Rogue waves on the double-periodic background for a nonlinear Schrödinger equation with higher-order effects

In this paper, we study the dynamics of rogue waves on the double-periodic background for a nonlinear Schrödinger equation with higher-order effects. First, we consider two types of double-periodic solutions in terms of the Jacobi elliptic functions. When the elliptic modulus approaches 1, such both double-periodic wave solutions can reduce to the Akhmediev breather solution. Second, with both double-periodic waves as seed solutions, we derive rogue wave solutions from the Darboux transformation method. Further, we demonstrate localized structures of rogue waves formed on double-periodic waves for two sets of different eigenvalues. In addition, we discuss how the higher-order effect affects double-periodic background waves and rogue waves. We expect that our obtained results may help understand rogue waves manifestations on the double-periodic background occurring in hydrodynamics and nonlinear optics with higher-order effects.


Introduction
Originally, rogue waves are large-amplitude waves on the ocean surfaces surprisingly appearing from nowhere and disappearing without a trace [1,2]. As a special type of nonlinear wave phenomenon, rogue waves are also observed in various physical contexts such as in the optical fibers [3,4] and in the Bose-Einstein condensates [5]. In the past few decades, the physical mechanisms of rogue waves have been intensively studied by different theoretical approaches and laboratory experiments [6][7][8]. From the mathematical point of view, as the simplest fundamental rational solution of the nonlinear Schrödinger (NLS) equation, the Peregrine soliton with a peaked hump and two side valleys has been considered as a prototype of rogue waves [9]. This type of soliton solution exhibits a doubly localized peak on a finite background. Analytically, the Peregrine soliton can be regarded as the limiting case of either Kuznetsov-Ma breather or Akhmediev breather when the period tends to infinity [10,11]. Rogue wave solutions of the NLS equation can be obtained by the Darboux transformation method [12,13] or the Hirota bilinear method [14] on the nonzero plane wave background. Therefore, rogue waves are considered as excitation patterns from the plane waves. In addition to the plane waves, the periodic wave background can also lead to rogue wave excitation in the NLS equation [15,16]. Such rogue waves are formed on the background of modulational insta-bility of the Jacobian elliptic periodic waves [17][18][19][20][21][22]. Furthermore, rogue waves on the double-periodic background in the NLS equation have also been derived by using an algebraic method [23].
As a fundamental mathematical model in nonlinear science, the NLS equation contains the lowest order dispersion and the cubic nonlinearity. However, due to the diversity of the nonlinear media, other significant physical higher-order effects should be taken into account in realistic settings. For example, the higher-order dispersion, self-steepening and stimulated Raman scattering effects need to be considered for the ultrashort pulse propagation in highly dispersive optical fibers [24,25]. Therefore, various extensions of the NLS equation have been proposed to model the nonlinear phenomena in many fields of physics. Recently, Ankiewicz et al. have proposed an extension of the NLS equation to the infinite NLS hierarchy with an arbitrary number of higherorder terms and free real coefficients [26]. This extension can be used to model various physical problems of nonlinear wave evolution with a large degree of flexibility. Particularly, under different reduced coefficients, besides the NLS equation, this extension also includes many important integrable nonlinear equations such as the Hirota equation [27], the Lakshmanan-Porsezian-Daniel (LPD) equation [28] and the quintic NLS equation [29,30].
In this paper, we study the following NLS equation with higher-order effects [28] i q t + q xx + 2|q| 2 q which is also referred to as the LPD equation. Hereby, q(x, t) denotes the complex packet envelope, the asterisk * represents complex conjugation, and γ is an arbitrary real parameter standing for the strength of higherorder linear and nonlinear effects. Obviously, Eq. (1) involves the fourth-order dispersion, quintic nonlinearity and higher-order nonlinearity dispersions. Therefore, Eq. (1) can be thought of as a more appropriate prototype of the wave evolution in the real world. In fact, Eq. (1) was originally derived as a model for the nonlinear spin excitations in one-dimensional isotropic biquadratic Heisenberg ferromagnetic spin [28,[31][32][33]. Also, this equation is relevant for the propagation of ultrashort optical pulses in high-speed optical transmission system [34,35]. As the third member of the NLS hierarchy of equations, its many integrable properties and various exact solutions have been investigated by different methods [33,[36][37][38][39][40][41][42]. It has been shown that Eq. (1) possesses the solitons, breathers and rogue waves, simultaneously. In Ref. [21], we construct rogue wave solutions of Eq. (1) on the spaceperiodic background, i.e., the Jacobi elliptic dn-and cn-functions. In particular, when the elliptic modulus approaches one, these rogue wave solutions can reduce to multi-pole soliton solutions. In contrast with the standard NLS equation, the higher-order effects have been made toward the dynamical behaviors of localized waves like the compression of the breather and rogue waves [40,41]. In Ref. [23], researchers have developed an algebraic method for constructing rogue wave solutions for the NLS equation on the double-periodic wave background. In this method, by imposing a constraint between the potential and squared eigenfunctions, one can identify the Hamiltonian system associated with the spatial and temporal parts of spectral problem. Then, the particular eigenvalues and eigenfunctions in spectral problem are characterized from closed differential equations. In order to generate rogue waves on the double-periodic wave background, one should consider a second linearly independent solution of spectral problem which is unbounded and non-periodic. Finally, by the substitution of the second solution into the one-fold Darboux transformation formula, rogue wave solutions can be derived from the double-periodic wave solutions.
In this paper, we adopt an algebraic method developed in Ref. [23] to construct rogue wave solutions on the double-periodic background for Eq. (1). Compared with the standard NLS equation, this model involves the fourth-order dispersion and high-order nonlinearity and emerges very often in various nonlinear physical problems such as in the molecular systems and nonlinear optics. In particular, this equation can reduce to the standard NLS equation when the coefficient γ is zero. Our first aim is to show that rogue wave solutions on the double-periodic background exist in this equation in the presence of higher-order effects. Then, we focus on discussing how the higher-order effect affects double-periodic background waves and rogue waves by varying the higher-order nonlinear and dispersion parameter.

Double-periodic solutions
The exact doubly periodic solutions of the focusing NLS equation were constructed in Ref. [11] by separating the variables. The family of doubly periodic solutions is controlled by three parameters which are the three roots of a fourth-order polynomial. The doubly periodic solutions (2) and (3) of Eq. (1) can also be obtained by reducing Eq. (1) to the first-order quadrature which is similar to that of the NLS equation. Recently, the double-periodic solutions of the infinite NLS hierarchy with an arbitrary number of free real coefficients have been constructed in Ref. [43]. As far as we know, such solutions have been experimentally observed in a loss-compensated optical fiber [44]. These solutions play an important role in the theoretical background for nonlinear experiments in many physical contexts [45]. Lately, the double-periodic waves of the NLS equation have been experimentally observed in a super water wave tank [46]. For Eq. (1), two types of double-periodic solutions are given in terms of the Jacobi elliptic functions [43] (i). Type-A solution where sn, dn and cn are the Jacobian elliptic functions, k ∈ (0, 1) is the elliptic modulus, and where It can be seen that solutions (2) and (3) are both doubly periodic about x and t. The fundamental periods B for the solution (3) with K being the complete elliptic integral of the first kind. Figure 1a, b present the amplitudes plots of the double-periodic waves via solutions (2) and (3). It can be seen that two double-periodic solutions are qualitatively different since they are located on different sides of the separatrix [43]. The maxima of the solution (2) periodically appear with a shift of half period, whereas the maxima in the solution (3) repeatedly appear at the same position. The difference can be seen clearly from the false color plots of two double-periodic solutions, as shown in Fig. 2. As can be seen from solutions (2) and (3), both temporal periods depend on the higherorder parameter γ .
In the limit as the modulus k → 1, both doubleperiodic solutions (2) and (3) reduce to the Akhmediev breather of Eq. (1) which is periodic in x and homoclinic in t, as shown in Fig. 3.

Spectral problem
The spectral problem associated with Eq. (1) can be written in the form with where ϕ = (φ, ψ) T (The superscript T denotes matrix transpose) is the vector eigenfunction and λ is the spectral parameter. It is known that the one-fold Darboux transformation of Eq. (1) has the following explicit expression where ϕ = (φ 1 , ψ 1 ) T is a nonzero solution of spectral problem (6) with λ = λ 1 and q 1 (x, t) denotes the first iterated solution of Eq. (1). With a seed solution q(x, t), Eq. (7) can be used to generate a new explicit solution In what follows, the key step to generate rogue wave solutions by the formula (7) is to solve the linear Eqs. (6a) and (6b) with both double-periodic solutions (2) and (3) as the seed solutions. However, since the seed solutions involve many elliptic functions, it is not easy to obtain exact solutions from the spectral problem (6). Recently, there are some contributions on this issue by the nonlinearization of spectral problem [23]. As Eq. (1) and the NLS equation belong to the same hierarchy, the method developed in Ref. [23] can be applied similarly to Eq. (1).
Assume that the solution q(x, t) of Eq. (1) is related to the squared eigenfunctions by Plugging the above constraint into the space part of spectral problem (6), we obtain the following finitedimensional Hamiltonian system By differentiating Eq. (8) successively in x from the first order to the fourth order, one can obtain a set of ordinary differential equations with respect to q(x, t). The last differential equation can be closed on q(x, t) as the fourth-order Lax-Novikov equation for the hierarchy of stationary NLS equation [23]. Furthermore, the squared eigenfunctions appearing in the one-fold Darboux transformation (7) can be derived from the third-order Lax-Novikov equation (the details can be seen in Ref. [23]) as follows: 3) are all constants. Similar expressions for φ 2 2 , ψ 2 2 and φ 2 ψ 2 can be obtained by the transformation λ 1 ↔ λ 2 .
As the third member of NLS hierarchy, the Lax spectral problem for Eq. (1) has the same spatial part but the different temporal part. Therefore, the computational details of the eigenvalues of spectral problem (6) are similar. The spatial part of the spectral problem (6) with q(x, t) given by the double-periodic waves via solutions (2) and (3) admits three pairs of eigenvalues ±λ 1 , ±λ 2 , ±λ 3 (refer to Ref. [23]), namely: (i). For the type-A solution (2), one eigenvalue is real and the other two eigenvalues are complex where τ 1 = 2α and α 2 + β 2 = 0.25 with α and β as two real constants. (ii). For the type-B solution (3), three eigenvalues are all real where τ 1 varies from 0 to 1, τ 2 = τ 3 − τ 1 and τ 3 = 1.
In order to obtain rogue wave solutions on the double-periodic background, we need to construct two sets of solutions of spectral problem (6) with the same eigenvalue. The first solution of spectral problem (6) is bounded and periodic, whereas the second linearly dependent solution is unbounded and non-periodic. Let ϕ = (φ 1 , ψ 1 ) T be a solution of Lax pair (6) for λ = λ 1 . The second, linearly independent solution ϕ = (φ 1 ,ψ 1 ) T of the same equations is obtained in the form: where θ is a undetermined function of x and t. It is easy to check that these two sets of solutions are linearly dependent. Substituting Eq. (15) into Eq. (6a) and Eq. (6b), we have where The system of first-order Eqs. (16) and (17) is compatible in the sense that F t = G x . Therefore, it can be solved by the following explicit integration formula: where (x 0 , t 0 ) is arbitrarily fixed. It is not easy to obtain the explicit expression of θ(x, t) from the above integration formula. Therefore, we carry out the numerical computation to evaluate the expression (18).

Rogue waves on the double-periodic background
Substituting the double-periodic solutions q(x, t) (expressed in solutions (2) and (3)) and the eigenfunctions of the second solution ϕ = (φ 1 ,ψ 1 ) T of spectral problem with λ = λ 1 into the one-fold Darboux transformation formula (7), we can obtain a new solution of Eq. (1) Furthermore, the use of Eq. (18) and the identities (10)- (12), rogue waves can be derived from the solution (19) on the double-periodic background. Figure 4 displays the spatiotemporal structure of the rogue wave solution (19) formed on the background of the type-A double-periodic solution (2) with k = 0.8 and γ = 0.1 for (a) the real eigenvalue λ 1 = τ 1 , τ 1 = 2α and (b) the complex eigenvalue λ 2 = √ α + iβ, α = 0.4, and β = 0.3. As shown in Fig. 4, the central part is a rogue wave and the background is periodic in both space and time.
The conventional definition of the rogue wave's magnification factor is the ratio of the maximal value of the wave amplitude to the mean value of the wave background. In particular, the magnification factor of a rogue wave can be computed as the ratio of the maximal value of the wave amplitude to the maximal value of the wave background [15,23]. The first definition of the magnification factor is higher than the second definition. In the following, we adopt the second definition and show numerically the magnification factor for the rogue waves constructed on the double-periodic background.
In Fig. 4, the maximum amplitude of the rogue wave is reached at origin. Our calculation shows the rogue wave is actually enhanced in the center position up to 1.28 [ Fig. 4a], 1.77 [Fig. 4b] times as high as the maximum height of the type-A double-periodic background, respectively. These enhancement values are all below 3. For comparison, when the value of γ becomes bigger and the eigenvalues are the same as those in Fig. 4a, b, we also provide in Fig. 5a, b the surface plots of rogue waves on the double-periodic solution (2). We observe that the amplitudes of rogue waves also keep max-ima at origin, but the number of peaks of background waves increases. Rogue waves, respectively, have the enhancement values 1.49 [ Fig. 5a] and 1.92 [Fig. 5b].
In Fig. 6a-c, we show the surface plots of rogue waves on the background of the type-B double-periodic solution (3) with k = 0.9 and γ = 0.01 for the three real eigenvalues λ 1 = √ τ 1 = 1 2 , λ 2 = √ τ 2 = 1 and 2 . Rogue waves are located at origins, and the maximum values of their amplitudes are 2.88, 3.11 and 3.09, respectively. By carrying out the numerical computation, we find that the rogue wave for the eigenvalue λ 2 is the highest wave of the three. This property is the same as that in the NLS equation [23] and the Hirota equation [47]. For comparison, we also provide another set of surface plots of rogue waves in Fig. 7 with the same parameters except for γ = 0.1 in Fig. 6. The maximum heights of rogue waves are 2.90, 3.43 and 3.27, respectively. By comparison with Fig. 6, it is seen that each rogue wave has an enhancement due to the increase in the value of γ .

Conclusions
In this paper, we have presented rogue wave solutions on the background of two types of double-periodic waves in the higher-order NLS equation involving higher-order dispersion and nonlinearity. With these solutions, we have revealed nonlinear dynamics of rogue waves on the top of two types of double-periodic waves for two sets of different eigenvalues. Further, we have discussed how the parameter γ denoting the strength of higher-order linear and nonlinear effect affects the double-periodic waves and rogue waves from the following three aspects: (1) The parameter γ is associated with a frequency of the modulation along the time; (2) The number of peaks of both double-periodic background waves in the same time interval is increasing when the value of γ increases; (3) Rogue waves can be enhanced in the center position due to the increase in the value of γ .