1 Introduction

In the 1960s, Zabusky and Kruskal utilized the finite difference method to investigate the numerical solutions of the Korteweg-de Vries (KdV) equation and discovered that solitons possess the characteristic of elastic collisions property, where their shapes and velocities remain unchanged after collisions [1]. In addition to the KdV equation, solitons can be found in various other \((1+1)\)-dimensional continuous and discrete equations, such as the modified KdV equation, nonlinear Schrödinger equation, Burgers equation, sine-Gordon equation, and the nonlinear Schrödinger equations with non local characteristics [2,3,4,5,6]. These equations have found extensive applications in modeling nonlinear phenomena across diverse fields including mathematics, physics, optics, fluid dynamics, chemistry, biology, and more. Consequently, the investigation of solitons has emerged as a prominent and highly regarded research area [7,8,9,10,11]. With the advancement of soliton theory, researchers have discovered various types of soliton solutions, including line solitons, algebraic solitons, breathers, rogue waves, lumps, and peakon solutions, among others.

The significant contributions of soliton solutions to the understanding of nonlinear phenomena have led to extensive research on the mathematical structures and potential algebraic properties of soliton equations. The study of solutions for nonlinear soliton equations continues to be a thriving and ongoing subject of research. These investigations aim to deepen our knowledge of the rich dynamics and behaviors exhibited by solitons, thereby advancing our understanding of nonlinear systems and their applications.

Scholars have given various methods to derive soliton solutions, including inverse scattering transform [12], Darboux transform [13,14,15], Hirota bilinear method [16,17,18,19,20,21,22,23], Painlevé analysis [24, 25], and Riemann-Hilbert method [26]. Another effective method is the KP hierarchy reduction technique [27] based on Hirota’s bilinear method.

The Kyoto school has developed a technique that has proven successful in deriving soliton solutions for numerous integrable equations. This method has been widely applied and has yielded significant results in obtaining soliton solutions for a variety of integrable equations [28, 29]. Ohta and Yang have shown that this method is also applicable for obtaining rogue wave solutions of equations such as the nonlocal nonlinear Schrödinger equation (NLSE) [30], Davey-Stewartson (DS) equation [31], and Ablowitz-Ladik equation [32]. Furthermore, this method can be used to derive ( semi-) rational solutions of the third-type DS equation [33]. In 2018, Feng [34] further improved the KP hierarchy reduction method and obtained general soliton solutions for the NLSE with zero and nonzero boundary conditions using this method. Several soliton solutions for nonlocal integrable equations have also been constructed using similar approaches [35]. Compared to other methods for deriving soliton solutions and (semi-) rational solutions, the expressions obtained through the KP hierarchy method are more general and concise.

The aim of this paper is to make a study on the \((2+1)\)-dimensional KD equation [36] in the stratified shear flow, internal and shallow-water waves, plasmas and other fields, with the help of KP hierarchy reduction technique and symbolic computation

$$\begin{aligned}&U_{t}-U_{xxx}-6\beta U U_{x}+\frac{3}{2}\alpha ^{2}U^{2}U_{x}-3 V_{y}\nonumber \\ {}&\quad +3\alpha U_{x}V=0,\nonumber \\&U_{y}=V_{x}, \end{aligned}$$
(1.1)

where \(\alpha \) and \(\beta \) are arbitrary constants, U and V are the analytic functions of the variables x, y and t, the subscripts denote the partial differential derivatives. Eq. (1.1) is a highly significant nonlinear partial differential equation, which has been attributed to the renowned Russian mathematicians Konopelchenko and Dubrovsky. It serves as a powerful mathematical tool for analyzing and understanding a wide range of nonlinear wave phenomena, encompassing sound waves, water waves, and light waves. Its applications extend to the exploration of various wave-related phenomena, including wave propagation, interference, and scattering. In the domain of solid mechanics, Eq. (1.1) assumes a crucial role in characterizing the vibrations and waves exhibited by elastic bodies. Specifically, it facilitates the comprehensive description of the interplay between stress and strain within elastic materials, as well as the propagation behavior of elastic waves. In the field of optics, Eq. (1.1) proves instrumental in studying the intricate nonlinear propagation dynamics of light. It enables the analysis of phenomena such as self-focusing, self-phase modulation, and optical solitons, which bear significant implications for essential applications in optical fiber communications and lasers. Eq. (1.1) can be reduced to the Gardner [37,38,39,40], KP [41,42,43,44] and modified KP (mKP) [45, 46] in ocean dynamics, fluid mechanics and plasma physics. For example, (1) when \(V=0\), Eq. (1.1) are reduced to the Gardner equation, which is used to analyze the solitary waves, interfacial waves in the stratified shear flow and internal waves in a stratified ocean; (2) when \(\alpha =0\), Eq. (1.1) are reduced to the KP equation to model the shallow-water waves, propagation of weakly nonlinear dispersive long waves, etc; (3) when \(\beta =0\), Eq. (1.1) can be reduced to the mKP equation that can describe the evolution of solitary waves and the propagation of ionic acoustic and electromagnetic waves.

KD equation (1.1) have attracted a lot of attention because of its practical physical significance. In [47], abundant new exact non-travelling wave solutions of KD equation were presented based on improved tanh function method. Wang [48] propose the further improved F-expansion method to get Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions of KD equation. Multiple lump solutions of KD equation are obtained by the Hirota bilinear method [49]. Via the Sato theory and Hirota method, Yuan and Tian [50] present the soliton solutions in terms of the Gram determinant which can yield the bright, depression and kink solitons. Breather waves [51], Wronskian solutions [52] and Painlevé property [53] of KD equation were studied. Recently, Grinevich and Santini [54] conducted research and demonstrated that rogue wave solutions or Akhmediev breathers can be formed in the presence of non-zero backgrounds, represented by elliptic functions. They also investigated the breathers and rogue wave solutions in the context of the Hirota equation and the nonlinear Schrödinger equation with elliptic function backgrounds [55]. The construction of soliton solutions on periodic function backgrounds presents a more challenging task. In this paper, the aim is to construct (semi-) rational solutions, such as solitons, linear breathers and algebraic soliton solutions, of KD equation on the zero background and periodic background. Compared with asymptotic analysis of exponential-type solitons, asymptotic behavior of algebraic soliton solutions is much more difficult to be conducted. We analyze the asymptotic behaviors of algebraic solitons of KD equation.

The organization of this article is as follows. In Sect. 2, we provided a detailed exposition on the derivation and simplification of the solution to KD equations from a modified KP equation. Subsequently, in Sect. 3, we conducted an analysis of the dynamics of solitons in both constant and periodic backgrounds. The Sect. 4 focused on the investigation of breathing solutions. Moving forward, in Sect. 5, we explored various dynamic behaviors, including algebraic solitons, multiple rational solutions. Lastly, in Sect. 6, we summarized the research findings and engaged in a discussion.

2 The determinant structure

Through the dependent variable transformations

$$\begin{aligned} U=\frac{2}{\alpha }\left( \ln \frac{G}{F}\right) _{x},~~~~~~~~ V=\frac{2}{\alpha }\left( \ln \frac{G}{F}\right) _{y}, \end{aligned}$$
(2.1)

then the Eq. (1.1) can be transformed into the following bilinear forms

$$\begin{aligned}&(D_{t}-D_{x}^{3}+3 D_{x}D_{y}+6{\mathbb {A}}D_{y}) G\cdot F=0,\nonumber \\&(D_{y}+D_{x}^{2}+2{\mathbb {A}}D_{x}) G\cdot F=0, \end{aligned}$$
(2.2)

where \({\mathbb {A}}=-\frac{\beta }{\alpha }\), G and F are real functions of variables x, y and t, D is the Hirota derivative [16] defined by

$$\begin{aligned}&D_{x}^{m_{1}}D_{y}^{m_{2}}D_{t}^{m_{3}} g(x,y,t)\cdot f(x,y,t)\equiv \\&\left( \frac{\partial }{\partial x}-\frac{\partial }{\partial x^{'}}\right) ^{m_{1}} \left( \frac{\partial }{\partial y}-\frac{\partial }{\partial y^{'}}\right) ^{m_{2}}\\&\left( \frac{\partial }{\partial t}-\frac{\partial }{\partial t^{'}}\right) ^{m_{3}}g(x,y,t) f(x^{'},y^{'},t^{'})\mid _{x=x^{'},y=y^{'},t=t^{'}}, \end{aligned}$$

with f being a differentiable function of x, y, and t; g being a differentiable function of the formal variables \(x^{'}\), \(y^{'}\), and \(t^{'}\), while \(m_{1}\), \(m_{2}\), and \(m_{3}\) being the non-negative integers.

We start from the bilinear equations of the KP hierarchy [30]

$$\begin{aligned}&(D_{x}^{3}+3 D_{x}D_{y}-4D_{t}+3a(D_{x}^{2}+D_{y})+6a^{2}D_{x})\nonumber \\ {}&\quad \tau _{n+1}\cdot \tau _{n}=0,\nonumber \\&(D_{x}^{2}-D_{y}+2 a D_{x}) \tau _{n+1}\cdot \tau _{n}=0. \end{aligned}$$
(2.3)

Under the variable transformation \(x_{1}=x\), \(x_{2}=-y\), \(x_{-1}=4t\), \(G=\tau _{n+1}\) and \(F=\tau _{n}\), the modified KP equation (2.3) turns into bilinear KD equations (2.2)

$$\begin{aligned}&(D_{x_{1}}^{3}+3 D_{x_{1}}D_{x_{2}}-4D_{x_{-1}}+6{\mathbb {A}}D_{x_{2}}) \tau _{n+1}\cdot \tau _{n}=0,\nonumber \\&(D_{x_{1}}^{2}-D_{x_{2}}+2{\mathbb {A}} D_{x_{1}}) \tau _{n+1}\cdot \tau _{n}=0, \end{aligned}$$
(2.4)

and taking \(F=\tau _{0}\) and \(G=\tau _{1}\), the above bilinear equation become

$$\begin{aligned}&(D_{x_{1}}^{3}+3 D_{x_{1}}D_{x_{2}}-4D_{x_{-1}}+6{\mathbb {A}}D_{x_{2}}) g\cdot f=0,\nonumber \\&(D_{x_{1}}^{2}-D_{x_{2}}+2{\mathbb {A}} D_{x_{1}}) g\cdot f=0, \end{aligned}$$
(2.5)

in the algebraic solutions, the bilinear equations (2.5) are reduced to the bilinear equations (2.2). For simplicity of calculation, let parameter \(\alpha =2\) in the following.

2.1 Gram determinant solution for the bilinear equation

In this subsection, we first derive the Gram solutions for the higher-dimensional bilinear system. According to lemma 3.1 of [28], we get the following lemma 1:

Lemma 1

The KP hierarchy have been proved to have the Gram-type determinant solution

$$\begin{aligned} \tau _{n}=\det \limits _{1\le i,j \le N} (m_{ij}^{(n)}) \end{aligned}$$
(2.6)

where N is any positive integer. Which are satisfied the next differential and difference relations,

$$\begin{aligned} \partial _{x_{1}}m_{ij}^{(n)}&= \phi _{i}^{(n)}\varphi _{j}^{(n)},\\ \partial _{x_{2}}m_{ij}^{(n)}&= (\partial _{ x_{1}}\phi _{i}^{(n)})\varphi _{j}^{(n)}-\phi _{i}^{(n)}) (\partial _{ x_{1}}\varphi _{j}^{(n)}),\\ \partial _{x_{3}}m_{ij}^{(n)}&= (\partial _{ x_{1}}^{2}\phi _{i}^{(n)})\varphi _{j}^{(n)}+ \phi _{i}^{(n)} (\partial _{ x_{1}}^{2}\varphi _{j}^{(n)})\\&\quad - (\partial _{ x_{1}}\phi _{i}^{(n)})(\partial _{ x_{1}}\varphi _{i}^{(n)}), \\ m_{ij}^{(n+1)}&= m_{ij}^{(n)} +\phi _{i}^{(n)}\varphi _{j}^{(n+1)},\\ \phi _{i}^{(n+1)}&= (\partial _{x_{1}}-{\mathbb {A}}))\phi _{i}^{(n)},\\ \varphi _{i}^{(n-1)}&=- (\partial _{x_{1}}+{\mathbb {A}}))\varphi _{j}^{(n)},\\ \partial _{x_{2}} \phi _{i}^{(n)}&= \partial _{x_{1}}^{2}\phi _{i}^{(n)},\\ \partial _{x_{2}} \varphi _{j}^{(n)}&=-\partial _{x_{1}}^{2} \varphi _{j}^{(n)},\\ \partial _{x_{3}} \phi _{i}^{(n)}&= \partial _{x_{1}}^{3}\phi _{i}^{(n)},\\ \partial _{x_{3}} \varphi _{j}^{(n)}&=\partial _{x_{1}}^{3}\varphi _{j}^{(n)}, \end{aligned}$$

with \(m_{ij}^{(n)}\), \(\phi _{i}^{(n)}\) and \(\varphi _{j}^{(n)}\) as the functions of \(x_{1}\), \(x_{2}\) and \(x_{3}\), i and j as the integers, and the superscript “(n)” as a note for distinguishing the n-th power. Substituting elements \( m_{ij}^{(n)}\) into Gram determinant, we obtain N-soliton solutions for KD equations.

In order to construct soliton solutions and semi-rational soluions to the bilinear equations defined in (2.5), we choose the special \(\phi \) and \(\varphi \) functions

$$\begin{aligned} \phi _{i}^{(n)}&=(u_{i}-{\mathbb {A}})^{n} e^{\xi _{i}} \varphi _{j}^{(n)}=(-v_{j}+{\mathbb {A}})^{-n} e^{\eta _{j}} \end{aligned}$$
(2.7)
$$\begin{aligned} m_{ij}^{(n)}&=c_{j}\delta _{ij}+\int ^{x_{1}}\phi _{i}^{(n)} \varphi _{j}^{(n)} dx_{1}=c_{j}\delta _{ij}\nonumber \\&\quad +\frac{1}{u_{i}+v_{j}}\left( -\frac{u_{i}-{\mathbb {A}}}{v_{j}+{\mathbb {A}}}\right) ^{n} e^{\xi _{i}+\eta _{j}} \end{aligned}$$
(2.8)

where

$$\begin{aligned} \xi _{i}= & {} u_{i}x+u_{i}^{2}x_{2}+u_{i}^{3} x_{3}+\xi _{i0},\nonumber \\ \eta _{j}= & {} v_{j}x-v_{j}^{2}x_{2}+v_{j}^{3}x_{3}+\eta _{j0}, \end{aligned}$$
(2.9)

\(u_{i}\), \(v_{j}\) as the constants, \(\delta _{ij}\) as the Kronecker delta notation. By substituting the above elements \(m_{ij}^{(n)}\) into the Gram determinant, the n-soliton solution of the KD equation is obtained.

According Ref. [30], the differential operators \(A_{i}\) and \(B_{j}\) are introduced

$$\begin{aligned} A_{i}= & {} \mathop {\sum }\limits _{k=0}^{n_{i}} a_{ik} \left( u_{i}\partial _{u_{i}}\right) ^{n_{i}-k},\nonumber \\ B_{j}= & {} \mathop {\sum }\limits _{h=0}^{n_{j}} b_{jh} \left( v_{j}\partial _{v_{j}}\right) ^{n_{j}-h}, \end{aligned}$$
(2.10)

to generate rational solutions. Here \(a_{ik}\) and \(b_{jl}\) are arbitrary complex constants, and \(n_{i}\) and \(n_{j}\) are arbitrary positive integers. We apply the differential \(A_{i}\) and \(B_{j}\) to \(m_{ij}^{(n)}\) and denote

$$\begin{aligned}&{\widetilde{m}}_{ij}^{(n)}=A_{i}B_{j}m_{ij}^{(n)}\nonumber \\&=c_{j}\delta _{ij}+(-\frac{u_{i}-{\mathbb {A}}}{v_{j}+{\mathbb {A}}})e^{\xi _{i}+\eta _{j}}\mathop {\sum }\limits _{k=0}^{n_{i}} a_{ik} (u_{i}\partial _{u_{i}}\nonumber \\ {}&\quad +\frac{n u_{i}}{u_{i}-{\mathbb {A}}}+{\widetilde{\xi }}_{i})^{n_{i}-k}\nonumber \\&\mathop {\sum }\limits _{h=0}^{n_{j}} b_{jh} (v_{j}\partial _{v_{j}}-\frac{n v_{j}}{v_{j}+{\mathbb {A}}}+{\widetilde{\eta }}_{j} )^{n_{j}-h}\frac{1}{u_{i}+v_{j}}, \end{aligned}$$
(2.11)

where

$$\begin{aligned} {\widetilde{\xi }}_{i} = u_{i}x_{1}+2u_{i}^{2}x_{2}+3 u_{i}^{3} x_{3},{\widetilde{\eta }}_{j} = v_{j}x_{1}-2v_{j}^{2}x_{2}+3 v_{j}^{3} x_{3}. \end{aligned}$$

Since these operators apply only to the parameters \(u_{1}\) and \(v_{j}\), the determinant \( \tau _{n}=\det \limits _{1\le i,j \le N} (m_{ij}^{(n)})\) also solves bilinear equations (2.4).

When \(c_{j}\) is zero in equation (2.11), the determinant elements are polynomials in x, y, and t, such that \(\tau _{n}\) is a polynomial, resulting in a rational solution for the KD equation. When \(c_{j}\) is not zero, we can obtain semi-rational solutions expressed as a combination of exponential functions and polynomials. According to the above analysis, we have the following theorem.

Theorem 2.1

KD equations (1.1) adimts solutions transformations

$$\begin{aligned} U=\frac{2}{\alpha }\left( \ln \frac{\tau _{n+1}}{\tau _{n}}\right) _{x},V=\frac{2}{\alpha }\left( \ln \frac{\tau _{n+1}}{\tau _{n}}\right) _{y}, \end{aligned}$$
(2.12)

where the determinant \(\tau _{n}= \det \nolimits _{1\le i,j \le N} ({\widetilde{m}}_{ij}^{(n)})\) has elements

$$\begin{aligned}&{\widetilde{m}}_{ij}^{(n)}=c_{j}\delta _{ij}+(-\frac{u_{i}-{\mathbb {A}}}{v_{j}+{\mathbb {A}}})e^{\xi _{i}+\eta _{j}}\nonumber \\&\mathop {\sum }\limits _{k=0}^{n_{i}} a_{ik} (u_{i}\partial _{u_{i}}+\frac{n u_{i}}{u_{i}-{\mathbb {A}}}+{\widetilde{\xi }}_{i})^{n_{i}-k}\nonumber \\&\mathop {\sum }\limits _{h=0}^{n_{j}} b_{jh} (v_{j}\partial _{v_{j}}-\frac{n v_{j}}{v_{j}+{\mathbb {A}}}+{\widetilde{\eta }}_{j} )^{n_{j}-h}\frac{1}{u_{i}+v_{j}}, \end{aligned}$$
(2.13)

and

$$\begin{aligned} {\widetilde{\xi }}_{i}&= u_{i}x_{1}+2u_{i}^{2}x_{2}+3 u_{i}^{3} x_{3} ,\\ {\widetilde{\eta }}_{j}&=v_{j}x_{1}-2v_{j}^{2}x_{2}+3 v_{j}^{3} x_{3} ,\\ \xi _{i}&=u_{i}x+u_{i}^{2}x_{2}+u_{i}^{3} x_{3}+\xi _{i0} ,\\ \eta _{j}&=v_{j}x-v_{j}^{2}x_{2}+v_{j}^{3}x_{3}+\eta _{j0} {,} \end{aligned}$$

where \(u_{i}\), \(v_{j}\), and \(c_{j}\) are arbitrary complex constants, and \(n_{i}\) and \(n_{j}\) are the arbitrary positive integers. By choosing different types of functions of \(m_{ij}^{(n)}\), \(\phi _{i} \), \(\varphi _{j} \), and \({\widetilde{m}}_{ij}^{(n)}\), the soliton, breather, and (semi-)rational solutions to KD equations (1.1) can be constructed.

3 Dynamics of the soliton solutions

In this section, we construct and analyze the dynamics of soliton solutions for the KD equation (1.1) separately with constant and periodic background.

3.1 One-soliton solutions

Setting \(N=1\) in equation (2.7) and get the first order determinant

$$\begin{aligned} m_{ij}^{(n)}&=c_{j} \delta _{ij}+\frac{1}{u_{i}+v_{j}}\left( -\frac{u_{i}-{\mathbb {A}}}{v_{j}+{\mathbb {A}}}\right) ^{n} e^{\xi _{i}+\eta _{j}},\nonumber \\ \xi _{1}+\eta _{1}&=(u_{1}+v_{1}) x+(v_{1}^{2}-u_{1}^{2})y\nonumber \\&\quad +4 (u_{1}^{3}+v_{1}^{3}) t+\xi _{10} +\eta _{10} , \end{aligned}$$
(3.1)

here \(u_{1}\), \(v_{1}\), \(c_{1}\), \(\xi _{10}\), \(\eta _{10} \) are all parameters.

Case 1 On the zero background

If we take \(c_{1}=1\), \(u_{1}\), \(v_{1}\), \(\xi _{10}\), and \(\eta _{10} \) are arbitrary constants, functions F and G have the form

$$\begin{aligned}&F=1+\frac{1}{v_{1}+u_{1}} e^{\xi _{1}+\eta _{1}},\nonumber \\&G=1+\left( -\frac{u_{1}-{\mathbb {A}}}{v_{1}+{\mathbb {A}}}\right) \frac{1}{v_{1}+u_{1}} e^{\xi _{1}+\eta _{1}}, \end{aligned}$$
(3.2)

and the potential u and v are the form

$$\begin{aligned}&U=\frac{2}{\alpha }\left( \ln \frac{g}{f}\right) _{x}=-\frac{(u_{1}+v_{1})^{3}e^{ \xi _{1}+\eta _{1}}}{(v_{1}+{\mathbb {A}})(u_{1}+v_{1} + e^{ \xi _{1}+\eta _{1}})^{2}},\nonumber \\&V=\frac{2}{\alpha }\left( \ln \frac{g}{f}\right) _{y}=\frac{(u_{1}+v_{1})^{3}e^{ \xi _{1}+\eta _{1}}(u_{1}-v_{1})}{(v_{1}+{\mathbb {A}})(u_{1}+v_{1}+ e^{ \xi _{1}+\eta _{1}})^{2}}. \end{aligned}$$
(3.3)

Plots of one-bright-soliton solution and one-dark-soliton solution are shown in Fig. 1.

Fig. 1
figure 1

One-soliton solution u and v (3.3) with parameters \(c=1\), \({\mathbb {A}}=-1\), \(u_{1}=1\), \(v_{1}=\frac{1}{2}\), \(\xi _{10}=\eta _{10}=0\), \(t=0\)

Full size image
Fig. 2
figure 2

One periodic solution u (3.5) with suitable parameters \(c=1\), \(\vartheta =\frac{1}{2}\), \(\theta =1\), \({\mathbb {A}}=\frac{1}{2}\), \(\xi _{10}=\eta _{10}=0\), \(t=0\)

Full size image

Case 2 On the periodic background

If we take \(c_{1}=\text {i} c\), \(u_{1}=\vartheta +\text {i}\theta \) and \(v_{1}=-u_{1}^{*}\), where \(\vartheta \), \(\theta \), c are real numbers and \(c \theta \ne 0\), then functions F and G have the form

$$\begin{aligned} F&=c_{1}+\frac{1}{v_{1}+u_{1}} e^{\xi _{1}+\eta _{1}}\nonumber \\&=\text {i} c-\frac{i}{2 \theta } e^{2 \text {i} \theta x-4 \text {i} \vartheta \theta y +8 (\vartheta +\text {i} \theta )^{3} t+\xi _{10}+\eta _{10}},\\ G&=c_{1}+\left( -\frac{u_{1}-{\mathbb {A}}}{v_{1}+{\mathbb {A}}}\right) \frac{1}{v_{1}+u_{1}} e^{\xi _{1}+\eta _{1}}\nonumber \\&=\text {i} c+\frac{\text {i} (\vartheta + \text {i} \theta -{\mathbb {A}})}{2 (-\vartheta +\text {i} \theta +{\mathbb {A}}) \theta }\nonumber \\ {}&\quad e^{2 \text {i} \theta x-4 \text {i} \vartheta \theta y+8 (\vartheta +\text {i} \theta )^{3} t+\xi _{10}+\eta _{10}},\nonumber \end{aligned}$$
(3.4)

substituting (3.4) into (2.1), we obtain the one-soliton solution

$$\begin{aligned} U&=\frac{2}{\alpha }\left( \ln \frac{G}{F}\right) _{x}\nonumber \\&=\frac{8 \theta ^{3} c e^{2 \text {i} \theta x-4 \text {i} \vartheta \theta y +8 (\vartheta +\text {i} \theta )^{3} t+\xi _{10}+\eta _{10}}}{(-\vartheta +\text {i} \theta +{\mathbb {A}})(-2 c \theta +e^{2\text {i} \theta x-4 \text {i} \vartheta \theta y+8 (\vartheta +\text {i} \theta )^{3} t+\xi _{10}+\eta _{10}})^{2}} \end{aligned}$$
(3.5)

when \(u_{1}\) is a pure imaginary number, the solutions (3.5) are periodic in both x and t with periods \(-\frac{\text {i}}{\theta }\) and \(\frac{\text {i} \pi }{4 \theta ^{3}}\), respectively. It will be seen later that it plays an important role when constructing soliton and (semi-) rational solutions of KD equation (1.1) in the periodic context. Plot of this periodic one-soliton solutions is depicted in Fig. 2. In this paper, regular solutions (3.5) provide the periodic background for higher-order soliton solution (see Fig. 2).

Fig. 3
figure 3

Two soliton solution of U with suitable parameters \({\mathbb {A}}=-1\), \(u_{1}=1\), \(u_{2}=3\), \(v_{1}=2\), \(v_{2}=4\), \(t=0\)

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3.2 Two-soliton solutions

Case 1 On the constant background

In order to derive two-soliton solution, by taking \(N=2\). For the convenience of calculation, we choose \(c_{1}=c_{2}=1\). The F and G in \(2\times 2\) determinant

$$\begin{aligned} F&=\left| \begin{array}{ccc} 1+\frac{1}{u_{1}+v_{1}} e^{\xi _{1}+\eta _{1}} &{}\frac{1}{u_{1}+v_{2}} e^{\xi _{1}+\eta _{2}} \\ \frac{1}{u_{2}+v_{1}} e^{\xi _{2}+\eta _{1}} &{} 1+\frac{1}{u_{2}+v_{2}} e^{\xi _{2}+\eta _{2}} \\ \end{array}\right| \nonumber \\&=1+\frac{1}{v_{1}+u_{1}} e^{\varsigma _{1}}+\frac{1}{v_{2}+u_{2}} e^{\varsigma _{2}}\nonumber \\&\quad +\frac{(u_{1}-u_{2})(v_{1}-v_{2})}{(v_{1}+u_{1})(v_{2}+u_{1})(v_{1}+u_{2})(v_{2}+u_{2})} e^{\varsigma _{1}+\varsigma _{2}}, \end{aligned}$$
(3.6)
$$\begin{aligned} G&=\left| \begin{array}{ccc} 1+\frac{1}{u_{1}+v_{1}} (-\frac{u_{1}-{\mathbb {A}}}{v_{1}+{\mathbb {A}}})e^{\xi _{1}+\eta _{1}} &{}\frac{1}{u_{1}+v_{2}} (-\frac{u_{1}-{\mathbb {A}}}{v_{2}+{\mathbb {A}}}) e^{\xi _{1}+\eta _{2}} \nonumber \\ \frac{1}{u_{2}+v_{1}} (-\frac{u_{2}-{\mathbb {A}}}{v_{1}+{\mathbb {A}}})e^{\xi _{2}+\eta _{1}} &{} 1+\frac{1}{u_{2}+v_{2}} (-\frac{u_{2}-{\mathbb {A}}}{v_{2}+{\mathbb {A}}})e^{\xi _{2}+\eta _{2}} \\ \end{array}\right| \\&=1-\frac{u_{1}-{\mathbb {A}}}{(v_{1}+{\mathbb {A}})(v_{1}+u_{1})} e^{\varsigma _{1}}-\frac{u_{2}-{\mathbb {A}}}{(v_{2}+{\mathbb {A}})(v_{2}+u_{2})} e^{\varsigma _{2}}\nonumber \\&\quad +\frac{(u_{1}\!-\!{\mathbb {A}})(u_{2}\!-\!{\mathbb {A}})(u_{1}\!-\!u_{2})(v_{1}\!-\! v_{2})}{(v_{1}\!+\!{\mathbb {A}})(v_{2}\!+\!{\mathbb {A}})(u_{1}\!+\!v_{1})(u_{1}\!+\!v_{2})(u_{2}\!+\!v_{1})(u_{2}\!+\!v_{2})} e^{\varsigma _{1}+\varsigma _{2}}, \end{aligned}$$
(3.7)

in which

$$\begin{aligned} \varsigma _{1}&=\xi _{1}+\eta _{1}= (u_{1}+v_{1}) x+(v_{1}^{2}-u_{1}^{2})y\\&\quad +4 (u_{1}^{3}+v_{1}^{3}) t+\xi _{10} +\eta _{10} ,\\ \varsigma _{2}&=\xi _{2}+\eta _{2}= (u_{2}+v_{2}) x+(v_{2}^{2}-u_{2}^{2})y\\&\quad +4 (u_{2}^{3}+v_{2}^{3}) t+\xi _{20} +\eta _{20}. \end{aligned}$$

After simple algebra calculation, we obtain two solitons. We analyzed the collision of two solitons and provided the asymptotic behavior of the collision when \(t=0\). The two-soliton solution of U is expressed as

$$\begin{aligned} U=\frac{-N_{112}e^{2\zeta _{1}+\zeta _{2}}-N_{122}e^{\zeta _{1}+2\zeta _{2}} -N_{12} e^{\zeta _{1}+\zeta _{2}}- N_{1} e^{\zeta _{1}}- N_{2} e^{\zeta _{2}}}{D_{1}e^{\zeta _{1}}+D_{2}e^{\zeta _{2}}+D_{22}e^{2\zeta _{2}} +D_{12}e^{\zeta _{1}+\zeta _{2}}+D_{122}e^{\zeta _{1}+2\zeta _{2}}+D_{112}e^{2\zeta _{1}+\zeta _{2}} +D_{1122}e^{2\zeta _{1}+2\zeta _{2}}+D_{11}e^{2\zeta _{1}}+1}, \end{aligned}$$
(3.8)

where

$$\begin{aligned}&N_{1}=\frac{-u_{1}-v_{1}}{v_{1}+{\mathbb {A}}}, N_{2}=\frac{u_{2}+v_{2}}{v_{2}+{\mathbb {A}}},\\&N_{112}=-\frac{(v_{1}-v_{2})(u_{1}-u_{2})(-u_{1}+{\mathbb {A}})(u_{2}+v_{2})}{(u_{2}+v_{1})(u_{1}+v_{2})(v_{2}+{\mathbb {A}})(v_{1}+{\mathbb {A}})(u_{1}+v_{1})^{2}},\\&N_{122}=-\frac{(v_{1}-v_{2})(u_{1}-u_{2})(-u_{2}+{\mathbb {A}})(u_{1}+v_{1})}{(u_{2}+v_{1})(u_{1}+v_{2})(v_{2}+{\mathbb {A}})(v_{1}+{\mathbb {A}})(u_{2}+v_{2})^{2}},\\&N_{12}=\frac{(-u_{1}+{\mathbb {A}})(u_{1}-u_{2}+v_{1}-v_{2})}{(u_{1}+v_{1})(v_{1}+{\mathbb {A}})(u_{2}+v_{2})}\\&\quad +\frac{(u_{1}\!+\!u_{2}\!+\!v_{1}\!+\!v_{2})(u_{1}\!-\!u_{2})(-u_{2}\!+\!{\mathbb {A}})(-u_{1}\!+\!{\mathbb {A}})(v_{1}-v_{2})}{(u_{1}+v_{1})(v_{1} +{\mathbb {A}})(u_{2}+v_{2})(v_{2}+{\mathbb {A}})(u_{1}+v_{2})(u_{2}+v_{1})}\\&\quad -\frac{(-u_{2}+{\mathbb {A}})(u_{1}-u_{2}+v_{1}-v_{2})}{(u_{2}+v_{2})(v_{2}+{\mathbb {A}})(u_{1}+v_{1})}\\&\quad -\frac{(u_{1}+u_{2}+v_{1}+v_{2})(v_{1}-v_{2})(u_{1}-u_{2})}{(u_{1}+v_{1})(u_{2}+v_{2})(u_{1}+v_{2})(u_{2}+v_{1})},\\&D_{1}=\frac{2{\mathbb {A}}-u_{1}+v_{1}}{(u_{1}+v_{1})(v_{1}+{\mathbb {A}})},\quad D_{2}=\frac{2{\mathbb {A}}-u_{2}+v_{2}}{(u_{2}+v_{2})(v_{2}+{\mathbb {A}})},\\&D_{22}=\frac{-u_{2}+{\mathbb {A}}}{(u_{2}+v_{2})^{2}(v_{2}+{\mathbb {A}})},\quad D_{11}=\frac{-u_{1}+{\mathbb {A}}}{(u_{1}+v_{1})^{2}(v_{1}+{\mathbb {A}})},\\&D_{122}=\frac{2{\mathbb {A}}-u_{2}+v_{2}}{(u_{2}+v_{2})(v_{2}+{\mathbb {A}})},\quad D_{112}=\frac{-u_{2}+{\mathbb {A}}}{(u_{2}+v_{2})^{2}(v_{2}+{\mathbb {A}})},\\&D_{12}= \frac{(u_{1}-u_{2})(-u_{2}+{\mathbb {A}})(-u_{1}+{\mathbb {A}})(v_{1}-v_{2})}{(u_{1}+v_{1})(v_{1}+{\mathbb {A}})(u_{2}+v_{2})(v_{2}+{\mathbb {A}})(u_{1}+v_{2})(u_{2}+v_{1})}\\&\quad +\frac{(-u_{2}+{\mathbb {A}})}{(u_{2}+v_{2})(v_{2}+{\mathbb {A}})(u_{1}+v_{1})}\\&\quad +\frac{(-u_{1}+{\mathbb {A}})}{(u_{1}+v_{1})(v_{1}+{\mathbb {A}})(u_{2}+v_{2})}\\&\quad +\frac{(v_{1}-v_{2})(u_{1}-u_{2})}{(u_{1}+v_{1})(u_{2}+v_{2})(u_{1}+v_{2})(u_{2}+v_{1})}, \end{aligned}$$
$$\begin{aligned}&D_{1122}=\frac{(u_{1}-u_{2})^{2}(-u_{1}+{\mathbb {A}})(-u_{2}+{\mathbb {A}})(v_{1}-v_{2})^{2}}{(u_{1}\!+\!v_{1})^{2}(v_{1}\!+\!{\mathbb {A}}) (u_{2}\!+\!v_{2})^{2}(v_{2}\!+\!{\mathbb {A}})(u_{1}\!+\!v_{2})^{2}(u_{2}\!+\!v_{1})^{2}}. \end{aligned}$$
Fig. 4
figure 4

Two soliton solution of U with suitable parameters \({\mathbb {A}}=-1\), \(u_{1}=1\), \(u_{2}=\frac{3}{2}\), \(v_{1}=2\), \(v_{2}=4\), \(t=0\)

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Fig. 5
figure 5

Two soliton solution of U with suitable parameters \({\mathbb {A}}=-1\), \(u_{1}=1\), \(u_{2}=3\), \(v_{1}=-\frac{1}{3}\), \(v_{2}=-\frac{9}{4}\), \(t=0\)

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Figures 3, 4, and 5 respectively depict the interaction and corresponding density of two solitons under different parameters in a zero background. Figures 3 and 4 illustrate the interaction between a bright soliton and a dark soliton for different values of parameter \(u_{2}\). Figure 5 shows the interaction between two bright solitons. We define the left-moving soliton as soliton 1, located on the plane \(\zeta _{1} = 0\), and the right-moving soliton as soliton 2, located on the plane \(\zeta _{2} = 0\). Subsequently, we present the asymptotic forms of the solutions for these two solitons.

We have the following asymptotic forms for the two-soliton solutions.

  • Before collision (\(y\rightarrow -\infty \))

    Soliton 1 (\(\zeta _{1}\approx 0, \zeta _{2}\rightarrow +\infty \))

    $$\begin{aligned}&U\rightarrow 2\frac{-N_{122}e^{\zeta _{1}}}{D_{22}+D_{122}e^{\zeta _{1}}+D_{1122}e^{2\zeta _{1}}}\\&\quad =-2 \frac{(u_{1}+v_{1})^{2}}{(2{\mathbb {A}}-u_{1}+v_{1})+2\sqrt{(-u_{1}+{\mathbb {A}})(v_{1}+{\mathbb {A}}) cosh(\varsigma _{1}-\varrho _{1})}},\\&e^{\varrho _{1}}=\frac{(u_{2}+v_{1})(u_{1}+v_{2})(u_{1}+v_{1})}{(u_{1}-u_{2})(v_{1}-v_{2})}\sqrt{\frac{v_{1}+{\mathbb {A}}}{-u_{1}+{\mathbb {A}}}}. \end{aligned}$$

    Soliton 2 (\(\zeta _{2}\approx 0, \zeta _{1}\rightarrow -\infty \))

    $$\begin{aligned}&U\rightarrow 2\frac{- N_{2} e^{\zeta _{2}}}{D_{2}e^{\zeta _{2}}+D_{22}e^{2\zeta _{2}}+1}\\&\quad =-2 \frac{(u_{2}+v_{2})^{2}}{(2{\mathbb {A}}-u_{2}+v_{2})+2\sqrt{(-u_{2}+{\mathbb {A}})(v_{2}+{\mathbb {A}}) cosh(\varsigma _{2}-\varrho _{2})}},\\&e^{\varrho _{2}}=(u_{2}+v_{2})\sqrt{\frac{v_{2}+{\mathbb {A}}}{-u_{2}+{\mathbb {A}}}}. \end{aligned}$$

  • After collision (\(y\rightarrow +\infty \))

    Soliton 1 (\(\zeta _{1}\approx 0, \zeta _{2}\rightarrow -\infty \))

    $$\begin{aligned}&U\rightarrow 2\frac{- N_{1} e^{\zeta _{1}}}{D_{1}e^{\zeta _{1}}++D_{11}e^{2\zeta _{1}}+1}\\&\quad =-2 \frac{(u_{1}+v_{1})^{2}}{(2{\mathbb {A}}-u_{1}+v_{1})+2\sqrt{(-u_{1}+{\mathbb {A}})(v_{1}+{\mathbb {A}}) cosh(\varsigma _{1}-\varrho _{3})}},\\&e^{\varrho _{3}}=(u_{1}+v_{1})\sqrt{\frac{v_{1}+{\mathbb {A}}}{-u_{1}+{\mathbb {A}}}}. \end{aligned}$$
    Fig. 6
    figure 6

    Two soliton solution of U on the periodic background with suitable parameters \({\mathbb {A}}=-1\), \(u_{1}=1\), \(u_{2}=3\), \(v_{1}=-\frac{1}{3}\), \(v_{2}=-2\), \(t=0\), \(\vartheta =-1\), \(\theta =1\), \(c_{1}=c_{2}=1\), \(c_{3}=\text {i}\), \(\xi _{i}=\eta _{j}=0~(i=1,2,3)\)

    Full size image

    Soliton 2 (\(\zeta _{2}\approx 0, \zeta _{1}\rightarrow +\infty \))

    $$\begin{aligned}&U\rightarrow 2\frac{-N_{112}e^{\zeta _{2}}}{D_{112}e^{\zeta _{2}}+D_{1122}e^{2\zeta _{2}}+D_{11}}\\&\quad =-2 \frac{(u_{2}+v_{2})^{2}}{(2{\mathbb {A}}-u_{2}+v_{2})+2\sqrt{(-u_{2}+{\mathbb {A}})(v_{2}+{\mathbb {A}})}cosh(\varsigma _{2}-\varrho _{4})},\\&e^{\varrho _{4}}=\frac{(u_{2}+v_{1})(u_{1}+v_{2})(u_{2}+v_{2})}{(u_{1}-u_{2})(v_{1}-v_{2})}\sqrt{\frac{v_{2}+{\mathbb {A}}}{-u_{2}+{\mathbb {A}}}}. \end{aligned}$$

According to the analysis of asymptotic behavior, when two solitons collide, their interaction is completely elastic. This means that after the collision, the shape and velocity of the two solitary waves remain unchanged. In other words, they retain their original waveform and motion state without any deformation or energy loss.

Case 2 On the periodic background

Taking \(N=3\) and \(v_{3}=-u_{3}^{*}\) in Eq.  (2.9) to construct two-soiton solutions on the periodic background. The tau functions F and G have the next determinant expressions

$$\begin{aligned} F= & {} \det \limits _{1\le i,j \le 3} (m_{ij}^{(0)}),\nonumber \\ G= & {} \det \limits _{1\le i,j \le 3} (m_{ij}^{(1)}) \end{aligned}$$
(3.9)

where the entries are defined by

$$\begin{aligned}&m_{ij}^{(n)}=c_{j}\delta _{ij}+\frac{1}{u_{i}+v_{j}}\left( -\frac{u_{i}-{\mathbb {A}}}{v_{j}+{\mathbb {A}}}\right) ^{n} e^{\xi _{i}+\eta _{j}},~~~n=0,1, \end{aligned}$$
(3.10)

in which \(u_{3}=\vartheta +\text {i}\theta \), \(v_{3}=\text {i}p_{3}^{*}\), \(c_{1}=c_{2}=1\), \(\vartheta =a\), \( \xi _{i}+\eta _{i} =\xi _{i}+\eta _{i}= (u_{i}+v_{i}) x+(q_{i}^{2}-u_{i}^{2})y+4 (v_{i}^{3}+v_{i}^{3}) t+\xi _{i0} +\eta _{i0}. ~~i=1,2,\) \( \xi _{3}+\eta _{3} =-8 \text {i} \theta ^{3} t+(24 \text {i} t \vartheta ^{2}-4 \text {i} y \vartheta + 2 \text {i} x)\theta +\xi _{30}+\eta _{30} \).

Parameters \(u_{1}\), \(u_{2}\), \(v_{1}\), \(v_{2}\), \(\vartheta \), \(\theta \), and c are all real numbers. Substituting the functions F and G into Eq. (2.1), we obtain the two soliton solution in a periodic background. Figure 6 depicts the interaction of two solitons against the periodic background and their corresponding densities.

4 Line breather solutions

In this section, we provide linear breather solutions and their propagation evolution diagrams by selecting different values of N and parameters.

Case 1 On the constant background

To construct the traveling breather solutions of the KD equation under a constant background, we first set the size of the determinant to be even in Theorem 2.1. Assuming \(N=2\), \({\tilde{u}}_{1}={\tilde{v}}_{2}=\text {i}u\), \({\tilde{v}}_{1}={\tilde{u}}_{2}=\text {i}v\), \(c_{1}=\vartheta + \text {i} \theta \), \(c_{2}=-c_{1}^{*}\), and choosing \({\tilde{u}}_{i}={\tilde{v}}_{i}\) are pure imaginary, u, v, \(\vartheta \), and \(\theta \) are real. In particular, \(\vartheta \) and \(\theta \) are not equal to zero. Then, U and V are rewritten as follows

$$\begin{aligned} U&=\frac{2}{\alpha }\left( \ln \frac{G}{F}\right) _{x},\nonumber \\ V&=\frac{2}{\alpha }\left( \ln \frac{G}{F}\right) _{y}, \end{aligned}$$
(4.1)

in which

$$\begin{aligned} F&=\left| \begin{array}{ccc} (\vartheta + \text {i} \theta ) e^{-\xi _{1}-\eta _{1}} +\frac{1}{\text {i} u+\text {i} v} &{}-\frac{\text {i}}{2 u} \\ -\frac{\text {i}}{2v} &{} (-\vartheta +\text {i} \theta ) e^{-\xi _{2}-\eta _{2}}+\frac{1}{\text {i} u+\text {i} v} \\ \end{array}\right| \nonumber \\&=\frac{(u-v)^{2}}{4 (u+v)^{2} u v} e^{\xi _{1}+\eta _{1}+\xi _{2}+\eta _{2}}+\frac{\text {i} \vartheta +\theta }{p+q}e^{\xi _{1}+\eta _{1}}\nonumber \\&\quad +\frac{-\text {i} \vartheta +\theta }{u+v}e^{\xi _{2}+\eta _{2}}-\vartheta ^{2}-\theta ^{2}, \end{aligned}$$
(4.2)
$$\begin{aligned} G&=\left| \begin{array}{ccc} (\vartheta + \text {i} \theta ) e^{-\xi _{1}-\eta _{1}} -\frac{u}{(\text {i} u+\text {i} v) v}&{}\frac{ \text {i} }{2 u} \\ \frac{\text {i}}{2 v} &{} (-\vartheta + \text {i} \theta ) e^{-\xi _{2}-\eta _{2}} -\frac{v}{(\text {i} u+\text {i} v) u} \\ \end{array}\right| \nonumber \\&=\frac{(\text {i} au +\text {i} av -{\mathbb {A}}^{2}+uv)(u-v)^{2}}{4 (u+v)^{2}(\text {i} v+{\mathbb {A}})(\text {i}u +{\mathbb {A}})u v}e^{\xi _{1}+\eta _{1}+\xi _{2}+\eta _{2}},\nonumber \\&\quad + \frac{(\text {i} \vartheta +\theta )({\mathbb {A}}^{2}+u^{2})}{ (u+v)(\text {i} v+{\mathbb {A}})(\text {i}u +{\mathbb {A}})}e^{\xi _{1}+\eta _{1}}\nonumber \\&\quad -\frac{(\text {i} \vartheta -\theta )({\mathbb {A}}^{2}+v^{2})}{ (u+v)(\text {i} v+{\mathbb {A}})(\text {i} u +{\mathbb {A}})}e^{\xi _{2}+\eta _{2}}\nonumber \\&\quad -\frac{(\text {i} {\mathbb {A}}u+\text {i} {\mathbb {A}} v+{\mathbb {A}}^{2}-u v)(\vartheta ^{2}+\theta ^{2})}{(\text {i} v+{\mathbb {A}})(\text {i} u +{\mathbb {A}})}, \end{aligned}$$
(4.3)

and

$$\begin{aligned}&\xi _{1}+\eta _{1}=-4\text {i}(u^{3}+v^{3})t+\text {i}(u+v)x\\&\quad +(u^{2}-v^{2})y+\xi _{10}+\eta _{10},\\&\xi _{2}+\eta _{2}=-4\text {i}(u^{3}+v^{3})t+\text {i}(u+v)x\\&\quad +(-u^{2}+v^{2})y+\xi _{20}+\eta _{20}. \end{aligned}$$

Line breather solution of U at \(t = 0\) on the constant background is depicted in Fig. 7.

Fig. 7
figure 7

Line breather solution u on the constant background with suitable parameters \(\vartheta =0\), \(\theta =-2\), \(u=\frac{1}{10}\), \(v=\frac{2}{5}\), \(t=0\), \({\mathbb {A}}=1\), \(\xi _{i0}=\eta _{j0}=0~(i,j=1,2)\)

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Case 2 On the periodic background

In order to derive line breathing solutions on a periodic background, we impose that the determinant in Theorem 2.1 should be an odd number. For simplicity, let’s assume the determinant size is \(N = 3\). Assuming \(N=3\), \({\tilde{u}}_{1}={\tilde{v}}_{2}=\text {i} u\), \({\tilde{v}}_{1}={\tilde{u}}_{2}=\text {i} v\), \({\tilde{u}}_{3}={\tilde{v}}_{3}=\text {i} m\), \(c_{1}=\vartheta +\text {i} \theta \), \(c_{2}=-c_{1}^{*}\), we can obtain line breathing solutions on the periodic background with the tau function

$$\begin{aligned}&F=\left| \begin{array}{cccc} (\vartheta + \text {i} \theta ) e^{-\xi _{1}-\eta _{1}} +\frac{1}{\text {i} p+\text {i} q} &{}-\frac{\text {i}}{2 p} &{}\frac{1}{\text {i} p+\text {i} m} \\ -\frac{\text {i}}{2q} &{} (-\vartheta + i \theta ) e^{-\xi _{2}-\eta _{2}}+\frac{1}{\text {i} p+\text {i} q} &{}\frac{1}{\text {i} q+\text {i} m} \\ \frac{1}{\text {i} q+\text {i} m} &{}\frac{1}{\text {i} p+\text {i} m} &{} (-\vartheta + \text {i} \theta ) e^{-\xi _{3}-\eta _{3}}+\frac{\text {i}}{2 m} \\ \end{array}\right| , \end{aligned}$$
(4.4)
$$\begin{aligned}&G=\left| \begin{array}{ccc} (\vartheta + \text {i} \theta ) e^{-\xi _{1}-\eta _{1}} -\frac{p}{(\text {i} p+\text {i} q) q}&{}\frac{\text {i}}{2 p}&{}-\frac{p}{(\text {i} p+\text {i} m) m} \\ \frac{\text {i}}{2 q} &{} (-\vartheta + i \theta ) e^{-\xi _{2}-\eta _{2}} -\frac{q}{(\text {i} p+\text {i} q) p} &{}-\frac{q}{(\text {i} q+\text {i} m) m}\\ -\frac{m}{(\text {i} q+\text {i} m) q} &{}-\frac{m}{(\text {i} p+\text {i} m) p} &{} (-\vartheta + \text {i} \theta ) e^{-\xi _{3}-\eta _{3}} +\frac{\text {i}}{2 m} \\ \end{array}\right| , \end{aligned}$$
(4.5)

then we obtain the line breather solution on the periodic background and through the graphic display their transmission dynamics behavior.

Fig. 8
figure 8

Line breather solution u on the periodic background with suitable parameters \(\vartheta =0\), \(\theta =-1\), \(p=\frac{1}{10}\), \(q=\frac{2}{5}\), \(t=0\), \({\mathbb {A}}=1\), \(m=1\), \(c_{3}=-\text {i}\), \(\xi _{i0}=\eta _{j0}=0~(i,j=1,2,3)\)

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Line breather solution of U on the constant background is depicted in Fig. 8 with suitable parameters. Through Figs. 7 and 8, we find that the amplitude of the line respirator may be higher than that on a constant background due to the appearance of the periodic background.

5 Dynamics of the rational solutions

5.1 Fundamental algebraic solitons

Taking \(N=1\), \(n=0\), \(c_{1}=0\) in Theorem 2.1. Then, the KD equation (1.1) has the solution (2.1) with

$$\begin{aligned} F&={\widetilde{m}}_{11}^{(0)}=\frac{1}{u_{1}+v_{1}} \left[ ({\widetilde{\xi }}_{1}+c_{11}-\frac{u_{1}}{u_{1}+v_{1}}) ({\widetilde{\eta }}_{1}+d_{11}\right. \nonumber \\&\quad \left. -\frac{v_{1}}{u_{1}+v_{1}}) +\frac{u_{1}v_{1}}{(u_{1}+v_{1})^{2}}) \right] e^{\xi _{1}+\eta _{1}}, \end{aligned}$$
(5.1)
$$\begin{aligned} G&={\widetilde{m}}_{11}^{(1)}=-\frac{u_{1}-{\mathbb {A}}}{v_{1}+{\mathbb {A}}}\frac{1}{u_{1}+v_{1}}\nonumber \\&\quad \times \left[ \left( \frac{u_{1}}{u_{1}-{\mathbb {A}}}+{\widetilde{\xi }}_{1}+c_{11}-\frac{u_{1}}{u_{1}+v_{1}}\right) \right. \nonumber \\&\quad \times \left. \left( -\frac{v_{1}}{v_{1}+{\mathbb {A}}}+{\widetilde{\eta }}_{1} +d_{11}-\frac{v_{1}}{u_{1}+v_{1}}\right) \right. \nonumber \\&\quad \left. +\frac{u_{1}v_{1}}{(u_{1}+v_{1})^{2}}\right] e^{\xi _{1}+\eta _{1}}. \end{aligned}$$
(5.2)

where

$$\begin{aligned}&{\widetilde{\xi }}_{1} =u_{1}x-2u_{1}^{2}y+12u_{1}^{3} t , \\&{\widetilde{\eta }}_{1} =v_{1}x+2v_{1}^{2}y+12v_{1}^{3} t , \end{aligned}$$

Here, choosing parameters \(c_{11}=\text {i} c \), \(d_{11}=\text {i} d \) and \({\widetilde{u}}_{1}\), \({\widetilde{v}}_{1}\), \({\mathbb {A}}\) as real. After a simple calculation, we obtain

$$\begin{aligned} |F|&= \frac{1}{u_{1}+v_{1}} \left| \left( {\widetilde{\xi }}_{1}+\text {i} c -\frac{u_{1}}{u_{1}+v_{1}}\right) \right. \\&\quad \left. \left( {\widetilde{\eta }}_{1}+d i-\frac{v_{1}}{u_{1}+v_{1}}\right) +\frac{u_{1}v_{1}}{(u_{1}+v_{1})^{2}} \right| e^{\xi _{1}+\eta _{1}},\\&=\frac{1}{u_{1}+v_{1}} \sqrt{F_{1}^{2}+F_{2}^{2}}e^{\xi _{1}+\eta _{1}},\\ F_{1}&=\left( {\widetilde{\xi }}_{1}-\frac{u_{1}}{u_{1}+v_{1}}\right) \left( {\widetilde{\eta }}_{1}-\frac{v_{1}}{u_{1}+v_{1}}\right) \\ {}&\quad -cd+\frac{u_{1}v_{1}}{(u_{1}+v_{1})^{2}},\\ F_{2}&=\left( {\widetilde{\xi }}_{1}-\frac{u_{1}}{u_{1}+v_{1}}\right) d+ \left( {\widetilde{\eta }}_{1}-\frac{v_{1}}{u_{1}+v_{1}}\right) c.\\ |G|&=\frac{u_{1}-{\mathbb {A}}}{v_{1}+{\mathbb {A}}}\frac{1}{u_{1}+v_{1}}\\&\quad \left| \left( \frac{u_{1}}{u_{1}-{\mathbb {A}}}+{\widetilde{\xi }}_{1}+\text {i} c -\frac{u_{1}}{u_{1}+v_{1}}\right) \right. \\&\quad \left. \times \left( -\frac{v_{1}}{v_{1}+{\mathbb {A}}}+{\widetilde{\eta }}_{1}+\text {i} d-\frac{v_{1}}{u_{1}+v_{1}}\right) \right. \\&\quad \left. +\frac{u_{1}v_{1}}{(u_{1}+v_{1})^{2}}\right| e^{\xi _{1}+\eta _{1}},\\&=\left| \frac{u_{1}-{\mathbb {A}}}{v_{1}+{\mathbb {A}}}\right| \frac{1}{u_{1}+v_{1}} \sqrt{G_{1}^{2}+G_{2}^{2}}e^{\xi _{1}+\eta _{1}},\\ G_{1}&=\left( \frac{u_{1}}{u_{1}-{\mathbb {A}}}+{\widetilde{\xi }}_{1}-\frac{u_{1}}{u_{1}+v_{1}}\right) \\&\quad \times \left( -\frac{v_{1}}{v_{1}+{\mathbb {A}}}+{\widetilde{\eta }}_{1}-\frac{v_{1}}{u_{1}+v_{1}}\right) ,\\ G_{2}&=\left( \frac{u_{1}}{u_{1}-{\mathbb {A}}}+{\widetilde{\xi }}_{1}-\frac{u_{1}}{u_{1}+v_{1}}\right) d\\&\quad + \left( -\frac{v_{1}}{v_{1}+{\mathbb {A}}}+{\widetilde{\eta }}_{1}-\frac{v_{1}}{u_{1}+v_{1}}\right) c. \end{aligned}$$

To ensure the regularity of |U| and |V|, we need to take parameters so that \(F_{1}^{2}+F_{2}^{2}\ne 0\) and \(G_{1}^{2}+G_{2}^{2}\ne 0\). The elastic collision of the two-soliton is demonstrated in Fig. 9.

Fig. 9
figure 9

Algebraic two-soliton solutions of U and V with suitable parameters \(u_{1}=\frac{1}{2}\), \(v_{1}=1\), \(c=d=2\), \({\mathbb {A}}=1\), \(\xi _{10}=\eta _{10}=0\), \(t=0\)

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Fig. 10
figure 10

Algebraic three-soliton solutions U with suitable parameters \(u_{1}=-\frac{1}{2}\), \(v_{1}=\frac{3}{2}\), \(u_{2}=-\frac{5}{2}\), \(v_{2}=2\), \(c_{11}=\frac{1}{2}\text {i}\), \(d_{11}=2\text {i}\), \(c_{21}=\text {i}\), \(d_{21}=40 \text {i}\), \({\mathbb {A}}=1\), \(\xi _{10}=\eta _{10}=0\), \(\xi _{20}=\eta _{20}=0\), (a, d) \(t=\frac{1}{4}\), (b, e), \(t=0\), (c, f) \(t=-\frac{1}{6}\)

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5.2 Algebraic multi-soliton solutions

Taking \(N=2\), \(n_{1}=1\), \(n=0\), \(c_{1}=0\) in Theorem 2.1. Thus we get

$$\begin{aligned}&F=\left| \begin{array}{ccc} {\widetilde{m}}_{11}^{(0)}&{}{\widetilde{m}}_{12}^{(0)} \\ {\widetilde{m}}_{21}^{(0)} &{} {\widetilde{m}}_{22}^{(0)} \\ \end{array}\right| \nonumber \\&G=\left| \begin{array}{ccc} {\widetilde{m}}_{11}^{(1)}&{}{\widetilde{m}}_{12}^{(1)} \\ {\widetilde{m}}_{21}^{(1)} &{} {\widetilde{m}}_{22}^{(1)} \\ \end{array}\right| , \end{aligned}$$
(5.3)

where

$$\begin{aligned} {\widetilde{m}}_{ij}^{(0)}&=\frac{1}{u_{i}+v_{j}}\nonumber \\&\quad \times \left[ ({\widetilde{\xi }}_{i}+c_{i1}-\frac{u_{i}}{u_{i}+v_{j}}) ({\widetilde{\eta }}_{j}+d_{j1}-\frac{v_{j}}{u_{i}+v_{j}})\right. \nonumber \\ {}&\quad \left. +\frac{u_{i}v_{j}}{(u_{i}+v_{j})^{2}}\right] e^{\xi _{i}+\eta _{j}}, \end{aligned}$$
(5.4)
$$\begin{aligned} {\widetilde{m}}_{ij}^{(1)}&=-\frac{u_{i}-{\mathbb {A}}}{v_{j}+{\mathbb {A}}}\frac{1}{u_{i}+v_{j}} \nonumber \\&\quad \times \left[ (\frac{u_{i}}{u_{i}-{\mathbb {A}}}+{\widetilde{\xi }}_{i}+c_{i1}-\frac{u_{i}}{u_{i}+v_{j}})\right. \nonumber \\&\left. \quad \times (-\frac{v_{j}}{v_{j}+{\mathbb {A}}}+{\widetilde{\eta }}_{j}+d_{j1}-\frac{v_{j}}{u_{i}+v_{j}})\right. \nonumber \\ {}&\quad \left. +\frac{u_{i}v_{j}}{(u_{i}+v_{j})^{2}}\right] e^{\xi _{i}+\eta _{j}}, \end{aligned}$$
(5.5)

On the wave plane, there are three types of waves with different propagation directions and speeds. As time progresses (refer to Fig. 10, the position of the collision center changes. This can be visualized in a two-dimensional image. Figure 10 illustrates that waves with larger amplitudes will catch up to waves with smaller amplitudes. After the collision, the waves will separate from each other due to variations in their heights. If we choose parameters as \(u_{1}=v_{2}=-\frac{1}{4}\), \(v_{1}=u_{2}=\frac{5}{4}\), \(c_{11}=d_{11}=2\text {i}\), \(c_{21}=4\text {i}\), \(d_{21}=10 \text {i}\), \(\xi _{10}=\eta _{10}=0\), \(\xi _{20}=\eta _{20}=0\), the image of the potential U degenerates into that of algebraic two-soliton solution in Section 3.2 (see Fig. 11 ). Plot of the degeneration algebraic two-soliton U is depicted in Fig. 5.

Fig. 11
figure 11

Algebraic soliton solution U with suitable parameters \(u_{1}=v_{2}=-\frac{1}{4}\), \(v_{1}=u_{2}=\frac{5}{4}\), \(c_{11}=d_{11}=2\text {i}\), \(c_{21}=4 \text {i}\), \(d_{21}=10 \text {i}\), \(\xi _{10}=\eta _{10}=0\), \(\xi _{20}=\eta _{20}=0\), \(t=0\)

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5.3 Multiple rational solutions on periodic background

Taking \(N=3\), \(n_{1}=1\), \(n=0\), \(c_{1}=0\) in Theorem 2.1.

$$\begin{aligned}&f=\left| \begin{array}{cccc} {\widetilde{m}}_{11}^{(0)}&{}{\widetilde{m}}_{12}^{(0)} &{}{\widetilde{m}}_{13}^{(0)} \\ {\widetilde{m}}_{21}^{(0)} &{} {\widetilde{m}}_{22}^{(0)} &{}{\widetilde{m}}_{23}^{(0)} \\ {\widetilde{m}}_{31}^{(0)} &{} {\widetilde{m}}_{32}^{(0)} &{}{\widetilde{m}}_{33}^{(0)} \\ \end{array}\right| \nonumber \\&g=\left| \begin{array}{cccc} {\widetilde{m}}_{11}^{(1)}&{}{\widetilde{m}}_{12}^{(1)}&{}{\widetilde{m}}_{13}^{(1)} \\ {\widetilde{m}}_{21}^{(1)} &{} {\widetilde{m}}_{22}^{(1)} &{} {\widetilde{m}}_{23}^{(1)}\\ {\widetilde{m}}_{31}^{(1)} &{} {\widetilde{m}}_{32}^{(1)} &{} {\widetilde{m}}_{33}^{(1)}\\ \end{array}\right| , \end{aligned}$$
(5.6)

where

$$\begin{aligned} {\widetilde{m}}_{ij}^{(n)}&=\frac{1}{u_{i}+v_{j}}\left( -\frac{u_{i}-{\mathbb {A}}}{v_{j}+{\mathbb {A}}}\right) ^{n}\\&\quad \times \left( {\widetilde{\xi }}_{i}+c_{i1}+\frac{n u_{i}}{u_{i}-{\mathbb {A}}}-\frac{u_{i}}{u_{i}+v_{j}}\right) e^{\xi _{i}+\eta _{j}},\\ {\widetilde{m}}_{ij}^{(n)}&=\frac{1}{u_{i}+v_{j}}\left( -\frac{u_{i}-{\mathbb {A}}}{v_{j}+{\mathbb {A}}}\right) ^{n}\\&\quad \times \left( {\widetilde{\eta }}_{i}+c_{i1}-\frac{n v_{j}}{v_{j}+{\mathbb {A}}}-\frac{u_{i}}{u_{i}+v_{j}}\right) e^{\xi _{i}+\eta _{j}},\\ {\widetilde{m}}_{ij}^{(n)}&=c_{j}+\frac{1}{u_{3}+v_{3}}\left( -\frac{u_{3}-{\mathbb {A}}}{v_{3}+{\mathbb {A}}}\right) e^{\xi _{i}+\eta _{j}}, \end{aligned}$$

Substituting F and G into Eq. (2.1) yields semi-rational solutions U and V. Since the explicit expressions of U and V are very complicated, we omit the specific forms of U and V here. To better investigate the dynamics of this class of semi-intelligible, we set the parameter to \(u_{1}=v_{2}=-\frac{1}{2}\), \(v_{1}=u_{2}=\frac{5}{2}\), \(v_{3}=u_{3}=-\text {i}\), \(c_{11}=2\text {i}\), \(d_{11}=6\text {i}\), \(c_{21}=4\text {i}\), \(d_{21}=8 \text {i}\), \(\xi _{i0}=\eta _{j0}=0,~~~(i,j=1,2,3)\), \(c_{1}=c_{2}=1\), \(c_{3}=-5\text {i}\), \(t=0\)

Fig. 12
figure 12

Semi-rational soliton solution U with suitable parameters \(u_{1}=v_{2}=-\frac{1}{2}\), \(v_{1}=u_{2}=\frac{5}{2}\), \(v_{3}=u_{3}=-\text {i}\), \(c_{11}=2\text {i}\), \(d_{11}=6\text {i}\), \(c_{21}=4 \text {i}\), \(d_{21}=8 \text {i}\), \(\xi _{i0}=\eta _{j0}=0,~~~(i,j=1,2,3)\), \(c_{1}=c_{2}=1\), \(c_{3}=-5i\), \(t=0\)

Full size image
Fig. 13
figure 13

Semi-rational soliton solution U with suitable parameters \(u_{1}=u_{2}=-\frac{1}{2}\), \(v_{1}=v_{2}=\frac{5}{2}\), \(v_{3}=u_{3}=-\text {i}\), \(c_{11}=2\text {i}\), \(d_{11}=6\text {i}\), \(c_{21}=4 \text {i}\), \(d_{21}=8 \text {i}\), \(\xi _{i0}=\eta _{j0}=0,~~~(i,j=1,2,3)\), \(c_{1}=c_{2}=1\), \(c_{3}=-5\text {i}\), \(t=0\)

Full size image

From Fig. 12(a), it can be seen that three periodic waves are displayed on the periodic background surface, of which two waves are anti-dark waves and the third wave is dark waves. When the parameter is taken as \(u_{1}=u_{2}=-\frac{1}{2}\), \(v_{1}=v_{2}=\frac{5}{2}\), \(v_{3}=u_{3}=-\text {i}\), \(c_{11}=2\text {i}\), \(d_{11}=6\text {i}\), \(c_{21}=4\text {i}\), \(d_{21}=8 \text {i}\), \(\xi _{i0}=\eta _{j0}=0, (i,j=1,2,3)\), \(c_{1}=c_{2}=1\), \(c_{3}=-5\text {i}\), \(t=0\), Fig. 13(a) shows two periodic waves on a periodic background, one is anti-dark and the other is dark. It can be seen that the amplitude of the wave increases after the collision, while the propagation direction does not change.

6 Conclusions and discussions

KP hierarchy reduction technique is an effective method for studying the soliton structure of integrable systems. Based on this method, we provide general soliton solutions and (semi-)rational solutions of the \(N \times N\) determinant of the KD equation using bilinear methods. Soliton solutions, line breather solutions, algebraic soliton solutions, and multiple rational solutions are constructed for constant and periodic backgrounds, respectively. The dynamical characteristics of the solutions under different parameters are analyzed. We find that these solutions are located on a constant background for \(N = 2n\) and on a periodic background for \(N = 2n+1\). We analyze in detail the local dynamics of rational and semi-rational solutions for \(N = 1\), 2, and 3 cases. Using KP reduction to study integrable equations is an interesting direction and a topic to be explored in progress.