Rogue waves on the periodic wave background in the Kadomtsev–Petviashvili I equation

Rogue waves arising on the background of two families of periodic standing waves in the Kadomtsev–Petviashvili I (KPI) equation are investigated. By the nonlinearization of spectral problem and Darboux transformation approach, the rogue wave solutions of the KPI equation on the Jacobian elliptic functions dn and cn background are derived. Darboux transformation is also used to introduce trigonometric function periodic background for the higher-order rogue wave in the KPI equation. On the plane wave background, some rogue waves, breathers and hybrid waves in the KPI equation are obtained as a byproduct in this paper. Moreover, the results represented in this paper enrich the dynamics of (2+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2+1)$$\end{document} dimensional nonlinear wave equations.


Introduction
In 1895, Korteweg and de Vries (KdV) derived a nonlinear evolution equation governing one dimensional, small amplitude, surface gravity wave propagating in a shallow channel of water [1] where subscripts denote partial differentiations. It is well known that the KdV equation (1) possesses the solitary wave solution, and the interaction of two solitary wave solutions is a remarkable property of elasticity. The solitary wave solutions of the KdV equation (1) propagate in one direction in shallow water. In 1970, Kadomtsev B.B. and Petviashvili V.I. presented a twodimensional generalization of the KdV equation, which is the Kadomtsev-Petviashvili (KP) equation [2,3] (u t + 6uu x + u x x x ) x + 3σ 2 u yy = 0, where σ 2 = ±1 and subscripts denote partial differentiations. The evolution described by the KP equation is weakly nonlinear, weakly dispersive and weakly twodimensional with all three effects being of the same order. If σ 2 = −1 and σ 2 = 1, then we have the KPI equation and the KP II equation respectively. The KPI equation (3) describes the long waves in water waves with weak surface tension, and the KPII equation (4) describes waves with strong surface tension. The KPI equation (3) can also describe the two-dimensional matter-wave pulses in Bose-Einstein condensates [4]. The KPI equation (3) can be decomposed into the focusing nonlinear Schrödinger equation (NLS) equation and the complex modified KdV (mKdV) equation by constraint [5]. The KPII equation (4) can also be decomposed into a (1+1)-dimensional Broer-Kaup (BK) and (1+1)-dimensional higher-order BK equation by using the symmetry constraint [6].
As far as we know, the rogue periodic wave solutions of the KPI equation (3) have not been explored. The purpose of this paper is to construct the rogue periodic waves for the KPI equation (3) by the nonlinearization of spectral problem and DT. The structure of this paper is given as follows. In Sect. 2, we are divided into the following three parts: (1) the DT of KPI equation (3) and the (1+1)-dimensional complex mKdV equation where q = q(x, y, t) is a compatible solution of (5) and (6). We consider the following constraint A decomposition of the KPI equation (3) is exactly related to (5) and (6). These mean that if q(x, y, t) solves (5) and (6), then the corresponding solution of the KPI equation (3) can be generated in terms of the constraint (7).
The equations (5) and (6) have the Lax representations as follows (2) , where and q * is the conjugate of q. The zero curvature equations U y − V (5) and (6), respectively.

Traveling periodic solutions
In order to obtain periodic wave solutions of the KPI equation (3), we take where Q(x, t) is the real periodic function to be determined and b is a real constant. Substituting (22) into (5) and (6), we have the following second-and third-order ordinary differential equation and respectively. Now, suppose that then from (23) and (24), we have and Integrating (27), one obtains Comparing (26) and (28), we have b = c 4 . (28) is rewritten as Integrating Eq. (29), we have where d is a constant. There are two kinds of the traveling periodic wave solutions in (30) expressed by the Jacobian elliptic function dn and cn as and where k ∈ (0, 1) is elliptic modulus.
With the help of (22), it is easy to know that the left-hand side of (19) becomes zero. Namely, we have Comparing (28) and (30) with (20) and (21), respectively, we have The second equation in (33) can be satisfied with two choices, either μ = 0 or μ = 0 and 1 = 4μ 2 .
In order to obtain the dn-periodic wave solution of the KPI equation (3), we apply the one-fold DT (60) to the Jacobian elliptic function dn and set the seed solution q = Q(ξ )e 1 4 ict , λ = η = λ + . In order to obtain the cn-periodic wave solution of the KPI equation (3), we use the one-fold DT to the Jacobian elliptic function cn and set the seed solution q = Q(ξ )e

Hybrid nonlinear waves
In this section, we will construct hybrid nonlinear wave solutions for the KPI equation (3) by the elementary DT. The hybrid higher order rogue wave and onebreather solutions are discussed in detail, and some figures are plotted.
For the solution (67), fixing A = 1 2 , the time evolution for the first-order rogue wave (67) is illustrated through Fig. 4, where (a) with t = − 1 5 , (b) with t = 0 and (c) with t = 1 5 . The shape of the first-order rogue wave is similar to [29], and this adjustment is in order to observe the interaction between the breath wave and rogue wave more clearly.
For the solution (71), fixing t = 0 and A = 1 2 , the wave shape of the second-order rogue wave (71) is given in Fig. 5, where (a) with a 1 = b 1 = 0, (b) with a 1 = b 1 = 1 and (c) with a 1 = b 1 = 4. The shape of the second-order rogue wave is similar to [29], and this adjustment is in order to observe the interaction between the breath wave and rogue wave more clearly.
(2) For the solution (73), fixing A = λ 2 = 1 2 , the time evolution for the first-order rogue wave and breather is shown through Fig. 6, where (a) with t = − 1 3 , (b) with t = 0 and (c) with t = 1 3 . In Fig.  6, the first-order rogue wave and the one-breather have an overtaking collision in the process of transmission. The research shows that breather's amplitude descent and band width growth with the eigenvalue λ 2 descent. Figure 7 shown that the breather becomes a periodic wave background with A = 1 2 , λ 2 = 1 15 . In Fig. 7, the time evolution of the firstorder rogue wave (73) on a periodic wave background, where (a) with t = − 1 3 , (b) with t = 0 and (c) with t = 1 3 . These results are similar to [35,36].
(3) For the solution (75), fixing A = 1 2 , a 1 = b 1 = 5, λ 2 = λ * 2 = 1 6 , the time evolution for the hybrid second-order rogue wave and breather solution in (75) is illustrated in Fig. 8, where (a) with t = − 1 For the solution (75), fixing A = 1 2 , a 1 = b 1 = 5, λ 2 = λ * 2 = 1 15 , the time evolution for the second-order rogue wave on a periodic wave background (75) is given in Fig. 9, where (a) with t = − 1 3 , (b) with t = 0 and (c) with t = 1 3 , this is similar to the last case. By constructing the hybrid first-order rogue wave and breather and the hybrid second-order rogue wave and breather solution, we can come to a conclusion that no energy conversion occurs during the interaction between the breather and rogue waves, which different from [35,36].

Discussions
(1) By comparison, we find that the periodic backgrounds of different equations have different effects on rogue waves on the elliptic dn or cn background. For example, in [19,20], the periodic background has no effect on the shape of the rogue wave on the elliptic dn background. However, for the rogue wave solution on the elliptic cn background, the periodic background affects the shape of rogue wave and makes it periodic. In [14][15][16], we find that the periodic background has no effect on the shape of the rogue wave on the elliptic cn background. However, for the rogue wave solution on the elliptic dn background, the periodic background affects the shape of rogue wave and makes it periodic. The reason for the difference between the two needs to be studied. (2) By adjusting the wave center parameters, we find that the breather and rogue waves have no influence on each other, which is different from the conclusion obtained by [35,36].

Conclusions
In this paper, we have systematically investigated exact solutions for rogue waves arising on the background of periodic standing waves in the KPI equation (

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Declarations
Conflict of interest The authors declare that they have no conflict of interest.
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