Breathers and rogue waves on the double-periodic background for the reverse-space-time derivative nonlinear Schr\"odinger equation

In the present investigation, the solutions on the periodic and double-periodic background are successfully constructed by Darboux transformation using a plane wave seed solution. Firstly, the Darboux transformation for the reverse-space-time DNLS equation is constructed. Secondly, periodic solutions, breathers, double-periodic solutions, breathers on the periodic background and double-periodic background are studied. Thirdly, the higher-order rogue waves on the periodic and double-periodic background are constructed by semi-degenerate Darboux transformations. In addition, the dynamic behaviors of the solutions are plotted to show some interesting new solution structures.

The derivative nonlinear Schrödinger equation (DNLS) [17][18][19][20] is given by where the complex function q = q(x, t) denotes the wave envelope and * denotes the complex conjugation. Eq. (1.1) arises in the study of circular polarized Alfvén waves in plasma [21], propagating parallel to the magnetic field [22], which is one of the most important integrable systems in mathematics and physics. Recently, the equation is also used to describe large-amplitude magnetohydrodynamic waves [23,24] of plasmas, nonlinear optics, the sub-picosecond and femtosecond pulses in single-mode optical fiber [25][26][27]. The DNLS and nonlocal DNLS equations are reduced from the Kaup-Newell system [28,29] and are Lax integrable. There generate many new physical phenomena and have important physical significance when nonlocal terms are added to nonlinear equations. In recent years, many researchers have studied nonlocal DNLS equations from different viewpoints and perspectives. For example, in [30], the global bounded solutions of the nonlocal DNLS equation have been obtained from zero seed solution by Darboux transformation (DT) [31][32][33][34][35][36]. Furthermore, solutions and connections of nonlocal DNLS equations have been studied in [37]. In [38], the periodic bounded solutions of the second-type nonlocal DNLS equation from zero seed solutions have been studied. The P T -symmetric, reverse-time, and reverse-space-time nonlocal DNLS equations are integrable infinite dimensional Hamiltonian dynamical systems, which were first proposed by Ablowitz and Musslimani [39,41]. The general N-solitons in these three nonlocal nonlinear Schrödinger equations are presented by Yang in [40]. To investigate the connections between solutions at reverse-space-time points (x, t) and (−x, −t), we need to consider the reverse-space-time reduction. The reverse-space-time DNLS equation is as follows here the symmetry reductions are nonlocal both in space and time. The reverse-space-time DNLS equation has many physical applications in optics, ocean water waves, quantum entanglement and an unconventional system of magnetics etc [42][43][44]. For eq. (1.2), the evolution of the solution at location (x, t), depends both on the local position (x, t) and the distant position (−x, −t). This implies that the states of the solution at distant opposite locations are directly related, reminiscent of quantum entanglement in pairs of particles [40]. The solution of reverse-space-time DNLS equation can extend the solution of the local equation to a more general case and deepen the physical research on the mechanism of ocean rogue waves. These results would also be useful for understanding the corresponding rational soliton phenomena in many fields of nonlocal nonlinear dynamical systems such as nonlinear optics, Bose-Einstein condensates, ocean and other relevant fields [45,46]. In general, it is extremely nontrivial to construct the rogue waves on a periodic background which is usually associated with complicated Jacobi elliptic functions [47][48][49][50][51][52][53][54], PT symmetry [55], integrable equations with variable coefficients [56,57], or vector integrable equations [58]. In [59,60], the rogue waves on the periodic background have been constructed by using odd-order semi-degenerate DT. In this article, we mainly study the breathers and the rogue waves on the double periodic background by using even-fold DT and even-order semi-degenerate DT respectively. This is an effective new method to construct the solutions on double-periodic background without using Jacobi elliptic functions. It is of great physical significance to study rogue waves on a double-periodic background. For example, the rogue waves on the double-periodic background in the hydrodynamical experiments are possible due to the rogue waves on the continuous wave background observed in laser optics and water tanks [61]. Rogue waves on the double-periodic background could be relevant to diagnostics of rogue waves on the ocean surface and understanding the formation of random waves due to modulation instability [62].
In this work, we construct the breathers and rogue waves on the periodic background by odd-fold DT and odd-order semi-degenerate DT respectively. This is the first time to extend this method to reverse-space-time nonlocal equations. Remarkably, using a plane wave seed solution, the breathers and rogue waves on the double-periodic background are first successfully constructed by even-fold DT and even-order semi-degenerate DT respectively. By taking the dynamics analysis of the first-order rogue waves on double-periodic background, we show two types of structures: the two peaks and four peaks. The interesting thing can also be seen from the dynamic figures of the second-order rogue waves on the double-periodic background. There are two types of structures: one peak and two peaks. We shall also show the transformation process of double-periodic background and plane wave background in this study. These results have not been reported to our best knowledge.
The organizational structure of this paper is as follows. In section 2, the determinant representation of the nfold DT formula is given. In section 3, using a plane wave seed solution, the periodic solution, breathers on the periodic background are given by odd-fold DT. The double-periodic solution, breathers and breathers on the doubleperiodic background solution are given by even-fold DT. In section 4, we construct higher-order rogue waves on the periodic background and double-periodic background by semi-degenerate DT formula. The final section is devoted to conclusion.

DT of the reverse-space-time DNLS equation
Starting from the Kaup-Newell system [63,64], when the reduction condition is r(x, t) = −q(−x, −t), the Lax pair of the reverse-space-time DNLS equation can be obtained as follows.
ã , j = 1, 2, . . . , which is the eigenfunction of the Lax pair (2.1) and (2.2) associated with λ = λ j . The eigenfunction admit the following symmetry condition: DT has unique advantages in constructing solutions due to pure algebraic construction. In this section, the DT for the Eq. (1.2) will be introduced. Under gauge transformation After derivation, we get the following conclusion.
Furthermore, the following identity can be deduced U [1] t − V [1] x + [U [1] , V [1] Due to the matrix T is nonsingular, the zero curvature equation x + [U [1] , V [1] ] = 0. This implies that, in order to make spectral problem Eq. (2.1) is invariant under the gauge transformation Eq. (2.4), it is important to find a matrix T so that U [1] and V [1] have the same forms as U and V . At the same time, the old solutions (q, q(−x, −t)) in spectral matrixes U and V are mapped into new solutions (q [1], q [1](−x, −t)) in transformed spectral matrixes U [1] and V [1] .
In general, if the Darboux matrix T is a polynomial of the parameter λ, for simplicity, we take T as Substituting the Darboux matrix T into Eq. (2.6) and Eq. (2.7), the one-fold DT formula can be derived by comparing the coefficient in terms of λ i We also can deduced that b 1 = c 1 = 0, a 1 and d 1 are undetermined functions about x and t. a 0 , b 0 , c 0 and d 0 are constants. In order to obtain the specific expression of the elements in the matrix T , for simplicity, let a 0 = d 0 = 0 then In particular, taking b 0 = c 0 = λ 1 , the one-fold DT formula is given by the eigenfunction Ψ 1 associated with λ 1 as follows After n times iterations based on the one-fold Darboux matrix (2.10), the form of n-fold Darboux matrix is as follows where P 0 is a constant matrix and P i (1 ≤ i ≤ n) is a matrix function about x and t. Using the same derivation method as the one-fold DT formula, yields Furthermore, the determinant representation of a n , d n and b n−1 can be given by the kernel problem of DT matrix T n . i.e., Then the concrete expression of the new solutions q[n] can be seen in the following.
where W 11 , W 12 , and W 21 have different forms depending on the parity of n. When n = 2k, When n = 2k + 1,

Breathers on the periodic and double-periodic background
Starting from the seed solutions q(x, t) = ce i(ax+bt) and q(−x, −t) = ce −i(ax+bt) , where b = −ac 2 + a 2 and c denoting the background height. In this section, the periodic solution, breathers on the periodic background are given by odd-fold DT. In addition, the double-periodic solution, breathers and breathers on the double-periodic background solution are given by even-fold DT.
For n=2: The two-fold DT formula (2.14) of the reverse-space-time DNLS equation implies a solution We can derive breathers and double-periodic solution according to different reduced methods of spectrum parameter λ 1 and λ 2 as follows. Case 1: λ 2 = ±λ * 1 , now q[2] is a breathers. For simplicity, we take λ 2 = −λ * 1 = −α 1 +iβ 1 and Im −a 2 − 4λ 4 1 − 4λ 2 1 c 2 − a = 0. Then lim x→∞ t→∞ |q[2]| 2 = c 2 . and the center trajectory equation of solution q [2] can be calculated as x = 4(β 2 1 − α 2 1 )t. We can know that the solution evolves periodically along the straight line with a certain angle of x axis and t axis when β 2 1 = α 2 1 from the above analysis (see Fig. 2(a)). And when β 2 1 = α 2 1 , the classical Ma breathers (time periodic breather) can be seen in Fig. 2(b) and the Akhmediev breathers (space periodic breather) can be seen in Fig. 2(c). Next, we construct rogue wave q r by letting the period of the breathers tend to be infinity. Let c → −2β 1 in q [2], then x, m 10 = 256(α 4 1 β 4 1 + 2α 2 1 β 6 1 + β 8 1 )x 4 + 32(α 2 1 β 2 1 + 3β 4 1 )x 2 + 1. After calculation and analysis, we know that lim x→∞ t→∞ |q r | 2 = (2β 1 ) 2 . The maximum amplitude of |q r | 2 equals to (6β 1 ) 2 occurs at x = 0 and t = 0. This means that the maximum amplitude of the rogue wave q r is 3 times compared with the asymptotic plane wave at infinity. The min amplitude of |q r | 2 occurs at ( The rogue wave |q r | with specific parameter α 1 = 1 2 and β 1 = 1 is plotted in Fig. (2(d)). From the graph of the rogue wave |q r | , we can see that the rogue wave q r has a single peak with two caves on both sides of the peak. The optical pulse q r only exists locally with all variables and disappears as time and space go far.
Case 2: λ 1 = iβ 1 , λ 2 = iβ 2 and β 2 = ±β 1 , q [2] is represented as a double-periodic wave solution which is similar to the Jacobi elliptic function-type seed solution. From the Fig. 3, we can see clearly that there are two periodic waves with different directions in the double-periodic wave solution, and when the two waves with different directions are superimposed on each other, a higher wave peak can be generated. From a visual perspective, it seems that several parallel breathers are generated under the period background. We find that the periodic solution can be generated by first-fold DT. The breathers and double-periodic solutions can be constructed respectively according to second-fold DT. Therefore, we consider constructing the breathers on the periodic background by odd-fold DT, and constructing the breathers on the double-periodic background by even-fold DT.
For n=3: Set λ 2 = −λ * 1 = −α 1 + iβ 1 , λ 3 = iβ 3 and a = 2α 2 1 − 2β 2 1 + c 2 . Parameter values have a great influence on the propagation direction of the breathers. When β 2 1 = α 2 1 , solution q[3] is a general breather solution on periodic background (see Fig. 4(a)). When β 2 1 = α 2 1 , the classical Ma breathers on the periodic background can be seen in Fig.  4(b) and the Akhmediev breathers on the periodic background can be seen in Fig. 4(c). There are some interesting phenomenons: Under the perturbation of the periodic background, the crest of the Ma breathers is cut and the phase shift occurs at the center of the breathers.
For n=4: Set λ 2 = −λ * 1 = −α 1 + iβ 1 , λ 3 = iβ 3 , λ 4 = iβ 4 and β 4 = ±β 3 . The breathers on a double-periodic background generated by formula (2.14). Similar to n = 2, we also let a = 2α 2 1 − 2β 2 1 + c 2 . When β 2 1 = α 2 1 , we can construct the general breathers on the double-periodic (see Fig. 5(a)). Under the disturbance of double-periodic background, the propagation direction of general breathers usually produces shift. When β 2 1 = α 2 1 , the Ma breathers and Akhmediev breathers can be constructed by adjusting spectrum parameters. As for the Ma breathers on the double-periodic, due to the great influence of the double-periodic background, the image of Ma breathers solution looks like it disappears in the double-periodic background (see Fig. 5(b)). The Akhmediev breathers on the doubleperiodic background is plotted in Fig. 5(c). Visually, it looks like a breathers with a higher amplitude is generated under the several parallel breathers background.

Higher-order Rogue waves on the double-periodic background
Note that the eigenfunction is degenerated when λ = ± 1 2 √ 2a − c 2 − 1 2 ic. In this case, the higher-order rogue waves can be derived. Combined with the methods of constructing the periodic and double-periodic background in the previous section, we will give the solutions of higher-order rogue waves on the periodic and double-periodic background in this section. Since periodic can be derived by odd-fold DT, both breathers and double-periodic solution can be obtained by even-fold DT. We can construct higher-order rogue waves on the periodic by odd-order semi-degenerate DT and higher-order rogue waves on the double-periodic by even-order semi-degenerate DT.
The Higher-order rogue waves on periodic background can be generated by odd-order semi-degenerate DT. For n = 3, Here, set a = 1 and c = 1 for convenience, the first-order rogue waves q 3 on the periodic background for Eq. (1.2) can be obtained. The patterns of q 3 are displayed in Fig. 7(a) and Fig. 7(b). For β 3 > 0, the rogue wave pattern locates on the area where the periodic pattern reaches its amplitude. However, for β 3 < 0, the rogue wave pattern locates in the middle of two amplitude trajectories of the periodic pattern, which looks like that the rogue wave is generated by the interaction of two waves of the periodic pattern.
For n = 5, ic and λ 5 = iβ 5 , the second-order rogue waves on the periodic background for Eq. (1.2) was shown in Fig. 8. The second-order rogue waves have a high amplitude peak on the center distributed with some lower peaks and four caves. Same with the case n = 3, for β 5 > 0, the rogue wave pattern locates on the area where the periodic pattern reaches its amplitude (see Fig. 8(a)). For β 5 < 0, the rogue wave pattern locates in the middle of two amplitude trajectories of the periodic pattern (see Fig. 8(b)). And the periodic background can influence the peak value of the rogue wave. When n = 4, we can obtain the first-order rogue waves on the double-periodic background by (4.1). The selection of parameters have effect both on the amplitude of the double-periodic background and the amplitude of the rogue waves. The interesting thing is that there are two peaks rogue wave on the double-periodic background when we take a = 1, c = 1, β 3 = 0.1 and β 4 = √ 2 2 (see Fig. 9(a)). There are four peaks rogue wave on the double-periodic background when a = 1, c = 1 2 , β 3 = 0.1 and β 4 = √ 2 2 (see Fig. 9(b)). Significantly, when β 3 =−β 4 , the rogue waves on the double-periodic background will convert to the rogue waves on the plane wave (see Fig. 9(c)). Due to the reverse-space-time reduction conditions of the eq.(1.2), the positions of the rogue wave solutions show the connections between reverse-space-time points (x, t) and (−x, −t). We can verify this intuitively by observing the positions of two peaks rogue wave and four peaks rogue wave.  When n = 6, we can obtain the second-order rogue waves on the double-periodic background by (4.1). Similar to the first-order rogue wave on double-periodic background, the selection of parameters also have effect both on the amplitude of the double-periodic background and rogue waves. The positions of the second-rogue waves also show the connections between reverse-space-time points (x, t) and (−x, −t). Compared with first-order rogue waves on the double-periodic background, the difference is that there are one peaks on the double-periodic background when we take a = 1, c = 1, β 5 = 0.1 and β 6 = √ 2 2 (see Fig. 10(a)). And there are two peaks on the double-periodic background when we take a = 1, c = 1 2 , β 5 = 0.1 and β 6 = √ 2 2 (see Fig. 10(b)). See it visually in three dimensions, we can find that the energy centered on the rogue wave and gradually dissipates to a steady state. When β 5 =−β 6 , the second-order rogue waves on the double-periodic background will convert to the second-order rogue waves on the plane wave (see Fig. 10(c)). In addition, compared with the first-order rogue waves, second-order rogue waves have higher amplitude.

Conclusion
In our present investigation, we constructed the breathers and rogue waves on the double-periodic background for Eq. (1.2), which are first generated by plane wave seed solution. The general breathers, Ma breathers and Akmediev breathers on double-periodic background were generated by even-fold DT. Due to the influence of the double-periodic background, the image of Ma breathers solution looks like it disappears in the double-periodic background and the propagation direction of general breathers produces shift.
By using the even-order semi-degenerate DT, we derived the first-order and second-order rogue waves on the doubleperiodic background. Due to the reverse-space-time reduction conditions, the positions of rogue wave solutions show connections between reverse-space position and reverse-time points (x, t) and (−x, −t). For the first-order rogue waves on the double-periodic background, we find that there are two peaks or four peaks when we adjust the parameters. There are one peak or two peaks on the double-periodic background when adjusting parameters of second-order rogue waves. Second-order rogue waves have higher amplitude than first-order rogue waves. Significantly, the double-periodic background will convert to the plane wave background when β n−1 = −β n .
We generated the general breathers, Ma breathers and Akhmediev breathers by odd-fold DT. There are some interesting phenomenons: the crest of the Ma breathers is cut and the phase shift occurs at the center of the general breathers with the perturbation of periodic background. The first-order and second-order rogue waves on the periodic background were derived by odd-order semi-degenerate DT formula, respectively. When β n > 0, the rogue wave patterns are located in the area where the periodic pattern reaches its amplitude. However, when β n < 0, the rogue wave patterns locate in the middle of two amplitude trajectories of the periodic pattern, which looks like that the rogue wave is generated by the interaction of two waves of the periodic pattern. The higher-order rogue waves have a high amplitude peak on the center distributed with some lower peaks and even numbers of caves.
Moreover, as we remarked in the introduction, rogue waves on the periodic and double-periodic background have some important uses and applications in many diverse areas of mathematics and physics. Therefore, the results which are presented in this article are also of great physical significance. For example, the rogue waves on the periodic and double-periodic background can be relevant to diagnostics of rogue waves on the ocean surface and understanding the formation of random waves due to modulation instability.