Kadomtsev–Petviashvili equation with self-consistent sources: breathers, lumps and their interactions

Abstract

Under investigation in this paper is the Kadomtsev–Petviashvili (KP) equation with self-consistent sources. We adopt two different bilinear systems which require different KP hierarchy reductions to construct the corresponding breather solutions in Gramian determinant. Then, we construct two distinct categories of breather solutions, and show that both types are the same only under the condition $a=\alpha =0$. Furthermore, employing the long-wave limits to breather solutions, we obtain some rational and exp-rational solutions, which reveal various nonlinear interactions, including between two lumps, between a lump and a breather and between a lump and a soliton. Space-localized “dark” and tetrapetalous-type breathers and lumps are also addressed. Interactions between the above hybrid waves is of fundamental importance in the understanding of high-dimensional nonlinear phenomena of extreme ocean waves and in other fields such as solid state physics and plasma physics.

1 Introduction

The Kadomtsev–Petviashvili (KP) equation

$$\begin{aligned} 4u_{T}+u_{XXX}+6uu_{X}+3\sigma ^2\partial ^{-1}u_{YY}=0 \end{aligned}$$

(1)

is perhaps the prototypical of the $(2+1)$-dimensional integrable system which has been proposed to study the stability of the one-soliton solution of the Korteweg–de Vries (KdV) equation under the influence of weak transverse perturbations [1]. Here, u is a function of the scaled time coordinate T and the spatial coordinates X and Y. The case $\sigma ^2=-1$ denotes the KP-I equation, which is a govern model to describe the waves in thin films with high surface tension, while the case $\sigma ^2=1$ is known as the KP-II equation, which models the water waves with small surface tension [2]. Experiment in wave basic has demonstrated the existence of a family of periodic gravitational water waves that propagate without practically any change in form on the surface of shallow water with uniform depth [3], asymmetric versions of such waves have also been studied [4]. On flat beaches, nonlinear shallow ocean-wave soliton interactions can be simulated by the interactions between the multi-soliton solutions of the KP equation [5]. Ocean waves are generated by winds and storms, and the majority of the ocean waves are approximately periodic. The two-dimensional nature of the water surface makes the KP equation being a natural place to seek solutions that might describe approximately periodic waves in shallow water [6]. KP equation also has possible connections to tsunami dynamics [7].

The integral systems with self-consistent sources was first constructed by Mel’Nikov [8], and such systems can be derived from the Lax pair [8, 9]. Mathematically, such systems of equations arise via relating the sources to the singular part of the dispersion law [10], or based on the high-order constrained flows of soliton equations [11, 12]. In the physical contexts, the KdV equation with self-consistent sources has been applied to describe the interaction of short and long capillary-gravity waves [13]; The nonlinear Schrödinger equation (NLS) with self-consistent sources represents the nonlinear interaction of an electrostatic high-frequency wave with the ion acoustic wave in a two component homogeneous plasma [14, 15].

This work will concentrate on the following KP equation with self-consistent sources (KPESCS) [16,17,18,19,20],

$$\begin{aligned} \begin{array}{l} \displaystyle -4u_{t}+u_{xxx}+6uu_{x}+3\sigma ^2\partial ^{-1}u_{yy}\\ -\sum _{\varsigma =1}^{K}(\Phi _\varsigma \Psi _\varsigma )_x=0,\\ \displaystyle \sigma \Phi _{\varsigma ,y}=\Phi _{\varsigma ,xx}+u\Phi _{\varsigma },\\ \displaystyle \hspace{0.08cm}-\sigma \Psi _{\varsigma ,y}=\Psi _{\varsigma ,xx}+u\Psi _{\varsigma },\ \ \sigma ^2=\pm 1, \end{array} \end{aligned}$$

(2)

where $\Phi _\varsigma$ and $\Psi _\varsigma$ are complex functions of the scaled spatial coordinates x, y and temporal coordinate t, u is a real function, the integer $1\le \varsigma \le K$ denotes the degree of the self-consistent sources. KPESCS goes back to KP-IESCS and KP-IIESCS, respectively, when $\sigma ^2=-1$ and $\sigma ^2=1$. Equation (2) has been used to describe the interaction of a long wave with a short-wave packet propagating at an angle to each other in solid state physics, hydrodynamics, plasma physics [16]. Physically, the sources can result in solitary waves moving with a non-constant velocity and therefore lead to a variety of dynamics of physical models. If the sources becomes zero, then Eq. (2) degenerates into Eq. (1), and hence Eq. (2) can be regarded as a coupled form of Eq. (1) as a consequence. Hereinafter we discuss the case $\sigma ^2=-1$, i.e., the KP-IESCS.

Consider the following transformation

$$\begin{aligned} u=2(\textrm{ln}f)_{xx},\ \ \Phi _\varsigma =\rho _0e^{i\theta }\frac{g_\varsigma }{f},\ \ \Psi _\varsigma =\rho _0e^{-i\theta }\frac{h_\varsigma }{f},\nonumber \\ \end{aligned}$$

(3)

with $\theta =\alpha x+ \alpha ^2 y+ w(t)$, and $\alpha$ being a real constant. Equation (2) has the following bilinear form [20]:

$$\begin{aligned} \begin{array}{l} \displaystyle (D_{x}^4-4D_{x}D_{t}-3D_{y}^2+\rho _0^2) f\cdot f =\rho _0^2gh^{\mathrm{{T}}}, \\ \displaystyle (D_{x}^2+2i\alpha D_{x}-iD_{y}) g_\varsigma \cdot f =0,\\ \displaystyle (D_{x}^2-2i\alpha D_{x}+iD_{y}) h_\varsigma \cdot f =0, \end{array} \end{aligned}$$

(4)

where $D_{x}$, $D_{y}$ and $D_{t}$ are the Hirota bilinear differential operators defined by [21], and $g=(g_1,g_2,\ldots ,g_K)$, $h=(h_1,h_2,\ldots ,h_K)$ are the row vectors, the superscript “T” means the transposition.

Via the generalized binary Darboux transformation [17], Wronskian technique and Hirota method [18], N-soliton solutions for Eq. (2) have been derived. Lie symmetries for Eq. (2) have been constructed [19]. The Rogue wave solutions for Eq. (2) have been constructed by the KP hierarchy reduction procedure [20]. The discrete versions of the Eq. (2) have been also studied, see Refs. [22, 23] and references therein. However, breathers, lumps and interactions between lumps and breathers for Eq. (2) have not been investigated before. Breather, coherent nonlinear solitary wave structure on an unstable background, is related to the modulation instability phenomena [24, 25]. There are typically three types of the breathers: Kuznetsov-Ma, Akhmediev and Tajiri-Watanabe breathers [26]. Breathers can be used in various physical scenes such as the ocean [27], optics [28] and plasmas [29]. The Fermi-Pasta-Ulam recurrence can be induced by breathers in an optical microresonator [30, 31]. Breathers are known to model extreme waves in several nonlinear dispersive media in which the initial underlying process is assumed to be narrow banded [32], and breather-collision dynamics have practicalness for nonlinear fiber optics [33]. Lumps are the localized algebraically decaying solutions which have applications in ocean [34, 35], optics [36, 37] and Bose condensate [38].

There are many powerful tools to construct the various exact solutions of the integral systems, such as the Kudryashov’s simplest equation approach [39], the generalised exponential rational function approach [39, 40], the $(G'/G)$-expansion method [37, 41] and KP hierarchy reduction procedure [42]. Herein, we will employ the KP hierarchy reduction procedure to study the hybrid waves which include solitons, breathers and lumps of Eq. (2). From Sato theory, the bilinear forms of integrable systems belong to the KP hierarchy and its extensions [43]. In Sect. 2, two categories of the breather solutions in Gramian determinant for Eq. (2) will be constructed. Exp-rational and rational solutions are also addressed by employing the long-wave limits on such breather solutions. In Sect. 3, we adopt a different bilinear equation of the KP hierarchy to derive another two categories of the breather solutions in Gramian determinant for Eq. (2), the corresponding exp-rational and rational solutions are also addressed. Section 4 is comprised of our conclusions.

2 Breathers for Eq.  (2) through the KP hierarchy reduction procedure

Firstly, the bilinear equations in the KP hierarchy

$$\begin{aligned} \begin{array}{l} \displaystyle (D^4_{x_{1}}-4D_{x_{1}}D_{x_{3}}+3D^2_{x_{2}}) \varvec{\tau _{n}} \cdot \varvec{\tau ^{\mathrm{{T}}}_{n}}=0,\\ \displaystyle (D_{x_{1}}D_{x_{-1}}-2)\varvec{\tau _{n}} \cdot \varvec{\tau ^{\mathrm{{T}}}_{n}}=-2\varvec{\tau _{n+1}}\cdot \varvec{\tau ^{\mathrm{{T}}}_{n-1}},\\ \displaystyle (D_{x_{1}}^2-D_{x_{2}}+2aD_{x_{1}})\tau _{n+1,\varsigma }\cdot \tau _{n,\varsigma }=0,\\ \displaystyle (D_{x_{1}}^2+D_{x_{2}}-2aD_{x_{1}})\tau _{n-1,\varsigma }\cdot \tau _{n,\varsigma }=0, \end{array} \end{aligned}$$

(5)

have the following $N \times N$ Gram determinant solutions

$$\begin{aligned}{} & {} \tau _{n,\varsigma }=\det _{1\le r,j\le N} [m_{rj,\varsigma }^{(n)}], \end{aligned}$$

(6)

where the matrix elements are given by

$$\begin{aligned} \partial _{x_1}m^{(n)}_{rj,\varsigma }= & {} \psi _{r,\varsigma }^{(n)}\phi _{j,\varsigma }^{(n)},\\ \partial _{x_2}m^{(n)}_{rj,\varsigma }= & {} (\partial _{x_1}\psi _{r,\varsigma }^{(n)})\phi _{j,\varsigma }^{(n)}-\psi _{r,\varsigma }^{(n)}(\partial _{x_1}\phi _{j,\varsigma }^{(n)}),\\ \partial _{x_3}m^{(n)}_{rj,\varsigma }= & {} (\partial ^2_{x_1}\psi _{r,\varsigma }^{(n)})\phi _{j,\varsigma }^{(n)}-(\partial _{x_1}\psi _{r,\varsigma }^{(n)})(\partial _{x_1}\phi _{j,\varsigma }^{(n)})\\ {}{} & {} +\psi _{r,\varsigma }^{(n)}(\partial ^2_{x_1}\phi _{j,\varsigma }^{(n)}),\\ \partial _{x_{-1}}m^{(n)}_{rj,\varsigma }= & {} -\psi _{r,\varsigma }^{(n-1)}\phi _{j,\varsigma }^{(n+1)},\\ m^{(n+1)}_{rj,\varsigma }= & {} m^{(n)}_{rj,\varsigma }+\psi _{r,\varsigma }^{(n)}\phi _{j,\varsigma }^{(n+1)},\\ \partial _{x_2}\psi _{r,\varsigma }^{(n)}= & {} \partial ^2_{x_1}\psi _{r,\varsigma }^{(n)},\\ \partial _{x_2}\phi _{j,\varsigma }^{(n)}= & {} -\partial ^2_{x_1}\phi _{j,\varsigma }^{(n)},\\ \partial _{x_3}\psi _{r,\varsigma }^{(n)}= & {} \partial ^3_{x_1}\psi _{r,\varsigma }^{(n)},\\ \partial _{x_3}\phi _{j,\varsigma }^{(n)}= & {} \partial ^3_{x_1}\phi _{j,\varsigma }^{(n)},\ \ r,j\in \mathbb {Z^+}, \end{aligned}$$

and $\varvec{\tau _{n}}=(\tau _{n,1},\tau _{n,2},\ldots ,\tau _{n,K})$ is the row vector. In order to construct breather solutions, we set

$$\begin{aligned}{} & {} m^{(n)}_{rj,\varsigma }=\delta _{rj,\varsigma }+\frac{1}{p_{r,\varsigma }+q_{j,\varsigma }}\psi _{r,\varsigma }^{(n)}\phi _{j,\varsigma }^{(n)},\\{} & {} \psi _{r,\varsigma }^{(n)}=(p_{r,\varsigma }-a)^{n}(p_{r,\varsigma }+q_{r,\varsigma })e^{\xi _{r,\varsigma }},\\{} & {} \phi _{j,\varsigma }^{(n)}={\left( -\frac{1}{q_{j,\varsigma }+a}\right) }^ne^{\eta _{j,\varsigma }}, \end{aligned}$$

with

$$\begin{aligned} \xi _{r,\varsigma }= & {} (p_{r,\varsigma }-a)^{-1}x_{-1}+p_{r,\varsigma }x_{1}+p_{r,\varsigma }^2x_{2}\\ {}{} & {} +p_{r,\varsigma }^3x_{3}+\xi _{r,\varsigma }^0,\\ \eta _{j,\varsigma }= & {} (q_{j,\varsigma }+a)^{-1}x_{-1}+q_{j,\varsigma }x_{1}-q_{j,\varsigma }^2x_{2}\\ {}{} & {} +q_{j,\varsigma }^3x_{3}+\eta _{j,\varsigma }^0. \end{aligned}$$

Under the context of the KP hierarchy reduction procedure, we introduce an auxiliary variable s and reconstruct the first equation of Bilinear Form (4) as

$$\begin{aligned} \begin{array}{l} \displaystyle (D_{x}^4+D_{x}D_{s}-3D_{y}^2) f\cdot f =0, \\ \displaystyle (4D_{x}D_{t}+D_{x}D_{s}) f\cdot f =\rho _0^2\left( f^2-gh^{\mathrm{{T}}}\right) . \end{array} \end{aligned}$$

(7)

Under the assumptions

$$\begin{aligned} x_{1}=ix,\ \ x_{2}=iy,\ \ x_{3}=-it+4is,\ \ x_{-1}=-i\frac{\rho _0^2}{8}t, \end{aligned}$$

and $\alpha =a$, we further introduce

$$\begin{aligned} f=\tau _{0,\varsigma },\ \ g_\varsigma =\tau _{1,\varsigma },\ \ h_\varsigma =\tau _{-1,\varsigma }, \end{aligned}$$

then

$$\begin{aligned} \begin{array}{l} \displaystyle f=\left| \delta _{rj}+\frac{p_{r,\varsigma }+q_{r,\varsigma }}{p_{r,\varsigma }+q_{j,\varsigma }}e^{\xi _{r,\varsigma }+\eta _{j,\varsigma }} \right| _{1\le r,j \le M}, \ \ M\in \mathbb {Z^+},\\ \displaystyle g_\varsigma =\left| \delta _{rj}+\left( -\frac{p_{r,\varsigma }-a}{q_{j,\varsigma }+a}\right) \frac{p_{r,\varsigma }+q_{r,\varsigma }}{p_{r,\varsigma }+q_{j,\varsigma }}e^{\xi _{r,\varsigma }+\eta _{j,\varsigma }} \right| _{1\le r,j \le M},\\ \displaystyle h_\varsigma =\left| \delta _{rj}+\left( -\frac{q_{j,\varsigma }+a}{p_{r,\varsigma }-a}\right) \frac{p_{r,\varsigma }+q_{r,\varsigma }}{p_{r,\varsigma }+q_{j,\varsigma }}e^{\xi _{r,\varsigma }+\eta _{j,\varsigma }} \right| _{1\le r,j \le M}, \end{array}\nonumber \\ \end{aligned}$$

(8)

with

$$\begin{aligned} \xi _{r,\varsigma }= & {} ip_{r,\varsigma }x+ip_{r,\varsigma }^2y-i\left[ p_{r,\varsigma }^3+\frac{\rho _0^2}{8(p_{r,\varsigma }-a)} \right] t\\ {}{} & {} +\xi _{r,\varsigma }^0,\\ \eta _{j,\varsigma }= & {} iq_{j,\varsigma }x-iq_{j,\varsigma }^2y-i\left[ q_{j,\varsigma }^3+\frac{\rho _0^2}{8(q_{j,\varsigma }+a)} \right] t\\ {}{} & {} +\eta _{j,\varsigma }^0, \end{aligned}$$

where $p_{r,\varsigma }$’s, $q_{j,\varsigma }$’s, $\xi _{r,\varsigma }^0$’s and $\eta _{j,\varsigma }^0$’s are the complex constants, $\delta _{rj}$ is the Kronecker delta notation. Without loss of generality, s can be set to zero.

Remark 2.1

From Expression (8), $g_\varsigma$ depends on the values of $p_{r,\varsigma }$ and $q_{j,\varsigma }$, and the expressions of $g_\varsigma$’s share the same format, the situation is also agreement with $h_\varsigma$ and $f_\varsigma$. Therefore, we omit the subscript “$\varsigma$” in the following discussions for convenience.

2.1 Breather-I for Eq. (2)

2.1.1 Breather solutions

We choose the following conditions

where $\Omega _k\in {\mathbb {C}}$ and $\theta _k\in {\mathbb {R}}$, therefore f is a real function. Summarizing the above results, Eq. (2) has the N-th order breather solutions:

$$\begin{aligned} u=2(\textrm{ln}f)_{xx},\ \ \Phi _\varsigma =\rho _0e^{i\theta }\frac{g_\varsigma }{f},\ \ \Psi _\varsigma = \rho _0e^{-i\theta }\frac{h_\varsigma }{f},\nonumber \\ \end{aligned}$$

(9)

where

$$\begin{aligned}{} & {} f=\Delta \left| {\begin{array}{*{5}{c}} {\Theta ^0_{1,1}} &{} {\Theta ^0_{1,2}} &{} {{...}} &{} {\Theta ^0_{1,N}} \\ {\Theta ^0_{2,1}} &{} {\Theta ^0_{2,2}} &{} {{...}} &{} {\Theta ^0_{2,N}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {\Theta ^0_{N,1}} &{} {\Theta ^0_{N,2}} &{} {{...}} &{} {\Theta ^0_{N,N}} \\ \end{array}} \right| ,\\{} & {} g_\varsigma =h_\varsigma ^*=\Delta \left| {\begin{array}{*{5}{c}} {\Theta ^1_{1,1}} &{} {\Theta ^1_{1,2}} &{} {{...}} &{} {\Theta ^1_{1,N}} \\ {\Theta ^1_{2,1}} &{} {\Theta ^1_{2,2}} &{} {{...}} &{} {\Theta ^1_{2,N}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {\Theta ^1_{N,1}} &{} {\Theta ^1_{N,2}} &{} {{...}} &{} {\Theta ^1_{N,N}} \\ \end{array}} \right| , \end{aligned}$$

with

Remark 2.2

Since $\Delta$ is a function independent of x and $(\ln {f})_{xx}=\frac{ff_{xx}-f_x^2}{f^2}$, $\Delta$ can be eliminated.

The first-order breather solutions can be derived by setting $N=1$ in Solutions (9),

$$\begin{aligned} \begin{array}{l} \displaystyle f=1+e^{\vartheta _{1}}+e^{\vartheta _{1}^*}+A_1e^{\vartheta _{1}+\vartheta _{1}^*},\\ \displaystyle g_\varsigma =1+A_2e^{\vartheta _{1}}+(A_2^*)^{-1}e^{\vartheta _{1}^*}+A_1A_2(A_2^*)^{-1}e^{\vartheta _{1}+\vartheta _{1}^*},\\ \displaystyle h_\varsigma =1+(A_2)^{-1}e^{\vartheta _{1}}+A_2^*e^{\vartheta _{1}^*}+A_1A_2^*(A_2)^{-1}e^{\vartheta _{1}+\vartheta _{1}^*}, \end{array} \end{aligned}$$

(10)

where

$$\begin{aligned}{} & {} A_1=1-\left( \frac{\theta _{1}}{\Omega _{1}^*-\Omega _{1}}\right) ^2,\ \ A_2=\frac{\Omega _{1}-\frac{\theta _{1}}{2}+a}{\Omega _{1}+\frac{\theta _{1}}{2}+a},\\{} & {} \vartheta _{1}=i\theta _{1}x-2i\theta _{1}\Omega _{1}y\\{} & {} -\!i\theta _{1}\left[ \frac{\theta _{1}^2}{4}\! +\!3\Omega _{1}^2\!+\!\frac{\rho _0^2}{2\theta _{1}^2-8(\Omega _{1}+a)^2}\right] t+\vartheta _{1}^0. \end{aligned}$$

Reconstructing Solutions (10) by taking $\vartheta _{1}=\vartheta _{1R}+i\vartheta _{1I}$, $\Omega _{1}=\Omega _{1R}+i\Omega _{1I}$ and $\vartheta _{1}^0=\vartheta _{1R}^0+i\vartheta _{1I}^0$, we obtain

$$\begin{aligned}{} & {} f=e^{\vartheta _{1R}}\left[ \sqrt{A_1}\cosh \left( \vartheta _{1R}+\frac{\ln A_1}{2}\right) +2\cos {\vartheta _{1I}} \right] ,\\{} & {} g_\varsigma =1\!+\!\vartheta _{1R}[A_2\!+\!(A_2^*)^{-1}]\cos (\vartheta _{1I})\!\\{} & {} +\!i\vartheta _{1R}[A_2\!-\!(A_2^*)^{-1}]\sin (\vartheta _{1I})\!+\!A_1A_2(A_2^*)^{-1}e^{2\vartheta _{1R}},\\ \displaystyle{} & {} h_\varsigma =1+\vartheta _{1R}(A_2^{-1}+A_2^*)\cos (\vartheta _{1I})\\{} & {} +i\vartheta _{1R}(A_2^{-1}-A_2^*)\sin (\vartheta _{1I})+A_1A_2^*A_2^{-1}e^{2\vartheta _{1R}}, \end{aligned}$$

with

$$\begin{aligned} \vartheta _{1R}= & {} 2\Omega _{1I}\theta _{1}y+\left[ 6\Omega _{1R}\Omega _{1I}\theta _{1} +\frac{\rho _0^2\theta _{1}\lambda _2}{8(\lambda _1^2+\lambda _2^2)}\right] t\\ {}{} & {} +\vartheta _{1R}^0,\\ \vartheta _{1I}= & {} \theta _{1}x-2\theta _{1}\Omega _{1R}y-\left[ \frac{\theta _{1}^3}{4}+3\theta _{1}(\Omega _{1R}^2-\Omega _{1I}^2)\right. \\{} & {} \left. +\frac{\rho _0^2\theta _{1}\lambda _1}{8(\lambda _1^2+\lambda _2^2)}\right] t+\vartheta _{1I}^0,\\ \lambda _1= & {} \frac{\theta _{1}^2}{4}-(\Omega _{1R}+a)^2+\Omega _{1I}^2,\\ \lambda _2= & {} 2\Omega _{1I}(\Omega _{1R}+a). \end{aligned}$$

Since Solutions (11) consists of trigonometric and hyperbolic functions, it leads breathers to traverse along the curve $L_1$: $\vartheta _{1R}+\frac{\ln A_1}{2}=0$ and periodic along the curve $L_2$: $\vartheta _{1I}=0$, which means the position of the breather depends on the line $L_1$ and the shape of the breather depends on the line $L_2$, and $\vartheta _{1}^0$ affects the phase shift. The velocities of $L_1$ and $L_2$ are $(\infty , -\frac{24\Omega _{1R}(\lambda _1^2+\lambda _2^2)+\rho _0^2(\Omega _{1R}+a)}{8(\lambda _1^2+\lambda _2^2)})$ and $(V_1,-\frac{V_1}{2\Omega _{1R}})$ with $V_1=\frac{\theta _{1}^2}{4}+3(\Omega _{1R}^2-\Omega _{1I}^2) +\frac{\rho _0^2\lambda _1}{8(\lambda _1^2+\lambda _2^2)}$, respectively. The period of the breather in the x-direction is expressed as $\frac{\pi }{|\theta _{1}\Omega _{1I}|}$. When $\Omega _1$ is not purely imaginary, the angle between $L_1$ and $L_2$ is $\arctan \frac{1}{2\Omega _{1R}}$. While $L_1$ is perpendicular to $L_2$ with $\Omega _1$ being purely imaginary. It should be noted that the density plot of each interval of the breather in $\Phi (\Psi )$ component is tetrapetalous with $\Omega _{1R}=0$, as seen in Fig. 1.

Fig. 1

The first-order breathers via Solutions (10) with $\rho _0=1$, $a=1$, $\theta _{1}=1$, $\vartheta _{1}^0=0$ and $t=0$

By taking $N=2$ in Solutions (9), we can present explicit form of the second-order breather solutions. If we set $N=3$, $N=4$, ..., we will derive the third-order, fourth-order breather solutions, ....

It should be noted that to our knowledge, in Fig. 1, the “dark”-type and tetrapetalous breathers in $\Phi (\Psi )$ component have never been studied in KP equation. While, the breathers for the coupled nonlinear NLS-type equations have excited the similar phenomenon, such as the Manakov system [44] and the coherently coupled NLS equations with negative coupling [45]. Propagation of the four-petal Gaussian beams also have been applied in strongly nonlocal nonlinear media [46].

2.1.2 Exp-rational solutions

In order to construct the exp-rational solutions, we follow the long-wave limit procedure [47] by setting $M={\widetilde{N}}+\widetilde{N'}$ and take $\theta _{k_1}\rightarrow 0$, $(1\le k_1\le {\widetilde{N}})$ in Solutions (9), then the exp-rational solutions addresses

$$\begin{aligned}{} & {} f=\begin{array}{llll}A_{2{\widetilde{N}},2{\widetilde{N}}} &{} B_{2{\widetilde{N}},2\widetilde{N'}}\\ C_{2\widetilde{N'},2{\widetilde{N}}} &{} D_{2\widetilde{N'},2\widetilde{N'}}\end{array},\nonumber \\{} & {} g_\varsigma =h_\varsigma ^*=\begin{array}{llll}{\mathfrak {A}}_{2{\widetilde{N}},2{\widetilde{N}}} &{} {\mathfrak {B}}_{2{\widetilde{N}},2\widetilde{N'}}\\ {\mathfrak {C}}_{2\widetilde{N'},2{\widetilde{N}}} &{} {\mathfrak {D}}_{2\widetilde{N'},2\widetilde{N'}}\end{array},\ N,N'\in \mathbb {Z^+}, \end{aligned}$$

(11)

where

$$\begin{aligned}&A_{k_1,k_1}=\left( \begin{array}{llll} \chi _{k_1} &{}\frac{1}{\Omega _{k_1}^*-\Omega _{k_1}}\\ \frac{1}{\Omega _{k_1}^*-\Omega _{k_1}} &{} \chi _{k_1}^* \end{array}\right) ,\\&A_{k_1,l_1}=\left( \begin{array}{llll}\frac{1}{\Omega _{l_1}-\Omega _{k_1}} &{} \frac{1}{\Omega _{l_1}^*-\Omega _{k_1}} \\ \frac{1}{\Omega _{k_1}^*-\Omega _{l_1}} &{} \frac{1}{\Omega _{k_1}^*-\Omega _{l_1}^*}\end{array}\right) ,\\&B_{k_1,{l_2}}={\left( \begin{array}{llll}\frac{1}{\Omega _{l_2}-\Omega _{k_1}+\frac{\theta _{l_2}}{2}} &{} \frac{1}{\Omega _{l_2}^*-\Omega _{k_1}-\frac{\theta _{l_2}}{2}} \\ \frac{1}{\Omega _{k_1}^*-\Omega _{l_2}-\frac{\theta _{l_2}}{2}} &{} \frac{1}{\Omega _{k_1}^*-\Omega _{l_2}^*+\frac{\theta _{l_2}}{2}}\end{array}\right) , } \\&C_{{k_2},{l_1}}=\left( \begin{array}{llll}\frac{1}{\Omega _{l_1}-\Omega _{k_2}+\frac{\theta _{k_2}}{2}} &{} \frac{1}{\Omega _{l_1}^*-\Omega _{k_2}+\frac{\theta _{k_2}}{2}}\\ \frac{1}{\Omega _{k_2}^*-\Omega _{l_1}+\frac{\theta _{k_2}}{2}} &{} \frac{1}{\Omega _{k_2}^*-\Omega _{l_1}^*+\frac{\theta _{k_2}}{2}}\end{array}\right) ,\\&D_{k_2,k_2}=\left( \begin{array}{llll}{\theta _{k_2}}^{-1}(1+e^{-\vartheta _{k_2}}) &{} (\Omega _{k_2}^*-\Omega _{k_2})^{-1}\\ (\Omega _{k_2}^*-\Omega _{k_2})^{-1} &{} {\theta _{k_2}}^{-1}(1+e^{-\vartheta _{k_2}^*})\end{array}\right) , \\&D_{{k_2},{l_2}}=\left( \begin{array}{llll}\frac{1}{\Omega _{l_2}-\Omega _{k_2}+\frac{\theta _{k_2}+\theta _{l_2}}{2}} &{} \frac{1}{\Omega _{l_2}^*-\Omega _{k_2}+\frac{\theta _{k_2}-\theta _{l_2}}{2}} \\ \frac{1}{\Omega _{k_2}^*-\Omega _{l_2}+\frac{\theta _{k_2}-\theta _{l_2}}{2}} &{} \frac{1}{\Omega _{k_2}^*-\Omega _{l_2}^*+\frac{\theta _{k_2}+\theta _{l_2}}{2}}\end{array}\right) ,\\&{\mathfrak {A}}_{k_1,k_1}=\left( \begin{array}{llll} \chi _{k_1}-\frac{1}{\Omega _{k_1}+a} &{}\frac{1}{\Omega _{k_1}^*-\Omega _{k_1}}\frac{\Omega _{k_1}+a}{\Omega _{k_1}^*+a}\\ \frac{1}{\Omega _{k_1}^*-\Omega _{k_1}}\frac{\Omega _{k_1}^*+a}{\Omega _{k_1}+a} &{} \chi _{k_1}^*+\frac{1}{\Omega _{k_1}^*+a} \end{array}\right) ,\\&{\mathfrak {A}}_{k_1,l_1}=\left( \begin{array}{llll}\frac{\Omega _{k_1}+a}{\Omega _{l_1}+a}\frac{1}{\Omega _{l_1}-\Omega _{k_1}} &{} \frac{\Omega _{k_1}+a}{\Omega _{l_1}^*+a}\frac{1}{\Omega _{l_1}^*-\Omega _{k_1}} \\ \frac{\Omega _{k_1}^*+a}{\Omega _{l_1}+a}\frac{1}{\Omega _{k_1}^*-\Omega _{l_1}} &{} \frac{\Omega _{k_1}^*+a}{\Omega _{l_1}^*+a}\frac{1}{\Omega _{k_1}^*-\Omega _{l_1}^*}\end{array}\right) ,\\&{\mathfrak {B}}_{k_1,{l_2}}={\left( \begin{array}{llll}\frac{\Omega _{k_1}+a}{\Omega _{l_2} +\frac{\theta _{l_2}}{2}+a}\frac{1}{\Omega _{l_2}-\Omega _{k_1}+\frac{\theta _{l_2}}{2}} &{} \frac{\Omega _{k_1}+a}{\Omega _{l_2}^*-\frac{\theta _{l_2}}{2}+a}\frac{1}{\Omega _{l_2}^* -\Omega _{k_1}-\frac{\theta _{l_2}}{2}} \\ \frac{\Omega _{k_1}^*+a}{\Omega _{l_2}+\frac{\theta _{l_2}}{2}+a}\frac{1}{\Omega _{k_1}^* -\Omega _{l_2}-\frac{\theta _{l_2}}{2}} &{} \frac{\Omega _{k_1}^*+a}{\Omega _{l_2}^*-\frac{\theta _{l_2}}{2}+a}\frac{1}{\Omega _{k_1}^* -\Omega _{l_2}^*+\frac{\theta _{l_2}}{2}}\end{array}\right) ,}\ \\ \end{aligned}$$

$$\begin{aligned}{} & {} {\mathfrak {C}}_{{k_2},{l_1}}=\left( \begin{array}{llll}\frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}+a}{\Omega _{l_1}+a}\frac{1}{\Omega _{l_1}-\Omega _{k_2}+\frac{\theta _{k_2}}{2}} &{} \frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}+a}{\Omega _{l_1}^*+a}\frac{1}{\Omega _{l_1}^*-\Omega _{k_2}+\frac{\theta _{k_2}}{2}}\\ \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}+a}{\Omega _{l_1}+a}\frac{1}{\Omega _{k_2}^*-\Omega _{l_1}+\frac{\theta _{k_2}}{2}} &{} \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}+a}{\Omega _{l_1}^*+a}\frac{1}{\Omega _{k_2}^*-\Omega _{l_1}^*+\frac{\theta _{k_2}}{2}}\end{array}\right) ,\\{} & {} {\mathfrak {D}}_{k_2,k_2}=\left( \begin{array}{llll}{\theta _{k_2}}^{-1}(\frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}+a}{\Omega _{k_2}+\frac{\theta _{k_2}}{2}+a}+e^{-\vartheta _{k_2}}) &{} \frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}+a}{\Omega _{k_2}^*-\frac{\theta _{k_2}}{2}+a}(\Omega _{k_2}^*-\Omega _{k_2})^{-1}\\ \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}+a}{\Omega _{k_2}+\frac{\theta _{k_2}}{2}+a}(\Omega _{k_2}^*-\Omega _{k_2})^{-1} &{} {\theta _{k_2}}^{-1}(\frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}+a}{\Omega _{k_2}^*-\frac{\theta _{k_2}}{2}+a}+e^{-\vartheta _{k_2}^*})\end{array}\right) ,\\{} & {} {\mathfrak {D}}_{{k_2},{l_2}}=\left( \begin{array}{llll}\frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}+a}{\Omega _{l_2} +\frac{\theta _{l_2}}{2}+a}\frac{1}{\Omega _{l_2}-\Omega _{k_2}+\frac{\theta _{k_2}+\theta _{l_2}}{2}} &{} \frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}+a}{\Omega _{l_2}^*-\frac{\theta _{l_2}}{2}+a}\frac{1}{\Omega _{l_2}^*-\Omega _{k_2}+\frac{\theta _{k_2}-\theta _{l_2}}{2}} \\ \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}+a}{\Omega _{l_2}+\frac{\theta _{l_2}}{2}+a}\frac{1}{\Omega _{k_2}^*-\Omega _{l_2}+\frac{\theta _{k_2}-\theta _{l_2}}{2}} &{} \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}+a}{\Omega _{l_2}^*-\frac{\theta _{l_2}}{2}+a}\frac{1}{\Omega _{k_2}^*-\Omega _{l_2}^*+\frac{\theta _{k_2}+\theta _{l_2}}{2}}\end{array}\right) , \end{aligned}$$

with

$$\begin{aligned}{} & {} \chi _{k_1}=ix-2i\Omega _{k_1}y-i\left[ 3\Omega _{k_1}^2-\frac{\rho _0^2}{8(\Omega _{k_1}+a)^2}\right] t, \\{} & {} \vartheta _{k_2}=\!i\theta _{k_2}x\!-\!2i\theta _{k_2}\Omega _{k_2}y\!\\{} & {} \qquad -\!i\theta _{2}\left[ \frac{\theta _{2}^2}{4}\! + 3\Omega _{2}^2\!+\!\frac{\rho _0^2}{2\theta _{2}^2-8(\Omega _{2}+a)^2}\right] t\!+\!\vartheta _{k_2}^0. \end{aligned}$$

Remark 2.3

To ensure the existence of the limit

$$\begin{aligned} \chi _{k_1}=\lim _{\theta _{k_1}\rightarrow 0}\frac{1}{\theta _{k_1}}(1+e^{-\vartheta _{k_1}}), \end{aligned}$$

it is necessary to fulfill the condition $\vartheta _{k_1}^0=i\pi$ during the application of the long-wave limit.

Exp-rational solutions which depict the interaction between a lump and the first-order breather can be constructed by setting ${\widetilde{N}}=\widetilde{N'}=1$ in Solutions (11):

$$\begin{aligned}{} & {} f=\left| {\begin{array}{*{5}{c}} {A_{11}} &{} B_{12}\\ {C_{21}} &{} {D_{22}} \end{array}} \right| ,\ \ \nonumber \\{} & {} g_\varsigma =h_\varsigma ^*=\left| {\begin{array}{*{5}{c}} {{\mathfrak {A}}_{11}} &{} {\mathfrak {B}}_{12}\\ {{\mathfrak {C}}_{21}} &{} {{\mathfrak {D}}_{22}} \end{array}} \right| . \end{aligned}$$

(12)

There are four cases for the interaction between the lump and breather corresponding with $\Omega _1$ and $\Omega _2$, each case shows the elastic interaction, as depicted in Fig. 2.

Fig. 2

Interactions between one lump and one breather via Solutions (12) with $\rho _0=1$, $a=1$, $\theta _{2}=1$, $\vartheta _{2}^0=0$: Case 1: $\Omega _{1}=\Omega _{2}=i$; Case 2: $\Omega _{1}=1+i, \Omega _{2}=i$; Case 3: $\Omega _{1}=i, \Omega _{2}=1+i$; Case 4: $\Omega _{1}=\Omega _{2}=1+i$

2.1.3 Rational solutions

To derive the rational solutions that give rise to lumps, we further apply the long-wave limit [47] $(\theta _k\rightarrow 0)$ in Solutions (9),

$$\begin{aligned}{} & {} f=\left| {\begin{array}{*{5}{c}} {\Lambda ^0_{1,1}} &{} {\Lambda ^0_{1,2}} &{} {{...}} &{} {\Lambda ^0_{1,M}} \\ {\Lambda ^0_{2,1}} &{} {\Lambda ^0_{2,2}} &{} {{...}} &{} {\Lambda ^0_{2,M}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {\Lambda ^0_{M,1}} &{} {\Lambda ^0_{M,2}} &{} {{...}} &{} {\Lambda ^0_{M,M}} \\ \end{array}} \right| ,\ \nonumber \\{} & {} \ g_\varsigma =h_\varsigma ^*=\left| {\begin{array}{*{5}{c}} {\Lambda ^1_{1,1}} &{} {\Lambda ^1_{1,2}} &{} {{...}} &{} {\Lambda ^1_{1,M}} \\ {\Lambda ^1_{2,1}} &{} {\Lambda ^1_{2,2}} &{} {{...}} &{} {\Lambda ^1_{2,M}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {\Lambda ^1_{M,1}} &{} {\Lambda ^1_{M,2}} &{} {{...}} &{} {\Lambda ^1_{M,M}} \\ \end{array}} \right| , \end{aligned}$$

(13)

with

$$\begin{aligned}{} & {} \Lambda ^0_{k,k}=\left( \begin{array}{llll} \chi _{k} &{} \frac{1}{\Omega _{k}^*-\Omega _{k}}\\ \frac{1}{\Omega _{k}^*-\Omega _{k}} &{} \chi _{k}^* \end{array}\right) ,\ \ \\{} & {} \Lambda ^0_{k,l}=\left( \begin{array}{llll}\frac{1}{\Omega _{l}-\Omega _{k}} &{} \frac{1}{\Omega _{l}^*-\Omega _{k}}\\ \frac{1}{\Omega _{k}^*-\Omega _{l}} &{} \frac{1}{\Omega _{k}^*-\Omega _{l}^*} \end{array}\right) ,\\{} & {} \Lambda ^1_{k,k}=\left( \begin{array}{llll} \chi _{k}-\frac{1}{\Omega _{k}+a} &{}\frac{1}{\Omega _{k}^*-\Omega _{k}}\frac{\Omega _{k}+a}{\Omega _{k}^*+a}\\ \frac{1}{\Omega _{k}^*-\Omega _{k}}\frac{\Omega _{k}^*+a}{\Omega _{k}+a} &{} \chi _{k}^*+\frac{1}{\Omega _{k}^*+a} \end{array}\right) ,\ \ \\{} & {} \Lambda ^1_{k,l}=\left( \begin{array}{llll}\frac{\Omega _{k}+a}{\Omega _{l}+a}\frac{1}{\Omega _{l}-\Omega _{k}} &{} \frac{\Omega _{k}+a}{\Omega _{l}^*+a}\frac{1}{\Omega _{l}^*-\Omega _{k}} \\ \frac{\Omega _{k,\varsigma }^*+a}{\Omega _{l}+a}\frac{1}{\Omega _{k}^*-\Omega _{l}} &{} \frac{\Omega _{k}^*+a}{\Omega _{l}^*+a}\frac{1}{\Omega _{k}^*-\Omega _{l}^*}\end{array}\right) ,\\{} & {} \chi _{k}=ix-2i\Omega _{k}y-i\left[ 3\Omega _{k}^2-\frac{\rho _0^2}{8(\Omega _{k}+a)^2}\right] t. \end{aligned}$$

The first-order rational solutions can be constructed by setting $M=1$ in Solutions (13):

$$\begin{aligned}{} & {} u=4\frac{\chi _{1R}^2-\chi _{1I}^2+(2\Omega _{1I})^{-2}}{[\chi _{1R}^2+\chi _{1I}^2+(2\Omega _{1I})^{-2}]^2}, \end{aligned}$$

(14a)

$$\begin{aligned}{} & {} \Phi _\varsigma ={\Psi _\varsigma }^*=\rho _0e^{i\theta }\left[ 1+\frac{2i(\Omega _{1R}\chi _{1R}+\Omega _{1I}\chi _{1I}+a\chi _{1I})}{[\chi _{1R}^2+\chi _{1I}^2+(2\Omega _{1I})^{-2}] (\Omega _{1R}^2+\Omega _{1I}^2+2a\Omega _{1R}+a^2)}\right] , \end{aligned}$$

(14b)

where $\chi =\chi _{1R}+i\chi _{1I}$, $\Omega _{1}=\Omega _{1R}+i\Omega _{1I}$ and

$$\begin{aligned}{} & {} \chi _{1R}=2\Omega _{1I}y+\left[ 6\Omega _{1R}\Omega _{1I}+\frac{b_1\rho _0^2}{8(a_{1}^2+b_{1}^2)}\right] t,\\{} & {} \chi _{1I}=x-2\Omega _{1R}y+\left[ 3(\Omega _{1I}^2-\Omega _{1R}^2)+\frac{a_1\rho _0^2}{8(a_{1}^2+b_{1}^2)}\right] t,\\{} & {} a_1=(\Omega _{1R}+a)^2-\Omega _{1I}^2,\\{} & {} b_1=2\Omega _{1I}(\Omega _{1R}+a). \end{aligned}$$

From Expression (14a), u is the permanent lump with one hump and two valleys moving on the constant background along the trajectories $\chi _{1R}=0$ and $\chi _{1I}=0$. The amplitude of u is $16\Omega _{1I}^2$. It should be noted that there is no line rogue wave under this situation. There are three cases for $\Phi$ component: two humps-two valleys, one humps-two valleys and two humps-one valley.

Fig. 3

One lump via Solutions (14) with $t=0$, $\rho _0=a=1$

Second-order rational solutions can be constructed by setting $M=2$ in Solutions (13):

$$\begin{aligned}{} & {} f=\left| {\begin{array}{*{5}{c}} {\Lambda ^0_{1,1}} &{} {\Lambda ^0_{1,2}}\\ {\Lambda ^0_{2,1}} &{} {\Lambda ^0_{2,2}}\\ \end{array}} \right| ,\nonumber \\{} & {} g_\varsigma =h_\varsigma ^*=\left| {\begin{array}{*{5}{c}} {\Lambda ^1_{1,1}} &{} {\Lambda ^1_{1,2}}\\ {\Lambda ^1_{2,1}} &{} {\Lambda ^1_{2,2}}\\ \end{array}} \right| . \end{aligned}$$

(15)

Figure 4 presents the interactions between two lumps on the (x, y) plane. In the u component, the two lump maintain their shapes unchanged during the interaction. While in the $\Phi$ component, firstly there is an inelastic interaction between a single-hump lump and two-hump-one-valley lump, then two lumps merge into one lump and finally the shapes of two lumps changes and there is the interaction between a two-hump lump and two-hump-two-valley lump and a two-hump-one-valley lump.

Fig. 4

Interaction between two lumps via Solutions (15) with $\rho _0=a=1$, $\Omega _1=\frac{3}{2}i$, $\Omega _2=\frac{i}{2}$

2.2 Breather-II for Eq. (2)

2.2.1 Breather solutions

Considering

$$\begin{aligned} \begin{array}{l} \displaystyle p_{2k-1}=-i{\tilde{\Omega }}_k+\frac{i{\tilde{\theta }}_k}{2},\\ p_{2k}=i{\tilde{\Omega }}_k^*+\frac{i{\tilde{\theta }}_k}{2},\\ {\tilde{\xi _{2k}^0}}^*={\tilde{\xi }}_{2k-1}^{0},\\ \displaystyle q_{2k-1}=i{\tilde{\Omega }}_k+\frac{i{\tilde{\theta }}_k}{2},\\ q_{2k}=-i{\tilde{\Omega }}_k^*+\frac{i{\tilde{\theta }}_k}{2},\\ {\tilde{\eta _{2k}^0}}^*={\tilde{\eta }}_{2k-1}^0, \end{array} \end{aligned}$$

(16)

with ${\tilde{\Omega }}_k\in {\mathbb {C}}$ and ${\tilde{\theta }}_k\in {\mathbb {R}}$, we have $f=f^*$. Following the procedure of the KP hierarchy, we construct another type breather solutions for Eq. (2):

$$\begin{aligned} u=2(\textrm{ln}f)_{xx},\ \ \Phi _\varsigma =\rho _0e^{i\theta }\frac{g_\varsigma }{f},\ \ \Psi _\varsigma = \rho _0e^{-i\theta }\frac{h_\varsigma }{f},\nonumber \\ \end{aligned}$$

(17)

where

$$\begin{aligned}{} & {} f={\widetilde{\Delta }}\left| {\begin{array}{*{5}{c}} {{\widetilde{\Theta }}^0_{1,1}} &{} {{\widetilde{\Theta }}^0_{1,2}} &{} {{...}} &{} {{\widetilde{\Theta }}^0_{1,N}} \\ {{\widetilde{\Theta }}^0_{2,1}} &{} {{\widetilde{\Theta }}^0_{2,2}} &{} {{...}} &{} {{\widetilde{\Theta }}^0_{2,N}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {{\widetilde{\Theta }}^0_{N,1}} &{} {{\widetilde{\Theta }}^0_{N,2}} &{} {{...}} &{} {{\widetilde{\Theta }}^0_{N,N}} \\ \end{array}} \right| ,\ \ \\{} & {} g_\varsigma =h_\varsigma ^*={\widetilde{\Delta }}\left| {\begin{array}{*{5}{c}} {{\widetilde{\Theta }}^1_{1,1}} &{} {{\widetilde{\Theta }}^1_{1,2}} &{} {{...}} &{} {{\widetilde{\Theta }}^1_{1,N}} \\ {{\widetilde{\Theta }}^1_{2,1}} &{} {{\widetilde{\Theta }}^1_{2,2}} &{} {{...}} &{} {{\widetilde{\Theta }}^1_{2,N}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {{\widetilde{\Theta }}^1_{N,1}} &{} {{\widetilde{\Theta }}^1_{N,2}} &{} {{...}} &{} {{\widetilde{\Theta }}^1_{N,N}} \\ \end{array}} \right| , \\{} & {} {\widetilde{\Theta }}^0_{k,k}=\left( \begin{array}{llll}-\frac{i}{{\tilde{\theta }}_k}(1+e^{-{\tilde{\vartheta }}_{k}}) &{} \frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_k^*-{\tilde{\theta }}_k}\\ \frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_k^*+{\tilde{\theta }}_k} &{} \frac{i}{{\tilde{\theta }}_k}(1+e^{-{\tilde{\vartheta }}^*_k})\end{array}\right) ,\ \end{aligned}$$

$$\begin{aligned}{} & {} {\widetilde{\Theta }}^0_{k,l}=\left( \begin{array}{llll}\frac{i}{{\tilde{\Omega }}_k-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} &{} \frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_l^*-\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}}\\ \frac{i}{{\tilde{\Omega }}_k^*+{\tilde{\Omega }}_l+\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} &{} \frac{i}{{\tilde{\Omega }}_k^*-{\tilde{\Omega }}_l^*+\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} \end{array}\right) , (k\ne l),\\{} & {} {\widetilde{\Theta }}^1_{k,k}=\left( \begin{array}{llll}-\frac{i}{{\tilde{\theta }}_k}(e^{-{\tilde{\vartheta }}_k}+\frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}-ia}{{\tilde{\Omega }}_k+\frac{{\tilde{\theta }}_k}{2}-ia}) &{} \frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}-ia}{-{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}-ia}\frac{i}{{\tilde{\Omega }}_k^*+{\tilde{\Omega }}_k-{\tilde{\theta }}_k}\\ \frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}+ia}{-{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}+ia}\frac{i}{{\tilde{\Omega }}_k^*+{\tilde{\Omega }}_k+{\tilde{\theta }}_k} &{} \frac{i}{{\tilde{\theta }}_k}(e^{-{\tilde{\vartheta }}^*_k}+\frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}+ia}{{\tilde{\Omega }}^*_k-\frac{{\tilde{\theta }}_k}{2}+ia})\end{array}\right) ,\\{} & {} {\widetilde{\Theta }}^1_{k,l}=\left( \begin{array}{llll}\frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}-ia}{{\tilde{\Omega }}_l+\frac{{\tilde{\theta }}_l}{2}-ia}\frac{i}{({\tilde{\Omega }}_k-{\tilde{\Omega }}_l)-\frac{({\tilde{\theta }}_k+{\tilde{\theta }}_l)}{2}} &{} \frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}-ia}{-{\tilde{\Omega }}^*_l+\frac{{\tilde{\theta }}_l}{2}-ia}\frac{i}{({\tilde{\Omega }}_k+{\tilde{\Omega }}_l^*)-\frac{({\tilde{\theta }}_k+{\tilde{\theta }}_l)}{2}}\\ \frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}+ia}{-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_l}{2}+ia}\frac{i}{({\tilde{\Omega }}_k^*+{\tilde{\Omega }}_l)+\frac{({\tilde{\theta }}_k+{\tilde{\theta }}_l)}{2}} &{} \frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}+ia}{{\tilde{\Omega }}^*_l-\frac{{\tilde{\theta }}_l}{2}+ia}\frac{i}{({\tilde{\Omega }}_k^*-{\tilde{\Omega }}_l^*)+\frac{({\tilde{\theta }}_k+{\tilde{\theta }}_l)}{2}} \end{array}\right) , (k\ne l),\\{} & {} {\tilde{\vartheta }}_k=-{\tilde{\theta }}_kx+2i{\tilde{\theta }}_k{\tilde{\Omega }}_ky-{\widetilde{\theta }}_{k} \left[ \frac{{\widetilde{\theta }}_{k}^2}{4}\!+\!3{\tilde{\Omega }}_{k}^2\!+\!\frac{\rho _0^2}{2{\tilde{\theta }}_{k}^2+8(i{\tilde{\Omega }}_{k}+a)^2}\right] t+{\tilde{\vartheta }}_k^0,\\{} & {} {\tilde{\Delta }}=e^{\Sigma _{k=1}^{N}{\tilde{\vartheta }}_k+{\tilde{\vartheta }}_k^*} \prod \limits _{k=1}^{N}{\tilde{\theta }}_k^2,\ {\tilde{\vartheta }}_k^0={\tilde{\xi }}_k^0+{\tilde{\eta }}_k^0, \end{aligned}$$

${\tilde{\Delta }}$ can also be omitted. Indeed, it should be pointed out that Solutions (17) are equivalent to Solutions (9) under $p_{2k-1}\rightarrow ip_{2k-1}$, $p_{2k} \rightarrow -ip_{2k-1}$, $q_{2k-1} \rightarrow iq_{2k-1}$ and $q_{2k} \rightarrow -iq_{2k}$.

First-order breather solutions can be constructed by taking $N=1$ in Solutions (17):

$$\begin{aligned} \begin{array}{l} \displaystyle f=1+e^{{\tilde{\vartheta }}_1}+e^{{\tilde{\vartheta }}_1^*}+B_1e^{{\tilde{\vartheta }}_1+{\tilde{\vartheta }}_1^*},\\ \displaystyle g=1+B_2e^{{\tilde{\vartheta }}_1}+B_3e^{{\tilde{\vartheta }}_1^*}+B_1B_2B_3e^{{\tilde{\vartheta }}_1+{\tilde{\vartheta }}_1^*}, \end{array} \end{aligned}$$

(18)

where

$$\begin{aligned}{} & {} {\tilde{\vartheta }}_1=-{\tilde{\theta }}_1x+2i{\tilde{\theta }}_1{\tilde{\Omega }}_1y\\{} & {} -{\widetilde{\theta }}_{1} \left[ \frac{{\widetilde{\theta }}_{1}^2}{4}\!+\!3{\tilde{\Omega }}_{1}^2\!+\!\frac{\rho _0^2}{2{\tilde{\theta }}_{1}^2+8(i{\tilde{\Omega }}_{1}+a)^2}\right] t+{\tilde{\vartheta }}_1^0,\\{} & {} B_1=\frac{({\tilde{\Omega }}_{1}+{\tilde{\Omega }}_{1}^*)^2}{({\tilde{\Omega }}_{1}+{\tilde{\Omega }}_{1}^*)^2-{\tilde{\theta }}_1^2},\ \\{} & {} B_2=-\frac{-i{\tilde{\Omega }}_1+\frac{i{\tilde{\theta }}_1}{2}-a}{i{\tilde{\Omega }}_1+\frac{i{\tilde{\theta }}_1}{2}+a},\ \ B_3=- \frac{i{\tilde{\Omega }}^*_1+\frac{i{\tilde{\theta }}_1}{2}-a}{-i{\tilde{\Omega }}^*_1+\frac{i{\tilde{\theta }}_1}{2}+a}. \end{aligned}$$

Similarly, we reconstruct f by taking ${\tilde{\vartheta }}_{1}={\tilde{\vartheta }}_{1R}+i{\tilde{\vartheta }}_{1I}$, ${\tilde{\Omega }}_{1}={\tilde{\Omega }}_{1R}+i{\tilde{\Omega }}_{1I}$, ${\tilde{\vartheta }}_{1}^0={\tilde{\vartheta }}_{1R}^0+i{\tilde{\vartheta }}_{1I}^0$ and

$$\begin{aligned} f=e^{{\tilde{\vartheta }}_{1R}}\left[ \sqrt{B_1}\cosh \left( {\tilde{\vartheta }}_{1R}+\frac{\ln B_1}{2}\right) +2\cos {{\tilde{\vartheta }}_{1I}} \right] , \end{aligned}$$

with

$$\begin{aligned}{} & {} {\tilde{\vartheta }}_{1R}=-2{\tilde{\theta }}_{1}x-2{\tilde{\theta }}_{1}{\tilde{\Omega }}_{1I}y-{\tilde{\theta }}_{1}\\{} & {} \quad \left[ \frac{{\tilde{\theta }}_{1}^2}{4}+3(\Omega _{1R}^2-\Omega _{1I}^2)+\frac{\rho _0^2{\tilde{\lambda }}_1}{{\tilde{\lambda }}_1^2+{\tilde{\lambda }}_2^2}\right] t+{\tilde{\vartheta }}_{1R}^0,\\{} & {} {\tilde{\vartheta }}_{1I}=2{\tilde{\theta }}_{1}{\tilde{\Omega }}_{1R}y-{\tilde{\theta }}_{1} \left( 6{\tilde{\Omega }}_{1R}{\tilde{\Omega }}_{1I}+\frac{\rho _0^2{\tilde{\lambda }}_2}{{\tilde{\lambda }}_1^2+{\tilde{\lambda }}_2^2}\right) t\\ {}{} & {} +{\tilde{\vartheta }}_{1I}^0,\\{} & {} {\tilde{\lambda }}_1=2\theta _{1}^2+8[(a-\Omega _{1I})^2-\Omega _{1R}^2],\\ {}{} & {} {\tilde{\lambda }}_2=16\Omega _{1R}(a-\Omega _{1I}). \end{aligned}$$

The first-order breather has exhibited the periodicity both in x- and y- directions, which is different from the first-order breather in Fig. 1. When ${\tilde{\Omega }}_1$ is not purely imaginary, the angle between $L_1$ and $L_2$ is $\arctan (-\frac{1}{2{\tilde{\Omega }}_{1I}})$. The velocities of $L_1$ and $L_2$ parts are $(V_2, \frac{V_2}{\Omega _{1I}})$ with $V_2= -[\frac{{\tilde{\theta }}_{1}^2}{8}+\frac{3}{2}(\Omega _{1R}^2-\Omega _{1I}^2)+\frac{\rho _0^2{\tilde{\lambda }}_1}{2({\tilde{\lambda }}_1^2+{\tilde{\lambda }}_2^2)}]$ and $(\infty , 3{\tilde{\Omega }}_{1R}{\tilde{\Omega }}_{1I}+\frac{8\rho _0^2(a-\Omega _{1I})}{{\tilde{\lambda }}_1^2+{\tilde{\lambda }}_2^2})$, respectively. The periods of the breather in x- and y- directions are expressed as $\frac{\pi }{|{\tilde{\theta }}_{1}|}$ and $\frac{\pi }{|{\tilde{\theta }}_{1}{\tilde{\Omega }}_{1I}|}$. If we choose ${\tilde{\Omega }}_{1}$ is purely imaginary, then $L_2$ diminishes. Further, we find that the breather degenerates into the soliton: u component is a bright soliton while $\Phi$ component is dark soliton. The soliton moves without shape changing.

Fig. 5

The first-order breathers via Solutions (18) with ${\tilde{\theta }}_{1}=1$, $\rho _0=a=1$, ${\tilde{\vartheta }}_{1}^0=0$ and $t=0$

Second-order breather solutions can be constructed by taking $N=2$ in Solutions (17):

$$\begin{aligned}{} & {} f=\left| {\begin{array}{*{5}{c}} {{\widetilde{\Theta }}^0_{1,1}} &{} {{\widetilde{\Theta }}^0_{1,2}} \\ {{\widetilde{\Theta }}^0_{2,1}} &{} {{\widetilde{\Theta }}^0_{2,2}} \\ \end{array}} \right| ,\nonumber \\{} & {} g_\varsigma =h_\varsigma ^*=\left| {\begin{array}{*{5}{c}} {{\widetilde{\Theta }}^1_{1,1}} &{} {{\widetilde{\Theta }}^1_{1,2}} \\ {{\widetilde{\Theta }}^1_{2,1}} &{} {{\widetilde{\Theta }}^1_{2,2}} \\ \end{array}} \right| . \end{aligned}$$

(19)

Similarly, the second-order breather can be treated as the superposition of the first-order ones.

2.2.2 Exp-rational solutions

In order to construct the exp-rational solutions, if we follow the long-wave limit procedure [47] by setting $M={\widetilde{N}}+\widetilde{N'}$ and taking $(\theta _{k_1}\rightarrow 0, 1\le k_1\le {\widetilde{N}})$ in Solutions (9), then the exp-rational solutions addresses

$$\begin{aligned}{} & {} f=\left( \begin{array}{llll}A_{2{\widetilde{N}},2{\widetilde{N}}} &{} B_{2{\widetilde{N}},2\widetilde{N'}}\\ C_{2\widetilde{N'},2{\widetilde{N}}} &{} D_{2\widetilde{N'},2\widetilde{N'}}\end{array}\right) ,\nonumber \\{} & {} g_\varsigma =h_\varsigma ^*=\left( \begin{array}{llll}{\mathfrak {A}}_{2{\widetilde{N}},2{\widetilde{N}}} &{} {\mathfrak {B}}_{2{\widetilde{N}},2\widetilde{N'}}\\ {\mathfrak {C}}_{2\widetilde{N'},2{\widetilde{N}}} &{} {\mathfrak {D}}_{2\widetilde{N'},2\widetilde{N'}}\end{array}\right) , \end{aligned}$$

(20)

where

$$\begin{aligned}{} & {} A_{k_1,k_1}=\left( \begin{array}{llll} {\widetilde{\chi }}_{k_1} &{} \frac{i}{\Omega _{k_1}+\Omega _{k_1}^*}\\ \frac{i}{\Omega _{k_1}+\Omega _{k_1}^*} &{} {\widetilde{\chi }}_{k_1}^* \end{array}\right) ,\ \\{} & {} A_{k_1,l_1}=\left( \begin{array}{llll}\frac{i}{{\tilde{\Omega }}_{k_1}-{\tilde{\Omega }}_{l_1}} &{} \frac{i}{{\tilde{\Omega }}_{k_1}+{\tilde{\Omega }}_{l_1}^*}\\ \frac{i}{{\tilde{\Omega }}_{k_1}^*+{\tilde{\Omega }}_{l_1}} &{} \frac{i}{{\tilde{\Omega }}_{k_1}^*-{\tilde{\Omega }}_{l_1}^*} \end{array}\right) ,\\{} & {} B_{k_1,{l_2}}={\left( \begin{array}{llll}\frac{i}{(\Omega _{k_1}-\Omega _{l_2})-\frac{{\tilde{\theta }}_{l_2}}{2}} &{} \frac{i}{(\Omega _{k_1}+\Omega _{l_2}^*)-\frac{{\tilde{\theta }}_{l_2}}{2}} \\ \frac{i}{(\Omega _{k_1}^*+\Omega _{l_2})+\frac{{\tilde{\theta }}_{l_2}}{2}} &{} \frac{i}{(\Omega _{k_1}^*-\Omega _{l_2}^*)+\frac{{\tilde{\theta }}_{l_2}}{2}}\end{array}\right) ,}\ \\{} & {} \ C_{{k_2},{l_1}}={\left( \begin{array}{llll}\frac{i}{\Omega _{k_2}-\Omega _{l_1}-\frac{{\tilde{\theta }}_{k_2}}{2}} &{} \frac{i}{\Omega _{l_1}^*+\Omega _{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}}\\ \frac{i}{\Omega _{k_2}^*+\Omega _{l_1}+\frac{{\tilde{\theta }}_{k_2}}{2}} &{} \frac{i}{\Omega _{k_2}^*-\Omega _{l_1}^*+\frac{{\tilde{\theta }}_{k_2}}{2}}\end{array}\right) ,}\\{} & {} D_{k_2,k_2}=\left( \begin{array}{llll}-\frac{i}{{\tilde{\theta }}_{k_2}}(1+e^{-{\tilde{\vartheta }}_{k_2}}) &{} \frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_{k_2}^*-{\tilde{\theta }}_{k_2}}\\ \frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_{k_2}^*+{\tilde{\theta }}_{k_2}} &{} \frac{i}{{\tilde{\theta }}_{k_2}}(1+e^{-{\tilde{\vartheta }}^*_{k_2}})\end{array}\right) ,\ \\{} & {} D_{{k_2},{l_2}}=\left( \begin{array}{llll}\frac{i}{{\tilde{\Omega }}_{k_2}-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_{l_2}}{2}} &{} \frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_{l_2}^*-\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_{l_2}}{2}}\\ \frac{i}{{\tilde{\Omega }}_{k_2}^*+{\tilde{\Omega }}_{l_2}+\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_{l_2}}{2}} &{} \frac{i}{{\tilde{\Omega }}_{k_2}^*-{\tilde{\Omega }}_{l_2}^*+\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_{l_2}}{2}} \end{array}\right) ,\\{} & {} {\mathfrak {A}}_{k_1,k_1}=\left( \begin{array}{llll} {\widetilde{\chi }}_{k_1}-\frac{1}{i\Omega _{k_1}+a} &{}\frac{i}{\Omega _{k_1}+\Omega _{k_1}^*}\frac{-\Omega _{k_1}+ia}{\Omega _{k_1}^*+ia}\\ \frac{i}{\Omega _{k_1}+\Omega _{k_1}^*}\frac{-\Omega _{k_1}^*-ia}{\Omega _{k_1}-ia} &{} {\widetilde{\chi }}_{k_1}^*-\frac{1}{i\Omega _{k_1}^*-a} \end{array}\right) ,\ \\{} & {} {\mathfrak {A}}_{k_1,l_1}=\left( \begin{array}{llll}\frac{\Omega _{k_1}-ia}{\Omega _{l_1}-ia}\frac{i}{\Omega _{k_1}-\Omega _{l_1}} &{} \frac{\Omega _{k_1}-ia}{-\Omega _{l_1}^*-ia}\frac{i}{\Omega _{k_1}+\Omega _{l_1}^*} \\ \frac{\Omega _{k_1}^*+ia}{-\Omega _{l_1}+ia}\frac{i}{\Omega _{k_1}^*+\Omega _{l_1}} &{} \frac{\Omega _{k_1}^*+ia}{\Omega _{l_1}^*+ia}\frac{i}{\Omega _{k_1}^*-\Omega _{l_1}^*}\end{array}\right) ,\\ \end{aligned}$$

$$\begin{aligned}{} & {} {\mathfrak {B}}_{{k_1},{l_2}}\\{} & {} ={\left( \begin{array}{llll}\frac{{\tilde{\Omega }}_{k_1}-ia}{{\tilde{\Omega }}_{l_2}+\frac{{\tilde{\theta }}_{l_2}}{2}-ia}\frac{i}{{\tilde{\Omega }}_{k_1}-{\tilde{\Omega }}_{l_2} -\frac{{\tilde{\theta }}_{l_2}}{2}} &{} \frac{{\tilde{\Omega }}_{k_1}-ia}{-{\tilde{\Omega }}^*_{l_2}+\frac{{\tilde{\theta }}_{l_2}}{2}-ia}\frac{i}{{\tilde{\Omega }}_{k_1}+{\tilde{\Omega }}_{l_2}^*-\frac{{\tilde{\theta }}_{l_2}}{2}}\\ \frac{{\tilde{\Omega }}^*_{k_1}+ia}{-{\tilde{\Omega }}_{l_2}-\frac{{\tilde{\theta }}_{l_2}}{2}+ia}\frac{i}{{\tilde{\Omega }}_{k_1}^*+{\tilde{\Omega }}_{l_2}+\frac{{\tilde{\theta }}_{l_2}}{2}} &{} \frac{{\tilde{\Omega }}^*_{k_1}+ia}{{\tilde{\Omega }}^*_{l_2}-\frac{{\tilde{\theta }}_{l_2}}{2}+ia}\frac{i}{{\tilde{\Omega }}_{k_1}^*-{\tilde{\Omega }}_{l_2}^*+\frac{{\tilde{\theta }}_{l_2}}{2}} \end{array}\right) , }\ \\{} & {} {\mathfrak {C}}_{{k_2},{l_1}}\\{} & {} =\left( \begin{array}{llll}\frac{{\tilde{\Omega }}_{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}-ia}{{\tilde{\Omega }}_{l_1}-ia} \frac{i}{{\tilde{\Omega }}_{k_2}-{\tilde{\Omega }}_{l_1}-\frac{{\tilde{\theta }}_{k_2}}{2}} &{} \frac{{\tilde{\Omega }}_{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}-ia}{-{\tilde{\Omega }}^*_{l_1}-ia}\frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_{l_1}^*-\frac{{\tilde{\theta }}_{k_2}}{2}}\\ \frac{{\tilde{\Omega }}^*_{k_2}+\frac{{\tilde{\theta }}_{k_2}}{2}+ia}{-{\tilde{\Omega }}_{l_1}+ia}\frac{i}{{\tilde{\Omega }}_{k_2}^*+{\tilde{\Omega }}_{l_1}+\frac{{\tilde{\theta }}_{k_2}}{2}} &{} \frac{{\tilde{\Omega }}^*_{k_2}+\frac{{\tilde{\theta }}_{k_2}}{2}+ia}{{\tilde{\Omega }}^*_{l_1}+ia}\frac{i}{{\tilde{\Omega }}_{k_2}^*-{\tilde{\Omega }}_{l_1}^*+\frac{{\tilde{\theta }}_{k_2}}{2}} \end{array}\right) ,\\{} & {} {\mathfrak {D}}_{k_2,k_2}\\ {}{} & {} =\left( \begin{array}{llll}-\frac{i}{{\tilde{\theta }}_k}(e^{-{\tilde{\vartheta }}_k}+\frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}-ia}{{\tilde{\Omega }}_k+\frac{{\tilde{\theta }}_k}{2}-ia}) &{} \frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}-ia}{-{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}-ia}\frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_k^*-{\tilde{\theta }}_k}\\ \frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}+ia}{-{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}+ia}\frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_k^*+{\tilde{\theta }}_k} &{} \frac{i}{{\tilde{\theta }}_k}(e^{-{\tilde{\vartheta }}^*_k}+\frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}+ia}{{\tilde{\Omega }}^*_k-\frac{{\tilde{\theta }}_k}{2}+ia})\end{array}\right) ,\\{} & {} {\mathfrak {D}}_{{k_2},{l_2}}\\ {}{} & {} =\left( \begin{array}{llll}\frac{{\tilde{\Omega }}_{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}-ia}{{\tilde{\Omega }}_l +\frac{{\tilde{\theta }}_l}{2}-ia}\frac{i}{{\tilde{\Omega }}_{k_2}-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_l}{2}} &{} \frac{{\tilde{\Omega }}_{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}-ia}{-{\tilde{\Omega }}^*_l+\frac{{\tilde{\theta }}_l}{2}-ia} \frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_l^*-\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_l}{2}}\\ \frac{{\tilde{\Omega }}^*_{k_2}+\frac{{\tilde{\theta }}_{k_2}}{2}+ia}{-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_l}{2}+ia}\frac{i}{{\tilde{\Omega }}_{k_2}^*+{\tilde{\Omega }}_l+\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} &{} \frac{{\tilde{\Omega }}^*_{k_2}+\frac{{\tilde{\theta }}_{k_2}}{2}+ia}{{\tilde{\Omega }}^*_l-\frac{{\tilde{\theta }}_l}{2}+ia} \frac{i}{{\tilde{\Omega }}_{k_2}^*-{\tilde{\Omega }}_l^*+\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_l}{2}} \end{array}\right) , \end{aligned}$$

with

$$\begin{aligned}{} & {} \chi _{k_1}=-x+2i\Omega _{k_1}y-\left[ 3\Omega _{k_1}^2+\frac{\rho _0^2}{8(i\Omega _{k_1}+a)^2}\right] t, \\{} & {} \vartheta _{k_2}=-{\tilde{\theta }}_{k_2}x+2i{\tilde{\theta }}_{k_2}{\tilde{\Omega }}_{k_2}y-{\widetilde{\theta }}_{{k_2}} \left[ \frac{{\widetilde{\theta }}_{{k_2}}^2}{4}\!\right. \\{} & {} \left. +\!3{\tilde{\Omega }}_{{k_2}}^2\!+\!\frac{\rho _0^2}{2{\tilde{\theta }}_{{k_2}}^2+8(i{\tilde{\Omega }}_{{k_2}}+a)^2}\right] t+{\tilde{\vartheta }}_{k_2}^0. \end{aligned}$$

If we set ${\widetilde{N}}$=$\widetilde{N'}$=1 in Solutions (20), then there are two cases under this circumstance. One case is the interaction between the breather and the lump, another one is the interaction between the soliton and the lump with $\Omega _{1}$ purely imaginary.

Fig. 6

Solutions (20) with $\rho _0=a=1$, $\theta _{1}=1$, $\vartheta _{2}^0=10$ and $t=0$. (a) and (b): $\Omega _{1}=1+i$, (c) and (d): $\Omega _{1}=i$

2.2.3 Rational solutions

Since the result is equivalent to the one in Sect. 2.1.3 with $\Omega _{k}\rightarrow -i{\tilde{\Omega }}^*_{k}$, we do not present it here again.

3 Breathers through a different KP hierarchy reduction procedure

Now, we introduce another auxiliary variable v and reconstruct the first equation of Bilinear Form (4) [20]:

$$\begin{aligned} \begin{array}{l} \displaystyle (D_{x}^4+D_{x}D_{v}-3D_{y}^2-12\alpha ^2D_{x}^2+12\alpha D_{x}D_{y})\\ f\cdot f =0, \\ \displaystyle (D_{x}D_{v}-12\alpha ^2D_{x}^2+12\alpha D_{x}D_{y}+4D_{x}D_{t}) \\ f\cdot f =\rho _0^2\left( f^2-gh^{\mathrm{{T}}}\right) . \end{array} \end{aligned}$$

(21)

The bilinear equations in the KP hierarchy

$$\begin{aligned} \begin{array}{l} \displaystyle (D^4_{x_{1}}-4D_{x_{1}}D_{x_{3}}+3D^2_{x_{2}}) \varvec{\tau _{n}} \cdot \varvec{\tau ^{\mathrm{{T}}}_{n}}=0,\\ \displaystyle (D_{x_{1}}D_{x_{-1}}-2)\varvec{\tau _{n}} \cdot \varvec{\tau ^{\mathrm{{T}}}_{n}}=-2\varvec{\tau _{n+1}}\cdot \varvec{\tau ^{\mathrm{{T}}}_{n-1}},\\ \displaystyle (D_{x_{1}}^2-D_{x_{2}})\tau _{n+1,\varsigma }\cdot \tau _{n,\varsigma }=0,\\ \displaystyle (D_{x_{1}}^2+D_{x_{2}})\tau _{n-1,\varsigma }\cdot \tau _{n,\varsigma }=0, \end{array} \end{aligned}$$

(22)

have the following $N \times N$ Gram determinant solutions

$$\begin{aligned}{} & {} \tau _{n,\varsigma }=\det _{1\le r,j\le N} [m_{rj,\varsigma }^{(n)}], \end{aligned}$$

(23)

where the matrix elements are given by

$$\begin{aligned}{} & {} \partial _{x_1}m^{(n)}_{rj,\varsigma }=\psi _{r,\varsigma }^{(n)}\phi _{j,\varsigma }^{(n)},\\{} & {} \partial _{x_2}m^{(n)}_{rj,\varsigma }=(\partial _{x_1}\psi _{r,\varsigma }^{(n)})\phi _{j,\varsigma }^{(n)}-\psi _{r,\varsigma }^{(n)}(\partial _{x_1}\phi _{j,\varsigma }^{(n)}),\\{} & {} \partial _{x_3}m^{(n)}_{rj,\varsigma }=(\partial ^2_{x_1}\psi _{r,\varsigma }^{(n)})\phi _{j,\varsigma }^{(n)}-(\partial _{x_1}\psi _{r,\varsigma }^{(n)})(\partial _{x_1}\phi _{j,\varsigma }^{(n)})\\ {}{} & {} +\psi _{r,\varsigma }^{(n)}(\partial ^2_{x_1}\phi _{j,\varsigma }^{(n)}),\\{} & {} \partial _{x_{-1}}m^{(n)}_{rj,\varsigma }=-\psi _{r,\varsigma }^{(n-1)}\phi _{j,\varsigma }^{(n+1)},\\{} & {} m^{(n+1)}_{rj,\varsigma }=m^{(n)}_{rj,\varsigma }+\psi _{r,\varsigma }^{(n)}\phi _{j,\varsigma }^{(n+1)},\\{} & {} \partial _{x_2}\psi _{r,\varsigma }^{(n)}=\partial ^2_{x_1}\psi _{r,\varsigma }^{(n)},\ \ \partial _{x_2}\phi _{j,\varsigma }^{(n)}=-\partial ^2_{x_1}\phi _{j,\varsigma }^{(n)},\\{} & {} \partial _{x_3}\psi _{r,\varsigma }^{(n)}=\partial ^3_{x_1}\psi _{r,\varsigma }^{(n)},\ \ \partial _{x_3}\phi _{j,\varsigma }^{(n)}=\partial ^3_{x_1}\phi _{j,\varsigma }^{(n)}. \end{aligned}$$

Then, we choose

$$\begin{aligned}{} & {} m^{(n)}_{rj,\varsigma }=\delta _{rj,\varsigma }+\frac{1}{p_{r,\varsigma }+q_{j,\varsigma }}\psi _{r,\varsigma }^{(n)}\phi _{j,\varsigma }^{(n)},\\{} & {} \psi _{r,\varsigma }^{(n)}=(p_{r,\varsigma })^{n}(p_{r,\varsigma }+q_{r,\varsigma })e^{\xi _{r,\varsigma }},\\{} & {} \phi _{j,\varsigma }^{(n)}={\left( -\frac{1}{q_{j,\varsigma }}\right) }^ne^{\eta _{j,\varsigma }}, \end{aligned}$$

with

$$\begin{aligned}{} & {} \xi _{r,\varsigma }=\frac{1}{p_{r,\varsigma }}x_{-1}+p_{r,\varsigma }x_{1}+p_{r,\varsigma }^2x_{2}+p_{r,\varsigma }^3x_{3}+\xi _{r,\varsigma }^0,\\{} & {} \eta _{j,\varsigma }=\frac{1}{q_{j,\varsigma }}x_{-1}+q_{j,\varsigma }x_{1}-q_{j,\varsigma }^2x_{2}+q_{j,\varsigma }^3x_{3}+\eta _{j,\varsigma }^0. \end{aligned}$$

Remark 3.1

KP hierarchy (22) is different from KP hierarchy (5).

Under the context of the KP hierarchy reduction procedure, we introduce the changes of the variables:

$$\begin{aligned} x_{1}=i(x+2\alpha y-3\alpha ^2 t),\ \ x_{2}=i(y-3\alpha t),\ \\ x_{3}=-it+4iv,\ \ x_{-1}=-i\frac{\rho _0^2}{8}t. \end{aligned}$$

Further, we introduce

$$\begin{aligned} f=\tau _{0,\varsigma },\ \ g_\varsigma =\tau _{1,\varsigma },\ \ h_\varsigma =\tau _{-1,\varsigma }, \end{aligned}$$

then

$$\begin{aligned} \begin{array}{l} \displaystyle f=\left| \delta _{rj}+\frac{p_{r,\varsigma }+q_{r,\varsigma }}{p_{r,\varsigma }+q_{j,\varsigma }}e^{\xi _{r,\varsigma }+\eta _{j,\varsigma }} \right| _{1\le r,j \le M},\\ \displaystyle g_\varsigma =\left| \delta _{rj}+\left( -\frac{p_{r,\varsigma }}{q_{j,\varsigma }}\right) \frac{p_{r,\varsigma } +q_{r,\varsigma }}{p_{r,\varsigma }+q_{j,\varsigma }}e^{\xi _{r,\varsigma }+\eta _{j,\varsigma }} \right| _{1\le r,j \le M},\\ \displaystyle h_\varsigma =\left| \delta _{rj}\!+\!\left( -\frac{q_{j,\varsigma }}{p_{r,\varsigma }}\right) \!\frac{p_{r,\varsigma } +q_{r,\varsigma }}{p_{r,\varsigma }\!+\!q_{j,\varsigma }}e^{\xi _{r,\varsigma }\!+\!\eta _{j,\varsigma }} \right| _{1\le r,j \le M}, \end{array}\nonumber \\ \end{aligned}$$

(24)

with

$$\begin{aligned}{} & {} \xi _{r,\varsigma }=ip_{r,\varsigma }x+i(2\alpha p_{r,\varsigma }+p_{r,\varsigma }^2)y-i\\ {}{} & {} \left( 3\alpha ^2 p _{r,\varsigma }+3\alpha p_{r,\varsigma }^2+p_{r,\varsigma }^3+\frac{\rho _0^2}{8}\frac{1}{p_{r,\varsigma }} \right) t+\xi _{r,\varsigma }^0,\\{} & {} \eta _{j,\varsigma }=iq_{j,\varsigma }x+i(2\alpha q_{j,\varsigma }-q_{j,\varsigma }^2)y\\{} & {} \quad -i\left( 3\alpha ^2 q _{j,\varsigma }-3\alpha q_{j,\varsigma }^2+q_{j,\varsigma }^3+\frac{\rho _0^2}{8}\frac{1}{q_{j,\varsigma }} \right) t+\eta _{j,\varsigma }^0, \end{aligned}$$

where v can be set to zero without loss of generality, $p_{r,\varsigma }$’s, $q_{j,\varsigma }$’s, $\xi _{r,\varsigma }^0$’s and $\eta _{j,\varsigma }^0$’s are the complex constants. In fact, the index $\varsigma$ can be also omitted in the following discussions.

Using the same argument in as Sec. 2, we make the parameter constraints of $p_{r}$, $q_{j}$, $\xi _{r}^0$ and $\eta _{j}^0$ to satisfy $f=f^*$.

3.1 Breather-III for Eq. (2)

3.1.1 Breather solutions

Set

$$\begin{aligned}{} & {} p_{2k-1}=-\Omega _{k}+\frac{\theta _{k}}{2},\ \ p_{2k}=-\Omega _{k}^*-\frac{\theta _{k}}{2},\ \ {\xi _{2k}^0}^*=\xi _{2k-1}^{0},\\{} & {} q_{2k-1}=\Omega _{k}+\frac{\theta _{k}}{2},\ \ q_{2k}=\Omega _{k}^*-\frac{\theta _{k}}{2},\ \ {\eta _{2k,\varsigma }^0}^*=\eta _{2k-1}^0, \end{aligned}$$

with $\Omega _k\in {\mathbb {C}}$ and $\theta _k\in {\mathbb {R}}$, which ensures $f=f^*$. According to Solutions (24), we construct the breather solutions for Eq. (2) as follows:

$$\begin{aligned} u=2(\textrm{ln}f)_{xx},\ \ \Phi _\varsigma =\rho _0e^{i\theta }\frac{g_\varsigma }{f},\ \ \Psi _\varsigma = \rho _0e^{-i\theta }\frac{h_\varsigma }{f}, \nonumber \\ \end{aligned}$$

(25)

where

$$\begin{aligned}{} & {} f=\Upsilon \left| {\begin{array}{*{5}{c}} {\Gamma ^0_{1,1}} &{} {\Gamma ^0_{1,2}} &{} {{...}} &{} {\Gamma ^0_{1,N}} \\ {\Gamma ^0_{2,1}} &{} {\Gamma ^0_{2,2}} &{} {{...}} &{} {\Gamma ^0_{2,N}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {\Gamma ^0_{N,1}} &{} {\Gamma ^0_{N,2}} &{} {{...}} &{} {\Gamma ^0_{N,N}} \\ \end{array}} \right| ,\ \ \\{} & {} g_\varsigma =h_\varsigma ^*=\Upsilon \left| {\begin{array}{*{5}{c}} {\Gamma ^1_{1,1}} &{} {\Gamma ^1_{1,2}} &{} {{...}} &{} {\Gamma ^1_{1,N}} \\ {\Gamma ^1_{2,1}} &{} {\Gamma ^1_{2,2}} &{} {{...}} &{} {\Gamma ^1_{2,N}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {\Gamma ^1_{N,1}} &{} {\Gamma ^1_{N,2}} &{} {{...}} &{} {\Gamma ^1_{N,N}} \\ \end{array}} \right| , \end{aligned}$$

with

$$\begin{aligned}{} & {} \Gamma ^0_{k,k}=\left( \begin{array}{llll}{\theta _{k}}^{-1}(1+e^{-\vartheta _{k}}) &{} (\Omega _{k}^*-\Omega _{k})^{-1}\\ (\Omega _{k}^*-\Omega _{k})^{-1} &{} {\theta _{k}}^{-1}(1+e^{-\vartheta _{k}^*})\end{array}\right) ,\\{} & {} {\Gamma ^0_{k,l}=\left( \begin{array}{llll}\frac{1}{\Omega _{l}-\Omega _{k}+\frac{\theta _{k}+\theta _{l}}{2}} &{} \frac{1}{\Omega _{l}^*-\Omega _{k}+\frac{\theta _{k}-\theta _{l}}{2}} \\ \frac{1}{\Omega _{k}^*-\Omega _{l}+\frac{\theta _{k}-\theta _{l}}{2}} &{} \frac{1}{\Omega _{k}^*-\Omega _{l}^*+\frac{\theta _{k}+\theta _{l}}{2}}\end{array}\right) , }(k\ne l), \\{} & {} {\Gamma ^1_{k,k}=\left( \begin{array}{llll}{\theta _{k}}^{-1}\bigg (e^{-\vartheta _{k}}+\frac{\Omega _{k}-\frac{\theta _{k}}{2}}{\Omega _{k}+\frac{\theta _{k}}{2}}\bigg ) &{} (\Omega _{k}^*-\Omega _{k})^{-1}\frac{\Omega _{k}-\frac{\theta _{k}}{2}}{\Omega _{k}^*-\frac{\theta _{k}}{2}}\\ (\Omega _{k}^*-\Omega _{k})^{-1}\frac{\Omega _{k}^*+\frac{\theta _{k}}{2}}{\Omega _{k}+\frac{\theta _{k}}{2}} &{} {\theta _{k}}^{-1}\bigg (e^{-\vartheta _{k}^*}+\frac{\Omega _{k}^*+\frac{\theta _{k}}{2}}{\Omega _{k}^* -\frac{\theta _{k}}{2}}\bigg )\end{array}\right) },\\{} & {} { \Gamma ^1_{k,l}=\left( \begin{array}{llll}\left( \frac{\Omega _{k}-\frac{\theta _{k}}{2}}{\Omega _{l} +\frac{\theta _{l}}{2}}\right) \frac{1}{\Omega _{l}-\Omega _{k}+\frac{\theta _{k}+\theta _{l}}{2}} &{} \left( \frac{\Omega _{k}-\frac{\theta _{k}}{2}}{\Omega _{l}^*-\frac{\theta _{l}}{2}}\right) \frac{1}{\Omega _{l}^* -\Omega _{k}+\frac{\theta _{k}-\theta _{l}}{2}} \\ \left( \frac{\Omega _{k}^*+\frac{\theta _{k}}{2}}{\Omega _{l}+\frac{\theta _{l}}{2}}\right) \frac{1}{\Omega _{k}^* -\Omega _{l}+\frac{\theta _{k}-\theta _{l}}{2} } &{} \left( \frac{\Omega _{k}^*+\frac{\theta _{k}}{2}}{\Omega _{l}^*-\frac{\theta _{l}}{2}}\right) \frac{1}{\Omega _{k}^* -\Omega _{l}^*+\frac{\theta _{k}+\theta _{l}}{2}} \end{array}\right) , }\\{} & {} (k\ne l), \\{} & {} \vartheta _{k}\!=\!i\theta _{k}x\!+\!2i\theta _{k}(\alpha -\Omega _{k})y\!\\{} & {} -\!i\theta _{k}\left( 3\alpha ^2-6\alpha \Omega _{k,\varsigma }+3\Omega _{k}^2+\frac{1}{4}\theta _{k}^2+\frac{\rho _0^2}{2\theta _{k}^2-8\Omega _{k}^2} \right) t\!+\!\vartheta _{k}^0, \\{} & {} \Upsilon =e^{\Sigma _{k=1}^{N}\vartheta _{k}+\vartheta _{k}^*} \prod \limits _{k=1}^{N}\theta _{k}^2. \end{aligned}$$

We mention that $\Upsilon$ can be ignored in the following sections.

Remark 3.2

Under the condition $a=\alpha =0$, Solutions (9) equals to Solutions (25).

First-order breather solutions can be constructed by setting $N=1$ in Solutions (25),

$$\begin{aligned} \begin{array}{l} \displaystyle f=1+e^{\vartheta _{1}}+e^{\vartheta _{1}^*}+C_1e^{\vartheta _{1}+\vartheta _{1}^*},\\ \displaystyle g_\varsigma =1+C_2e^{\vartheta _{1}}+(C_2^*)^{-1}e^{\vartheta _{1}^*}+C_1C_2(C_2^*)^{-1}e^{\vartheta _{1}+\vartheta _{1}^*},\\ \displaystyle h_\varsigma =1+(C_2)^{-1}e^{\vartheta _{1}}+C_2^*e^{\vartheta _{1}^*}+C_1C_2^*(C_2)^{-1}e^{\vartheta _{1}+\vartheta _{1}^*}, \end{array} \nonumber \\ \end{aligned}$$

(26)

where

$$\begin{aligned}{} & {} C_1=1-\left( \frac{\theta _{1}}{\Omega _{1}^*-\Omega _{1}}\right) ^2,\ \ C_2=\frac{\Omega _{1}-\frac{\theta _{1}}{2}}{\Omega _{1}+\frac{\theta _{1}}{2}},\\{} & {} \vartheta _{1}\!=\!i\theta _{1}x\!+\!2i\theta _{1}(\alpha -\Omega _{1})y\! -\!i\theta _{1}\\{} & {} \qquad \times \left( 3\alpha ^2-6\alpha \Omega _{1}+3\Omega _{1}^2+\frac{1}{4}\theta _{1}^2+\frac{\rho _0^2}{2\theta _{1}^2-8\Omega _{1}^2} \right) \\{} & {} \qquad \times t\!+\!\vartheta _{1}^0. \end{aligned}$$

Reconstructing Solutions (26) by taking $\vartheta _{1}=\vartheta _{1R}+i\vartheta _{1I}$, $\Omega _{1}=\Omega _{1R}+i\Omega _{1I}$, we obtain that

$$\begin{aligned}{} & {} f=e^{\vartheta _{1R}}\left[ \sqrt{C_1}\cosh \left( \vartheta _{1R}+\frac{\ln C_1}{2}\right) +2\cos {\vartheta _{1I}} \right] ,\\{} & {} g_\varsigma =1\!+\!\vartheta _{1R}(C_2+(C_2^*)^{-1})\cos (\vartheta _{1I})\!\\{} & {} \qquad +\!i\vartheta _{1R}(C_2-(C_2^*)^{-1})\sin (\vartheta _{1I})\!+\!C_1C_2(C_2^*)^{-1}e^{2\vartheta _{1R}},\\{} & {} h_\varsigma =1\!+\!\vartheta _{1R}(C_2^{-1}+C_2^*)\cos (\vartheta _{1I})\!\\{} & {} \qquad +\!i\vartheta _{1R}(C_2^{-1}-C_2^*)\sin (\vartheta _{1I})\!+\!C_1C_2^*C_2^{-1}e^{2\vartheta _{1R}}, \end{aligned}$$

with

$$\begin{aligned} \vartheta _{1R}= & {} 2\theta _{1}\Omega _{1I}y\\{} & {} +2\theta _{1}\Omega _{1I} \left[ 3(\Omega _{1R}-\alpha )+\frac{8\rho _0^2\Omega _{1R}}{\beta _1^2+256\Omega _{1R}^2\Omega _{1I}^2} \right] t+\vartheta _{1R}^0,\\ \vartheta _{1I}\!=\! & {} \theta _{1}x\!+\!2\theta _{1}(\alpha \!-\!\Omega _{1R})y\!\\{} & {} -\!\theta _{1}\left[ \frac{\theta _{1}^2}{4}\!+\!3 \alpha (\alpha \!-\!2\Omega _{1R})\!+\!3(\Omega _{1R}^2\!-\!\Omega _{1I}^2)\!+\!\frac{\rho _0^2\beta _1}{\beta _1^2+256\Omega _{1R}^2\Omega _{1I}^2}\right] t\!+\!\vartheta _{1I}^0,\\ \beta _1= & {} 2\theta _{1}^2+8(\Omega _{1I}^2-\Omega _{1R}^2), \end{aligned}$$

and $\vartheta _{1}^0=\vartheta _{1R}^0+i\vartheta _{1I}^0$. Since Solutions (27) consists of trigonometric and hyperbolic functions, it leads breathers to traverse along the curve $\vartheta _{1R}+\frac{\ln A_1}{2}=0$ and periodic along the curve $\vartheta _{1I}=0$.

Remark 3.3

From the expression of $\vartheta _{1I}$, we know that if $\alpha =\Omega _{1R}$, the coefficient of y becomes zero, which makes the solutions exciting more phenomenon in $x-t$ plane.

Fig. 7

Periodic wave and breather via Solutions (26)with $\rho _0=1$, $\Omega _{1}=1+i$, $\theta _{1}=1$, $\vartheta _{1}^0=0$ and $y=1$. Periodic wave: $\alpha =1$; Breather: $\alpha =2$

Fig. 8

The first-order breathers via Solutions (26) with $\rho _0=\alpha =1$, $\theta _{1}=1$, $\vartheta _{1}^0=0$ and $t=0$

Periodic plane waves in shallow water can also been observed in natural world, and the periodic wave for Eq. (2) looks very much like the wave pattern in Fig. 3 of Ref. [6].

When $\Omega _1$ is purely imaginary, the angle between $L_1$ and $L_2$ is $\arctan \frac{1}{2(\alpha \!-\!\Omega _{1R})}$. The velocities of $L_1$ and $L_2$ parts are $(\infty , -3(\Omega _{1R}-\alpha )-\frac{8\rho _0^2\Omega _{1R}}{\beta _1^2+256\Omega _{1R}^2\Omega _{1I}^2})$ and $(V_3,\frac{V_3}{2(\Omega _{1R}-\alpha )})$ with $V_3=\frac{\theta _{1}^2}{4}\!+\!3 \alpha (\alpha \!-\!2\Omega _{1R})\!+\!3(\Omega _{1R}^2\!-\!\Omega _{1I}^2)\!+\!\frac{\rho _0^2\beta _1}{\beta _1^2+256\Omega _{1R}^2\Omega _{1I}^2}$, respectively. The period of the breather in the x-direction is expressed as $\frac{\pi }{|\theta _{1}\Omega _{1I}|}$. The results show that $\alpha$ dose not affect the period of the breather.

Second-order breather solutions can be constructed by taking $N=2$ in Solutions (25),

$$\begin{aligned} u=2(\textrm{ln}f)_{xx},\ \ \Phi _\varsigma =\rho _0e^{i\theta }\frac{g_\varsigma }{f},\ \ \Psi _\varsigma = \rho _0e^{-i\theta }\frac{h_\varsigma }{f}, \nonumber \\ \end{aligned}$$

(27)

with

$$\begin{aligned} f=\left( \begin{array}{llll}\Gamma ^0_{1,1} &{} \Gamma ^0_{1,2}\\ \Gamma ^0_{2,1} &{} \Gamma ^0_{2,2}\end{array}\right) ,\ \ g_\varsigma =h_\varsigma ^*=\left( \begin{array}{llll}\Gamma ^1_{1,1} &{} \Gamma ^1_{1,2}\\ \Gamma ^1_{2,1} &{} \Gamma ^1_{2,2}\end{array}\right) . \end{aligned}$$

3.1.2 Exp-rational solutions

Exp-rational solutions can be addressed by taking the long-wave limit technique [47] $(\theta _{k_1}\rightarrow 0, 1\le k_1\le {\widetilde{N}})$ in Solutions (25),

$$\begin{aligned}{} & {} f=\left( \begin{array}{llll}A_{2{\widetilde{N}},2{\widetilde{N}}} &{} B_{2{\widetilde{N}},2\widetilde{N'}}\\ C_{2\widetilde{N'},2{\widetilde{N}}} &{} D_{2\widetilde{N'},2\widetilde{N'}}\end{array}\right) ,\nonumber \\{} & {} g_\varsigma =h_\varsigma ^*=\left( \begin{array}{llll}{\mathfrak {A}}_{2{\widetilde{N}},2{\widetilde{N}}} &{} {\mathfrak {B}}_{2{\widetilde{N}},2\widetilde{N'}}\\ {\mathfrak {C}}_{2\widetilde{N'},2{\widetilde{N}}} &{} {\mathfrak {D}}_{2\widetilde{N'},2\widetilde{N'}}\end{array}\right) , \end{aligned}$$

(28)

where

$$\begin{aligned}{} & {} A_{k_1,k_1}=\left( \begin{array}{llll} \chi _{k_1} &{}\frac{1}{\Omega _{k_1}^*-\Omega _{k_1}}\\ \frac{1}{\Omega _{k_1}^*-\Omega _{k_1}} &{} \chi _{k_1}^* \end{array}\right) ,\ \\{} & {} A_{k_1,l_1}=\left( \begin{array}{llll}\frac{1}{\Omega _{l_1}-\Omega _{k_1}} &{} \frac{1}{\Omega _{l_1}^*-\Omega _{k_1}} \\ \frac{1}{\Omega _{k_1}^*-\Omega _{l_1}} &{} \frac{1}{\Omega _{k_1}^*-\Omega _{l_1}^*}\end{array}\right) ,\\{} & {} B_{k_1,{l_2}}={\left( \begin{array}{llll}\frac{1}{\Omega _{l_2}-\Omega _{k_1}+\frac{\theta _{l_2}}{2}} &{} \frac{1}{\Omega _{l_2}^*-\Omega _{k_1}-\frac{\theta _{l_2}}{2}} \\ \frac{1}{\Omega _{k_1}^*-\Omega _{l_2}-\frac{\theta _{l_2}}{2}} &{} \frac{1}{\Omega _{k_1}^*-\Omega _{l_2}^*+\frac{\theta _{l_2}}{2}}\end{array}\right) , }\ \\{} & {} C_{{k_2},{l_1}}=\left( \begin{array}{llll}\frac{1}{\Omega _{l_1}-\Omega _{k_2}+\frac{\theta _{k_2}}{2}} &{} \frac{1}{\Omega _{l_1}^*-\Omega _{k_2}+\frac{\theta _{k_2}}{2}}\\ \frac{1}{\Omega _{k_2}^*-\Omega _{l_1}+\frac{\theta _{k_2}}{2}} &{} \frac{1}{\Omega _{k_2}^*-\Omega _{l_1}^*+\frac{\theta _{k_2}}{2}}\end{array}\right) ,\\{} & {} D_{k_2,k_2}=\left( \begin{array}{llll}{\theta _{k_2}}^{-1}(1+e^{-\vartheta _{k_2}}) &{} (\Omega _{k_2}^*-\Omega _{k_2})^{-1}\\ (\Omega _{k_2}^*-\Omega _{k_2})^{-1} &{} {\theta _{k_2}}^{-1}(1+e^{-\vartheta _{k_2}^*})\end{array}\right) ,\\{} & {} D_{{k_2},{l_2}}=\left( \begin{array}{llll}\frac{1}{\Omega _{l_2}-\Omega _{k_2}+\frac{\theta _{k_2}+\theta _{l_2}}{2}} &{} \frac{1}{\Omega _{l_2}^*-\Omega _{k_2}+\frac{\theta _{k_2}-\theta _{l_2}}{2}} \\ \frac{1}{\Omega _{k_2}^*-\Omega _{l_2}+\frac{\theta _{k_2}-\theta _{l_2}}{2}} &{} \frac{1}{\Omega _{k_2}^*-\Omega _{l_2}^*+\frac{\theta _{k_2}+\theta _{l_2}}{2}}\end{array}\right) ,\\{} & {} {\mathfrak {A}}_{k_1,k_1}=\left( \begin{array}{llll} \chi _{k_1}-\frac{1}{\Omega _{k_1}} &{}\frac{1}{\Omega _{k_1}^*-\Omega _{k_1}}\frac{\Omega _{k_1}}{\Omega _{k_1}^*}\\ \frac{1}{\Omega _{k_1}^*-\Omega _{k_1}}\frac{\Omega _{k_1}^*}{\Omega _{k_1}} &{} \chi _{k_1}^*+\frac{1}{\Omega _{k_1}^*} \end{array}\right) ,\\{} & {} {\mathfrak {A}}_{k_1,l_1}=\left( \begin{array}{llll}\frac{\Omega _{k_1}}{\Omega _{l_1}}\frac{1}{\Omega _{l_1}-\Omega _{k_1}} &{} \frac{\Omega _{k_1}}{\Omega _{l_1}^*}\frac{1}{\Omega _{l_1}^*-\Omega _{k_1}} \\ \frac{\Omega _{k_1}^*}{\Omega _{l_1}}\frac{1}{\Omega _{k_1}^*-\Omega _{l_1}} &{} \frac{\Omega _{k_1}^*}{\Omega _{l_1}^*}\frac{1}{\Omega _{k_1}^*-\Omega _{l_1}^*}\end{array}\right) ,\\{} & {} {\mathfrak {B}}_{k_1,{l_2}}={\left( \begin{array}{llll}\frac{\Omega _{k_1}}{\Omega _{l_2}+\frac{\theta _{l_2}}{2}}\frac{1}{\Omega _{l_2}-\Omega _{k_1}+\frac{\theta _{l_2}}{2}} &{} \frac{\Omega _{k_1}}{\Omega _{l_2}^*+\frac{\theta _{l_2}}{2}}\frac{1}{\Omega _{l_2}^*-\Omega _{k_1}-\frac{\theta _{l_2}}{2}} \\ \frac{\Omega _{k_1}^*}{\Omega _{l_2}+\frac{\theta _{l_2}}{2}}\frac{1}{\Omega _{k_1}^*-\Omega _{l_2}-\frac{\theta _{l_2}}{2}} &{} \frac{\Omega _{k_1}^*}{\Omega _{l_2}^*+\frac{\theta _{l_2}}{2}}\frac{1}{\Omega _{k_1}^*-\Omega _{l_2}^*+\frac{\theta _{l_2}}{2}}\end{array}\right) , \ }\ \\ {}{} & {} {\mathfrak {C}}_{{k_2},{l_1}}=\left( \begin{array}{llll}\frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}}{\Omega _{l_1}}\frac{1}{\Omega _{l_1}-\Omega _{k_2}+\frac{\theta _{k_2}}{2}} &{} \frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}}{\Omega _{l_1}^*}\frac{1}{\Omega _{l_1}^*-\Omega _{k_2}+\frac{\theta _{k_2}}{2}}\\ \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}}{\Omega _{l_1}}\frac{1}{\Omega _{k_2}^*-\Omega _{l_1}+\frac{\theta _{k_2}}{2}} &{} \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}}{\Omega _{l_1}^*}\frac{1}{\Omega _{k_2}^*-\Omega _{l_1}^*+\frac{\theta _{k_2}}{2}}\end{array}\right) ,\\{} & {} {\mathfrak {D}}_{k_2,k_2}=\left( \!\begin{array}{llll}{\theta _{k_2}}^{-1}(\frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}}{\Omega _{k_2}+\frac{\theta _{k_2}}{2}}+e^{-\vartheta _{k_2}}) &{} \frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}}{\Omega _{k_2}^*-\frac{\theta _{k_2}}{2}}(\Omega _{k_2}^*-\Omega _{k_2})^{-1}\\ \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}}{\Omega _{k_2}+\frac{\theta _{k_2}}{2}}(\Omega _{k_2}^*-\Omega _{k_2})^{-1} &{} {\theta _{k_2}}^{-1}(\frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}}{\Omega _{k_2}^*-\frac{\theta _{k_2}}{2}}+e^{-\vartheta _{k_2}^*})\end{array}\!\right) \!, \\ {}{} & {} {\mathfrak {D}}_{{k_2},{l_2}}=\left( \!\begin{array}{llll}\frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}}{\Omega _{l_2}+\frac{\theta _{l_2}}{2}}\frac{1}{\Omega _{l_2}-\Omega _{k_2} +\frac{\theta _{k_2}+\theta _{l_2}}{2}} &{} \frac{\Omega _{k_2}-\frac{\theta _{k_2}}{2}}{\Omega _{l_2}^*-\frac{\theta _{l_2}}{2}}\frac{1}{\Omega _{l_2}^*-\Omega _{k_2}+\frac{\theta _{k_2}-\theta _{l_2}}{2}} \\ \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}}{\Omega _{l_2}+\frac{\theta _{l_2}}{2}}\frac{1}{\Omega _{k_2}^*-\Omega _{l_2}+\frac{\theta _{k_2}-\theta _{l_2}}{2}} &{} \frac{\Omega _{k_2}^*+\frac{\theta _{k_2}}{2}}{\Omega _{l_2}^*-\frac{\theta _{l_2}}{2}}\frac{1}{\Omega _{k_2}^*-\Omega _{l_2}^* +\frac{\theta _{k_2}+\theta _{l_2}}{2}}\end{array}\!\right) \!, \end{aligned}$$

with

$$\begin{aligned}{} & {} \chi _{k_1}=ix+2i(\alpha -\Omega _{k_1})y-i\left( 3\alpha ^2-6\alpha \Omega _{k_1}\right. \\{} & {} \left. +3\Omega _{k_1}^2-\frac{\rho _0^2}{8\Omega ^2_{k_1}}\right) t, \\{} & {} \vartheta _{k_2}\!=\!i\theta _{k_2}x\!+\!2i\theta _{k_2}(\alpha -\Omega _{k_2})y\! -\!i\theta _{k_2}\\{} & {} \times \left( 3\alpha ^2-6\alpha \Omega _{k_2}+3\Omega _{k_2}^2+\frac{1}{4}\theta _{k_2}^2+\frac{\rho _0^2}{2\theta _{k_2}^2-8\Omega _{k_2}^2} \right) \\{} & {} \times \,t\!+\!\vartheta _{k_2}^0. \end{aligned}$$

When we choose ${\widetilde{N}}=\widetilde{N'}=1$, there will be a interaction between a lump and a breather. It should be noted that if $\Omega _{1}$ is purely imaginary and the imaginary part of $\Omega _{2}$ equals to $\Omega _{1}$, the interaction between a lump and a breather is inelastic. Likewise, if $\Omega _{2}$ is purely imaginary and the imaginary part of $\Omega _{1}$ equals to $\Omega _{2}$, there will also be an inelastic interaction between a lump and a breather.

Fig. 9

Inelastic interaction between a lump and a breather via Solutions (28) with $\Omega _1=1+i$, $\Omega _2=i$, $\rho _0=\alpha =1$, $\theta _{1}=1$, $\vartheta _{2}^0=0$

3.1.3 Rational solutions

In order to derive the rational solutions, we take the long-wave limit technique [47] $(\theta _k\rightarrow 0)$ in Solutions (25):

$$\begin{aligned}{} & {} f=\left| {\begin{array}{*{5}{c}} {{\widetilde{\Lambda }}^0_{1,1}} &{} {{\widetilde{\Lambda }}^0_{1,2}} &{} {{...}} &{} {{\widetilde{\Lambda }}^0_{1,M}} \\ {{\widetilde{\Lambda }}^0_{2,1}} &{} {{\widetilde{\Lambda }}^0_{2,2}} &{} {{...}} &{} {{\widetilde{\Lambda }}^0_{2,M}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {{\widetilde{\Lambda }}^0_{M,1}} &{} {{\widetilde{\Lambda }}^0_{M,2}} &{} {{...}} &{} {{\widetilde{\Lambda }}^0_{M,M}} \\ \end{array}} \right| ,\ \ \nonumber \\{} & {} g_\varsigma =h_\varsigma ^*=\left| {\begin{array}{*{5}{c}} {{\widetilde{\Lambda }}^1_{1,1}} &{} {{\widetilde{\Lambda }}^1_{1,2}} &{} {{...}} &{} {{\widetilde{\Lambda }}^1_{1,M}} \\ {{\widetilde{\Lambda }}^1_{2,1}} &{} {{\widetilde{\Lambda }}^1_{2,2}} &{} {{...}} &{} {{\widetilde{\Lambda }}^1_{2,M}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {{\widetilde{\Lambda }}^1_{M,1}} &{} {{\widetilde{\Lambda }}^1_{M,2}} &{} {{...}} &{} {{\widetilde{\Lambda }}^1_{M,M}} \\ \end{array}} \right| , \end{aligned}$$

(29)

with

$$\begin{aligned}{} & {} {\widetilde{\Lambda }}^0_{k,k}=\left( \begin{array}{llll} \chi _{k} &{} \frac{1}{\Omega _{k}^*-\Omega _{k}}\\ \frac{1}{\Omega _{k}^*-\Omega _{k}} &{} \chi _{k}^* \end{array}\right) ,\ \ \\{} & {} {\widetilde{\Lambda }}^0_{k,l}=\left( \begin{array}{llll}\frac{1}{\Omega _{l}-\Omega _{k}} &{} \frac{1}{\Omega _{l}^*-\Omega _{k}}\\ \frac{1}{\Omega _{k}^*-\Omega _{l}} &{} \frac{1}{\Omega _{k}^*-\Omega _{l}^*} \end{array}\right) ,\\{} & {} {\widetilde{\Lambda }}^1_{k,k}=\left( \begin{array}{llll} \chi _{k_1}-\frac{1}{\Omega _{k_1}} &{}\frac{1}{\Omega _{k_1}^*-\Omega _{k_1}}\frac{\Omega _{k_1}}{\Omega _{k_1}^*}\\ \frac{1}{\Omega _{k_1}^*-\Omega _{k_1}}\frac{\Omega _{k_1}^*}{\Omega _{k_1}} &{} \chi _{k_1}^*+\frac{1}{\Omega _{k_1}^*} \end{array}\right) ,\ \ \\{} & {} {\widetilde{\Lambda }}^1_{k,l}=\left( \begin{array}{llll}\frac{\Omega _{k_1}}{\Omega _{l_1}}\frac{1}{\Omega _{l_1}-\Omega _{k_1}} &{} \frac{\Omega _{k_1}}{\Omega _{l_1}^*}\frac{1}{\Omega _{l_1}^*-\Omega _{k_1}} \\ \frac{\Omega _{k_1}^*}{\Omega _{l_1}}\frac{1}{\Omega _{k_1}^*-\Omega _{l_1}} &{} \frac{\Omega _{k_1}^*}{\Omega _{l_1}^*}\frac{1}{\Omega _{k_1}^*-\Omega _{l_1}^*}\end{array}\right) , \\{} & {} \chi _{k_1}=ix+2i(\alpha -\Omega _{k_1})y\\{} & {} \qquad \quad -i\left( 3\alpha ^2-6\alpha \Omega _{k_1}+3\Omega _{k_1}^2-\frac{\rho _0^2}{8\Omega ^2_{k_1}}\right) t, \end{aligned}$$

First-order rational solutions can be constructed by setting $M=1$ in Solutions (29),

$$\begin{aligned}{} & {} u=4\frac{\chi _{1R}^2-\chi _{1I}^2+(2\Omega _{1I})^{-2}}{[\chi _{1R}^2+\chi _{1I}^2+(2\Omega _{1I})^{-2}]^2}, \end{aligned}$$

(30a)

$$\begin{aligned}{} & {} \Phi _\varsigma ={\Psi _\varsigma }^*=\rho _0e^{i\theta }\nonumber \\{} & {} \quad \left[ 1+\frac{2i(\Omega _{1R}\chi _{1R}+\Omega _{1I}\chi _{1I})}{[\chi _{1R}^2+\chi _{1I}^2+(2\Omega _{1I})^{-2}] (\Omega _{1R}^2+\Omega _{1I}^2)}\right] , \nonumber \\ \end{aligned}$$

(30b)

where $\chi =\chi _{1R}+i\chi _{1I}$, $\Omega _{1}=\Omega _{1R}+i\Omega _{1I}$ and

$$\begin{aligned}{} & {} \chi _{1R}=2\Omega _{1I}y+\left[ 6\Omega _{1R}\Omega _{1I}-6\alpha \Omega _{1I}+\frac{b_1\rho _0^2}{8({\widetilde{a}}_{1}^2+{\widetilde{b}}_{1}^2)}\right] t,\\{} & {} \chi _{1I}=x+2(\alpha -\Omega _{1R})y\\{} & {} \qquad +\left[ 3(\Omega _{1I}^2-\Omega _{1R}^2)+6\alpha \Omega _{1R}-3\alpha ^2+\frac{{\tilde{a}}_1\rho _0^2}{8(a_{1}^2+{\tilde{b}}_{1}^2)}\right] t,\\{} & {} {\widetilde{a}}_1=\Omega _{1R}^2-\Omega _{1I}^2,\\{} & {} {\widetilde{b}}_1=2\Omega _{1I}\Omega _{1R}. \end{aligned}$$

The first-order solutions are similar with the ones in Fig. 3, and $\alpha$ can not change the type of the lump. In fact, $\alpha$ relates to the characteristic line [48] of the lump, so that the lump seems to rotate as $\alpha$ changing.

Fig. 10

Effects of $\alpha$ on lumps via Solutions (30) with $\rho _0=1$, $\Omega _1=1+i$, $t=0$

3.2 Breather-IV for Eq. (2)

3.2.1 Breather solutions

Choosing

$$\begin{aligned}{} & {} p_{2k-1}=-i{\tilde{\Omega }}_k+\frac{i{\tilde{\theta }}_k}{2},\ \ p_{2k}=i{\tilde{\Omega }}_k^*+\frac{i{\tilde{\theta }}_k}{2},\\{} & {} {\tilde{\xi _{2k}^0}}^*={\tilde{\xi }}_{2k-1}^{0}, q_{2k-1}=i{\tilde{\Omega }}_k+\frac{i{\tilde{\theta }}_k}{2},\\{} & {} q_{2k}=-i{\tilde{\Omega }}_k^*+\frac{i\theta _k}{2},\ \ {\tilde{\eta _{2k}^0}}^*={\tilde{\eta }}_{2k-1}^0, \end{aligned}$$

with ${\tilde{\Omega }}_k\in {\mathbb {C}}$ and ${\tilde{\theta }}_k\in {\mathbb {R}}$, we have $f=f^*$. Thus, we obtain another type breather solutions for Eq. (2):

$$\begin{aligned} u=2(\textrm{ln}f)_{xx},\ \ \Phi _\varsigma =\rho _0e^{i\theta }\frac{g_\varsigma }{f},\ \ \Psi _\varsigma = \rho _0e^{-i\theta }\frac{h_\varsigma }{f},\nonumber \\ \end{aligned}$$

(31)

where

$$\begin{aligned}{} & {} f={\widetilde{\Upsilon }}\left| {\begin{array}{*{5}{c}} {{\widetilde{\Gamma }}^0_{1,1}} &{} {{\widetilde{\Gamma }}^0_{1,2}} &{} {{...}} &{} {{\widetilde{\Gamma }}^0_{1,N}} \\ {{\widetilde{\Gamma }}^0_{2,1}} &{} {{\widetilde{\Gamma }}^0_{2,2}} &{} {{...}} &{} {{\widetilde{\Gamma }}^0_{2,N}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {{\widetilde{\Gamma }}^0_{N,1}} &{} {{\widetilde{\Gamma }}^0_{N,2}} &{} {{...}} &{} {{\widetilde{\Gamma }}^0_{N,N}} \\ \end{array}} \right| ,\ \ \\{} & {} g_\varsigma =h_\varsigma ^*={\widetilde{\Upsilon }}\left| {\begin{array}{*{5}{c}} {{\widetilde{\Gamma }}^1_{1,1}} &{} {{\widetilde{\Gamma }}^1_{1,2}} &{} {{...}} &{} {{\widetilde{\Gamma }}^1_{1,N}} \\ {{\widetilde{\Gamma }}^1_{2,1}} &{} {{\widetilde{\Gamma }}^1_{2,2}} &{} {{...}} &{} {{\widetilde{\Gamma }}^1_{2,N}} \\ {{\vdots }} &{} {{\vdots }} &{} {{\ddots }} &{} {\vdots } \\ {{\widetilde{\Gamma }}^1_{N,1}} &{} {{\widetilde{\Gamma }}^1_{N,2}} &{} {{...}} &{} {{\widetilde{\Gamma }}^1_{N,N}} \\ \end{array}} \right| , \end{aligned}$$

with

$$\begin{aligned}{} & {} {\widetilde{\Gamma }}^0_{k,k}=\left( \begin{array}{llll}-\frac{i}{{\tilde{\theta }}_k}(1+e^{-{\tilde{\varsigma }}_k}) &{} \frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_k^*-{\tilde{\theta }}_k}\\ \frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_k^*+{\tilde{\theta }}_k} &{} \frac{i}{{\tilde{\theta }}_k}(1+e^{-{\tilde{\varsigma }}^*_k})\end{array}\right) ,\ \\{} & {} {\widetilde{\Gamma }}^0_{k,l}=\left( \begin{array}{llll}\frac{i}{{\tilde{\Omega }}_k-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} &{} \frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_l^*-\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}}\\ \frac{i}{{\tilde{\Omega }}_k^*+{\tilde{\Omega }}_l+\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} &{} \frac{i}{{\tilde{\Omega }}_k^*-{\tilde{\Omega }}_l^*+\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} \end{array}\right) , (k\ne l),\\{} & {} {\widetilde{\Gamma }}^1_{k,k}=\left( \begin{array}{llll}-\frac{i}{{\tilde{\theta }}_k}(e^{-{\tilde{\varsigma }}_k} +\frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}}{{\tilde{\Omega }}_k+\frac{{\tilde{\theta }}_k}{2}}) &{} \frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}}{-{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}}\frac{i}{{\tilde{\Omega }}_k^*+{\tilde{\Omega }}_k-{\tilde{\theta }}_k}\\ \frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}}{-{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}}\frac{i}{{\tilde{\Omega }}_k^*+{\tilde{\Omega }}_k+{\tilde{\theta }}_k} &{} \frac{i}{{\tilde{\theta }}_k}(e^{-{\tilde{\varsigma }}^*_k}+\frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}}{{\tilde{\Omega }}^*_k-\frac{{\tilde{\theta }}_k}{2}})\end{array}\right) ,\\{} & {} {\widetilde{\Gamma }}^1_{k,l}=\left( \begin{array}{llll}\frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}}{{\tilde{\Omega }}_l +\frac{{\tilde{\theta }}_l}{2}}\frac{i}{{\tilde{\Omega }}_k-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} &{} \frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}}{-{\tilde{\Omega }}^*_l+\frac{{\tilde{\theta }}_l}{2}}\frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_l^* -\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}}\\ \frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}}{-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_l}{2}}\frac{i}{{\tilde{\Omega }}_k^*+{\tilde{\Omega }}_l +\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} &{} \frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}}{{\tilde{\Omega }}^*_l-\frac{{\tilde{\theta }}_l}{2}}\frac{i}{{\tilde{\Omega }}_k^*-{\tilde{\Omega }}_l^* +\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} \end{array}\right) , (k\ne l),\\{} & {} {\tilde{\varsigma }}_k=-{\tilde{\theta }}_kx+2{\tilde{\theta }}_k(i{\tilde{\Omega }}_k-\alpha )y\\{} & {} + {\widetilde{\theta }}_{k}\left( 3\alpha ^2-6i\alpha {\tilde{\Omega }}_{k} -\frac{{\widetilde{\theta }}_{k}^2}{4}\!-\!3{\tilde{\Omega }}_{k}^2\! -\!\frac{\rho _0^2}{8{\tilde{\Omega }}_{k}^2-\!2{\tilde{\theta }}_{k}^2}\right) t+{\tilde{\varsigma }}_k^0,\\{} & {} {\tilde{\Upsilon }}=e^{\Sigma _{k=1}^{N}{\tilde{\varsigma }}_k+{\tilde{\varsigma }}_k^*} \prod \limits _{k=1}^{N}{\tilde{\theta }}_k^2, \end{aligned}$$

with ${\tilde{\varsigma }}_k^0={\tilde{\xi }}_k^0+{\tilde{\eta }}_k^0$. Similarly, ${\tilde{\Upsilon }}$ can be eliminated. Indeed, it should be pointed out that if we take $p_{2k-1}\rightarrow ip_{2k-1}$, $p_{2k} \rightarrow -ip_{2k-1}$, $q_{2k-1} \rightarrow iq_{2k-1}$ and $q_{2k} \rightarrow -iq_{2k}$, Solutions (31) are equivalent to Solutions (25).

Setting $N=1$ in Solutions (31), we derive the first-order breather solutions with

$$\begin{aligned} \begin{array}{l} f=e^{{\widetilde{\vartheta }}_{1R}}\left[ \sqrt{D_1}\cosh \left( {\widetilde{\vartheta }}_{1R}+\frac{\ln D_1}{2}\right) +2\cos {{\widetilde{\vartheta }}_{1I,\varsigma }} \right] ,\\ g_\varsigma \!=\!1\!+\!{\tilde{\vartheta }}_{1R}(D_2\!+\!(D_2^*)^{-1})\cos ({\tilde{\vartheta }}_{1I})\\ \quad +\! i{\tilde{\vartheta }}_{1R}(D_2\!-\!(D_2^*)^{-1})\sin ({\tilde{\vartheta }}_{1I})\!+\!D_1D_2(D_2^*)^{-1}e^{2{\tilde{\vartheta }}_{1R}},\\ h_\varsigma =1+{\tilde{\vartheta }}_{1R}(D_2^{-1}+D_2^*)\cos ({\tilde{\vartheta }}_{1I})\\ \quad +i{\tilde{\vartheta }}_{1R}(D_2^{-1}-D_2^*)\sin ({\tilde{\vartheta }}_{1I})+D_1D_2^*D_2^{-1}e^{2{\tilde{\vartheta }}_{1R}}, \end{array} \nonumber \\ \end{aligned}$$

(32)

with

$$\begin{aligned}{} & {} {\widetilde{\vartheta }}_{1R}\!=\!-\!{\tilde{\theta }}_{1}x\!-\!2{\tilde{\theta }}_{1}({\tilde{\Omega }}_{1I}\!+\!\alpha )y\!\\{} & {} +\!{\tilde{\theta }}_{1} \left[ 3\alpha ^2\!+\!\frac{{\tilde{\theta }}_{1}^2}{4}\!+\!6\alpha {\tilde{\Omega }}_{1I}\!-\!3({\tilde{\Omega }}_{1R}^2\!\right. \\{} & {} \left. -\!{\tilde{\Omega }}_{1I}^2) -\frac{8\rho _0^2\mu _1}{\mu _1^2\!+\!{\tilde{\Omega }}_{1R}^2{\tilde{\Omega }}_{1I}^2} \right] t\!\\{} & {} +\!{\tilde{\vartheta }}_{1R}^0,\\{} & {} {\widetilde{\vartheta }}_{1I}=2{\tilde{\theta }}_{1}{\tilde{\Omega }}_{1R}y+{\tilde{\theta }}_{1}\left[ 6\alpha {\tilde{\Omega }}_{1R}+6{\tilde{\Omega }}_{1R}{\tilde{\Omega }}_{1I} \right. \\{} & {} \left. -\frac{\rho _0^2{\tilde{\Omega }}_{1R}{\tilde{\Omega }}_{1I}}{4(\mu _1^2+4{\tilde{\Omega }}_{1R}^2{\tilde{\Omega }}_{1I}^2)}\right] t+{\tilde{\vartheta }}_{1I}^0,\\{} & {} \mu _1={\tilde{\Omega }}_{1R}^2-{\tilde{\Omega }}_{1I}^2-\frac{{\tilde{\theta }}_{1}^2}{4},\\{} & {} D_1=\frac{-2{\tilde{\Omega }}_{1R}}{{\tilde{\theta }}_{1}^2-4{\tilde{\Omega }}_{1R}^2}, \\{} & {} \ D_2=\frac{-i{\tilde{\Omega }}_{1}+\frac{i{\tilde{\theta }}_{1}}{2}}{i{\tilde{\Omega }}_{1}+\frac{i{\tilde{\theta }}_{1}}{2}}. \end{aligned}$$

Figure 11 presents the first-order breather. When $\Omega _1$ is not purely imaginary, the angle between $L_1$ and $L_2$ is $\arctan (-\frac{1}{2(\Omega _{1I}+a)})$. The velocities of $L_1$ and $L_2$ parts are $(\infty , -[3\alpha +3{\tilde{\Omega }}_{1I} -\frac{\rho _0^2{\tilde{\Omega }}_{1I}}{4(\mu _1^2+4{\tilde{\Omega }}_{1R}^2{\tilde{\Omega }}_{1I}^2)}])$ and $(V_4,\frac{V_4}{2(\Omega _{1R}-\alpha )})$ with $V_4=3\alpha ^2\!+\!\frac{{\tilde{\theta }}_{1}^2}{4}\!+\!6\alpha {\tilde{\Omega }}_{1I}\!-\!3({\tilde{\Omega }}_{1R}^2\!-\!{\tilde{\Omega }}_{1I}^2) -\frac{8\rho _0^2\mu _1}{\mu _1^2\!+\!{\tilde{\Omega }}_{1R}^2{\tilde{\Omega }}_{1I}^2}$, respectively. The periods of the breather in the x- and y-directions are expressed as $\frac{2\pi }{|\theta _{1}|}$ and $\frac{\pi }{|\theta _{1}(\Omega _{1I}+\alpha |)}$. The results shows that $\alpha$ affects the period of the breather, which differs from the breather in Fig. 8 evidently. We note that when $\Omega _{1}$ is purely imaginary and $t=0$, the breather degenerates into one soliton, and there is a dark-soliton solution in $\Phi$ component, which is similar as in Fig. 5.

Fig. 11

The first-order breathers via Solutions (32) with $\rho _0=\alpha =1$, $\theta _{1}=1$, $\vartheta _{1}^0=0$ and $t=0$

Second-order breather solutions can be constructed by taking $N=2$ in Solutions (31),

$$\begin{aligned} u=2(\textrm{ln}f)_{xx},\ \ \Phi _\varsigma =\rho _0e^{i\theta }\frac{g_\varsigma }{f},\ \ \Psi _\varsigma = \rho _0e^{-i\theta }\frac{h_\varsigma }{f}, \nonumber \\ \end{aligned}$$

(33)

with

$$\begin{aligned} f=\left( \begin{array}{llll}\Gamma ^0_{1,1} &{} \Gamma ^0_{1,2}\\ \Gamma ^0_{2,1} &{} \Gamma ^0_{2,2}\end{array}\right) ,\ \ g_\varsigma =h_\varsigma ^*=\left( \begin{array}{llll}\Gamma ^1_{1,1} &{} \Gamma ^1_{1,2}\\ \Gamma ^1_{2,1} &{} \Gamma ^1_{2,2}\end{array}\right) . \end{aligned}$$

3.2.2 Exp-rational solutions

Fig. 12

Solutions (34) with $\rho _0=\alpha =1$, ${\tilde{\theta }}_{1}=1$, ${\tilde{\vartheta }}_{2}^0=10$ and $t=0$. (a) and (b): ${\tilde{\Omega }}_{1}=1+i$, (c) and (d): ${\tilde{\Omega }}_{1}=i$

Similarly, we take the long-wave limit [47] $(\theta _{k_1}\rightarrow 0, 1\le k_1\le {\widetilde{N}})$ in Solutions (31) and construct the exp-rational solutions with

$$\begin{aligned}{} & {} f=\left( \begin{array}{llll}A_{2{\widetilde{N}},2{\widetilde{N}}} &{} B_{2{\widetilde{N}},2\widetilde{N'}}\\ C_{2\widetilde{N'},2{\widetilde{N}}} &{} D_{2\widetilde{N'},2\widetilde{N'}}\end{array}\right) ,\nonumber \\{} & {} g=\left( \begin{array}{llll}{\mathfrak {A}}_{2{\widetilde{N}},2{\widetilde{N}}} &{} {\mathfrak {B}}_{2{\widetilde{N}},2\widetilde{N'}}\\ {\mathfrak {C}}_{2\widetilde{N'},2{\widetilde{N}}} &{} {\mathfrak {D}}_{2\widetilde{N'},2\widetilde{N'}}\end{array}\right) , \end{aligned}$$

(34)

where

$$\begin{aligned}{} & {} A_{k_1,k_1}=\left( \begin{array}{llll} {\widetilde{\chi }}_{k_1} &{} \frac{i}{\Omega _{k_1}+\Omega _{k_1}^*}\\ \frac{i}{\Omega _{k_1}+\Omega _{k_1}^*} &{} {\widetilde{\chi }}_{k_1}^* \end{array}\right) ,\\{} & {} A_{k_1,l_1}=\left( \begin{array}{llll}\frac{i}{{\tilde{\Omega }}_{k_1}-{\tilde{\Omega }}_{l_1}} &{} \frac{i}{{\tilde{\Omega }}_{k_1}+{\tilde{\Omega }}_{l_1}^*}\\ \frac{i}{{\tilde{\Omega }}_{k_1}^*+{\tilde{\Omega }}_{l_1}} &{} \frac{i}{{\tilde{\Omega }}_{k_1}^*-{\tilde{\Omega }}_{l_1}^*} \end{array}\right) ,\\{} & {} B_{k_1,{l_2}}={\left( \begin{array}{llll}\frac{i}{(\Omega _{k_1}-\Omega _{l_2})-\frac{\theta _{l_2}}{2}} &{} \frac{i}{(\Omega _{k_1}+\Omega _{l_2}^*)-\frac{\theta _{l_2}}{2}} \\ \frac{i}{(\Omega _{k_1}^*+\Omega _{l_2})+\frac{\theta _{l_2}}{2}} &{} \frac{i}{(\Omega _{k_1}^*-\Omega _{l_2}^*)+\frac{\theta _{l_2}}{2}}\end{array}\right) , }\ \\{} & {} C_{{k_2},{l_1}}=\left( \begin{array}{llll}\frac{i}{\Omega _{k_2}-\Omega _{l_1}-\frac{\theta _{k_2}}{2}} &{} \frac{i}{\Omega _{l_1}^*+\Omega _{k_2}-\frac{\theta _{k_2}}{2}}\\ \frac{i}{\Omega _{k_2}^*+\Omega _{l_1}+\frac{\theta _{k_2}}{2}} &{} \frac{i}{\Omega _{k_2}^*-\Omega _{l_1}^*+\frac{\theta _{k_2}}{2}}\end{array}\right) ,\\{} & {} D_{k_2,k_2}=\left( \begin{array}{llll}-\frac{i}{{\tilde{\theta }}_{k_2}}(1+e^{-{\tilde{\vartheta }}_{k_2}}) &{} \frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_{k_2}^*-{\tilde{\theta }}_{k_2}}\\ \frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_{k_2}^*+{\tilde{\theta }}_{k_2}} &{} \frac{i}{{\tilde{\theta }}_{k_2}}(1+e^{-{\tilde{\vartheta }}^*_{k_2}})\end{array}\right) ,\\{} & {} D_{{k_2},{l_2}}=\left( \begin{array}{llll}\frac{i}{{\tilde{\Omega }}_{k_2}-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_{l_2}}{2}} &{} \frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_{l_2}^*-\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_{l_2}}{2}}\\ \frac{i}{{\tilde{\Omega }}_{k_2}^*+{\tilde{\Omega }}_{l_2}+\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_{l_2}}{2}} &{} \frac{i}{{\tilde{\Omega }}_{k_2}^*-{\tilde{\Omega }}_{l_2}^*+\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_{l_2}}{2}} \end{array}\right) ,\\{} & {} {\mathfrak {A}}_{k_1,k_1}=\left( \begin{array}{llll} {\widetilde{\chi }}_{k_1}-\frac{1}{i\Omega _{k_1}} &{}\frac{i}{\Omega _{k_1}+\Omega _{k_1}^*}\frac{-\Omega _{k_1}}{\Omega _{k_1}^*}\\ \frac{i}{\Omega _{k_1}+\Omega _{k_1}^*}\frac{-\Omega _{k_1}^*}{\Omega _{k_1}} &{} {\widetilde{\chi }}_{k_1}^*-\frac{1}{i\Omega _{k_1}^*} \end{array}\right) ,\\{} & {} {\mathfrak {A}}_{k_1,l_1}=\left( \begin{array}{llll}\frac{\Omega _{k_1}}{\Omega _{l_1}}\frac{i}{\Omega _{k_1}-\Omega _{l_1}} &{} \frac{\Omega _{k_1}}{-\Omega _{l_1}^*}\frac{i}{\Omega _{k_1}+\Omega _{l_1}^*} \\ \frac{\Omega _{k_1}^*}{-\Omega _{l_1}}\frac{i}{\Omega _{k_1}^*+\Omega _{l_1}} &{} \frac{\Omega _{k_1}^*}{\Omega _{l_1}^*}\frac{i}{\Omega _{k_1}^*-\Omega _{l_1}^*}\end{array}\right) ,\\ \end{aligned}$$

$$\begin{aligned}{} & {} {\mathfrak {B}}_{{k_1},{l_2}}={\left( \begin{array}{llll}\frac{{\tilde{\Omega }}_{k_1}}{{\tilde{\Omega }}_{l_2}+\frac{{\tilde{\theta }}_{l_2}}{2}}\frac{i}{{\tilde{\Omega }}_{k_1}-{\tilde{\Omega }}_{l_2} -\frac{{\tilde{\theta }}_{l_2}}{2}} &{} \frac{{\tilde{\Omega }}_{k_1}}{-{\tilde{\Omega }}^*_{l_2}+\frac{{\tilde{\theta }}_{l_2}}{2}}\frac{i}{{\tilde{\Omega }}_{k_1}+{\tilde{\Omega }}_{l_2}^*-\frac{{\tilde{\theta }}_{l_2}}{2}}\\ \frac{{\tilde{\Omega }}^*_{k_1}}{-{\tilde{\Omega }}_{l_2}-\frac{{\tilde{\theta }}_{l_2}}{2}}\frac{i}{{\tilde{\Omega }}_{k_1}^*+{\tilde{\Omega }}_{l_2}+\frac{{\tilde{\theta }}_{l_2}}{2}} &{} \frac{{\tilde{\Omega }}^*_{k_1}}{{\tilde{\Omega }}^*_{l_2}-\frac{{\tilde{\theta }}_{l_2}}{2}}\frac{i}{{\tilde{\Omega }}_{k_1}^*-{\tilde{\Omega }}_{l_2}^*+\frac{{\tilde{\theta }}_{l_2}}{2}} \end{array}\right) , }\ \\{} & {} {\mathfrak {C}}_{{k_2},{l_1}}=\left( \begin{array}{llll}\frac{{\tilde{\Omega }}_{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}}{{\tilde{\Omega }}_{l_1}} \frac{i}{{\tilde{\Omega }}_{k_2}-{\tilde{\Omega }}_{l_1}-\frac{{\tilde{\theta }}_{k_2}}{2}} &{} \frac{{\tilde{\Omega }}_{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}}{-{\tilde{\Omega }}^*_{l_1}}\frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_{l_1}^*-\frac{{\tilde{\theta }}_{k_2}}{2}}\\ \frac{{\tilde{\Omega }}^*_{k_2}+\frac{{\tilde{\theta }}_{k_2}}{2}}{-{\tilde{\Omega }}_{l_1}}\frac{i}{{\tilde{\Omega }}_{k_2}^*+{\tilde{\Omega }}_{l_1}+\frac{{\tilde{\theta }}_{k_2}}{2}} &{} \frac{{\tilde{\Omega }}^*_{k_2}+\frac{{\tilde{\theta }}_{k_2}}{2}}{{\tilde{\Omega }}^*_{l_1}}\frac{i}{{\tilde{\Omega }}_{k_2}^*-{\tilde{\Omega }}_{l_1}^*+\frac{{\tilde{\theta }}_{k_2}}{2}} \end{array}\right) ,\\{} & {} {\mathfrak {D}}_{k_2,k_2}=\left( \begin{array}{llll}-\frac{i}{{\tilde{\theta }}_k}(e^{-{\tilde{\vartheta }}_k}+\frac{{\tilde{\Omega }}_k- \frac{{\tilde{\theta }}_k}{2}}{{\tilde{\Omega }}_k+\frac{{\tilde{\theta }}_k}{2}}) &{} \frac{{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}}{-{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}}\frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_k^*-{\tilde{\theta }}_k}\\ \frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}}{-{\tilde{\Omega }}_k-\frac{{\tilde{\theta }}_k}{2}}\frac{i}{{\tilde{\Omega }}_k+{\tilde{\Omega }}_k^*+{\tilde{\theta }}_k} &{} \frac{i}{{\tilde{\theta }}_k}(e^{-{\tilde{\vartheta }}^*_k}+\frac{{\tilde{\Omega }}^*_k+\frac{{\tilde{\theta }}_k}{2}}{{\tilde{\Omega }}^*_k-\frac{{\tilde{\theta }}_k}{2}})\end{array}\right) , \\{} & {} {\mathfrak {D}}_{{k_2},{l_2}}=\left( \begin{array}{llll}\frac{{\tilde{\Omega }}_{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}}{{\tilde{\Omega }}_l +\frac{{\tilde{\theta }}_l}{2}}\frac{i}{{\tilde{\Omega }}_{k_2}-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_l}{2}} &{} \frac{{\tilde{\Omega }}_{k_2}-\frac{{\tilde{\theta }}_{k_2}}{2}}{-{\tilde{\Omega }}^*_l+\frac{{\tilde{\theta }}_l}{2}} \frac{i}{{\tilde{\Omega }}_{k_2}+{\tilde{\Omega }}_l^*-\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_l}{2}}\\ \frac{{\tilde{\Omega }}^*_{k_2}+\frac{{\tilde{\theta }}_{k_2}}{2}}{-{\tilde{\Omega }}_l-\frac{{\tilde{\theta }}_l}{2}}\frac{i}{{\tilde{\Omega }}_{k_2}^*+{\tilde{\Omega }}_l+\frac{{\tilde{\theta }}_k+{\tilde{\theta }}_l}{2}} &{} \frac{{\tilde{\Omega }}^*_{k_2}+\frac{{\tilde{\theta }}_{k_2}}{2}}{{\tilde{\Omega }}^*_l-\frac{{\tilde{\theta }}_l}{2}} \frac{i}{{\tilde{\Omega }}_{k_2}^*-{\tilde{\Omega }}_l^*+\frac{{\tilde{\theta }}_{k_2}+{\tilde{\theta }}_l}{2}} \end{array}\right) , \end{aligned}$$

with

$$\begin{aligned}{} & {} {\tilde{\chi }}_{k_1}=-x+2(i{\tilde{\Omega }}_{k_1}-\alpha )y+\left( 3\alpha ^2-6i \alpha {\tilde{\Omega }}_{k_1}\right. \\{} & {} \left. -3{\tilde{\Omega }}_{k_1}^2+\frac{\rho _0^2}{8{\tilde{\Omega }}_{k_1}^2}\right) t, \\{} & {} {\tilde{\vartheta }}_{k_2}\!=\!-{\tilde{\theta }}_{k_2}x\!+\!2{\tilde{\theta }}_{k_2}(i{\tilde{\Omega }}_{k_2}-\alpha )y\\{} & {} \qquad \quad + {\widetilde{\theta }}_{k_2}\left( 3\alpha ^2-\frac{{\widetilde{\theta }}_{k_2}^2}{4}\!-\!3{\tilde{\Omega }}_{k_2}^2-6i\alpha {\tilde{\Omega }}_{k_2} \! -\!\frac{\rho _0^2}{8{\tilde{\Omega }}_{k_2}^2-\!2{\tilde{\theta }}_{k_2}^2}\right) t\!\\{} & {} \qquad \quad +\!{\tilde{\vartheta }}_{k_2}^0. \end{aligned}$$

According to Solutions (34), we can derive the interactions between breather and lump, and between lump and soliton with ${\tilde{\Omega }}_{1}$ purely imaginary.

3.2.3 Rational solutions

The result is equivalent to the one in Section 3.1.3 with $\Omega _{k}\rightarrow -i{\tilde{\Omega }}^*_{k}$, we do not present it here again.

4 Conclusions

In conclusion, we have investigated breather, exp-rational and rational solutions in Gramian determinant for the KPESCS equation via two procedures of the KP hierarchy reduction. Each procedure generates two categories of breathers. Exp-rational solutions containing breathers and lumps can be constructed by long-wave limit technique to the breather solutions. Superposition patterns have been demonstrated for different choices of the parameters. By implementing the long-wave limit, we also obtain rational solutions that lead to the lumps. The“dark” and tetrapetalous-type breathers and lumps are addressed, which have been usually observed in coupled NLS systems and have never been studied in KP equation to our knowledge. Furthermore, such breathers or lumps are on the $x-y$ planes, which unlike the ones in coupled NLS systems. Under the condition $a=\alpha =0$, Solutions (9) equal to Solutions (25). Nevertheless, a does not affect the periods of the Breather-I and Breather-II according to Solutions (9), while $\alpha$ affects the periods of the Breather-III and Breather-IV according to Solutions (25). Interactions between the above hybrid waves is of fundamental importance in the understanding of high-dimensional nonlinear phenomena of extreme ocean waves and in other fields such as solid state physics and plasma physics.