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Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg–de Vries equations
Journal Pre-proof Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg–de Vries equations Rusuo Ye, Yi Zhang, Qinyu Zhang, Xiaotong Chen
PII: DOI: Reference:
S0165-2125(19)30191-X https://doi.org/10.1016/j.wavemoti.2019.102425 WAMOT 102425
To appear in:
Wave Motion
Received date : 22 May 2019 Revised date : 18 September 2019 Accepted date : 19 September 2019 Please cite this article as: R. Ye, Y. Zhang, Q. Zhang et al., Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg–de Vries equations, Wave Motion (2019), doi: https://doi.org/10.1016/j.wavemoti.2019.102425. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg-de Vries equations Rusuo Yea , Yi Zhanga,∗, Qinyu Zhanga , Xiaotong Chena Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China
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Abstract
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With the aid of a generalized Darboux transformation (DT), we derive a hierarchy of rogue wave solutions to the coupled complex modified Korteweg-de Vries (mKdV) equations. Based on modulation instability (MI), two types of n-th order rational rogue wave solutions in compact determinant forms are presented. Especially, the rational rogue wave solutions up to the secondorder are performed explicitly and graphically. We find that there exist typical bright-dark
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composites and rogue wave doublets in first-order case and four or six fundamental rogue waves in second-order case. With appropriate choices of free parameters, the distribution shapes with four fundamental rogue waves admit triangular, quadrilateral, and line structures. Furthermore, triangular, ring, quadrilateral and line patterns can emerge with six fundamental rogue waves. Moreover, we exhibit the first-order semi-rational rogue wave solutions which can demonstrate the coexistence of one rational rogue wave and one breather. Our results can be applicable to the study of rogue wave manifestations in nonlinear optics.
Keywords: coupled complex mKdV equations, modulation instability, rational rogue waves,
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1. Introduction
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Darboux transformation, dynamics
Rogue waves (or freak waves), which are originally coined to describe short-lived gigantic surface gravity waves in the deep ocean, have been studied in various realms of science, nonlinear optical, Bose-Einstein condensates, versatile lasers, atmosphere and so on [1, 2, 3, 4]. Rogue waves are localized in both space and time, and they appear from nowhere and disappear without a trace, and are always expressed as rational form solutions mathematically. It is believed that
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the MI is the fundamental mechanism for the generation of the rogue waves [5, 6, 7]. Generally, ∗ Corresponding
author Email address: [email protected] (Yi Zhang)
Preprint submitted to Journal of LATEX Templates
September 21, 2019
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Akhmediev breathers or Kuznetsov-Ma solitons with infinite period tend to be rogue waves [8, 9]. It should be pointed out that the coexistence of the rogue wave and breather or soliton can also 10
emerge, which is the so-called semi-rational rogue wave [10, 11, 12, 13]. In recent years, vector rogue waves have been paid more and more attention since rogue waves of the multi-component systems are found to have various striking properties. Ling et al.
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[14] studied on dynamics of high-order rogue waves in the coupled nonlinear Schr¨ odinger (NLS) equations and the distribution patten for vector ones were much abundant than the ones for 15
scalar rogue waves. Wang et al. [15] discussed the coupled NLS equations by the generalized
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DT and many interesting high-order localized waves were shown. Further, Ling and Zhao [16] investigated some different types of complex excitations exactly and analytically based on the solutions and also showed that MI can be used to explain the structure of breathers and rogue wave in a quantitative way for the N -component NLS equations. 20
In this paper, we attempt to consider the following coupled complex mKdV equations [17,
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18, 19, 20, 21]
u1t + u1xxx + 3(|u1 |2 + |u2 |2 )u1x + 3(u1x u∗1 + u2x u∗2 )u1 = 0, u2t + u2xxx + 3(|u1 |2 + |u2 |2 )u2x + 3(u1x u∗1 + u2x u∗2 )u2 = 0,
(1)
where u1 (x, t) and u2 (x, t) are the complex envelopes of two field components and the symbol ∗ represents complex conjugation. Geng et al. [17] constructed algebro-geometric solutions of the coupled mKdV hierarchy associated with a 3 × 3 matrix spectral problem on the theory of 25
algebraic curves. Ma [18] obtained the multiple soliton solutions through a specific Riemann-
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Hilbert problem with an identity jump matrix for a coupled mKdV system. The objective of this paper is to investigate the dynamics of rogue wave solutions in the coupled complex mKdV system through the generalized DT approach. The formation of rogue wave solutions can be
The paper is organized as follows. In section 2, the Lax pair and DT of Eqs.(1) are given. In section 3, we discuss the linear stability of a continuous-wave solution regarding to MI. In section 4, we show that two types of n-th order rational rogue wave solutions. The rich dynamics of the rational and semi-rational rogue wave solutions are revealed. Finally, section 5 concludes the paper.
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explained in terms of MI theoretically.
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2. Lax pair and Darboux transformation The linear eigenvalue problem for Eqs. (1) can be expressed as 1 iλJ + iJP, 2 1 V (λ; P ) = iλ3 J + iλ2 JP + λV1 + V0 , 2
Φx = U (λ; P )Φ,
U (λ; P ) =
(2)
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Φt = V (λ; P )Φ, with
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V1 = iJP 2 + Px , V0 = Px P − P Px + 2iJP 3 − iJPxx , 0 u1 u2 1 0 0 J = 0 −1 0 , P = −u∗1 0 0 , −u∗2 0 0 0 0 −1
where Φ is the vector eigenfunction, λ is the spectral parameter. The compatibility condition Ut − Vx + [U, V ] = 0 exactly gives rise to Eqs. (1).
Φ[1]x = U (λ; P [1])Φ, Φ[1]t = V (λ; P [1])Φ, with V1 [1] = iJP [1]2 + P [1]x ,
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We can convert the system (2) into a new linear system
1 iλJ + iJP [1], 2 1 V (λ; P [1]) = iλ3 J + iλ2 JP [1] + λV1 [1] + V0 [1], 2 U (λ; P [1]) =
V0 [1] = P [1]x P [1] − P [1]P [1]x + 2iJP [1]3 − iJP [1]xx ,
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by the following elementary DT Φ[1] = T Φ,
T =I−
λ1 − λ∗1 Φ1 Φ†1 , λ − λ∗1 Φ†1 Φ1
λ1 − λ∗1 φ1 ψ1∗ , |φ1 |2 + |ψ1 |2 + |χ1 |2 λ1 − λ∗1 u2 [1] = u2 [0] + φ1 χ∗1 , |φ1 |2 + |ψ1 |2 + |χ1 |2
(3)
where Φ1 = (φ1 , ψ1 , χ1 )T is a special solution for system (2) at λ = λ1 and the symbol
†
denotes
Hermitian conjugation. Through the standard iterated steps for the above-mentioned Darboux matrix, we can establish a general n-fold Darboux matrix for Eqs. (1).
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u1 [1] = u1 [0] +
Theorem 1 Suppose that Φi = (φi , ψi , χi )T (i = 1, 2, · · ·, n) are n linearly independent solutions
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of the spectral problem (2), corresponding to λ = λi . Then, the n-fold DT is given by Tn = I − Y M −1 (λI − G)−1 Y † , M Y2† det Y1 0 u1 [n] = u1 [0] − , det(M ) M Y3† det Y1 0 , u2 [n] = u2 [0] − det(M ) 45
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(4)
where Y = (Φ1 , Φ2 , · · ·, Φn ), G = diag(λ∗1 , λ∗2 , · · ·, λ∗n ), M = (Mij )n×n , Mij = the i-th row of Y.
3. MI analysis
Φ†i Φj λj −λ∗ i
and Yi is
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Before deriving the high-order rogue wave solutions, we firstly study the MI of Eqs. (1). We start with the general seed potential functions in the form u1 [0] = c1 eiθ1 , 50
u2 [0] = c2 eiθ2 ,
(5)
where θj = aj x+[a3j −3aj (c21 +c22 )−3(a1 c21 +a2 c22 )]t and aj , cj (j = 1, 2) are all real parameters. By Fourier analysis [22] , the linearized stability of the above plane wave solutions can be achieved. For convenience, we define δ = a1 + a2 and ρ = a1 − a2 . We let c1 = c2 = c without loss of
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generality since choosing different background amplitudes will not generate new vector rogue waves’ structures. However, background frequency differences have important effects on the 55
structures of the vector rogue waves because it can not be erased by any trivial transformation,
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we can set a1 > a2 (i.e., ρ > 0).
To check whether the seed solutions are stable against small perturbations or not, we perturb the solutions with the following form
u1 = u1 [0](1 + q1 ),
u2 = u2 [0](1 + q2 ),
(6)
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where q1 and q2 are weak perturbations. Putting Eq. (6) into Eqs. (1) generates the linearized coupled complex mKdV equations q1t + q1xxx + 3ia1 q1xx + (9c2 − 3a21 )q1x + 6ia1 c2 (q1 + q1∗ ) + 3c2 q2x + 3ic2 δ(q2 + q2∗ ) = 0, q2t + q2xxx + 3ia2 q2xx + (9c2 − 3a22 )q2x + 6ia2 c2 (q2 + q2∗ ) + 3c2 q1x + 3ic2 δ(q1 + q1∗ ) = 0. 4
(7)
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By collecting the Fourier modes, the stability of the solution of the above linearized equations to wavenumber µ can be expressed as follows: ∗ q1 = f+ exp[iµ(x − Ω(t))] + f− exp[−iµ(x − Ω(t)∗ )], ∗ q2 = g+ exp[iµ(x − Ω(t))] + g− exp[−iµ(x − Ω(t)∗ )],
(8)
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where Ω(t) is a non-real function corresponding to instability. Substituting Eqs. (8) into Eqs. (7) and to guarantee the equations about {f+ , f− , g+ , g− } be solvable yields
which leads to
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3 3p 4 4c − 4c2 ρ2 + µ2 ρ2 Ω(t) =9c2 − (δ 2 + ρ2 ) − µ2 + 4 2 q p 3 + δ ρ2 + µ2 − 4c2 − 2 4c4 − 4c2 ρ2 + µ2 ρ2 , 2 2
(9)
(i) For 0 < c2 < ρ2 and µ2 < 4 ρc2 (ρ2 − c2 ), we have
q p −2ρ2 − 2µ2 + 8c2 + 2 (ρ2 − µ2 )(8c2 + ρ2 − µ2 ).
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3p 3 Im(Ω(t)) = −4c4 + 4c2 ρ2 − µ2 ρ2 + δ 2 4
(ii)For c2 ≥ ρ2 and ρ2 < µ2 < ρ2 + 8c2 , we determine q p 3 Im(Ω(t)) = δ 4c2 − ρ2 − µ2 + 2 4c4 − 4c2 ρ2 + µ2 ρ2 . 2
(10)
(11)
For cases (i) and (ii), the nonzero imaginary part of Ω(t) corresponds to linearly unstable modes and the perturbations q1 and q2 grow exponentially with t. This instability is called MI 70
since it yields a spontaneous modulation of the steady state. The existence of the above two
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cases of MI provides evidence for the occurrence of rogue waves in Eqs. (1). In the following section, we will derive two types of n-th order rogue wave solutions based on MI.
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4. n-th order rogue wave solutions
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where
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Substituting seed solutions (6) into 1 0 0 −iθ 1 Φ= 0 e 0 0 0 e−iθ2
Lax pair (2), we can obtain the fundamental solution 1 1 1 β1 eiη1 c c c iη2 (12) ξ −a , β e 2 1 1 ξ2 −a1 ξ3 −a1 c c c β3 eiη3 ξ1 −a2 ξ2 −a2 ξ3 −a2
1 1 ηj = (ξj − λ)x + (ξj3 − 6c2 ξ − 3c2 δ − λ3 )t, 2 2
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(13)
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βj (j = 1, 2, 3) are arbitrary complex constants, and ξj are the three roots of the cubic equation 1 1 ξ 3 − (δ + λ)ξ 2 + (δ 2 − ρ2 − 8c2 + 4λδ)ξ + c2 δ + λ(ρ2 − δ 2 ) = 0. 4 4
(14)
Below, we use a generalized DT scheme depends on the same spectral parameter and give the high-order rogue wave solutions directly without any iteration operation. The general procedure
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to obtain these solutions was presented in [23, 24, 25, 26, 27, 28, 29]. 4.1. The case of 0 < c2 < ρ2
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Under this condition, it can be pointed out that the algebraic equation (14) has no triple roots but allows two equal complex roots ξ1 , ξ2 and one single root ξ3 .
Without loss of generality, we choose a1 = −a2 = c = 2 and the spectral parameter can be written as λ1 = −
where be a small complex parameter. For notational brevity, we introduce eiηj (j = 1, 2, 3). Xj (λ1 ) = ξ 2−2 eiηj , j 2 iηj ξj +2 e
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√ √ √ √ √ 1√ √ 1√ √ 4 4 4 4 4 2( 27 − 3 3) + 2( 27 + 3 3)i + 2 3( 3 + 3)i2 , 2 2
(15)
(16)
In order to obtain adequate rational and semi-rational rogue wave solutions of the Eqs. (1),
we define
(17)
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Ψ1 () = mΞ1 + wΞ2 + lΞ3 , where
m =m1 + m2 2 + m3 4 + · · · + mn 2(n−1) ,
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w =w1 + w2 2 + w3 4 + · · · + wn 2(n−1) , l =l1 + l2 2 + l3 4 + · · · + ln 2(n−1) ,
and Ξ1 = X1 + X2 , Ξ2 = 1 (X1 − X2 ), Ξ3 = X3 with mj , wj , lj (1 ≤ j ≤ n) being 3n independent real constants. We find that the vector function Ψ1 () in Eq. (17) can be expanded around =0 as the following Taylor series
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Ψ1 () = Ψ[1] + Ψ[2] 2 + Ψ[3] 4 + · · · + Ψ[n] 2(n−1) + O(2n ).
(18)
It is proved that rogue wave solutions can be derived by taking limit → 0. Via the iterative generalized DT algorithm, we can have 6
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Theorem 2 The general rational rogue wave solutions for Eqs. (1) with the case 0 < c2 < ρ2 , {lj }j=1,2,··· ,n = 0 (more specifically, a1 = −a2 = c = 2) can be represented as
(19)
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1 u1 [n] = 2eiθ1 (1 − H1 S −1 H2† ), 2 1 u2 [n] = 2eiθ2 (1 − H1 S −1 H3† ), 2 with θ1 = 2x − 40t, θ2 = −2x + 40t, and Hj (j = 1, 2, 3) are 1 × n row vectors given by H1 H2 = [Ψ[1] , Ψ[2] , Ψ[3] , · · ·, Ψ[n] ], H3 and S can be determined by
+∞ X Ψ†1 Ψ1 = Sij (∗ )2(i−1) 2(j−1) . λ∗1 − λ1 i,j=1 95
Based on Eqs.(19), we find that when lj 6= 0, there are the combinations of exponential and
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polynomial function of x and t, which are the so-called semi-rational rogue wave solutions. Here, we only investigate the dynamics of the first-order vector solutions. For example, if one takes l1 = 0, the formation of the bright-dark rogue wave composites is displayed in Fig. 1. If we take l1 = 0.01, the coexistence of one rational rogue wave and one resonant breather is shown in Fig. 2. When we reduce the value |l1 |, the rational rogue wave and the breather will separate
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gradually.
Figure 1: Evolution plot of the first-order rational rogue wave with m1 = 0.1, w1 = 1, l1 = 0.
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4.2. The case of c2 ≥ ρ2
In the following, we pay attention to the special case c2 = ρ2 which is possible for cubic algebra equation (14) to have a triple root. 7
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Figure 2: Evolution plot of the first-order semi-rational rogue wave with m1 = 0.1, w1 = 1, l1 = 0.01.
As a simple example, we assume that a1 = 1, a2 = 12 , c = be choosen as λ2 =
and the spectral parameter can
3 3 √ + i 3(1 + 3 ). 4 4
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Introducing
1 2
Zj (λ2 ) =
eiηj
1 iηj 2(ξj −1) e 1 iηj 2ξj −1 e
and the following special form is given
,
(j = 1, 2, 3),
Ψ2 () = f Θ1 + gΘ2 + hΘ3 , where
(20)
(21)
(22)
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f = f1 + f2 3 + f3 6 + · · · + fn 3(n−1) ,
g = g1 + g2 3 + g3 6 + · · · + gn 3(n−1) ,
and
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h = h1 + h2 3 + h3 6 + · · · + hn 3(n−1) , 1 (Z1 + Z2 + Z3 ), 3√ 3 2 (Z1 + $∗ Z2 + $Z3 ), Θ2 = 3 √ 3 4 Θ3 = 2 (Z1 + $Z2 + $∗ Z3 ), 3
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Θ1 =
where $ = e2πi/3 and fj , gj , hj (1 ≤ j ≤ n) are 3n arbitrary real numbers. Afterwards, expanding 110
the vector function Ψ2 () in Eq. (22) at =0, we obtain Ψ2 () = Φ[1] + Φ[2] 3 + Φ[3] 6 + · · · + Φ[n] 3(n−1) + O(3n ). 8
(23)
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Similarly, through the limit process → 0 and by n-times iteration of the generalized DT, then we have the following theorem. Theorem 3 The general rational rogue wave solutions for Eqs. (1) with the case c2 ≥ ρ2 (more precisely, a1 = 1, a2 = 12 , c = 12 ) take the form
1 iθ1 e (1 − 2K1 Q−1 K2† ), 2 1 u2 [n] = eiθ2 (1 − 2K1 Q−1 K3† ). 2 u1 [n] =
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13 8 t,
θ2 = 12 x − 74 t, and Kj (j = 1, 2, 3) are 1 × n row vectors defined through K1 K2 = [Φ[1] , Φ[2] , Φ[3] , · · ·, Φ[n] ], K3
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Here θ1 = x −
(24)
and Q can be determined by
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+∞ X Ψ†2 Ψ2 = Qij (∗ )3(i−1) 3(j−1) . λ∗2 − λ2 i,j=1
In what follows, we demonstrate the pattern dynamics of the first-order and second-order rational rogue wave solutions.
4.2.1. The first-order vector rational rogue wave solutions
For n = 1, we derive the first-order vector rational rogue wave solutions involving three free parameters f1 , g1 and h1 through the compact formula (24) (see Figs. 3-4). One can observe from Fig. 3 that the first-order rogue wave solutions has multiple humps, which is very different
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from the classical single peak fundamental one. Fig. 4 shows that the rogue wave doublets, which is featured by two standard first-order fundamental rogue waves, appear on the temporal-spatial
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plane.
4.2.2. The second-order vector rational rogue wave solutions 125
For n = 2, the second-order vector rogue wave solutions with six free parameters fj , gj , and hj (j = 1, 2) are derived. It is readily to see that there are two types of rogue wave solutions corresponding to four or six fundamental rogue waves constructed by choosing h1 = 0 and h1 6= 0
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respectively.
When h1 = 0, we can classify rogue wave solutions which possess four fundamental rogue 130
waves into three kinds of patterns, including triangular, quadrilateral and line. When the parameters are chosen by f1 6= 0, g1 6= 0, we exhibit the second-order rogue wave solutions of triangular 9
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pattern which is formed by three rogue waves and one of them are fused by two fundamental rogue waves (see Fig. 5). Additionally, this structure of the triangular can change the shape by varying the parameters. As an example, we show one case for line pattern on the temporal-spatial 135
plane in Fig. 6. When choosing g1 6= 0, f2 6= 0, four fundamental rogue waves distribute forms an quadrilateral pattern in Fig. 7. Notice that quadrilateral structure are rarely obtained by
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other methods. Similar to the triangular, the shape of the quadrilateral can be varied through the parameters as well.
When h1 6= 0, there are mainly four kinds of structures, such as ring, triangular, quadrilateral and line patterns. By letting h1 6= 0 and the other parameters be zero, the fundamental second-
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order rogue wave solutions composed of six fundamental rogue waves are exhibited in Fig. 8. When h1 6= 0, f1 6= 0, g1 6= 0, the six fundamental rogue waves can be arranged with triangular pattern. By setting h1 6= 0, f2 6= 0, it is seen that five fundamental rogue waves distribute around one classical fundamental one in Fig. 9. When h1 6= 0, f1 6= 0, it is displayed that four fundamental rogue waves array with a quadrilateral and a second-order rogue wave in the center
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in Fig. 10. Moreover, the above distribute patterns can be changed by varying the parameters.
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Figure 3: Evolution plot of the first-order rational fundamental rogue wave with f1 = 0, g1 = h1 = 10.
5. Conclusions
In this paper, we have presented two types of n-th order rational rogue wave solutions under
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different relative frequencies for Eqs. (1) based on MI. We find that there are mainly two kinds of rational rogue wave solutions for the second-order corresponding to four fundamental rogue waves and six fundamental ones obtained by setting h1 = 0 and h1 6= 0 respectively. Through changing free parameters, the distribution patterns for Eqs. (1) contain different kinds of shapes, 10
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Figure 4: Evolution plot of the first-order rational rogue wave doublets with f1 = 50, g1 = 0, h1 = 1.
Figure 5: Evolution plot of the second-order rational rogue wave of triangle pattern with f1 = 100, g1 = 10, h1 =
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0, f2 = 0, g2 = 0, h2 = 0.
Figure 6: Evolution plot of the second-order rational rogue wave of line pattern with f1 = 1, g1 = 1, h1 = 0, f2 = 0, g2 = 0, h2 = 1000.
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Figure 7: Evolution plot of the second-order rational rogue wave of quadrilateral pattern 1 with f1 = 0, g1 =
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0.001, h1 = 0, f2 = 100, g2 = 0, h2 = 0.
Figure 8: Evolution plot of the second-order rational rogue wave of fundamental pattern with f1 = 0, g1 = 0, h1 =
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0.01, f2 = 0, g2 = 0, h2 = 0.
Figure 9: Evolution plot of the second-order rational rogue wave of ring pattern with f1 = 0, g1 = 0, h1 = 0.001, f2 = 100, g2 = 0, h2 = 0.
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Figure 10: Evolution plot of the second-order rational rogue wave of quadrilateral pattern 2 with f1 = 1000, g1 = 0, h1 = 0.1, f2 = 0, g2 = 0, h2 = 0.
including triangular, quadrilateral, ring, and line structures. Furthermore, we investigate the dynamics of first-order vector semi-rational rogue wave solutions. Additionally, the second-order vector semi-rational rogue wave solutions and the superposition of vector rogue wave solutions
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can be considered which are omitted here. We believe that these results are meaningful to study the complex rogue wave solutions phenomena.
Acknowledgements
The authors are highly thankful to the referees for their invaluable suggestions and comments 160
for the improvement of this paper. This work is supported by the National Natural Science
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Foundation of China (No.11371326 and No.11975145).
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[19] S. F. Tian, Initial-boundary value problems of the coupled modified Korteweg-de Vries equation on the half-line via the Fokas method, J. Phys. A-Math. Theor. 50 (39) (2017) 395204. [20] T. Tsuchida and M. Wadati. The coupled modified Korteweg-de Vries equations. J. Phys. Soc. Jpn. 67, 1175 (1998). [21] M. Iwao, R. Hirota. Soliton solutions of a coupled modified KdV equations. J. Phys. Soc. Jpn. 66, 577 (1997).
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[22] M. G. Forest, D. W. Mclaughlin, D. J. Muraki, et al, Nonfocusing instabilities in coupled, integrable
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nonlinear Schr¨ odinger PDEs, J. Nonlinear Sci. 10 (3) (2000) 291-331.
[23] B. Guo, L. Ling, Q. P. Liu. Nonlinear Schr¨ odinger equation: generalized Darboux transformation and rogue wave solutions, Phys. Rev. E 85 (2) (2012) 026607. 210
[24] S. Chen, D. Mihalache, Vector rogue waves in the Manakov system: diversity and compossibility,
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Phys. A: Math. Theor. 48 (21) (2015) 215202.
[25] X. Wang, C. Liu, L. Wang, Darboux transformation and rogue wave solutions for the variablecoefficients coupled Hirota equations, J. Math. Anal. Appl. 449 (2) (2017) 1534-1552. [26] T. Xu, W. H. Chan, Y. Chen, Higher-order rogue wave pairs in the coupled cubic-quintic nonlinear 215
Schr¨ odinger equations, Commun. Theor. Phys. 70 (2) (2018) 153-160. [27] Y. Zhang, J. W. Yang, K. W. Chow, et al, Solitons, breathers and rogue waves for the coupled Fokas-Lenells system via Darboux transformation, Nonlin. Anal. Real World Appl. 33 (2017) 237252.
the half-line, Math. Method. Appl. Sci. 41 (13) (2018) 5112-5123.
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[29] B. B. Hu, T. C. Xia, W. X. Ma, The Riemann-Hilbert approach to initial-boundary value problems for integrable coherently coupled nonlinear Schr¨ odinger systems on the half-line, E. Asian J. Appl. Math. 8 (3) (2018) 531-548.
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[28] B. B. Hu, T. C. Xia, A Fokas approach to the coupled modified nonlinear Schr¨ odinger equation on
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Journal Pre-proof Highlights: ● We derive a hierarchy of rogue wave solutions to the coupled complex modified
Korteweg-de
Vries
equations
by
virtue
of
Darboux
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transformation. ● Based on modulation instability, two types of n-th order rational rogue
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wave solutions in compact determinant forms are presented.
● The rich dynamics of the rational and semi-rational rogue wave
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solutions are revealed.
PII: DOI: Reference:
S0165-2125(19)30191-X https://doi.org/10.1016/j.wavemoti.2019.102425 WAMOT 102425
To appear in:
Wave Motion
Received date : 22 May 2019 Revised date : 18 September 2019 Accepted date : 19 September 2019 Please cite this article as: R. Ye, Y. Zhang, Q. Zhang et al., Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg–de Vries equations, Wave Motion (2019), doi: https://doi.org/10.1016/j.wavemoti.2019.102425. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Published by Elsevier B.V.
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Vector rational and semi-rational rogue wave solutions in the coupled complex modified Korteweg-de Vries equations Rusuo Yea , Yi Zhanga,∗, Qinyu Zhanga , Xiaotong Chena Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR China
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Abstract
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With the aid of a generalized Darboux transformation (DT), we derive a hierarchy of rogue wave solutions to the coupled complex modified Korteweg-de Vries (mKdV) equations. Based on modulation instability (MI), two types of n-th order rational rogue wave solutions in compact determinant forms are presented. Especially, the rational rogue wave solutions up to the secondorder are performed explicitly and graphically. We find that there exist typical bright-dark
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composites and rogue wave doublets in first-order case and four or six fundamental rogue waves in second-order case. With appropriate choices of free parameters, the distribution shapes with four fundamental rogue waves admit triangular, quadrilateral, and line structures. Furthermore, triangular, ring, quadrilateral and line patterns can emerge with six fundamental rogue waves. Moreover, we exhibit the first-order semi-rational rogue wave solutions which can demonstrate the coexistence of one rational rogue wave and one breather. Our results can be applicable to the study of rogue wave manifestations in nonlinear optics.
Keywords: coupled complex mKdV equations, modulation instability, rational rogue waves,
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1. Introduction
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Darboux transformation, dynamics
Rogue waves (or freak waves), which are originally coined to describe short-lived gigantic surface gravity waves in the deep ocean, have been studied in various realms of science, nonlinear optical, Bose-Einstein condensates, versatile lasers, atmosphere and so on [1, 2, 3, 4]. Rogue waves are localized in both space and time, and they appear from nowhere and disappear without a trace, and are always expressed as rational form solutions mathematically. It is believed that
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the MI is the fundamental mechanism for the generation of the rogue waves [5, 6, 7]. Generally, ∗ Corresponding
author Email address: [email protected] (Yi Zhang)
Preprint submitted to Journal of LATEX Templates
September 21, 2019
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Akhmediev breathers or Kuznetsov-Ma solitons with infinite period tend to be rogue waves [8, 9]. It should be pointed out that the coexistence of the rogue wave and breather or soliton can also 10
emerge, which is the so-called semi-rational rogue wave [10, 11, 12, 13]. In recent years, vector rogue waves have been paid more and more attention since rogue waves of the multi-component systems are found to have various striking properties. Ling et al.
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[14] studied on dynamics of high-order rogue waves in the coupled nonlinear Schr¨ odinger (NLS) equations and the distribution patten for vector ones were much abundant than the ones for 15
scalar rogue waves. Wang et al. [15] discussed the coupled NLS equations by the generalized
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DT and many interesting high-order localized waves were shown. Further, Ling and Zhao [16] investigated some different types of complex excitations exactly and analytically based on the solutions and also showed that MI can be used to explain the structure of breathers and rogue wave in a quantitative way for the N -component NLS equations. 20
In this paper, we attempt to consider the following coupled complex mKdV equations [17,
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18, 19, 20, 21]
u1t + u1xxx + 3(|u1 |2 + |u2 |2 )u1x + 3(u1x u∗1 + u2x u∗2 )u1 = 0, u2t + u2xxx + 3(|u1 |2 + |u2 |2 )u2x + 3(u1x u∗1 + u2x u∗2 )u2 = 0,
(1)
where u1 (x, t) and u2 (x, t) are the complex envelopes of two field components and the symbol ∗ represents complex conjugation. Geng et al. [17] constructed algebro-geometric solutions of the coupled mKdV hierarchy associated with a 3 × 3 matrix spectral problem on the theory of 25
algebraic curves. Ma [18] obtained the multiple soliton solutions through a specific Riemann-
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Hilbert problem with an identity jump matrix for a coupled mKdV system. The objective of this paper is to investigate the dynamics of rogue wave solutions in the coupled complex mKdV system through the generalized DT approach. The formation of rogue wave solutions can be
The paper is organized as follows. In section 2, the Lax pair and DT of Eqs.(1) are given. In section 3, we discuss the linear stability of a continuous-wave solution regarding to MI. In section 4, we show that two types of n-th order rational rogue wave solutions. The rich dynamics of the rational and semi-rational rogue wave solutions are revealed. Finally, section 5 concludes the paper.
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explained in terms of MI theoretically.
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35
2. Lax pair and Darboux transformation The linear eigenvalue problem for Eqs. (1) can be expressed as 1 iλJ + iJP, 2 1 V (λ; P ) = iλ3 J + iλ2 JP + λV1 + V0 , 2
Φx = U (λ; P )Φ,
U (λ; P ) =
(2)
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Φt = V (λ; P )Φ, with
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V1 = iJP 2 + Px , V0 = Px P − P Px + 2iJP 3 − iJPxx , 0 u1 u2 1 0 0 J = 0 −1 0 , P = −u∗1 0 0 , −u∗2 0 0 0 0 −1
where Φ is the vector eigenfunction, λ is the spectral parameter. The compatibility condition Ut − Vx + [U, V ] = 0 exactly gives rise to Eqs. (1).
Φ[1]x = U (λ; P [1])Φ, Φ[1]t = V (λ; P [1])Φ, with V1 [1] = iJP [1]2 + P [1]x ,
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We can convert the system (2) into a new linear system
1 iλJ + iJP [1], 2 1 V (λ; P [1]) = iλ3 J + iλ2 JP [1] + λV1 [1] + V0 [1], 2 U (λ; P [1]) =
V0 [1] = P [1]x P [1] − P [1]P [1]x + 2iJP [1]3 − iJP [1]xx ,
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by the following elementary DT Φ[1] = T Φ,
T =I−
λ1 − λ∗1 Φ1 Φ†1 , λ − λ∗1 Φ†1 Φ1
λ1 − λ∗1 φ1 ψ1∗ , |φ1 |2 + |ψ1 |2 + |χ1 |2 λ1 − λ∗1 u2 [1] = u2 [0] + φ1 χ∗1 , |φ1 |2 + |ψ1 |2 + |χ1 |2
(3)
where Φ1 = (φ1 , ψ1 , χ1 )T is a special solution for system (2) at λ = λ1 and the symbol
†
denotes
Hermitian conjugation. Through the standard iterated steps for the above-mentioned Darboux matrix, we can establish a general n-fold Darboux matrix for Eqs. (1).
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u1 [1] = u1 [0] +
Theorem 1 Suppose that Φi = (φi , ψi , χi )T (i = 1, 2, · · ·, n) are n linearly independent solutions
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of the spectral problem (2), corresponding to λ = λi . Then, the n-fold DT is given by Tn = I − Y M −1 (λI − G)−1 Y † , M Y2† det Y1 0 u1 [n] = u1 [0] − , det(M ) M Y3† det Y1 0 , u2 [n] = u2 [0] − det(M ) 45
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(4)
where Y = (Φ1 , Φ2 , · · ·, Φn ), G = diag(λ∗1 , λ∗2 , · · ·, λ∗n ), M = (Mij )n×n , Mij = the i-th row of Y.
3. MI analysis
Φ†i Φj λj −λ∗ i
and Yi is
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Before deriving the high-order rogue wave solutions, we firstly study the MI of Eqs. (1). We start with the general seed potential functions in the form u1 [0] = c1 eiθ1 , 50
u2 [0] = c2 eiθ2 ,
(5)
where θj = aj x+[a3j −3aj (c21 +c22 )−3(a1 c21 +a2 c22 )]t and aj , cj (j = 1, 2) are all real parameters. By Fourier analysis [22] , the linearized stability of the above plane wave solutions can be achieved. For convenience, we define δ = a1 + a2 and ρ = a1 − a2 . We let c1 = c2 = c without loss of
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generality since choosing different background amplitudes will not generate new vector rogue waves’ structures. However, background frequency differences have important effects on the 55
structures of the vector rogue waves because it can not be erased by any trivial transformation,
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we can set a1 > a2 (i.e., ρ > 0).
To check whether the seed solutions are stable against small perturbations or not, we perturb the solutions with the following form
u1 = u1 [0](1 + q1 ),
u2 = u2 [0](1 + q2 ),
(6)
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where q1 and q2 are weak perturbations. Putting Eq. (6) into Eqs. (1) generates the linearized coupled complex mKdV equations q1t + q1xxx + 3ia1 q1xx + (9c2 − 3a21 )q1x + 6ia1 c2 (q1 + q1∗ ) + 3c2 q2x + 3ic2 δ(q2 + q2∗ ) = 0, q2t + q2xxx + 3ia2 q2xx + (9c2 − 3a22 )q2x + 6ia2 c2 (q2 + q2∗ ) + 3c2 q1x + 3ic2 δ(q1 + q1∗ ) = 0. 4
(7)
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By collecting the Fourier modes, the stability of the solution of the above linearized equations to wavenumber µ can be expressed as follows: ∗ q1 = f+ exp[iµ(x − Ω(t))] + f− exp[−iµ(x − Ω(t)∗ )], ∗ q2 = g+ exp[iµ(x − Ω(t))] + g− exp[−iµ(x − Ω(t)∗ )],
(8)
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where Ω(t) is a non-real function corresponding to instability. Substituting Eqs. (8) into Eqs. (7) and to guarantee the equations about {f+ , f− , g+ , g− } be solvable yields
which leads to
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3 3p 4 4c − 4c2 ρ2 + µ2 ρ2 Ω(t) =9c2 − (δ 2 + ρ2 ) − µ2 + 4 2 q p 3 + δ ρ2 + µ2 − 4c2 − 2 4c4 − 4c2 ρ2 + µ2 ρ2 , 2 2
(9)
(i) For 0 < c2 < ρ2 and µ2 < 4 ρc2 (ρ2 − c2 ), we have
q p −2ρ2 − 2µ2 + 8c2 + 2 (ρ2 − µ2 )(8c2 + ρ2 − µ2 ).
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3p 3 Im(Ω(t)) = −4c4 + 4c2 ρ2 − µ2 ρ2 + δ 2 4
(ii)For c2 ≥ ρ2 and ρ2 < µ2 < ρ2 + 8c2 , we determine q p 3 Im(Ω(t)) = δ 4c2 − ρ2 − µ2 + 2 4c4 − 4c2 ρ2 + µ2 ρ2 . 2
(10)
(11)
For cases (i) and (ii), the nonzero imaginary part of Ω(t) corresponds to linearly unstable modes and the perturbations q1 and q2 grow exponentially with t. This instability is called MI 70
since it yields a spontaneous modulation of the steady state. The existence of the above two
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cases of MI provides evidence for the occurrence of rogue waves in Eqs. (1). In the following section, we will derive two types of n-th order rogue wave solutions based on MI.
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4. n-th order rogue wave solutions
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where
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Substituting seed solutions (6) into 1 0 0 −iθ 1 Φ= 0 e 0 0 0 e−iθ2
Lax pair (2), we can obtain the fundamental solution 1 1 1 β1 eiη1 c c c iη2 (12) ξ −a , β e 2 1 1 ξ2 −a1 ξ3 −a1 c c c β3 eiη3 ξ1 −a2 ξ2 −a2 ξ3 −a2
1 1 ηj = (ξj − λ)x + (ξj3 − 6c2 ξ − 3c2 δ − λ3 )t, 2 2
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(13)
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βj (j = 1, 2, 3) are arbitrary complex constants, and ξj are the three roots of the cubic equation 1 1 ξ 3 − (δ + λ)ξ 2 + (δ 2 − ρ2 − 8c2 + 4λδ)ξ + c2 δ + λ(ρ2 − δ 2 ) = 0. 4 4
(14)
Below, we use a generalized DT scheme depends on the same spectral parameter and give the high-order rogue wave solutions directly without any iteration operation. The general procedure
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to obtain these solutions was presented in [23, 24, 25, 26, 27, 28, 29]. 4.1. The case of 0 < c2 < ρ2
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Under this condition, it can be pointed out that the algebraic equation (14) has no triple roots but allows two equal complex roots ξ1 , ξ2 and one single root ξ3 .
Without loss of generality, we choose a1 = −a2 = c = 2 and the spectral parameter can be written as λ1 = −
where be a small complex parameter. For notational brevity, we introduce eiηj (j = 1, 2, 3). Xj (λ1 ) = ξ 2−2 eiηj , j 2 iηj ξj +2 e
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√ √ √ √ √ 1√ √ 1√ √ 4 4 4 4 4 2( 27 − 3 3) + 2( 27 + 3 3)i + 2 3( 3 + 3)i2 , 2 2
(15)
(16)
In order to obtain adequate rational and semi-rational rogue wave solutions of the Eqs. (1),
we define
(17)
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Ψ1 () = mΞ1 + wΞ2 + lΞ3 , where
m =m1 + m2 2 + m3 4 + · · · + mn 2(n−1) ,
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w =w1 + w2 2 + w3 4 + · · · + wn 2(n−1) , l =l1 + l2 2 + l3 4 + · · · + ln 2(n−1) ,
and Ξ1 = X1 + X2 , Ξ2 = 1 (X1 − X2 ), Ξ3 = X3 with mj , wj , lj (1 ≤ j ≤ n) being 3n independent real constants. We find that the vector function Ψ1 () in Eq. (17) can be expanded around =0 as the following Taylor series
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Ψ1 () = Ψ[1] + Ψ[2] 2 + Ψ[3] 4 + · · · + Ψ[n] 2(n−1) + O(2n ).
(18)
It is proved that rogue wave solutions can be derived by taking limit → 0. Via the iterative generalized DT algorithm, we can have 6
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Theorem 2 The general rational rogue wave solutions for Eqs. (1) with the case 0 < c2 < ρ2 , {lj }j=1,2,··· ,n = 0 (more specifically, a1 = −a2 = c = 2) can be represented as
(19)
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1 u1 [n] = 2eiθ1 (1 − H1 S −1 H2† ), 2 1 u2 [n] = 2eiθ2 (1 − H1 S −1 H3† ), 2 with θ1 = 2x − 40t, θ2 = −2x + 40t, and Hj (j = 1, 2, 3) are 1 × n row vectors given by H1 H2 = [Ψ[1] , Ψ[2] , Ψ[3] , · · ·, Ψ[n] ], H3 and S can be determined by
+∞ X Ψ†1 Ψ1 = Sij (∗ )2(i−1) 2(j−1) . λ∗1 − λ1 i,j=1 95
Based on Eqs.(19), we find that when lj 6= 0, there are the combinations of exponential and
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polynomial function of x and t, which are the so-called semi-rational rogue wave solutions. Here, we only investigate the dynamics of the first-order vector solutions. For example, if one takes l1 = 0, the formation of the bright-dark rogue wave composites is displayed in Fig. 1. If we take l1 = 0.01, the coexistence of one rational rogue wave and one resonant breather is shown in Fig. 2. When we reduce the value |l1 |, the rational rogue wave and the breather will separate
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gradually.
Figure 1: Evolution plot of the first-order rational rogue wave with m1 = 0.1, w1 = 1, l1 = 0.
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4.2. The case of c2 ≥ ρ2
In the following, we pay attention to the special case c2 = ρ2 which is possible for cubic algebra equation (14) to have a triple root. 7
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Figure 2: Evolution plot of the first-order semi-rational rogue wave with m1 = 0.1, w1 = 1, l1 = 0.01.
As a simple example, we assume that a1 = 1, a2 = 12 , c = be choosen as λ2 =
and the spectral parameter can
3 3 √ + i 3(1 + 3 ). 4 4
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Introducing
1 2
Zj (λ2 ) =
eiηj
1 iηj 2(ξj −1) e 1 iηj 2ξj −1 e
and the following special form is given
,
(j = 1, 2, 3),
Ψ2 () = f Θ1 + gΘ2 + hΘ3 , where
(20)
(21)
(22)
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f = f1 + f2 3 + f3 6 + · · · + fn 3(n−1) ,
g = g1 + g2 3 + g3 6 + · · · + gn 3(n−1) ,
and
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h = h1 + h2 3 + h3 6 + · · · + hn 3(n−1) , 1 (Z1 + Z2 + Z3 ), 3√ 3 2 (Z1 + $∗ Z2 + $Z3 ), Θ2 = 3 √ 3 4 Θ3 = 2 (Z1 + $Z2 + $∗ Z3 ), 3
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Θ1 =
where $ = e2πi/3 and fj , gj , hj (1 ≤ j ≤ n) are 3n arbitrary real numbers. Afterwards, expanding 110
the vector function Ψ2 () in Eq. (22) at =0, we obtain Ψ2 () = Φ[1] + Φ[2] 3 + Φ[3] 6 + · · · + Φ[n] 3(n−1) + O(3n ). 8
(23)
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Similarly, through the limit process → 0 and by n-times iteration of the generalized DT, then we have the following theorem. Theorem 3 The general rational rogue wave solutions for Eqs. (1) with the case c2 ≥ ρ2 (more precisely, a1 = 1, a2 = 12 , c = 12 ) take the form
1 iθ1 e (1 − 2K1 Q−1 K2† ), 2 1 u2 [n] = eiθ2 (1 − 2K1 Q−1 K3† ). 2 u1 [n] =
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13 8 t,
θ2 = 12 x − 74 t, and Kj (j = 1, 2, 3) are 1 × n row vectors defined through K1 K2 = [Φ[1] , Φ[2] , Φ[3] , · · ·, Φ[n] ], K3
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Here θ1 = x −
(24)
and Q can be determined by
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+∞ X Ψ†2 Ψ2 = Qij (∗ )3(i−1) 3(j−1) . λ∗2 − λ2 i,j=1
In what follows, we demonstrate the pattern dynamics of the first-order and second-order rational rogue wave solutions.
4.2.1. The first-order vector rational rogue wave solutions
For n = 1, we derive the first-order vector rational rogue wave solutions involving three free parameters f1 , g1 and h1 through the compact formula (24) (see Figs. 3-4). One can observe from Fig. 3 that the first-order rogue wave solutions has multiple humps, which is very different
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from the classical single peak fundamental one. Fig. 4 shows that the rogue wave doublets, which is featured by two standard first-order fundamental rogue waves, appear on the temporal-spatial
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plane.
4.2.2. The second-order vector rational rogue wave solutions 125
For n = 2, the second-order vector rogue wave solutions with six free parameters fj , gj , and hj (j = 1, 2) are derived. It is readily to see that there are two types of rogue wave solutions corresponding to four or six fundamental rogue waves constructed by choosing h1 = 0 and h1 6= 0
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respectively.
When h1 = 0, we can classify rogue wave solutions which possess four fundamental rogue 130
waves into three kinds of patterns, including triangular, quadrilateral and line. When the parameters are chosen by f1 6= 0, g1 6= 0, we exhibit the second-order rogue wave solutions of triangular 9
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pattern which is formed by three rogue waves and one of them are fused by two fundamental rogue waves (see Fig. 5). Additionally, this structure of the triangular can change the shape by varying the parameters. As an example, we show one case for line pattern on the temporal-spatial 135
plane in Fig. 6. When choosing g1 6= 0, f2 6= 0, four fundamental rogue waves distribute forms an quadrilateral pattern in Fig. 7. Notice that quadrilateral structure are rarely obtained by
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other methods. Similar to the triangular, the shape of the quadrilateral can be varied through the parameters as well.
When h1 6= 0, there are mainly four kinds of structures, such as ring, triangular, quadrilateral and line patterns. By letting h1 6= 0 and the other parameters be zero, the fundamental second-
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order rogue wave solutions composed of six fundamental rogue waves are exhibited in Fig. 8. When h1 6= 0, f1 6= 0, g1 6= 0, the six fundamental rogue waves can be arranged with triangular pattern. By setting h1 6= 0, f2 6= 0, it is seen that five fundamental rogue waves distribute around one classical fundamental one in Fig. 9. When h1 6= 0, f1 6= 0, it is displayed that four fundamental rogue waves array with a quadrilateral and a second-order rogue wave in the center
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in Fig. 10. Moreover, the above distribute patterns can be changed by varying the parameters.
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Figure 3: Evolution plot of the first-order rational fundamental rogue wave with f1 = 0, g1 = h1 = 10.
5. Conclusions
In this paper, we have presented two types of n-th order rational rogue wave solutions under
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different relative frequencies for Eqs. (1) based on MI. We find that there are mainly two kinds of rational rogue wave solutions for the second-order corresponding to four fundamental rogue waves and six fundamental ones obtained by setting h1 = 0 and h1 6= 0 respectively. Through changing free parameters, the distribution patterns for Eqs. (1) contain different kinds of shapes, 10
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Figure 4: Evolution plot of the first-order rational rogue wave doublets with f1 = 50, g1 = 0, h1 = 1.
Figure 5: Evolution plot of the second-order rational rogue wave of triangle pattern with f1 = 100, g1 = 10, h1 =
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0, f2 = 0, g2 = 0, h2 = 0.
Figure 6: Evolution plot of the second-order rational rogue wave of line pattern with f1 = 1, g1 = 1, h1 = 0, f2 = 0, g2 = 0, h2 = 1000.
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Figure 7: Evolution plot of the second-order rational rogue wave of quadrilateral pattern 1 with f1 = 0, g1 =
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0.001, h1 = 0, f2 = 100, g2 = 0, h2 = 0.
Figure 8: Evolution plot of the second-order rational rogue wave of fundamental pattern with f1 = 0, g1 = 0, h1 =
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0.01, f2 = 0, g2 = 0, h2 = 0.
Figure 9: Evolution plot of the second-order rational rogue wave of ring pattern with f1 = 0, g1 = 0, h1 = 0.001, f2 = 100, g2 = 0, h2 = 0.
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Figure 10: Evolution plot of the second-order rational rogue wave of quadrilateral pattern 2 with f1 = 1000, g1 = 0, h1 = 0.1, f2 = 0, g2 = 0, h2 = 0.
including triangular, quadrilateral, ring, and line structures. Furthermore, we investigate the dynamics of first-order vector semi-rational rogue wave solutions. Additionally, the second-order vector semi-rational rogue wave solutions and the superposition of vector rogue wave solutions
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155
can be considered which are omitted here. We believe that these results are meaningful to study the complex rogue wave solutions phenomena.
Acknowledgements
The authors are highly thankful to the referees for their invaluable suggestions and comments 160
for the improvement of this paper. This work is supported by the National Natural Science
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Foundation of China (No.11371326 and No.11975145).
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[28] B. B. Hu, T. C. Xia, A Fokas approach to the coupled modified nonlinear Schr¨ odinger equation on
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p ro
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Journal Pre-proof
Journal Pre-proof Highlights: ● We derive a hierarchy of rogue wave solutions to the coupled complex modified
Korteweg-de
Vries
equations
by
virtue
of
Darboux
of
transformation. ● Based on modulation instability, two types of n-th order rational rogue
p ro
wave solutions in compact determinant forms are presented.
● The rich dynamics of the rational and semi-rational rogue wave
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solutions are revealed.