1 Introduction

Nonlinear partial differential equations are applied to solving some complex problems in a variety of science and engineering [1,2,3,4,5,6,7,8,9]. Finding exact solutions plays an important role in nonlinear science. Among these exact solutions, solitary waves and lump solutions can be used to study natural phenomena appeared in fluids, engineering and nonlinear optics [10,11,12,13,14]. Lump waves which have attracted much attention are localized in all directions of spaces [12]. The study of this field is mainly by means of the Darboux transformation [15,16,17,18] and the Hirota bilinear method [19,20,21,22,23,24,25,26,27,28]. To describe complex physical phenomena, hybrid interaction solutions are widely investigated by combining different variable functions [29,30,31,32,33,34,35,36,37,38,39]. Interaction solutions among multi-soliton and other complicated waves are discussed by the localization procedure related to the nonlocal symmetry and the consistent tanh expansion method [29,30,31,32]. Interaction solutions between lump waves and multi-soliton are studied by using the Hirota bilinear method [33,34,35,36,37,38,39,40].

In this paper, we consider a (2 + 1)-dimensional coupled nonlinear partial differential equation (cNPDE)

$$\begin{aligned}&u_{xt} + \frac{3}{2} u_xu_{xx} + \frac{1}{4} u_{xxxx} + \delta _1 w_{x} + \delta _2u_{xy} \nonumber \\&\quad +\, \delta _3 u_{xx} + \frac{\delta _4}{4} (u_{xxxy} + 3u_xu_{xy} \nonumber \\&\quad + \,3u_{xx}u_y) + \frac{\delta _5}{4} (w_{xxy} + 3u_{xy}w + 3u_yw_x) \nonumber \\&\quad + \,\delta _6 \left( 3ww_x + \frac{1}{2}w_{xyy}\right) =0, \nonumber \\&u_{yy} - w_x = 0, \end{aligned}$$
(1)

where \(\delta _i \,(i=1,2,\ldots ,6)\) are arbitrary constants. Equation (1) reduces to a (2 + 1)-dimensional potential Kadomtsev–Petviashvili (pKP) equation by choosing \(\delta _2=\delta _3=\delta _4=\delta _5=\delta _6=0\), which describes the dynamics of a wave with a small amplitude. The periodic kink wave and the group-invariant solutions of the pKP equation have been derived [41, 42]. The nonlocal symmetry and interaction solutions of the pKP equation have been given by the localization procedure related to nonlocal symmetries [43].

This paper is organized as follows: in Sect. 2, we construct the Hirota bilinear form of Eq. (1) by using the Painlevé–Bäcklund transformation. In Sect. 3, we obtain two solitary waves by introducing a perturbation expansion. Lump waves are presented by solving the corresponding Hirota bilinear form in Sect. 4. In Sect. 5, interaction solutions between a lump and a one-kink soliton, and between a bi-lump and a one-soliton solution are derived by adding an exponential function to a quadratic function. The last section is a simple summary and discussion.

2 A bilinear form of a coupled nonlinear partial differential equation

Based on the Painlevé analysis [44], a Painlevé–Bäcklund transformation of Eq. (1) reads

$$\begin{aligned} u = \frac{u_0}{\phi } + u_1, \quad w = \frac{w_0}{\phi ^2} + \frac{w_1}{\phi } + w_2, \end{aligned}$$
(2)

where \(\phi \) is an auxiliary function of the variables x, y and t. The functions of \(u_1\) and \(w_2\) are arbitrary seed solution of Eq. (1). Substituting (2) into (1) and balancing the coefficients \(\phi ^{-5}\) and \(\phi ^{-3}\), we get

$$\begin{aligned} u_0= 2\phi _x, \quad w_0=-2\phi _y^2. \end{aligned}$$
(3)

Balancing the coefficient \(\phi ^{-4}\) gives

$$\begin{aligned} w_1 = 2\phi _{yy}. \end{aligned}$$
(4)

Substituting (3), (4) and the seed solution \(u_1=0, w_2=0\) into (2), we get

$$\begin{aligned} u = \frac{2\phi _x}{\phi }, \quad w = -\frac{2\phi _y^2}{\phi ^2} + \frac{2\phi _{yy}}{\phi }. \end{aligned}$$
(5)

A bilinear form of (1) is yielded

$$\begin{aligned}&2\phi \phi _{xt} - 2\phi _t\phi _x +\frac{1}{2} \phi \phi _{xxxx} - 2\phi _x\phi _{xxx} \nonumber \\&\quad +\, \frac{3}{2}\phi _{xx}^2 + 2\delta _1 (\phi \phi _{yy}-\phi _y^2) \nonumber \\&\quad +\, 2\delta _2(\phi \phi _{xy}-\phi _x\phi _y) + 2\delta _3(\phi \phi _{xx}-\phi _x^2) \nonumber \\&\quad +\, \delta _4 (\phi \phi _{xxxy} - \phi _{xxx}\phi _y - 3\phi _x\phi _{xxy} + 3\phi _{xx}\phi _{xy} ) \nonumber \\&\quad +\, \delta _5 (\phi \phi _{xyyy} - \phi _{x}\phi _{yyy} - 3\phi _y\phi _{xyy} + 3\phi _{xy}\phi _{yy}) \nonumber \\&\quad +\, \delta _6 (\phi \phi _{yyyy} - 4\phi _{y}\phi _{yyy} + 3\phi _{yy}^2) = 0. \end{aligned}$$
(6)

The bilinear equation (6) has the following equivalent formula:

$$\begin{aligned}&D_{t}D_{x} + \frac{1}{4} D_{x}^4 + \delta _1 D_{y}^2 + \delta _2 D_{x} D_{y} + \delta _3 D_{x}^2 \nonumber \\&\quad + \,\frac{\delta _4}{2} D_{x}^3D_{y} +\frac{\delta _5}{2} D_{x} D_{y}^3 + \frac{\delta _6}{2} D_{y}^4 = 0, \end{aligned}$$
(7)

with the D-operators defined by

$$\begin{aligned}&D_x^l D_y^n D_t^m f(x,y,t) \cdot g(x', y', t') \nonumber \\&\quad = \Bigl (\frac{\partial }{\partial x} - \frac{\partial }{\partial x'} \Bigr )^l \Bigl (\frac{\partial }{\partial y} - \frac{\partial }{\partial y'} \Bigr )^n \Bigl (\frac{\partial }{\partial t} - \frac{\partial }{\partial t'} \Bigr )^m \nonumber \\&\qquad f(x,y,t) \cdot g(x', y', t')|_{x=x', y=y', t=t'}. \end{aligned}$$
(8)
Fig. 1
figure 1

Profile of a two-front wave (12): a 3-dimensional plot of u with \(t=0\), b 3-dimensional plot of w with \(t=0\)

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Fig. 2
figure 2

Profile of a two-front wave (12): a 3-dimensional plot of u with \(t=0\), b 3-dimensional plot of w with \(t=0\)

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3 Solitary waves of a coupled nonlinear partial differential equation

The Hirota bilinear method has been widely used to solve a class of nonlinear evolution equations [45]. Based on the Hirota bilinear method, we assume that a two-front wave for \(\phi \) has a perturbation expansion

$$\begin{aligned} \phi = 1 + \exp (\theta _1) + \exp (\theta _2), \end{aligned}$$
(9)

where \(\theta _1 =a_1x+b_1y+c_1t\), \(\theta _2 =a_2x+b_2y+c_2t\), and \(a_1, b_1, c_1, a_2, b_2\) and \(c_2\) are arbitrary constants. Inserting (9) into (6) and solving the coefficients of different powers of the exponent functions, a relation among the arbitrary constants reads

$$\begin{aligned} c_1= & {} -\frac{a_1^3}{4} - \delta _1 \frac{b_1^2}{a_1} - \delta _2 b_1 - \delta _3 a_1 - \delta _4\frac{a_1^2b_1}{4} \nonumber \\&- \,\delta _5 \frac{b_1^3}{4} - \delta _6 \frac{b_1^4}{2a_1}, \nonumber \\ c_2= & {} -\frac{a_2^3}{4} - \delta _1 \frac{b_2^2}{a_2} - \delta _2 b_2 - \delta _3 a_2 - \delta _4\frac{a_2^2b_2}{4} \nonumber \\&-\, \delta _5 \frac{b_2^3}{4} - \delta _6 \frac{b_2^4}{2a_2}, \end{aligned}$$
(10)

where \(\delta _6\) satisfies

$$\begin{aligned} \delta _6= & {} \Bigl (\frac{3}{2}a_1^2a_2^2(a_1-a_2)^2 - 2\delta _1(a_1b_2-a_2b_1)^2 \nonumber \\&+\, \frac{1}{2}\delta _4a_1a_2(a_1-a_2)(a_1^2b_2 + 2a_1a_2(b_1-b_2)-a_2^2b_1) \nonumber \\&+\, \frac{3}{2}\delta _5a_1a_2b_1b_2(a_1-a_2)(b_1-b_2)\Bigr )/\bigl (a_1^2b_2^4\nonumber \\&-\,2a_1a_2b_1b_2(2b_1^2-3b_1b_2+2b_2^2)+a_2^2b_1^4\bigr ). \end{aligned}$$
(11)
Fig. 3
figure 3

Profile of two-soliton solution (16): a 3-dimensional plot of u with \(t=0\), b 3-dimensional plot of w with \(t=0\)

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Substituting (9) into (5) yields a two-front wave

$$\begin{aligned} u= & {} \frac{2(a_1\exp (\theta _1) + a_2 \exp (\theta _2))}{1 + \exp (\theta _1) + \exp (\theta _2)}, \nonumber \\ w= & {} -\frac{2(b_1\exp (\theta _1) + b_2\exp (\theta _2))^2}{(1+ \exp (\theta _1) + \exp (\theta _2))^2} \nonumber \\&+ \,\frac{2(b_1^2\exp (\theta _1) + b_2^2\exp (\theta _2))}{1 + \exp (\theta _1) + \exp (\theta _2)}, \end{aligned}$$
(12)

where \(c_1, c_2\) and \(\delta _6\) satisfy (10) and (11). We show a two-front wave for u and w with specific parameters \(a_1=-\frac{1}{2}, a_2=\frac{1}{2}, b_1=\frac{1}{2}, b_2=\frac{1}{2}, \delta _1=-1, \delta _2=1, \delta _3=2, \delta _4=1, \delta _5=2\) in Fig. 1, and another kind of a two-front wave with specific parameters \(a_1=-\frac{1}{3}, a_2=-\frac{1}{2}, b_1=1, b_2=\frac{1}{2}, \delta _1=-1, \delta _2=1, \delta _3=2, \delta _4=1, \delta _5=2\) in Fig. 2. The solution of w is shown as “U”-shaped and “Y”-shaped in Figs. 1b and 2b, respectively. Characteristics of two front waves are thus different by selecting different parameters.

For a two-soliton solution, we assume

$$\begin{aligned} \phi = 1 + \exp (\theta _1) + \exp (\theta _2) + a_{12}\exp (\theta _1 + \theta _2), \end{aligned}$$
(13)

where \(a_1, b_1, c_1, a_2, b_2, c_2\) and \(a_{12}\) are arbitrary parameters to be determined. Substituting (13) into (6) and solving the coefficients of different powers of the exponent functions, a relation among the arbitrary constants is

$$\begin{aligned} c_1= & {} -\frac{a_1^3}{4} - \delta _1 \frac{b_1^2}{a_1} - \delta _2 b_1 - \delta _3 a_1 - \delta _4\frac{a_1^2b_1}{4} \nonumber \\&- \,\delta _5 \frac{b_1^3}{4} - \delta _6 \frac{b_1^4}{2a_1}, \nonumber \\ c_2= & {} -\frac{a_2^3}{4} - \delta _1 \frac{b_2^2}{a_2} - \delta _2 b_2 - \delta _3 a_2 \nonumber \\&-\, \delta _4\frac{a_2^2b_2}{4} - \delta _5 \frac{b_2^3}{4} - \delta _6 \frac{b_2^4}{2a_2}, \end{aligned}$$
(14)

where \(\delta _5\) and \(\delta _6\) satisfy

$$\begin{aligned} \delta _5= & {} \frac{1}{A}\Bigl (\frac{2a_1^2a_2^2}{b_1b_2}\bigl ( b_1a_2(b_1^2+3b_2^2)- a_1b_2(3b_1^2+b_2^2) \bigr ) \nonumber \\&+ \,\delta _4\frac{a_1a_2}{b_1b_2} \bigl (a_2^2b_1(b_1^2+2b_2^2) \nonumber \\&+\, 2a_1a_2b_1b_2(b_2^2-b_1^2)-a_1^2b_2^2(2b_1^2+b_2^2) \bigr ) \nonumber \\&+ \,8\delta _1b_1b_2 (a_1b_2-a_2b_1) \Bigr ), \nonumber \\ \delta _6= & {} \frac{1}{A} \Bigl (\frac{3}{2}a_1^2a_2^2 (a_1^2-a_2^2) - 2\delta _1 (a_1^2b_2^2-a_2^2b_1^2) \nonumber \\&+\, \frac{\delta _4}{2}a_1a_2(a_1^2-a_2^2)(a_1b_2+a_2b_1)\Bigr ), \nonumber \\&A= a_1^2b_2^4+ 2a_1a_2b_1b_2(b_1^2-b_2^2) - a_2^2b_1^4. \end{aligned}$$
(15)

Substituting (13) into (5) yields a two-soliton solution

$$\begin{aligned} u= & {} \frac{2(a_1\exp (\theta _1) + a_2 \exp (\theta _2)+a_{12}(a_1+a_2)\exp (\theta _1+\theta _2))}{1 + \exp (\theta _1) + \exp (\theta _2)+ a_{12}\exp (\theta _1+\theta _2)}, \nonumber \\ w= & {} -\frac{2(b_1\exp (\theta _1) + b_2\exp (\theta _2) + a_{12}(b_1+b_2)\exp (\theta _1+\theta _2))^2}{(1+ \exp (\theta _1) + \exp (\theta _2) + a_{12}(a_1+a_2)\exp (\theta _1+\theta _2))^2} \nonumber \\&+ \,\frac{2(b_1^2\exp (\theta _1) + b_2^2\exp (\theta _2) + a_{12}(b_1+b_2)^2\exp (\theta _1+\theta _2) )}{1 + \exp (\theta _1) + \exp (\theta _2) + a_{12}(a_1+a_2)\exp (\theta _1+\theta _2)}.\nonumber \\ \end{aligned}$$
(16)

To illustrate this two-soliton solution (16), we select the parameters \(a_1=1, a_2=\frac{1}{4}, b_1=\frac{1}{2}, b_2=\frac{1}{2}, a_{12}=2, \delta _1=-1, \delta _2=2, \delta _3=1, \delta _4=3, \delta _5=2\). The interactions between two kink solitons and two solitons are shown in Fig. 3a, b, respectively.

4 Lump waves of a coupled nonlinear partial differential equation

To get lump waves of Eq. (1), we take a quadratic function \(\phi \) as

$$\begin{aligned}&\phi = g^2 + h^2 + a_9, \nonumber \\&g = a_1 x+a_2 y +a_3t + a_4, \nonumber \\&h = a_5 x+a_6y+a_7 t + a_8. \end{aligned}$$
(17)

where \(a_i \,(i=1,2,\ldots ,9)\) are arbitrary parameters. By substituting (17) into (7) and balancing different powers of x, y and t, we get the solutions of \(a_i\)’s

$$\begin{aligned} a_3= & {} - \frac{\delta _1(a_1a_2^2 + 2a_2a_5a_6 - a_1a_6^2)}{a_1^2+a_5^2} - \delta _2a_2 - \delta _3 a_1, \nonumber \\ a_9= & {} -\frac{3\delta _4(a_1^2+a_5^2)^2(a_1a_2+a_5a_6)}{4\delta _1(a_1a_6-a_2a_5)^2} \nonumber \\&-\,\frac{3\delta _5(a_1^2+a_5^2)(a_2^2+a_6^2)(a_1a_2+a_5a_6)}{4\delta _1(a_1a_6-a_2a_5)^2} \nonumber \\&- \,\frac{3\delta _6(a_1^2+a_5^2)(a_2^2+a_6^2)^2}{4\delta _1(a_1a_6-a_2a_5)^2} \nonumber \\&- \,\frac{3(a_1^2+a_5^2)^3}{4\delta _1(a_1a_6-a_2a_5)^2}, \nonumber \\ a_7= & {} - \frac{\delta _1(2a_1a_2a_6 - a_2^2a_5 + a_5a_6^2)}{a_1^2+a_5^2} - \delta _2a_6 -\delta _3a_5, \nonumber \\ \end{aligned}$$
(18)

which should satisfy the following constraint conditions:

$$\begin{aligned}&\delta _1 a_5 \ne 0,\quad a_1a_6-a_2a_5\ne 0, \nonumber \\&\delta _1 \Bigl [(a_1^2+a_5^2) (a_1^2+a_5^2 + \delta _4(a_1a_2+a_5a_6)) \nonumber \\&\quad +\, (a_2^2+a_6^2)(2\delta _6(a_2^2+a_6^2)\nonumber \\&\quad +\,\delta _5(a_1a_2+a_5a_6))\Bigr ] < 0, \end{aligned}$$
(19)

so that the localization of u and w in all directions of the (x, y)-plane is guaranteed. A class of lump waves of Eq. (1) is thus generated

$$\begin{aligned} u= & {} \frac{4a_1g + 4a_5h}{\phi }, \nonumber \\ w= & {} - \frac{8(a_2g+a_6h)^2}{\phi ^2} + \frac{4a_2^2+4a_6^2}{\phi }, \end{aligned}$$
(20)

where

$$\begin{aligned} \phi= & {} g^2 + h^2 -\frac{3\delta _4(a_1^2+a_5^2)^2(a_1a_2+a_5a_6)}{4\delta _1(a_1a_6-a_2a_5)^2} \nonumber \\&-\,\frac{3\delta _5(a_1^2+a_5^2)(a_2^2+a_6^2)(a_1a_2+a_5a_6)}{4\delta _1(a_1a_6-a_2a_5)^2}\nonumber \\&-\, \frac{3\delta _6(a_1^2+a_5^2)(a_2^2+a_6^2)^2}{4\delta _1(a_1a_6-a_2a_5)^2} - \,\frac{3(a_1^2+a_5^2)^3}{4\delta _1(a_1a_6-a_2a_5)^2}, \quad \nonumber \\ g= & {} a_1 x + a_2y \nonumber \\&- \left( \frac{\delta _1(a_1a_2^2 + 2a_2a_5a_6 - a_1a_6^2)}{a_1^2+a_5^2} + \,\delta _2a_2 + \delta _3 a_1\right) t + a_4, \nonumber \\ h= & {} a_5 x+a_6y \nonumber \\&- \left( \frac{\delta _1(2a_1a_2a_6 + a_2^2a_5 + a_5a_6^2)}{a_1^2+a_5^2} + \,\delta _2a_6 -\delta _3a_5 \right) t + a_8.\nonumber \\ \end{aligned}$$
(21)

To catch the moving path of the lump waves in (20), the critical point of the lump waves is calculated by solving \(\phi _x=\phi _y=0\). The exact moving path of the lump waves is written as

$$\begin{aligned}&x = x(t) = \frac{(a_2a_7 - a_3a_6)t-(a_2a_8-a_4a_6)}{a_1a_6-a_2a_5},\nonumber \\&y = y(t) = \frac{(a_1a_7 - a_3a_5)t-(a_1a_8-a_4a_5)}{a_1a_6-a_2a_5}, \end{aligned}$$
(22)

which can describe the traveling path of the lump waves along a straight line

$$\begin{aligned} y= \frac{a_3a_5-a_1a_7}{a_2a_7 -a_3a_6} x + \frac{a_3a_8-a_4a_7}{a_2a_7-a_3a_6}, \end{aligned}$$
(23)

with \(a_3, a_7\) and \(a_9\) satisfying (18). The parameters are selected as \(a_1=-1, a_2=2, a_4=-3, a_5=1, a_6=3, a_8=2, \delta _1=-1, \delta _2=2, \delta _3=3, \delta _4=2, \delta _5=1, \delta _6=1\) in Figs. 4 and 5. A lump wave of u is plotted in Fig. 4. The spatial structure of a lump wave is described in Fig. 4a. From Fig. 4a, we can easily know that the lump wave has a localized characteristic at \(t=0\). A bi-lump wave of w is plotted in Fig. 5. The spatial structure of a bi-lump wave is described in Fig. 5a at \(t=0\). Figures 4b and 5b represent the corresponding density plots of the lump wave. Figure 4c displays the contour plot of the lump wave at \(t=-35, t=0, t=36\). Figure 5c is the contour plot of the lump wave at \(t=-20, t=0, t=20\). The blue line of Figs. 4c and 5c is the relevant moving progress (23), i.e., \(y = \frac{2}{19} x + \frac{9}{19}\). The wave along x-axis of the lump wave is depicted in Figs. 4d and 5d.

Fig. 4
figure 4

Profile of a lump wave (20): a 3-dimensional plot with \(t=0\), b the corresponding density plot, c the red line is the contour plot of the lump wave at \(t=-35, t=0, t=36\), and the blue line is the relevant moving progress (23), i.e., \(y = \frac{2}{19} x + \frac{9}{19}\), d the wave propagation pattern of the wave along x-axis by selecting \(y=0\) and different time. (Color figure online)

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Fig. 5
figure 5

Profile of a bi-lump wave (20): a 3-dimensional plot of w with the time \(t=0\), b the corresponding density plot, c the red line is the contour plot of the lump wave at \(t=-20, t=0, t=20\), and the blue line is the relevant moving progress (23), i.e., \(y = \frac{2}{19} x + \frac{9}{19}\), d the wave propagation pattern of the wave along x-axis by selecting \(y=0\) and different time. (Color figure online)

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5 Interaction solution between a lump and a one-soliton solution

Interaction solutions between lumps and other type solutions can be constructed by mixing a quadratic function with other type functions. In order to find interaction solution between lump waves and a one-soliton solution, we assume that an interaction solution is determined by a sum of a quadratic function and an exponential function

$$\begin{aligned}&\phi = g^2 + h^2 + a_9 + k_1 \exp (k_2 x + k_3y + k_4t +k_5), \nonumber \\&g = a_1 x+a_2 y +a_3t +a_4, \nonumber \\&h = a_5 x+a_6y+a_7 t + a_8, \end{aligned}$$
(24)

with \(k_i\, (i=1,2,\ldots ,5)\) being five undetermined real parameters. By substituting (24) into (6) and vanishing the different powers of x, y and t, we obtain the following two cases of constraining relations for the parameters:

Fig. 6
figure 6

Profile of an interaction solution between a lump and a one-kink soliton solution (27): a 3-dimensional plot with \(t=0\), b the corresponding density plot, c the red line is contour plot at \(t=-42, t=0, t=42\) and the blue line is the relevant moving progress (23), i.e., \(y = -\frac{1}{15}x - \frac{17}{75}\), d the wave propagation pattern of the wave along x-axis by selecting \(y=0\) and different time t. (Color figure online)

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Fig. 7
figure 7

Profile of an interaction solution between a bi-lump and a one-soliton solution (27): a 3-dimensional plot with \(t=0\), b the corresponding density plot, c the red line is contour plot at \(t=-52, t=0, t=46\) and the blue line is the relevant moving progress (23), i.e., \(y = -\frac{1}{15}x - \frac{17}{75}\), d the wave propagation pattern of the wave along x-axis by selecting \(y=0\) and different time t

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Case I

$$\begin{aligned} a_3= & {} - \frac{\delta _1(a_1a_2^2 + 2a_2a_5a_6 - a_1a_6^2)}{a_1^2+a_5^2} - \delta _2a_2 - \delta _3 a_1, \\ a_7= & {} - \frac{\delta _1 (a_5a_6^2-a_5a_2^2+2a_1a_2a_6) }{a_1^2+a_5^2} - \delta _2a_6-\delta _3a_5, \nonumber \\ k_4= & {} -\frac{k_2^3}{4} - \delta _1\frac{k_3^2}{k_2} - \delta _2k_3 -\delta _3k_2 - \delta _4 \frac{k_3k_2^2}{4} \nonumber \\&-\, \delta _5 \frac{k_3^4}{4} -\delta _6\frac{k_3^4}{2k_2}, \nonumber \\ a_9= & {} \biggl [ 3\delta _1k_2^2A (k_2^2B - k_3^2A) \Bigl ((k_3^2A+k_2^2B)^2 \nonumber \\&+\,k_2^2k_3^2( D^2 - 3C^2) \Bigr ) + 8k_3^5A^2C(k_3C-2k_2B) \nonumber \\&+\, 16k_2^3k_3^2B D^2(2k_3C-k_2B) \nonumber \\&+\, 12k_2^2k_3^4 AB (D^2-C^2) \nonumber \\&+ \,4k_2^3B(k_2^3B^3 - 8k_2k_3^2BC^2+12k_3^3C^3) \biggr ] \nonumber \\&/\bigl (4k_2^2k_3^4\delta _1D^2E\bigr ) - \frac{3k_2 AC}{k_3^3D^2} (k_2^2B - k_3^2A), \nonumber \\ \delta _4= & {} -\frac{2k_2}{k_3} + \frac{8\delta _1(k_2B-k_3C)(k_3^2A-k_2^2B)}{3k_2^2k_3AE}, \nonumber \\ \delta _5= & {} \frac{2k_2^3}{k_3^3} + \frac{2C}{3k_3^2A} + \frac{k_3^4A+k_2^2B^2-2k_2^2k_3^2D^2 }{3k_2k_3^3AE}, \nonumber \\ \delta _6= & {} -\frac{k_2^4}{2k_3^4} \nonumber \\&+\, \frac{2\delta _1(k_2^2B - k_3^2A)(3k_3^2A-k_2^2B-2k_2k_3C)}{3k_3^4AE}, \nonumber \\ A= & {} a_1^2+a_5^2, \nonumber \\ B= & {} a_2^2 +a_6^2, \quad C=a_1a_2+a_5a_6, \nonumber \\ D= & {} a_1a_6-a_2a_5, \quad E=k_3^2A+k_2^2B-2k_2k_3C,\nonumber \end{aligned}$$
(25)

which should satisfy the following constraint conditions:

$$\begin{aligned} \delta _1 a_5k_2k_3\ne 0,\quad a_1a_6-a_2a_5\ne 0, \quad a_9>0, \end{aligned}$$
(26)

so that the localization of u and w in all directions of the (x, y)-plane is guaranteed. By substituting (24) into (5) and combining the parameters relations (25), we get the following interaction solution of Eq. (1):

$$\begin{aligned} u= & {} \frac{ 4a_1g + 4a_5h +2k_1k_2\exp (f)}{\phi }, \nonumber \\ w= & {} -\frac{2(2a_2g+2a_6h+k_1k_3\exp (f))^2}{\phi ^2} \nonumber \\&+ \,\frac{2(2a_2^2+2a_6^2+k_1k_3^2\exp (f))}{\phi }, \end{aligned}$$
(27)

where

$$\begin{aligned} \phi= & {} g^2 + h^2 + a_9 + k_1 \exp (f), \nonumber \\ g= & {} a_1 x + a_2y - \Biggl ( \frac{\delta _1(a_1a_2^2 + 2a_2a_5a_6 - a_1a_6^2)}{a_1^2+a_5^2} \nonumber \\&+\, \delta _2a_2 + \delta _3 a_1\Biggr ) t + a_4, \nonumber \\ h= & {} a_5 x+a_6y - \Biggl ( \frac{\delta _1 (a_5a_6^2-a_5a_2^2+2a_1a_2a_6) }{a_1^2+a_5^2} \nonumber \\&+ \,\delta _2a_6 + \delta _3a_5 \Biggr ) t + a_8, \nonumber \\ f= & {} k_2 x +k_3 y - \Bigl (\frac{k_2^3}{4} + \delta _1\frac{k_3^2}{k_2} + \delta _2k_3 + \delta _3k_2 \nonumber \\&+ \,\delta _4 \frac{k_3k_2^2}{4} + \delta _5 \frac{k_3^4}{4} + \,\delta _6\frac{k_3^4}{2k_2}\Bigr ) t +k_5. \end{aligned}$$
(28)

The parameters are selected as \(a_1=1, a_2=3, a_4=1, a_5=5, a_6=5, a_8=3, k_1=1, k_2=\frac{1}{3}, k_3=\frac{1}{2}, k_5=1, \delta _1=-1, \delta _2=2, \delta _3=1\) in Figs. 6 and 7. The interaction solution between a lump and a one-kink soliton of u is presented in Fig. 6a at \(t=0\). Figure 5b displays the corresponding density plot of the lump–kink wave. Figure 6c represents the homologous contour plot at time \(t=-42, t=0, t=42\). The interaction solution between a bi-lump and a one-soliton solution of w is presented in Fig. 7a at \(t=0\). The corresponding density is plotted in Fig. 7b. Figure 7c is the homologous contour plot at time \(t=-52, t=0, t=46\). The blue line shown in Figs. 6c and 7c is the relevant moving progress of the lump wave (23), i.e., \(y = -\frac{1}{15}x - \frac{17}{75}\). The wave along x-axis of the corresponding interaction solution is shown in Figs. 6d and 7d.

Case II

$$\begin{aligned} a_1= & {} \frac{2a_2\delta \sqrt{\delta _1} }{k_2}, \nonumber \\ a_3= & {} - \frac{\sqrt{\delta _1}a_2k_2(2a_5^2k_2^2 + 4\delta \delta _1 a_2^2 -\delta k_2^2a_5^2)}{2(a_5^2k_2^2+4\delta _1a_2^2)} \nonumber \\&- \frac{2\delta \sqrt{\delta _1}\delta _3a_2}{k_2} - \delta _2 a_2, \nonumber \\ a_6= & {} \frac{a_5k_2}{2\delta \sqrt{\delta _1}},\quad a_7 = - \frac{a_5k_2^2(a_5^2k_2^2 - 4\delta _1 a_2^2 + 8\delta \delta _1 a_2^2)}{4(a_5^2k_2^2+4\delta _1a_2^2)} \nonumber \\&-\, \frac{\delta _2k_2a_5 }{2\sqrt{\delta _1}} - \delta _3a_5, \nonumber \\ k_4= & {} -\frac{k_2^3}{4} - \delta _1\frac{k_3^2}{k_2} - \delta _2k_3 -\delta _3k_2 - \delta _4 \frac{k_3k_2^2}{4} \nonumber \\&- \,\delta _5 \frac{k_3^4}{4} -\delta _6\frac{k_3^4}{2k_2}, \nonumber \\ \delta _4= & {} -\frac{4\delta _1}{\delta \sqrt{\delta _1}k_2}, \quad \delta _5 = -\frac{8\delta _1^2-\delta _6k_2^4}{\delta \sqrt{\delta _1}k_2^3}, \end{aligned}$$
(29)

which \(\delta ^2=1\) and should satisfy the following constraint conditions:

$$\begin{aligned} \delta _1 a_5k_2 \ne 0,\quad a_9 > 0, \end{aligned}$$
(30)

so that localization of u and w in all directions of the (x, y)-plane is guaranteed. By substituting (24) into (5) and combining the parameters relations (29), we get the following interaction solution of Eq. (1):

$$\begin{aligned} u= & {} \frac{ 4a_1g + 4a_5h +2k_1k_2\exp (f)}{\phi }, \nonumber \\ w= & {} -\frac{2(2a_2g+2a_6h+k_1k_3\exp (f))^2}{\phi ^2} \nonumber \\&+\, \frac{2(2a_2^2+2a_6^2+k_1k_3^2\exp (f))}{\phi }, \end{aligned}$$
(31)

where

$$\begin{aligned} \phi= & {} g^2 + h^2 + a_9 + k_1 \exp (f), \nonumber \\ g= & {} a_1 x + a_2y \nonumber \\&-\Bigg ( \frac{\sqrt{\delta _1}a_2k_2(2a_5^2k_2^2 + 4\delta \delta _1 a_2^2 -\delta k_2^2a_5^2)}{2(a_5^2k_2^2+4\delta _1a_2^2)} \nonumber \\&+ \,\frac{2\delta \sqrt{\delta _1}\delta _3a_2}{k_2} + \delta _2 a_2\Bigg ) t + a_4, \nonumber \\ h= & {} a_5 x+a_6y - \Bigg (\frac{a_5k_2^2(a_5^2k_2^2 - 4\delta _1 a_2^2 + 8\delta \delta _1 a_2^2)}{4(a_5^2k_2^2+4\delta _1a_2^2)} \nonumber \\&+ \,\frac{\delta _2k_2a_5 }{2\sqrt{\delta _1}} + \delta _3a_5 \Bigg ) t + a_8, \nonumber \\ f= & {} k_2 x +k_3 y - \Bigg (\frac{k_2^3}{4} + \delta _1\frac{k_3^2}{k_2} + \delta _2k_3 + \delta _3k_2 \nonumber \\&+ \,\delta _4 \frac{k_3k_2^2}{4} + \delta _5 \frac{k_3^4}{4} + \delta _6\frac{k_3^4}{2k_2}\Bigg ) t +k_5. \end{aligned}$$
(32)

Similarly to the Case I, we can get interaction solutions between a lump and a one-kink soliton, and between a bi-lump and a one-soliton solution by using (31).

6 Conclusion

In this work, the Hirota bilinear form of Eq. (1) was derived by the truncated Painlevé analysis. Based on the obtained bilinear form, solitary waves were firstly constructed via a perturbative expansion (shown in Figs. 1, 2, 3). Then, some lump waves were found by using a positive quadratic function. Finally, the interaction solutions, between a lump wave and a one-kink soliton, and between a bi-lump wave and a one-soliton solution, were proposed by adding an additional exponential function to a positive quadratic function (shown in Figs. 4, 5, 6, 7).

In addition, we could also construct some new integrable systems by using the generalized bilinear operators [46], which are given by

$$\begin{aligned}&D_{p,t}D_{p,x} + \frac{1}{4} D_{p,x}^4 + \delta _1 D_{p,y}^2 + \delta _2 D_{p,x} D_{p,y} + \delta _3 D_{p,x}^2\nonumber \\&\quad + \frac{\delta _4}{2} D_{p,x}^3D_{p,y} +\frac{\delta _5}{2} D_{p,x} D_{p,y}^3 +\frac{\delta _6}{2} D_{p,y}^4 = 0, \end{aligned}$$
(33)

with the prime numbers \(p=3, 5,7, \cdots \). We are going to study hybrid solutions and integrable properties of Eq. (33).