Modulation instability, rogue waves and spectral analysis for the sixth-order nonlinear Schrodinger equation
MModulation instability, rogue waves and spectral analysis for the sixth-ordernonlinear Schr ¨odinger equation
Yunfei Yue a , b , Lili Huang a , b , Yong Chen a , b , c ∗ a Institute of Computer Theory, East China Normal University, Shanghai, 200062, China b Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, 200062, China c Department of Physics, Zhejiang Normal University, Jinhua, 321004, China
Abstract
Modulation instability, rogue wave and spectral analysis are investigated for the nonlinear Schr¨odinger equation withthe higher-order terms. The modulation instability distribution characteristics from the sixth-order to the eighth-ordernonlinear Schr¨odinger equations are studied. Higher-order dispersion terms are closely related to the distribution of mod-ulation stability regime, and n -order dispersion term corresponds to n − N th-order rogue wave is given. Dynamic phenomena of first-to third-order rogue waves are illustrated, which exhibit meaningful structures. Two arbitrary parameters play importantroles in the rogue wave solution. One can control deflection of crest of rogue wave and its width, while the other can causethe change of width and amplitude of rogue wave. When it comes to the third-order rogue wave, three typical nonlinearwave constructions, namely fundamental, circular and triangular are displayed and discussed. Through the spectral anal-ysis on first-order rogue wave, when these parameters satisfy certain conditions, it occurs a transition between W-shapedsoliton and rogue wave. Keywords:
Nonlinear Schr¨odinger equation; Modulation instability; Rogue wave; Darboux transformation.
1. Introduction
It has been extensive interests in studying rogue waves in recent years. Rogue wave was first put forward conceptuallyin the ocean [1]. Rogue waves are relatively large and spontaneous waves, whose appearance may result in catastrophicdamage [2]. Large amplitude, unexpected, coming out from nowhere without warning and suddenly vanishing awaywithout trace, are the basic characteristics [3]. In general, the nonlinear partial differential equation satisfying the fibercommunication model, for the stable solution, under the interaction of the dispersion term and the nonlinear term, therewill be instability, which is called modulation instability (MI). MI is a basic nonlinear process of exponential increasewith some small perturbations superimposed on the background of continuous waves in nonlinear dispersive media. MIof monochromatic nonlinear waves is a possible cause of the generation of rogue waves, which bursts sporadically morethan the average level of the water surface. It is considered that MI is a most ubiquitous kind of instability in the natureworld [4]. It exists both in the continuum and in the discrete nonlinear wave equations [5]. Since Benjamin and Feir’sgroundbreaking hydrodynamics construction [6], MI has played a prominent role in diverse areas of scientific research,for example, plasma physics [7], nonlinear optics [8], and fluid dynamics [9].In fact, the above mentioned instability can result in self-induced modulation of incoming continuous waves withsubsequent local pulses, which may be discovered in many physical systems. Due to the presence of this phenomenon,there are many interesting physical effects, such as break-up of deep water-gravity waves in the ocean, the formationof envelope solitons in electrical transmission lines and optical fibers, as well as the formation of cavitons in plasmas.Different distributions of the MI gain can lead to distinct pattern of nonlinear dynamic phenomena [10]. The dispersion ∗ Corresponding author.
Email address: [email protected] ( Yong Chen a , b , c ) Preprint submitted to Elsevier August 15, 2019 a r X i v : . [ n li n . S I] A ug erm and the nonlinear term are playing different roles in the nonlinear systems, but both of them affect the instability ofthe solutions for the nonlinear systems. Recently, some literatures have analyzed the importance of high-order dispersionterms, which is not only affect MI [11] but also induce some novel excited states [12, 13]. The study of MI regions innonlinear systems is crucial in many fields and is the basis for interpreting or regulating various models or phenomena indifferent fields.The nonlinear Schr¨odinger (NLS henceforth) equation has a prominent position in nonlinear physics. It has extensivephysical applications, especially in nonlinear optics [14], atmosphere [15], and water waves [16]. In 1983, Peregrine[17] gave the analytical expression of the rogue wave in the first-order as an outcome of MI on the constant wave back-ground. This type of rogue wave also has another name, that is Peregrine breather. In recent years, many authors [18–20]have reported the higher-order rogue wave solutions, some important physical properties and applications for the NLSequation. In addition, various extensions of NLS equation have also been studied, such as pair-transition-coupled NLSequation [21], variable coefficient NLS equation [22–24], three-component NLS equations[25], three-component coupledderivative NLS equations [26], and n -component NLS equations [27]. General high-order solitons of three different typesof nonlocal NLS equations in the reverse-time, PT-symmetric and reverse-space-time were derived by using a Riemann-Hilbert treatment [28].However, there exist only lowest-order terms (dispersion and nonlinearity) in the standard NLS equation [29]. Whenthe characteristics of the solutions exceed the simple approximation in deriving the NLS equation, the higher-order termswill hold the dominate role [30]. For instance, it may help to illustrate the physical properties of wave blow-up and collapse[31]. In 2016, Ankiewicz [32] et al . studied the following form of NLS equation containing higher-order nonlinear termsand dispersion terms, iq z + δ Γ ( q ) − i δ Γ ( q ) + δ Γ ( q ) − i δ Γ ( q ) + δ Γ ( q ) − i δ Γ ( q ) + δ Γ ( q ) + · · · = , (1)with Γ = q tt + q | q | , Γ = q ttt + q q t , Γ = q tttt + q ∗ q t + q | q t | + | q | q tt + q q ∗ tt + | q | q , Γ = q ttttt + | q | q ttt + | q | q t + qq t q ∗ tt + qq ∗ t q tt + q ∗ q t q tt + q t q ∗ t , Γ = q tttttt + q [60 | q t | q ∗ + q tt ( q ∗ ) + q ∗ tttt ] + q [12 q ∗ q tttt + q ∗ t q ttt + q t q ∗ ttt + q ∗ ) q t + | q tt | ] + q t [3 q ∗ q ttt + q ∗ q tt + q ∗ q tt ] + q [2 q ∗ q ∗ tt + ( q ∗ t ) ] + q ∗ q tt + q | q | , Γ = q ttttttt + q tt q ∗ t + q t | q tt | + | q t | q ttt + q { q t [2 q ∗ q ∗ tt + ( q ∗ t ) ] + q ∗ (2 q tt q ∗ t + q ttt q ∗ ) } + q t q ∗ ttt + q [ q ∗ (20 | q t | q t + q ttttt ) + q ttt q ∗ tt + q tt q ∗ ttt + q tttt q ∗ t + q t + q ∗ tttt + q t q tt ( q ∗ ) ] + | q | q t + q t ( q ∗ ) + q tt q ttt + q t q tttt ) q ∗ , Γ = q tttttttt + q [40 | q t | ( q ∗ ) + q ∗ ) q tt + q ∗ q ∗ tttt + q ∗ t q ∗ ttt + q ∗ tt ) ] + q [28 q ∗ (14 | q tt | + q t q ∗ ttt + q ∗ t q ttt + q tt ( q ∗ t ) + | q t | q ∗ tt + q t ( q ∗ ) + q tttt ( q ∗ ) + q ∗ tttttt ] + q { q t [9( q ∗ ) + q ∗ q ∗ tt ] + q t [728 q tt q ∗ t q ∗ + q ttt ( q ∗ ) + q ∗ ttttt ] + | q ttt | + q tttt q ∗ tt + q tt q ∗ tttt + q ttttt q ∗ t + q tt ( q ∗ ) + q tttttt q ∗ } + q tt | q tt | + q tt q ttt q ∗ t + q t q ttt q ∗ tt + q t q tt q ∗ ttt + q t q tttt q ∗ t + q t q ∗ tttt + q ∗ (30 q t q ∗ t + q ttttt q t + q ttt + q tt q tttt ) + q ∗ ) q t q tt + q q ∗ [ q ∗ q ∗ tt + ( q ∗ t ) ] + q | q | , where | q | = | q ( z , t ) | denotes envelope of the optical pulse with spatial coordinate z and scaled time coordinate t , δ i ( i = , , , , · · · ∞ ) represents the i -order real dispersion coefficient. Γ is the Hirota operator [33], Γ is the Lakshmanan-Porsezian-Daniel operator [34], Γ is known as the quintic operator [35], Γ is the sextic operator, Γ is the heptic operator, Γ is the octic operator. With an infinite number of arbitrary coefficients, these extensions are integrable. The arbitrariness2f coefficients enables us to go well beyond the single NLS equation.One of our goals in this paper is to investigate MI of a continuous wave for the NLS equation (1) with differenthigher-order terms. We discuss MI distribution characteristics from the sixth-order NLS equation to the eighth-order NLSequation. Comparing their MI gain functions of NLS equations with different order dispersion terms, it enables us tofind the distribution law of MS curves in the MI band. Then we focus on study rogue waves of the following reducedsixth-order NLS equation [36–38] iq z + δ Γ ( q ) + δ Γ ( q ) = . (2)Nowadays many methods have been developed to investigate rogue waves of the nonlinear systems, such as the Darbouxtransformation (DT) [39–44], Hirota method [45–48], nonlocal symmetry method [49]. Based on the generalized DT,higher-order rogue waves will be generated for Eq. (2). For the parameter of δ , many papers [50–52] choose δ = , thissetting has certain convenient features. Here, we also set δ = in Eq. (2). Via the analytical rational expressions and MIcharacteristics, the dynamics of rogue waves will be studied in detail.It is also our purpose here to investigate how to use the spectral features of the propagating wave envelope to reveal theexistence of nonlinearity and rogue wave in a short time before the occurrence of a special rogue wave event. To establishthe results, we apply the spectral analysis approach [53–57] to the first-order rogue wave solutions of Eq. (2).The remainder of our article is constructed as follows. In Section 2, MI distribution features of the NLS equationwith different higher-order nonlinear and dispersion terms will be discussed according to MI analysis theory. By virtueof the generalized DT of the sixth-order NLS equation, a concrete expression of the N-order rogue wave solutions will begiven in Section 3. In Section 4, Utilizing the expressions obtained in the previous section, the first-order, second-order,and third-order exact rogue wave solutions are presented, where their dynamic behavior are also analyzed. Section 5 isdevoted to spectral analysis on the first-order rogue wave. Finally, some conclusions are given.
2. Modulation instability
MI is observed in a time-averaged way and usually triggered from a continuous wave or quasi-continuous wave. Thecontinuous wave condition is corresponding to an effectively unbounded MI domain. Then it can yield information onaverage behavior of the nonlinear process and the general tendencies for instability, but usually prevents time-resolved ofthe stochastic dynamics. MI symmetry breaking can occur for the reason of higher-order dispersion [58]. MI is the basicmechanism for generating of rogue wave solutions. MI is an interactive gain procedure that generates priority frequencyintervals between patterns [59]. Studied here is the MI analysis on continuous waves for the NLS equation with differenthigher-order dispersion terms, in order to reveal the MI features considering of higher-order dispersion effects. The planewave solution of system (1) has the following form q cw = Ae i θ = Ae i ( kz + ω t ) . (3)There are three real constants, wave number k , amplitude A and frequency of background ω . Substituting Eq. (3) into Eq.(2), it can be obtained that k = A δ − A ω δ + A ω δ − ω δ + A − ω . (4)According to the MI theory, we add a small perturbation function p ( z , t ) to the plane-wave solution. Then a perturbationsolution can be derived as q pert = ( A + (cid:15) p ( z , t )) e i ( kz + ω t ) , (5)where p ( z , t ) = me i ( Kz +Ω t ) + ne − i ( Kz +Ω t ) , Ω indicates the disturbance frequency, and m , n are both small parameters. Substi-tuting the perturbation solution (5) into the sixth-order NLS equation (2), it can generate a system of linear homogeneousequations for m and n . Based on the existence conditions for solutions of linear homogeneous equations, that is, thedeterminant of coefficient matrix of m and n is equal to 0, which gives rise to a dispersion relation equation. By solving3his dispersion relation equation, MI gain can be obtained G = | Im ( K ) | = Im (cid:18) Ω (cid:113) ( Ω − A ) g (cid:19) , g = + (cid:104) Ω + ( − A + ω ) Ω + A − A ω + ω (cid:105) δ . (6)Similar to the above calculation process, we also obtain the MI gain functions of the seventh-order (i.e. δ = , δ (cid:44) , δ · · · δ = δ = , δ (cid:44) , δ · · · δ = G = | Im ( K ) | = Im (cid:18) Ω (cid:113) ( Ω − A ) g (cid:19) , g = + (cid:104) ω Ω + (cid:16) − A ω + ω (cid:17) Ω + A ω − A ω + ω (cid:105) δ , (7)and G = | Im ( K ) | = Im (cid:18) Ω (cid:113)(cid:0) Ω − A (cid:1) g (cid:19) , g = + (cid:104) − Ω + (cid:16) A − ω (cid:17) Ω + (cid:16) − A + A ω − ω (cid:17) Ω + A − A ω + A ω − ω (cid:105) δ . (8)From the above analysis, it appears that there exists two distinctive MI and modulation stability (MS) regions. In theregion of | Ω | < A , MI exists when g i (cid:44) , ( i = , , g i = , ( i = , , (a) (b) (c) (d)(e) (f) (g)Figure 1: Plots of the distribution of MI gain with perturbation frequency Ω and continuous background frequency ω , and A =
1. The dashed whitelines indicate the resonance lines, the dashed green lines mean boundary lines, and the solid green lines represent that perturbation is stable. In additionto MS curves and MS quasi-elliptic curves, the remaining areas are all non-zero MI gain in MI band. (a) The standard NLS equation [60]: no MS regionexists in the MI band. (b) The Hirota equation [61]: an MS curve exists in the MI band. (c) The Lakshmanan-Porsezian-Daniel equation [62] with δ =
0: an MS elliptic ring appears in the MI band. (d) The fifth-order NLS equation [63, 64] with δ = δ =
0: it not only has an MS curve, but alsohas an MI quasi-elliptic ring in MI band. (e) The sixth-order NLS equation: it has two MS quasi-elliptic rings in the MI band. (f) The seventh-orderNLS equation: here exists an MS curve and two MS quasi-elliptic rings in the MI band. (g) The eighth-order NLS equation: two MS curves and twoMS quasi-elliptic rings appear in the MI band. • When δ = and the remaining δ i = , i = , , , · · · , the system (1) is reduced to classical NLS equation. And thehighest power of g ( ω ) is equal to 0, namely, g ( ω ) =
1. Therefore, no MS region exists in the MI band ( | Ω | < A ), whichis described in Fig. 1(a). • When the above-described conditions wherein δ = δ (cid:44)
0, the system (1) is transformed into third-orderNLS equation. And the highest power of g ( ω ) is 1, i.e. a simple factor of ω . It appears that an MS curve exists in the MIband ( | Ω | < A ), which is described in Fig. 1(b). • When the above-described conditions wherein δ = δ (cid:44)
0, the system (1) can be degenerated to fourth-order NLS equation. The highest power of g ( ω ) is 2. There exists an MS elliptic ring in the MI band ( | Ω | < A ), whichis described in Fig. 1(c). • When the above-described conditions wherein δ = δ (cid:44)
0, then we can transform (1) into fifth-orderNLS equation. The highest power of g ( ω ) is 3. Both an MS curve and an MS quasi-elliptic ring occur in the MI band( | Ω | < A ), which is illustrated in Fig. 1(d). • When the above-described conditions wherein δ = δ (cid:44)
0, we can transform (1) into sixth-order NLSequation. And the highest power of g ( ω ) is 4. There are two MS quasi-elliptic rings in the MI band ( | Ω | < A ), which isillustrated in Fig. 1(e). • When the above-described conditions wherein δ = δ (cid:44)
0, and Eq. (1) is reduced to seventh-order NLSequation. The highest power of g ( ω ) is 5. There exists an MS curve and two MS quasi-elliptic rings in the MI band.( | Ω | < A ), see Fig. 1(f). • When the above-described conditions wherein δ = δ (cid:44)
0, we can transform (1) into eighth-order NLSequation. And the highest power of g ( ω ) is 6. Then it reveals that two MS curves and two MS quasi-elliptic rings existin the MI band ( | Ω | < A ). The distribution of this case is illustrated in Fig. 1(g).The MI distribution features of all above higher-order dispersion NLS equations are illustrated by Figs. 1(a-g). Ac-cording to the above analysis process, it is evident that there exist two arbitrary parameters, namely higher order dispersioncoefficient δ i , i = , , · · · and amplitude A . These parameters control the MS distribution of system (1) in MI band. Byadjusting the parameters, the MS quasi-elliptic and MS elliptic ring can be completely contained within the MI band orintersected at the MI boundary, the latter case yields two curves in MI band. By analyzing the expressions in Eq. (6), wecan discuss the MI distribution characteristics of the sixth-order NLS equation (2). Obviously, g is a polynomial about ω and its highest power is 4. If this polynomial factor is decomposed into the product form of a single factor, then we canget four solutions, which shows that G has four curves in the frequency plane ( ω, Ω ). Here, Fig. 1(e) illustrates the MIgain distribution features in the frequency plane ( ω, Ω ). It is clear that this frequency plane contains two different regions,namely, MI and MS. The expression Ω − A in G indicates that a low-perturbed frequency MI band ( | Ω | < A ) existsin the frequency plane ( ω, Ω ). Setting g =
0, it gives rise to MI gain G =
0, which represents two MS quasi-ellipticrings in frequency plane ( ω, Ω ) and demonstrated by Fig. 1(e). When selecting suitable parameters so that both ellipticalsemi-major axis is greater than 2, then there exist four curves in the MI band. With selecting of δ =
0, Eq. (2) is thentransformed into classical NLS equation and the corresponding MI gain G is reduced to Im (cid:16) Ω (cid:112) ( Ω − A ) (cid:17) . Fig. 1(a)displays the MI gain distribution features of classical NLS equation. Quite evidently, there are neither MS curves nor MSquasi-elliptic rings in the MI band.It is easy to find that Fig. 1 only shows one case where the MS ellipse is contained in the MI band, i.e. the semi-long axis of the MS ellipse less than 2. When choosing appropriate values of parameters to make the semi-long axis ofthe MS ellipse greater than 2, we can get another case that only the MS curves exists in the MI band. From Fig. 1(g),there appears that the MS elliptic ring in the middle degenerates into two MS curves. From the standard NLS equationto eighth-order dispersion NLS equation, the number of the MS curves are 0 , , , , , , , respectively; and the highestpower of g i ( ω ) , ( i = , , . . . ) are 0 , , , , , , , respectively. So we can obtain relation among them, which is listed inTable 1. 5 able 1: Relation between three parameters The Nonlinear Schr¨odingner equation with higher-order termsThe order of dispersion term 2 3 4 5 6 7 8 ... nThe highest power of ω in g i function 0 1 2 3 4 5 6 ... n-2The number of the MS curves in MI band 0 1 2 3 4 5 6 ... n-2
3. Gneralized Darboux Transformation for the sixth-order NLS equation
So as to study rational solution for the sixth-order NLS equation (2), we will construct a generalized DT in this section.Starting from the following linear spectral problems of Eq. (2), namely, Ψ t = i ( λσ + Q ) Ψ , Ψ z = (cid:88) c = i λ c V c , (9)where σ = , Q = q ∗ q , V c = A c B ∗ c B c − A c , with A = − | q | − δ | q | − δ [ q ( q ∗ t ) + ( q ∗ ) q t ] − δ | q | ( qq ∗ tt + q ∗ q tt ) − δ | q tt | + δ ( q t q ∗ ttt + q ∗ t q ttt − q ∗ q tttt − qq ∗ tttt ) , A = i δ | q | ( q t q ∗ − q ∗ t q ) + i δ ( q t q ∗ tt − q ∗ t q tt + q ∗ q ttt − q ∗ ttt q ) , A = + δ | q | − δ | q t | + δ ( q ∗ tt q + q tt q ∗ ) , A = i δ ( qq ∗ t − q ∗ q t ) , A = − δ | q | , A = , A = δ , B = − i δ | q | q t − i δ q ttt , B = i q t + i δ q ttttt + i δ ( qq ∗ t q tt + qq ∗ tt q t + | q | q ttt + | q | q t + q t | q t | + q ∗ q t q tt ) , B = q + δ q ∗ q t + δ | q | q tt + δ q q ∗ tt + δ q tttt + δ | q | q + δ q | q t | , B = − δ | q | q − δ q tt , B = i δ q t , B = δ q , B = , and eigenfunction Ψ = ( ψ , φ ) † , ψ and φ denote complex functions with z and t , † means matrix transpose, ∗ denotes thecomplex conjugation, λ is an spectral parameter of the linear spectral problem (9). It is clearly that U z − V t + UV − VU = Ψ = ( ψ , φ ) † is a fundamental solution based on the above spectral problem. Then the basic DT forEq. (2) has the form Ψ [1] = T [1] Ψ , T [1] = λ I − H [0] Λ H [0] − , q [1] = q [0] + λ ∗ − λ ) ψ [0] ∗ φ [0]( | ψ [0] | + | φ [0] | ) , (10)where φ [0] = φ , ψ [0] = ψ , and I = , H [0] = ψ [0] − φ [0] ∗ φ [0] ψ [0] ∗ , Λ = λ λ ∗ . In the general case, assuming similarly that Ψ l = ( ψ l , φ l ) † , 1 ≤ l ≤ N represents an elementary solution to (9) withq=q[0] at λ = λ l . Then N -fold basic DT for Eq. (2) is thereby inferred that6 [ N ] = T [ N ] T [ N − T [ N − · · · T [1] Ψ , T [ l ] = λ I − H [ l ] Λ l H [ l ] − , q [ N ] = q [ N − + λ ∗ N − λ N ) ψ N [ N − ∗ φ N [ N − | ψ N [ N − | + | φ N [ N − | ) , (11)where Ψ l [ l − = ( ψ l [ l − , φ l [ l − † , and Ψ l [ l − = T l [ l − T l [ l − T l [ l − · · · T l [1] Ψ l , T l [ k ] = T [ k ] | λ = λ l , ≤ l ≤ N , ≤ k ≤ l − , H [ l − = ψ l [ l − − φ l [ l − ∗ φ l [ l − ψ l [ l − ∗ , Λ l = λ l λ ∗ l . Considering an elementary solution cannot be iterated many times by the above method, it is necessary to constructthe generalized DT to overtake this difficulty. Therefore, suppose that Ψ = Ψ ( λ + (cid:15) ), which is a special solution of (9)with q[0] at λ = λ + (cid:15) , applying the taylor expansion on Ψ at (cid:15) = Ψ = Ψ [0]1 + Ψ [1]1 (cid:15) + Ψ [2]1 (cid:15) + Ψ [3]1 (cid:15) + · · · + Ψ [ N ]1 (cid:15) N + · · · , (12)with (cid:15) a small parameter and Ψ [ k ]1 = k ! ∂ k ∂λ k Ψ ( λ ) | λ = λ . It is obvious that Ψ [0]1 is the solution of (9) with q=q[0] at λ = λ . According to the basic DT (10), we can easily derive the 1-fold generalized DT formulas, that is Ψ [1] = T [1] Ψ , T [1] = λ I − H [0] Λ H [0] − , q [1] = q [0] + λ ∗ − λ ) ψ [0] ∗ φ [0]( | ψ [0] | + | φ [0] | ) , (13)with φ [0] = φ [0]1 , ψ [0] = ψ [0]1 , and H [0] = ψ [0] − φ [0] ∗ φ [0] ψ [0] ∗ , Λ = λ λ ∗ . Apparently, T [1] Ψ is the solution of (9) with q [1] at λ = λ + (cid:15) and T [1] Ψ [0]1 =
0. It is natural to draw the followingresult lim (cid:15) → T [1] | λ = λ + (cid:15) Ψ (cid:15) = lim (cid:15) → ( (cid:15) + T [1]) Ψ (cid:15) = Ψ [0]1 + T [1] Ψ [1]1 ≡ Ψ [1] , it gives a nonzero solution of the system (9) with q[1] at λ = λ . Hence, 2-fold generalized DT can be constructed, namely Ψ [2] = T [2] T [1] Ψ , T [2] = λ I − H [1] Λ H [1] − , q [2] = q [1] + λ ∗ − λ ) ψ [1] ∗ φ [1]( | ψ [1] | + | φ [1] | ) , (14)7ith Ψ [1] = ( ψ [1] , φ [1]) † , and H [1] = ψ [1] − φ [1] ∗ φ [1] ψ [1] ∗ , Λ = λ λ ∗ . Continuing the similar process above, we give 3-fold generalized DT. Under the following conditions T [1] Ψ [0]1 = , T [2]( Ψ [0]1 + T [1] Ψ [1]1 ) = , and applying the limit method,lim (cid:15) → [ T [2] T [1]] | λ = λ + (cid:15) Ψ (cid:15) = lim (cid:15) → ( T [2] + (cid:15) )( T [1] + (cid:15) ) Ψ (cid:15) = Ψ [0]1 + ( T [1] + T [2]) Ψ [1]1 + T [2] T [1] Ψ [2]1 ≡ Ψ [2] , thus a nontrivial solution can be obtained for the Lax pair (9) with q [2] at λ = λ . Then the 3-fold generalized DT isnaturally deduced as follows Ψ [3] = T [3] T [2] T [1] Ψ , T [3] = λ I − H [2] Λ H [2] − , q [3] = q [2] + λ ∗ − λ ) ψ [2] ∗ φ [2]( | ψ [2] | + | φ [2] | ) , (15)where Ψ [2] = ( ψ [2] , φ [2]) † , and H [2] = ψ [2] − φ [2] ∗ φ [2] ψ [2] ∗ , Λ = λ λ ∗ . Repeating N times of the above process, it naturally gives rise to the expression of N -fold generalized DT, which reads Ψ [ N − = Ψ [0]1 + N − (cid:88) l = T [ l ] Ψ [1]1 + l − (cid:88) k = N − (cid:88) l = T [ l ] T [ k ] Ψ [2]1 + · · · + T [ N − T [ N − · · · T [1] Ψ [ N − , Ψ [ N ] = T [ N ] T [ N − T [ N − · · · T [1] Ψ , T [ N ] = λ I − H [ N − Λ N H [ N − − , q [ N ] = q [ N − + λ ∗ − λ ) ψ [ N − ∗ φ [ N − | ψ [ N − | + | φ [ N − | ) , (16)where Ψ [ N − = ( ψ [ N − , φ [ N − † , and H [ l − = ψ [ l − − φ [ l − ∗ φ [ l − ψ [ l − ∗ , Λ l = λ λ ∗ , ≤ l ≤ N . Combining and simplifying the above formulas (13-16), it follows a compact formula for N -order rational solution of(2) that q [ N ] = q [0] + λ ∗ − λ ) N − (cid:88) j = ψ [ j ] ∗ φ [ j ]( | ψ [ j ] | + | φ [ j ] | ) . (17)In the following section, we can utilize the above formula to derive an arbitrary order rogue wave for (2). Then itfollows their dynamic behavior illustrations. 8 . Rogue wave solutions Having established the result of generalized DT, attention is now given to constructing higher-order rogue waves for(2). For this purpose, assuming the seed solution q [0] = e i θ , θ = at + ( − a δ + a δ − a δ − a + δ + z , a ∈ R . (18)The problem that a seed solution cannot be iterated by basic DT, can be solved by constructing its generalized DT. Andthen substituting Eq. (18) into Eq. (9), the corresponding fundamental vector solution can be obtained, that is Ψ = i ( C e M − C e − M ) e − θ ( C e M − C e − M ) e θ , (19)with C = ( a + λ + (cid:112) ( a + λ ) + (cid:112) ( a + λ ) + , C = ( a + λ + (cid:112) ( a + λ ) + (cid:112) ( a + λ ) + , M = (cid:112) ( a + λ ) + (cid:110) [ i ( − a + a λ − a λ + a λ − a λ + λ + a − a λ + a λ − λ − a + λ ) δ + i ( − a + λ )] z + it + N (cid:88) k = s k ξ k (cid:111) , s k = m k + in k , ( m k , n k ∈ R ) , where ξ is a small real parameter. Setting λ = − a + i + ξ and expanding Ψ at ξ =
0, it arrives at Ψ ( ξ ) = Ψ [0]1 + Ψ [1]1 ξ + Ψ [2]1 ξ + · · · . (20)Here, vector function Ψ [0]1 has the following explicit expression ψ [0]1 = + i η [0]1 e − i θ , φ [0]1 = − + i η [0]2 e i θ , (21)where η [0]1 = a − ia − a + ia ) δ + δ + a − i (60 δ + ,η [0]2 = ia + a − ia − a ) δ + i (180 δ + a + δ + . Clearly, Ψ [0]1 = ( ψ [0]1 , φ [0]1 ) † satisfies the system (9) with spectral parameter λ = − a + i . Therefore, utilizing the formula(17) with N =
1, it suffices to obtain first-order rogue wave of (2), that is q [1] = (cid:16) + D + iE F (cid:17) e i θ , (22)where F = a − a + a − a + a + z δ + a − a + z δ − a − a + azt δ + a + z − azt + t + , D = − a − a + a − a + a + z δ − a − a + z δ + a − a + azt δ − a + z + azt − t + , E = a − a + z δ + z . Obviously, there are two arbitrary parameters a and δ in the expression for q [1], the latter is the sixth-order dispersioncoefficient. Next, we fix δ to analyze the dynamic of rogue wave solution with changing of frequency a . Taking the caseof a = a <
0, the crest of rogue wave occurs counterclockwise deflection; while a >
0, the crest9 a) (b) (c)Figure 2: Plots of first-order rogue wave with a = − . , , .
5, from left to right, respectively and δ = . a = − . , − . , , . , .
5, from left to right, respectively and δ = . occurs clockwise deflection. In addition, the width of the crest changes. As | a | increases, the deflection angle of crest ofrogue wave increases, and so does its width. Figs. 2 and 3 illustrate the above dynamic characteristics. Now, we fix a tobe any particular constant and take limit on q [1] at δ → ∞ , that islim | δ |→∞ | q [1] | ≡ . (23)As the absolute value of δ increases, the modulus of q [1] gradually reverts to a constant background plane; the roguewave gradually disappeared and the energy gradually decreased. Without loss of generality, fix a =
0, Fig. 4 shows thisevolution process in the first-order rogue wave structure by varing parameter δ . For δ =
0, Eq.(1) degenerates into thestandard NLS equation, and it follows that the amplitude of | q [1] | is equal to 3. Figure 4: The evolution process of the rogue wave structure with the change of δ for a = Similar to the computational process of Section 3.2, taking limitlim ξ → T [1] | ξ = − a + i + ξ Ψ ξ = lim ξ → ( ξ + T [1]) Ψ ξ = Ψ [0]1 + T [1] Ψ [1]1 ≡ Ψ [1] , (24)10nd using the obtained formula (17) with N =
2, it is not hard to adduce the second-order rogue wave. Since theexpression of this solution is too cumbersome, we only show its dynamic behavior, which are illustrated by Figs. 5 and6. The corresponding contour map of Fig. 6 is demonstrated in Fig. 7. Substituting m = , n = ,
0) in the( t , z ) plane, see Fig. 5. However, when only changing a parameter m = a . (a) (b) (c)Figure 5: Plots of second-order rogue wave by choosing m = n = a = − . , , .
5, from left to right, respectively.(a) (b) (c)Figure 6: 3D graphics for second-order rogue wave by choosing δ = . m = n = a = − .
5, 0, 0 .
5, from left to right, respectively.(a) (b) (c) (d) (e)Figure 7: The corresponding contour graphics for the second-order rogue wave obtained in Fig. 6 with parameters: δ = . m = n = a = − . − .
3, 0,0 .
3, 0 .
5, from left to right, respectively.
Applying formula (17) with N =
3, it then follows the third-order rogue wave solution. Here, we just show three typesof third-order rogue wave solutions, fundamental pattern, triangular pattern and circular pattern rogue wave, respectively,see Fig. 8. The first row are the three-dimensional graphs, and the second row are the corresponding density maps.The amplitude of third-order fundamental rogue wave reaches maximum value 7 at point (0 ,
0) in the ( t , z ) plane. Obvi-ously, these rogue waves are symmetrical, which can be seen from Figs. 8(d-f). They also possess the above deflectioncharacteristics. 11 a) (b) (c)(d) (e) (f)Figure 8: Three kinds of third-order rogue wave structures for Eq. (2). Left columns: fundamental type structure at a = , δ = . , m i = n i = i = , m = m =
5. Spectral analysis of rogue waves
Our attention is now turned to spectral analysis on rogue wave solution for Eq. (22) in this section. In [53], it appearsthat the specific triangular spectrum for a Peregrine rogue wave could be applied to early warning of rogue waves byspectral measurements. The spectrum analysis is referred to as a useful method in predicting and exciting of rogue wavesolutions in the nonlinear fiber [54, 55]. For more conveniently to calculate the spectral of first-order rogue wave solution,we take δ = in Eq. (22). It then follows that q [1] = (cid:16) + iK ) K − (cid:17) exp( i θ ) , (25)where K = (5 a − a + z , θ = at + (cid:16) a − a + (cid:17) z , K = ( a − a + a − a + a + z − a − a + azt + t + . Now we perform spectrum analysis approach on the above derived first-order rogue wave solution by the Fourier trans-formation as follows F ( β, z ) = √ π (cid:90) + ∞−∞ q [1]( z , t ) exp( i β t ) dt . (26)From the solution (25), it is inferred that the rogue wave solution contains two parts, a plane wave and a variable signalpart. It is clear that the plane wave background becomes infinity and the integral is a δ function, so we omit the spectrumof plane wave background. The corresponding modulus of the rogue wave signal is given by | F ( β, z ) | = √ π exp (cid:16) − | β (cid:48) | (cid:112) + (5 a − a + z (cid:17) , (27)where β (cid:48) = β + a . 12irstly, from the perspective of the bottom row of Fig. 9 to analyze, it is clear that the spectrum of the solution (25) withdifferent a has strong symmetry properties. And then combined with the expression (27), when a (cid:44) ± (cid:113) ± √ a =
0, when compared to the case at a =
2. The corresponding density diagrams aredisplayed in Figs. 9(a) and 9(b). However, when a = ± (cid:113) ± √ N th-order rogue waves can be reduced to N th-order W-shaped solitons. This result in turn implies that MI analysis isconsistent with spectral analysis for the sixth-order Eq. (2) with δ = . From the perspective of MI gain function (6), itcan be adduced that g =
16 (30 A − Ω A − a A + Ω + a Ω + a + , (28)where A is the amplitude, Ω is the perturbed frequency and a is the frequency of background. By setting A = , Ω = g =
0, it follows g =
12 (5 a − a + = , (29)then there is a transformation of two sates here, which happens between the rogue wave and the W-shaped soliton in theregion of zero-frequency MS. (a) (b) (c)(d) (e) (f)Figure 9: The first row displays density figures of two first-order rogue waves and a W-shaped soliton solutions in Eq. (25) with a =
0, 2, and (cid:113) − √ | F ( β, z ) | in Eq.(27). In order to demonstrate the impact of the parameter δ , we will give the spectrum analysis of the rogue wave solutionby selecting a = q [1] = (cid:16) + i (1 + δ ) z )4 t + + δ ) z + − (cid:17) exp( i (1 + δ ) z ) , (30)13nd | F ( β, z ) | = √ π exp (cid:16) − | β | (cid:112) + + δ ) z (cid:17) . (31)Similarly, when δ (cid:44) − , the spectrum of the solution (30) also possesses specific triangular spectrum of a Peregrinerogue wave. In addition, their spectrum share the same features with different parameters as above Eq. (25). Obviously,a little change appears in δ , a big change presents in the corresponding triangular widening spectrum in contrast withabove condition. Setting δ = − , the solution (30) is a W-shaped soliton, presented in Fig. 10(c). Its correspondingspectrum appears in banded form, see Fig. 10. (a) (b) (c)Figure 10: Spectral dynamics of | F ( β, z ) | in Eq. (31) with δ =
0, 0 . − , from left to right, respectively.
6. Summary and discussions
In conclusion, MI of the continuous wave background has been investigated for the NLS equation with differenthigher-order dispersion term. The MI distribution characteristics from the sixth-order to the eighth-order NLS equationsare studied in detail. There are two arbitrary parameters, namely, higher-order dispersion term δ i , i = , , . . . and ampli-tude A . These parameters control the MS distribution of the NLS with different higher-order dispersion terms in the MIband. By adjusting the parameters, the MS quasi-elliptic and MS elliptic ring can be completely contained within the MIband or intersected at the MI boundary, the latter case yields two curves in the MI band. g i is a polynomial with ω , and itshighest power of ω is closely related to how many MS curves can exist in the MI band. It is adduced that the high-orderdispersion terms indeed affect the distribution of the MS regime, n -order dispersion term corresponds to n − N th-order rogue waves (17). Then the exact expression of first-orderrogue wave is demonstrated. Since expressions of high-order rogue waves are too cumbersome, we demonstrate its dy-namic behavior through pictures. There are two arbitrary parameters a and δ , the sign of the former determines thedirection of deflection and the magnitude of the absolute value affects the angle of deflection and the width of rogue wavesolution. While the latter can cause the change of the width and amplitude of rogue wave. For the first- to third-orderrogue waves, they all own the deflection properties mentioned above. For the third-order rogue wave solution, three kindsof structures, that is fundamental, triangular, and circular, are illustrated in Fig. 8, and their dynamic behavior features arediscussed in detail.Via the spectrum analysis approach on first-order rogue wave, It has been found that arbitrary parameters a and δ have effects on the spectrum of the solution (25). Fixing δ = , when a (cid:44) ± (cid:113) ± √ a is related to the size of the triangular spectrum; when a = ± (cid:113) ± √ a , when δ satisfies certain constraint, the spectrum of the solution (30) alsopresents the specific triangular spectrum or banded spectrum, which correspond to the rogue wave solution or W-shapedsoliton solution, respectively.Finally, it is worthy to mention that we will further study the excitation conditions and numerical analysis of variousnonlinear waves and their corresponding positions in the MI gain plane in the future. Acknowledgment
The project is supported by the National Natural Science Foundation of China (Nos. 11675054 and 11435005), GlobalChange Research Program of China (No. 2015CB953904), and Shanghai Collaborative Innovation Center of TrustworthySoftware for Internet of Things (No. ZF1213).
References [1] Draper L, Freak wave. Marine Observer 35 (1965) 193-195.[2] Kharif C, Pelinovsky E, Slyunyaev A, Quasi-linear wave focusing. Rogue waves in the ocean. Berlin: Springer (2009) 63-89.[3] Akhmediev N, Ankiewicz A, Taki M, Waves that appear from nowhere and disappear without a trace. Phys. Lett. A 373 (2009) 675-678.[4] Zakharov V E, Ostrovsky L A, Modulation instability: the beginning. Physica D 238 (2009) 540-548.[5] Rapti Z, Kevrekidis P G, Smerzi A, Bishop A R, Variational approach to the modulational instability. Phys. Rev. E 69 (2004) 017601.[6] Benjamin T B, Feir J E, The disintegration of wave trains on deep water Part 1. Theory. J. Fluid Mech. 27 (1967) 417-430 .[7] Taniuti T, Washimi H, Self-trapping and instability of hydromagnetic waves along the magnetic field in a cold plasma. Phys. Rev. Lett. 21 (1968)209.[8] Bespalov V I, Talanov V I, Filamentary structure of light beams in nonlinear liquids. Zh. Eksp. Teor. Fiz. Pis’ma Red., 3 (1966) 471, Englishtranslation: JETP Lett. 3 (1966) 307.[9] Witham G B, Non-linear dispersive waves. Proc. R. Soc. Lond. A 283 (1965) 238-261.[10] Zhao L C, Xin G G, Yang Z Y, Rogue-wave pattern transition induced by relative frequency. Phys. Rev. E 90 (2014) 022918.[11] Zhang J H, Wang L, Liu C, Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects.Proc. R. Soc. A 473 (2017) 20160681.[12] Wang X, Liu C, Wang L, Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations, J. Math. Anal.Appl. 449 (2017) 1534-1552.[13] Wang L, Zhang J H, Wang Z Q, Liu C, Li M, Qi F H, Guo R, Breather-to-soliton transitions, nonlinear wave interactions, and modulationalinstability in a higher-order generalized nonlinear Schr¨odinger equation. Phys. Rev. E 93 (2016) 012214.[14] Agrawal G P, Nonlinear Fiber Optics, Academic, New York, 2006.[15] Stenflo L, Marklund M, Rogue waves in the atmosphere. J. Plasma Phys. 76 (2010) 293-295.[16] Geist E L, Book review: Nonlinear ocean waves and the inverse scattering transform. 2011.[17] Peregrine D H, Water waves, nonlinear Schr¨odinger equations and their solutions. J. Aust. Math. Soc. 25 (1983) 16-43.[18] Akhmediev N, Ankiewicz A, Soto-Crespo J M, Rogue waves and rational solutions of the nonlinear Schr¨odinger equation. Phys. Rev. E 80 (2009)026601.[19] Ankiewicz A, Clarkson P A, Akhmediev N, Rogue waves, rational solutions, the patterns of their zeros and integral relations. J. Phys. A: Math.Theor. 43 (2010) 122002.[20] Ohta Y, Yang J K, General high-order rogue waves and their dynamics in the nonlinear Schr¨odinger equation. Proc. R. Soc. A 468 (2012) 1716-1740.[21] Zhang G Q, Yan Z Y, Wen X Y. Modulational instability, beak-shaped rogue waves, multi-dark-dark solitons and dynamics in pair-transition-coupled nonlinear Schr¨odinger equations. Proc. R. Soc. A 473 (2017) 20170243.[22] Gagnon L, Winternitz P, Symmetry classes of variable coefficient nonlinear Schr¨odinger equations. J. Phys. A: Math. Gen. 26 (1993) 7061.[23] Wang L, Zhang J H, Liu C, Li M, Qi F H, Breather transition dynamics, Peregrine combs and walls, and modulation instability in a variable-coefficient nonlinear Schr¨odinger equation with higher-order effects. Phys. Rev. E 93 (2016) 062217.[24] Yang Y Q, Wang X, Yan Z Y, Optical temporal rogue waves in the generalized inhomogeneous nonlinear Schr¨odinger equation with varyinghigher-order even and odd terms. Nonlinear Dyn. 81 (2015) 833-842.[25] Zhang G Q, Yan Z Y, Three-component nonlinear Schr¨odinger equations: Modulational instability, Nth-order vector rational and semi-rationalrogue waves, and dynamics. Commun. Nonlinear Sci. Numer. Simulat. 62 (2018) 117-133.[26] Xu T, Chen Y, Mixed interactions of localized waves in the three-component coupled derivative nonlinear Schr¨odinger equations. Nonlinear Dyn.92 (2018) 2133-2142.
27] Zhang G Q, Yan Z Y, The n-component nonlinear Schr¨odinger equations: dark-bright mixed N-and high-order solitons and breathers, and dynam-ics. Proc. R. Soc. A 474 (2018) 20170688.[28] Yang B, Chen Y, Dynamics of high-order solitons in the nonlocal nonlinear Schr¨odinger equations. Nonlinear Dyn. 94 (2018) 489-502.[29] Ankiewicz A, Wang Y, Wabnitz S, Akhmediev N, Extended nonlinear Schr¨odinger equation with higher-order odd and even terms and its roguewave solutions. Phys. Rev. E 89 (2014) 012907.[30] Cai L Y, Wang X, Wang L, Li M, Liu Y, Shi Y Y, Nonautonomous multi-peak solitons and modulation instability for a variable-coefficientnonlinear Schr¨odinger equation with higher-order effects. Nonlinear Dyn. 90 (2017) 2221-2230.[31] Berg´e L, Wave collapse in physics: principles and applications to light and plasma waves. Physics reports 303 (1998) 259-370.[32] Ankiewicz A, Kedziora D J, Chowdury A, Bandelow U, Akhmediev N, Infinite hierarchy of nonlinear Schr¨odinger equations and their solutions.Phys. Rev. E 93 (2016) 012206.[33] Hirota R, Exact envelope-soliton solutions of a nonlinear wave equation. J. Math. Phys. 14 (1973) 805-809.[34] Lakshmanan M, Porsezian K, Daniel M, Effect of discreteness on the continuum limit of the Heisenberg spin chain. Phys. Lett. A 133 (1988)483-488.[35] Chowdury A, Kedziora D J, Ankiewicz A, Akhmediev N, Soliton solutions of an integrable nonlinear Schr¨odinger equation with quintic terms.Phys. Rev. E 90 (2014) 032922.[36] Su J J, Gao Y T. Bilinear forms and solitons for a generalized sixth-order nonlinear Schr¨odinger equation in an optical fiber. Eur. Phys. J. Plus 132(2017) 53.[37] Sun W R. Breather-to-soliton transitions and nonlinear wave interactions for the nonlinear Schr¨odinger equation with the sextic operators in opticalfibers. Ann. Phys. 529 (2017) 1600227.[38] Lan Z Z, Guo B L, Conservation laws, modulation instability and solitons interactions for a nonlinear Schr¨odinger equation with the sexticoperators in an optical fiber. Optical and Quantum Electronics 50 (2018) 340.[39] Guo B L, Ling L M, Liu Q P, Nonlinear Schr¨odinger equation: generalized Darboux transformation and rogue wave solutions. Phys. Rev. E 85(2012) 026607.[40] He J S, Zhang H R, Wang L H, Fokas A S, Generating mechanism for higher-order rogue waves. Phys. Rev. E 87 (2013) 052914.[41] Wen X Y, Yan Z Y, Modulational instability and higher-order rogue waves with parameters modulation in a coupled integrable AB system via thegeneralized Darboux transformation. Chaos 25 (2015) 123115.[42] Wen X Y, Yan Z Y, Modulational instability and dynamics of multi-rogue wave solutions for the discrete Ablowitz-Ladik equation. J. Math. Phys.59 (2018) 073511.[43] Wei J, Wang X, Geng X G, Periodic and rational solutions of the reduced Maxwell-Bloch equations. Commun. Nonlinear Sci. Numer. Simulat. 59(2018) 1-14.[44] Ling L M, Feng B F, Zhu Z N, Multi-soliton, multi-breather and higher-order rogue wave solutions to the complex short pulse equation. PhysicaD 327 (2016) 13-29.[45] Liu Y B, Mihalache D, He J S, Families of rational solutions of the y-nonlocal Davey-Stewartson II equation. Nonlinear Dyn. 90 (2017) 2445-2455.[46] Chen J C, Ma Z Y, Hu Y H, Nonlocal symmetry, Darboux transformation and soliton-cnoidal wave interaction solution for the shallow water waveequation. J. Math. Anal. Appl. 460 (2018) 987-1003.[47] Huang L L, Yue Y F, and Chen Y, Localized waves and interaction solutions to a (3+1)-dimensional generalized KP equation. Comput. Math.Appl. 76 (2018) 831-844.[48] Yue Y F, Huang L L, Chen Y, Localized waves and interaction solutions to an extended (3+1)-dimensional Jimbo-Miwa equation. Appl. Math.Lett. 89 (2019) 70-77.[49] Huang L L, Chen Y, Localized excitations and interactional solutions for the re duce d Maxwell-Bloch equations. Commun. Nonlinear Sci Numer.Simulat. 67 (2019) 237-252.[50] Akhmediev N, Ankiewicz A, Solitons, nonlinear pulses and beams. Chapman and Hall, London 1997.[51] Ankiewicz A, Soto-Crespo J M, Akhmediev N, Rogue waves and rational solutions of the Hirota equation. Phys. Rev. E 81 (2010) 046602.[52] Ankiewicz A, Akhmediev N, Higher-order integrable evolution equation and its soliton solutions. Phys. Lett. A 378 (2014) 358-361.[53] Akhmediev N, Ankiewicz A, Soto-Crespo J M, Dudley J M, Rogue wave early warning through spectral measurements. Phys. Lett. A 375 (2011)541-544.[54] Akhmediev N, Soto-Crespo J M, Devine N, Hoffmann N P, Rogue wave spectra of the Sasa-Satsuma equation. Physica D 294 (2015) 37-42.[55] Bayindir C. Rogue wave spectra of the Kundu-Eckhaus equation. Phys. Rev. E 93 (2016) 062215.[56] Wang X, Liu C, Wang L, Rogue waves and W-shaped solitons in the multiple self-induced transparency system. Chaos 27 (2017) 093106.[57] Wang X, Liu C, W-shaped soliton complexes and rogue-wave pattern transitions for the AB system. Superlattices and Microstructures 107 (2017)299-309.[58] Droques M, Barviau B, Kudlinski A, Taki M, Boucon A, Sylvestre T, Mussot A, Symmetry-breaking dynamics of the modulational instabilityspectrum. Optics letters 36 (2011) 1359-1361.[59] Solli D R, Herink G, Jalali B, Ropers C, Fluctuations and correlations in modulation instability. Nature Photonics 6 (2012) 463-468.[60] Zhao L C, Ling L M, Quantitative relations between modulational instability and several well-known nonlinear excitations. J. Opt. Soc. Amer. B33 (2016) 850-856.[61] Liu C, Yang Z Y, Zhao L C, Duan L, Yang G Y, Yang W L, Symmetric and asymmetric optical multipeak solitons on a continuous wave backgroundin the femtosecond regime. Phys. Rev. E 94 (2016) 042221.[62] Duan L, Zhao L C, Xu W H, Liu C, Yang Z Y, Yang W L, Soliton excitations on a continuous-wave background in the modulational instability egime with fourth-order effects. Phys. Rev. E 95 (2017) 042212.[63] Yang Y Q, Yan Z Y, Malomed B A, Rogue waves, rational solitons, and modulational instability in an integrable fifth-order nonlinear Schr¨odingerequation. Chaos 25 (2015) 103112.[64] Li P, Wang L, Kong L Q, Wang X, Xie Z Y, Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schr¨odinger equation.Appl. Math. Lett. 85 (2018) 110-117.egime with fourth-order effects. Phys. Rev. E 95 (2017) 042212.[63] Yang Y Q, Yan Z Y, Malomed B A, Rogue waves, rational solitons, and modulational instability in an integrable fifth-order nonlinear Schr¨odingerequation. Chaos 25 (2015) 103112.[64] Li P, Wang L, Kong L Q, Wang X, Xie Z Y, Nonlinear waves in the modulation instability regime for the fifth-order nonlinear Schr¨odinger equation.Appl. Math. Lett. 85 (2018) 110-117.