair-transition-coupled nonlinear Schr¨ odinger equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 473 (2017) 201702. [23] S.F. Tian, T.T. Zhang, Long–time asymptotic behavior for the Gerdjikov–Ivanov type of derivative nonlinear Schr¨ odinger equation with time-periodic boundary condition, Proc. Amer. Math. Soc. 146 (2018) 1713–1729. [24] L. Ling, L.C. Zhao, Integrable pair-transition-coupled nonlinear Schr¨ odinger equations, Phys. Rev. E 92 (2015) 022924.
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