1 Introduction

Various fields of applied sciences, e.g., plasma physics, solid-state physics, and fluid mechanics depend greatly on nonlinear partial differential equations (NLPDEs). Today, symbolic packages allow more intricate algebraic computations to be performed in seeking exact solutions of NLPDEs. Various methods [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] have been found to be effective in the extraction of different wave structures. For example, Wazwaz [20] discovered multi-soliton waves of a new integrable equation of fifth-order using the simplified Hirota method. Feng and Zhang [21] through applying ansatz methods found lump and lump-soliton waves of the Fokas system. Hosseini et al. [22] utilized the homoclinic test method to construct breather waves of the Burgers equations.

Integrable PDEs are a class of PDEs that have some specific properties. Although there are several distinct formal definitions, an integrable equation must generally possess multi-soliton solutions, bilinear Bäcklund transformation, Lax pair, etc. A useful method for assessing the integrability of NLPDEs is the Painlevè analysis [23]. Wazwaz [24] checked the integrability of two 3D KdV equations using the Painlevè analysis. In another research designed by Wazwaz [25], the Painlevè analysis was employed to examine the integrability condition of 3D KP equations. Das et al. [26] investigated the integrability of the forced KdV equation using the Painlevè analysis.

Bell [27] in 1934 established an exponential polynomial known as the Bell polynomial. After that, Lambert and his coauthors [28,29,30,31] proposed a systematic method i.e. the Bell polynomial method that can be utilized to extract the HB form, BB transformation, Lax pair, and conservation laws of NLPDEs. In recent decades, there has been a significant amount of research [32,33,34,35,36,37,38,39,40] on NLPDEs using the Bell polynomial method. For instance, Guo [37] used the BPM to explore the Lax integrability of a 2D Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation. Pu and Chen [38] derived the BB transformation of a 2D KdV equation through the BPM. Fan and Bao [39] employed the BPM to seek the HB form of Hirota–Satsuma coupled KdV equations. Mandal et al. [40] found conservation laws of a 3D NLPDE using the BPM.

As a mathematical model for waves on shallow water surfaces, the KdV equation [41] has been established. Such a model [41] is indeed an integrable equation that involves a wide range of exact solutions, especially soliton solutions, despite the nonlinearity that usually makes PDEs intractable. In applied sciences, KdV-type equations are becoming increasingly popular due to their potential to model a wide variety of phenomena. Hosseini et al. [42] through applying the symbolic computation generated the positive multi-complexiton wave of a KdV equation of generalized type. Malik et al. [43] investigated the intgrability of a 2D KdV–mKdV equation and found its soliton waves using the new Kudryashov method. Huang and Lv [44] extracted soliton and rogue waves of a modified KdV equation of integrable type using the Darboux transformation.

In the current paper, a 3D generalized KdV equation, i.e.,

$${u}_{t}+{c}_{1}{u}_{x}+{c}_{2}{u}_{y}+{c}_{3}{u}_{z}+{u}_{5x}+15u{u}_{3x}+15{u}_{x}{u}_{2x}+45{u}^{2}{u}_{x}=0,$$
(1)

representing long-wave behavior in shallow water is considered and its HB form, BB transformation, Lax pair, integrability, multi-soliton waves, and lump, lump-soliton, and breather waves are constructed in detail. Special cases of Eq. (1) have gained a lot of interest during the last decades. For example,

  • multi-soliton waves derived by Sun et al. [45];

  • lump and mixed waves found by Lü et al. [46];

  • interaction waves extracted by Liu [47];

  • lump-periodic waves acquired by Yusuf and Sulaiman [48];

  • and complexiton waves obtained by Hosseini et al. [49].

This paper is organized as follows: Sect. 2 reviews some theorems regarding the Bell polynomial. Section 3 provides the HB form, BB transformation, and Lax pair of the governing equation by proving some theorems. Section 4 presents multi-soliton waves of the governing equation by applying the simplified Hirota method. Additionally, this section examines the integrability of the 3D-gKdV equation using the Painlevè test. Section 5 summarizes the achievements of the current paper.

2 Binary Bell polynomials

Bell [27] in 1934 established an exponential polynomial in the form

$${Y}_{nt}\left(y\right)={Y}_{n}\left({y}_{t},\dots ,{y}_{nt}\right)={e}^{-y}{\partial }_{t}^{n}{e}^{y},$$

where \(y={e}^{\alpha t}-{\alpha }_{0}\). Some results of such a definition are

$${Y}_{0}=1, {Y}_{1}={y}_{t}, {Y}_{2}={y}_{2t}+{y}_{t}^{2},\dots .$$

After this, Lambert and his coauthors [28,29,30,31] generalized Bell polynomials for multi-variable function \(f=f\left({x}_{1},\dots ,{x}_{l}\right)\) as follows

$${Y}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left(f\right)={e}^{-f}{\partial }_{{x}_{1}}^{{n}_{1}}\dots {\partial }_{{x}_{l}}^{{n}_{l}}{e}^{f}.$$

When \(f=f\left(x,t\right)\), the generalized Bell polynomials are defined as

$${Y}_{x}\left(f\right)={f}_{x}, {Y}_{2x}\left(f\right)={f}_{2x}+{f}_{x}^{2}, {Y}_{xt}\left(f\right)={f}_{xt}+{f}_{x}{f}_{t},\dots .$$

Accordingly, due to the above definition, the multi-dimensional binary Bell polynomials can be given as

$${\mathcal{Y}}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left(v,w\right)={\left.{Y}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left(f\right)\right|}_{f\begin{array}{c} \\ {r}_{1}{x}_{1},\dots ,{r}_{l}{x}_{l}=\left\{\begin{array}{c}{v}_{{r}_{1}{x}_{1},\dots ,{r}_{l}{x}_{l}}, \\ {w}_{{r}_{1}{x}_{1},\dots ,{r}_{l}{x}_{l}},\end{array}\right.\begin{array}{c} {r}_{1}+{r}_{2}+\dots +{r}_{l} \,\, \text{is \,odd},\\ {r}_{1}+{r}_{2}+\dots +{r}_{l} \,\, \text{is \; even},\end{array}\end{array}}$$

where \({r}_{k}=\text{0,1},\dots ,{n}_{k}, k=\text{0,1},\dots ,l\). Some results of such a definition are

$${\mathcal{Y}}_{x}\left(v\right)={v}_{x}, {\mathcal{Y}}_{2x}\left(v,w\right)={w}_{2x}+{v}_{x}^{2}, {\mathcal{Y}}_{xt}\left(v,w\right)={w}_{xt}+{v}_{x}{v}_{t},\dots .$$

Theorem 1

(See. [29]) The binary Bell polynomial \({\mathcal{Y}}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left(v,w\right)\) and the standard Hirota bilinear equation \({D}_{{x}_{1}}^{{n}_{1}}\dots {D}_{{x}_{l}}^{{n}_{l}}F.G\) have the following connection

$${\mathcal{Y}}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left(v=\text{ln}\left(F/G\right),w=\text{ln}\left(FG\right)\right)={\left(FG\right)}^{-1}{D}_{{x}_{1}}^{{n}_{1}}\dots {D}_{{x}_{l}}^{{n}_{l}}F\cdot G,$$
(2)

where \({n}_{1}+{n}_{2}+\dots +{n}_{l}\ge 1\), and Hirota’s bilinear operators are given by

$${D}_{{x}_{1}}^{{n}_{1}}\dots {D}_{{x}_{l}}^{{n}_{l}}F.G={\left({\partial }_{{x}_{1}}-{\partial }_{{x}_{1}^{\prime}}\right)}^{{n}_{1}}\dots {\left({\partial }_{{x}_{l}}-{\partial }_{{x}_{l}^{\prime}}\right)}^{{n}_{l}}F\left({x}_{1},{x}_{2},\dots ,{x}_{l}\right)\times {\left.G\left({x}_{1}^{\prime},{x}_{2}^{\prime},\dots ,{x}_{l}^{\prime}\right)\right|}_{{x}_{1}^{\prime}={x}_{1},\dots ,{x}_{l}^{\prime}={x}_{l}}.$$

In the special case, Eq. (2) for \(F=G\) becomes

$${G}^{-2}{D}_{{x}_{1}}^{{n}_{1}}\dots {D}_{{x}_{l}}^{{n}_{l}}G.G={\mathcal{Y}}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left(\text{0,2ln}\left(G\right)\right)=\left\{\begin{array}{c}0, \\ {P}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left(q\right),\end{array}\right.\begin{array}{c} {n}_{1}+{n}_{2}+\dots +{n}_{l} \,\, \text{is odd},\\ {n}_{1}+{n}_{2}+\dots +{n}_{l} \,\, \text{is even},\end{array}$$

in which the\(P\)-polynomials are

$${P}_{2x}\left(q\right)={q}_{2x}, {P}_{xt}\left(q\right)={q}_{xt}, {P}_{4x}\left(q\right)={q}_{4x}+3{q}_{2x}^{2}, {P}_{6x}\left(q\right)={q}_{6x}+15{q}_{2x}{q}_{4x}+15{q}_{2x}^{3},\dots .$$

Proposition 1

(See [29]) For \({\mathcal{Y}}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left(v,w\right)\), we have

$${\left(FG\right)}^{-1}{D}_{{x}_{1}}^{{n}_{1}}\dots {D}_{{x}_{l}}^{{n}_{l}}F.G={\mathcal{Y}}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}{\left.\left(v,w\right)\right|}_{v={\text{ln}}\left(F/G\right),w={\text{ln}}\left(FG\right)}={\mathcal{Y}}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}{\left.\left(v,v+q\right)\right|}_{v={\text{ln}}\left(F/G\right),q=2{\text{ln}}\left(G\right)}=\sum_{{n}_{1}+\dots +{n}_{l}= {\text{even}}}\sum_{{r}_{1}=0}^{{n}_{1}}\dots \sum_{{r}_{l}=0}^{{n}_{l}}\prod_{i=1}^{l}\left(\begin{array}{c}{n}_{i}\\ {r}_{i}\end{array}\right){P}_{{r}_{1}{x}_{1},\dots ,{r}_{l}{x}_{l}}\left(q\right){Y}_{\left({n}_{1}-{r}_{1}\right){x}_{1},\dots ,\left({n}_{l}-{r}_{l}\right){x}_{l}}\left(v\right).$$

Proposition 2

(See [29]) For \(v=\text{ln}\left(\psi \right)\) where \(\psi =F/G\), the \(Y\)-polynomial can be expressed as

$${{Y}_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}\left.\left(v\right)\right|}_{v=\text{ln}\left(\psi \right)}=\frac{{\psi }_{{n}_{1}{x}_{1},\dots ,{n}_{l}{x}_{l}}}{\psi }.$$

3 Governing equation and its HB form, BB transformation, and Lax pair

The present section formally constructs the HB form, BB transformation, and Lax pair of the governing equation.

Theorem 2

Using the transformation

$$u=2{\left(\text{ln}\left(G\right)\right)}_{xx},$$

Equation (1) can be bilinearized into

$$\left({D}_{x}{D}_{t}+{c}_{1}{D}_{x}^{2}+{c}_{2}{D}_{x}{D}_{y}+{c}_{3}{D}_{x}{D}_{z}+{D}_{x}^{6}\right)G\cdot G=0.$$

Proof

To discover the bilinear form of Eq. (1), a potential field \(q\) is considered as follows

$$u=c\left(t\right){q}_{2x},$$
(3)

where \(c(t)\) is an appropriate function that connects the governing equation with \(P\)-polynomials. Setting Eq. (3) in Eq. (1) leads to

$$c\left(t\right){q}_{2xt}+{c}_{1}c\left(t\right){q}_{3x}+{c}_{2}c\left(t\right){q}_{2xy}+{c}_{3}c\left(t\right){q}_{2xz}+{{c}^{\prime}}\left(t\right){q}_{2x}+c\left(t\right){q}_{7x}+15{c}^{2}\left(t\right){q}_{2x}{q}_{5x}+15{c}^{2}\left(t\right){q}_{3x}{q}_{4x}+45{c}^{3}\left(t\right){q}_{2x}^{2}{q}_{3x}=0.$$

Integrating with respect to \(x\) gives

$$c\left(t\right){q}_{xt}+{c}_{1}c\left(t\right){q}_{2x}+{c}_{2}c\left(t\right){q}_{xy}+{c}_{3}c\left(t\right){q}_{xz}+{{c}^{\prime}}\left(t\right){q}_{x}+15{c}^{3}\left(t\right){q}_{2x}^{3}+15{c}^{2}\left(t\right){q}_{2x}{q}_{4x}+c\left(t\right){q}_{6x}=0.$$

Comparing \(15{c}^{3}\left(t\right){q}_{2x}^{3}+15{c}^{2}\left(t\right){q}_{2x}{q}_{4x}+c\left(t\right){q}_{6x}\) with \({P}_{6x}\left(q\right)\) yields \(c(t)=1\). Now, the resulting equation can be written in \(P\)-polynomials as

$${P}_{xt}\left(q\right)+{c}_{1}{P}_{2x}\left(q\right)+{c}_{2}{P}_{xy}\left(q\right)+{c}_{3}{P}_{xz}\left(q\right)+{P}_{6x}\left(q\right)=0.$$
(4)

Using the transformation

$$q=2\text{ln}\left(G\right)\iff u={q}_{2x}=2{\left(\text{ln}\left(G\right)\right)}_{xx},$$

and Eq. (4), the bilinear form of Eq. (1) is represented as

$$\left({D}_{x}{D}_{t}+{c}_{1}{D}_{x}^{2}+{c}_{2}{D}_{x}{D}_{y}+{c}_{3}{D}_{x}{D}_{z}+{D}_{x}^{6}\right)G\cdot G=0,$$
(5)

or

$$2\left(G{G}_{xt}-{G}_{x}{G}_{t}\right)+2{c}_{1}\left(G{G}_{2x}-{G}_{x}^{2}\right)+2{c}_{2}\left(G{G}_{xy}-{G}_{x}{G}_{y}\right)+2{c}_{3}\left(G{G}_{xz}-{G}_{x}{G}_{z}\right)+2G{G}_{6x}-12{G}_{x}{G}_{5x}+30{G}_{2x}{G}_{4x}-20{G}_{3x}^{2}=0.$$
(6)

□

Theorem 3

Let \(G\) be a solution of the bilinear Eq. (5), then \(\overline{G }\) satisfies

$$\left({D}_{x}^{3}-\lambda \right)G\cdot \overline{G }=0,$$
$$\left(2{D}_{t}+2{c}_{1}{D}_{x}+2{c}_{2}{D}_{y}+2{c}_{3}{D}_{z}-15\lambda {D}_{x}^{2}-3{D}_{x}^{5}\right)G\cdot \overline{G }=0,$$

which is called the bilinear Bäcklund transformation of Eq. (1).

Proof

To construct the bilinear Bäcklund transformation of Eq. (1), let us consider

$$q=2\text{ln}\left(G\right), \overline{q }=2\text{ln}\left(\overline{G }\right),$$

and

$$E\left(q\right)={q}_{xt}+{c}_{1}{q}_{2x}+{c}_{2}{q}_{xy}+{c}_{3}{q}_{xz}+15{q}_{2x}^{3}+15{q}_{2x}{q}_{4x}+{q}_{6x}=0.$$

Due to the two-field condition, we have

$$E\left(\overline{q }\right)-E\left(q\right)={\left(\overline{q }-q\right)}_{xt}+{c}_{1}{\left(\overline{q }-q\right)}_{2x}+{c}_{2}{\left(\overline{q }-q\right)}_{xy}+{c}_{3}{\left(\overline{q }-q\right)}_{xz}+15{\overline{q} }_{2x}^{3}-15{q}_{2x}^{3}+15{\overline{q} }_{2x}{\overline{q} }_{4x}-15{q}_{2x}{q}_{4x}+{\left(\overline{q }-q\right)}_{6x}=0.$$

By considering new variables

$$v=\frac{1}{2}\left(\overline{q }-q\right)=\text{ln}\left(\frac{\overline{G}}{G }\right), w=\frac{1}{2}\left(\overline{q }+q\right)=\text{ln}\left(G\overline{G }\right),$$

the two-field condition becomes

$$E\left(\overline{q }\right)-E\left(q\right)=E\left(w+v\right)-E\left(w-v\right)=2({v}_{xt}+{c}_{1}{v}_{2x}+{c}_{2}{v}_{xy}+{c}_{3}{v}_{xz}+{v}_{6x}+15{v}_{2x}^{3}+15{v}_{2x}{w}_{4x}+15{v}_{4x}{w}_{2x}+45{v}_{2x}{w}_{2x}^{2})=2{\partial }_{x}\left({\mathcal{Y}}_{t}\left(v,w\right)+{c}_{1}{\mathcal{Y}}_{x}\left(v,w\right)+{c}_{2}{\mathcal{Y}}_{y}\left(v,w\right)+{c}_{3}{\mathcal{Y}}_{z}\left(v,w\right)+{\mathcal{Y}}_{5x}\left(v,w\right)\right)+R\left(v,w\right),$$

where

$$R\left(v,w\right)=30{v}_{2x}^{3}+20{v}_{2x}{w}_{4x}+10{v}_{4x}{w}_{2x}+60{v}_{2x}{w}_{2x}^{2}-10{v}_{2x}{v}_{x}^{4}-60{v}_{x}^{2}{v}_{2x}{w}_{2x}-20{v}_{x}^{3}{w}_{3x}-40{v}_{x}{v}_{2x}{v}_{3x}-20{v}_{x}^{2}{v}_{4x}-60{v}_{x}{w}_{2x}{w}_{3x}-10{v}_{x}{w}_{5x}-20{v}_{3x}{w}_{3x}.$$

Now, suppose that \({\mathcal{Y}}_{3x}=\lambda .\) This yields

$$R\left(v,w\right)=30{v}_{2x}^{3}+30{v}_{2x}{w}_{2x}^{2}+20{v}_{2x}{w}_{4x}+90{v}_{x}^{2}{w}_{2x}{v}_{2x}-30{v}_{x}{w}_{2x}{w}_{3x}+90{v}_{2x}{v}_{x}^{4}+60{v}_{x}^{3}{w}_{3x}-10{v}_{x}{w}_{5x}-20\lambda \left({w}_{3x}+2{v}_{x}{v}_{2x}\right),$$

or

$$R\left(v,w\right)=-5{\partial }_{x}\left({\mathcal{Y}}_{5x}+3\lambda {\mathcal{Y}}_{2x}\right).$$

Accordingly, the following coupled system of \(\mathcal{Y}\)-polynomials can be constructed

$${\mathcal{Y}}_{3x}\left(v,w\right)=\lambda ,$$
$$2{\mathcal{Y}}_{t}\left(v,w\right)+2{c}_{1}{\mathcal{Y}}_{x}\left(v,w\right)+2{c}_{2}{\mathcal{Y}}_{y}\left(v,w\right)+2{c}_{3}{\mathcal{Y}}_{z}\left(v,w\right)-15\lambda {\mathcal{Y}}_{2x}\left(v,w\right)-3{\mathcal{Y}}_{5x}\left(v,w\right)=0.$$

Finally, the bilinear Bäcklund transformation of Eq. (1) is

$$\left({D}_{x}^{3}-\lambda \right)G\cdot \overline{G }=0,$$
$$\left(2{D}_{t}+2{c}_{1}{D}_{x}+2{c}_{2}{D}_{y}+2{c}_{3}{D}_{z}-15\lambda {D}_{x}^{2}-3{D}_{x}^{5}\right)G\cdot \overline{G }=0.$$

□

Theorem 4

The governing equation admits the following Lax pair

$${L}_{1}\psi =\left({\partial }_{x}^{3}+3u{\partial }_{x}\right)\psi =\lambda \psi ,$$
$$\left({\partial }_{t}+{L}_{2}\right)\psi =\left({\partial }_{t}+{c}_{2}{\partial }_{y}+{c}_{3}{\partial }_{z}+\left(9{u}^{2}-3{u}_{2x}+{c}_{1}\right){\partial }_{x}+9\left({u}_{x}-\lambda \right){\partial }_{x}^{2}-18\lambda u\right)\psi =0.$$

Proof

Using \(v=\text{ln}\left(\psi \right)\) and the propositions, we find

$${\mathcal{Y}}_{t}=\frac{{\psi }_{t}}{\psi }, {\mathcal{Y}}_{x}=\frac{{\psi }_{x}}{\psi }, {\mathcal{Y}}_{y}=\frac{{\psi }_{y}}{\psi }, {\mathcal{Y}}_{z}=\frac{{\psi }_{z}}{\psi }, {\mathcal{Y}}_{2x}={q}_{2x}+\frac{{\psi }_{2x}}{\psi }, {\mathcal{Y}}_{3x}=3{q}_{2x}\frac{{\psi }_{x}}{\psi }+\frac{{\psi }_{3x}}{\psi }, {\mathcal{Y}}_{5x}=5\left({q}_{4x}+3{q}_{2x}^{2}\right)\frac{{\psi }_{x}}{\psi }+10{q}_{2x}\frac{{\psi }_{3x}}{\psi }+\frac{{\psi }_{5x}}{\psi }.$$

These findings generate the new coupled system

$${L}_{1}\psi =\left({\partial }_{x}^{3}+3{q}_{2x}{\partial }_{x}\right)\psi =\lambda \psi ,$$
$$\left({\partial }_{t}+{L}_{2}\right)\psi =\left({\partial }_{t}+{c}_{2}{\partial }_{y}+{c}_{3}{\partial }_{z}+\left(9{q}_{2x}^{2}-3{q}_{4x}+{c}_{1}\right){\partial }_{x}+9\left({q}_{3x}-\lambda \right){\partial }_{x}^{2}-18\lambda {q}_{2x}\right)\psi =0,$$

or the Lax pair of Eq. (1), i.e.

$${L}_{1}\psi =\left({\partial }_{x}^{3}+3u{\partial }_{x}\right)\psi =\lambda \psi ,$$
$$\left({\partial }_{t}+{L}_{2}\right)\psi =\left({\partial }_{t}+{c}_{2}{\partial }_{y}+{c}_{3}{\partial }_{z}+\left(9{u}^{2}-3{u}_{2x}+{c}_{1}\right){\partial }_{x}+9\left({u}_{x}-\lambda \right){\partial }_{x}^{2}-18\lambda u\right)\psi =0,$$

which \({q}_{2x}=u\). It can be checked that the integrability condition

$$\left({L}_{1}-\lambda ,{\partial }_{t}+{L}_{2}\right)\psi =0,$$

is satisfied if \(u\) is a solution of the governing equation.□

4 Governing equation and its different wave structures

In the present section, the authors deal with the governing equation to construct its different wave structures. Maple as a vital software has been employed to tackle the computations.

4.1 Multiple-soliton waves and integrability

To derive the single-soliton wave of Eq. (1), we define

$$u={e}^{{\theta }_{i}}, {\theta }_{i}={k}_{i}x+{r}_{i}y+{s}_{i}z+{\omega }_{i}t,$$

and substitute it into

$${u}_{t}+{c}_{1}{u}_{x}+{c}_{2}{u}_{y}+{c}_{3}{u}_{z}+{u}_{5x}=0.$$

This gives

$$\left({k}_{i}^{5}+{c}_{1}{k}_{i}+{c}_{2}{r}_{i}+{c}_{3}{s}_{i}+{\omega }_{i}\right){{e}}^{{k}_{i}x+{r}_{i}y+{s}_{i}z+{\omega }_{i}t}=0.$$

Thus, the dispersion relation \({\omega }_{i}\) is derived as

$${\omega }_{i}=-\left({k}_{i}^{5}+{c}_{1}{k}_{i}+{c}_{2}{r}_{i}+{c}_{3}{s}_{i}\right).$$

Now, the phase variable \({\theta }_{i}\) can be represented as

$${\theta }_{i}={k}_{i}x+{r}_{i}y+{s}_{i}z-\left({k}_{i}^{5}+{c}_{1}{k}_{i}+{c}_{2}{r}_{i}+{c}_{3}{s}_{i}\right)t.$$

Finally, a single-soliton wave to Eq. (1) can be established as

$$u=2{\left(\text{ln}\left(G\right)\right)}_{xx},$$

where

$$G=1+{e}^{{\theta }_{1}}, {\theta }_{1}={k}_{1}x+{r}_{1}y+{s}_{1}z-\left({k}_{1}^{5}+{c}_{1}{k}_{1}+{c}_{2}{r}_{1}+{c}_{3}{s}_{1}\right)t.$$

To detect the double-soliton wave, we consider

$$G=1+{e}^{{\theta }_{1}}+{e}^{{\theta }_{2}}+{a}_{12}{e}^{{\theta }_{1}+{\theta }_{2}},$$

where the phase variables are given as

$${\theta }_{1}={k}_{1}x+{r}_{1}y+{s}_{1}z-\left({k}_{1}^{5}+{c}_{1}{k}_{1}+{c}_{2}{r}_{1}+{c}_{3}{s}_{1}\right)t,$$
$${\theta }_{2}={k}_{2}x+{r}_{2}y+{s}_{2}z-\left({k}_{2}^{5}+{c}_{1}{k}_{2}+{c}_{2}{r}_{2}+{c}_{3}{s}_{2}\right)t,$$

and the phase shift \({a}_{12}\) can be acquired using some mathematical operations as

$${a}_{12}=\frac{{k}_{1}^{4}-3{k}_{2}{k}_{1}^{3}+4{k}_{1}^{2}{k}_{2}^{2}-3{k}_{1}{k}_{2}^{3}+{k}_{2}^{4}}{{k}_{1}^{4}+3{k}_{2}{k}_{1}^{3}+4{k}_{1}^{2}{k}_{2}^{2}+3{k}_{1}{k}_{2}^{3}+{k}_{2}^{4}}.$$

Accordingly, a double-soliton wave to Eq. (1) can be constructed as

$$u=2{\left(\text{ln}\left(G\right)\right)}_{xx}, G=1+{e}^{{\theta }_{1}}+{e}^{{\theta }_{2}}+{a}_{12}{e}^{{\theta }_{1}+{\theta }_{2}},$$

where

$${\theta }_{1}={k}_{1}x+{r}_{1}y+{s}_{1}z-\left({k}_{1}^{5}+{c}_{1}{k}_{1}+{c}_{2}{r}_{1}+{c}_{3}{s}_{1}\right)t,$$
$${\theta }_{2}={k}_{2}x+{r}_{2}y+{s}_{2}z-\left({k}_{2}^{5}+{c}_{1}{k}_{2}+{c}_{2}{r}_{2}+{c}_{3}{s}_{2}\right)t,$$
$${a}_{12}=\frac{{k}_{1}^{4}-3{k}_{2}{k}_{1}^{3}+4{k}_{1}^{2}{k}_{2}^{2}-3{k}_{1}{k}_{2}^{3}+{k}_{2}^{4}}{{k}_{1}^{4}+3{k}_{2}{k}_{1}^{3}+4{k}_{1}^{2}{k}_{2}^{2}+3{k}_{1}{k}_{2}^{3}+{k}_{2}^{4}}.$$

To find the triple-soliton wave, the authors check the three-soliton condition, i.e. [50, 51]

$$\sum_{{\mu }_{1},{\mu }_{2},{\mu }_{3}=\pm 1}P\left({\mu }_{1}{V}_{1}+{\mu }_{2}{V}_{2}+{\mu }_{3}{V}_{3}\right)P\left({\mu }_{1}{V}_{1}-{\mu }_{2}{V}_{2}\right)P\left({\mu }_{2}{V}_{2}-{\mu }_{3}{V}_{3}\right)P\left({\mu }_{1}{V}_{1}-{\mu }_{3}{V}_{3}\right)=2\sum_{\left({\mu }_{1},{\mu }_{2},{\mu }_{3}\right)\in S}P\left({\mu }_{1}{V}_{1}+{\mu }_{2}{V}_{2}+{\mu }_{3}{V}_{3}\right)P\left({\mu }_{1}{V}_{1}-{\mu }_{2}{V}_{2}\right)P\left({\mu }_{2}{V}_{2}-{\mu }_{3}{V}_{3}\right)P\left({\mu }_{1}{V}_{1}-{\mu }_{3}{V}_{3}\right),$$

in which

$$P=XT+{c}_{1}{X}^{2}+{c}_{2}XY+{c}_{3}XZ+{X}^{6},$$
$${V}_{i}=\left({k}_{i},{r}_{i},{s}_{i},{\omega }_{i}\right),$$
$$S=\left\{\left(\text{1,1},1\right),\left(\text{1,1},-1\right),\left(1,-\text{1,1}\right),\left(-\text{1,1},1\right)\right\},$$

is equal to zero or not. It is found that

$$2\sum_{\left({\mu }_{1},{\mu }_{2},{\mu }_{3}\right)\in S}P\left({\mu }_{1}{V}_{1}+{\mu }_{2}{V}_{2}+{\mu }_{3}{V}_{3}\right)P\left({\mu }_{1}{V}_{1}-{\mu }_{2}{V}_{2}\right)P\left({\mu }_{2}{V}_{2}-{\mu }_{3}{V}_{3}\right)P\left({\mu }_{1}{V}_{1}-{\mu }_{3}{V}_{3}\right)=2\left({e}_{1}+{e}_{2}+{e}_{3}+{e}_{4}\right)$$

where

$${e}_{1}=-625{k}_{3}^{2}{k}_{1}^{2}\left({k}_{1}^{2}-{k}_{1}{k}_{3}+{k}_{3}^{2}\right){\left({k}_{2}-{k}_{3}\right)}^{2}\left({k}_{2}^{2}-{k}_{3}{k}_{2}+{k}_{3}^{2}\right)\left({k}_{1}^{2}+\left({k}_{2}+{k}_{3}\right){k}_{1}+{k}_{2}^{2}+{k}_{3}{k}_{2}+{k}_{3}^{2}\right){\left({k}_{1}-{k}_{3}\right)}^{2}{\left({k}_{1}-{k}_{2}\right)}^{2}\left({k}_{1}^{2}-{k}_{1}{k}_{2}+{k}_{2}^{2}\right){k}_{2}^{2}\left({k}_{2}+{k}_{3}\right)\left({k}_{1}+{k}_{2}+{k}_{3}\right)\left({k}_{1}+{k}_{3}\right)\left({k}_{1}+{k}_{2}\right),$$
$${e}_{2}=-625\left({k}_{1}+{k}_{2}-{k}_{3}\right){k}_{3}^{2}{k}_{1}^{2}\left({k}_{2}-{k}_{3}\right)\left({k}_{2}^{2}+{k}_{3}{k}_{2}+{k}_{3}^{2}\right)\left({k}_{1}^{2}+{k}_{1}{k}_{3}+{k}_{3}^{2}\right)\left({k}_{1}-{k}_{3}\right){\left({k}_{1}-{k}_{2}\right)}^{2}\left({k}_{1}^{2}-{k}_{1}{k}_{2}+{k}_{2}^{2}\right){k}_{2}^{2}\left({k}_{1}^{2}+\left({k}_{2}-{k}_{3}\right){k}_{1}+{k}_{2}^{2}-{k}_{3}{k}_{2}+{k}_{3}^{2}\right){\left({k}_{2}+{k}_{3}\right)}^{2}{\left({k}_{1}+{k}_{3}\right)}^{2}\left({k}_{1}+{k}_{2}\right),$$
$${e}_{3}=625\left({k}_{1}^{2}+\left(-{k}_{2}+{k}_{3}\right){k}_{1}+{k}_{2}^{2}-{k}_{3}{k}_{2}+{k}_{3}^{2}\right){k}_{3}^{2}{k}_{1}^{2}\left({k}_{1}^{2}-{k}_{1}{k}_{3}+{k}_{3}^{2}\right)\left({k}_{2}-{k}_{3}\right)\left({k}_{2}^{2}+{k}_{3}{k}_{2}+{k}_{3}^{2}\right){\left({k}_{1}-{k}_{3}\right)}^{2}\left({k}_{1}-{k}_{2}\right)\left({k}_{1}^{2}+{k}_{1}{k}_{2}+{k}_{2}^{2}\right){k}_{2}^{2}{\left({k}_{2}+{k}_{3}\right)}^{2}\left({k}_{1}+{k}_{3}\right){\left({k}_{1}+{k}_{2}\right)}^{2}\left({k}_{1}-{k}_{2}+{k}_{3}\right),$$
$${e}_{4}=625{k}_{3}^{2}{k}_{1}^{2}{\left({k}_{2}-{k}_{3}\right)}^{2}\left({k}_{2}^{2}-{k}_{3}{k}_{2}+{k}_{3}^{2}\right)\left({k}_{1}^{2}+{k}_{1}{k}_{3}+{k}_{3}^{2}\right)\left({k}_{1}-{k}_{3}\right)\left({k}_{1}-{k}_{2}\right)\left({k}_{1}^{2}+{k}_{1}{k}_{2}+{k}_{2}^{2}\right){k}_{2}^{2}\left({k}_{2}+{k}_{3}\right){\left({k}_{1}+{k}_{3}\right)}^{2}\left({k}_{1}^{2}+\left(-{k}_{2}-{k}_{3}\right){k}_{1}+{k}_{2}^{2}+{k}_{3}{k}_{2}+{k}_{3}^{2}\right)\left({k}_{1}-{k}_{2}-{k}_{3}\right){\left({k}_{1}+{k}_{2}\right)}^{2},$$

is equal to zero. So, the three-soliton condition holds, and the triple-soliton wave exists for the governing equation.

Now, a triple-soliton wave to Eq. (1) is extracted as

$$u=2{\left(\text{ln}\left(G\right)\right)}_{xx}, G=1+{e}^{{\theta }_{1}}+{e}^{{\theta }_{2}}+{e}^{{\theta }_{3}}+{a}_{12}{e}^{{\theta }_{1}+{\theta }_{2}}+{a}_{13}{e}^{{\theta }_{1}+{\theta }_{3}}+{a}_{23}{e}^{{\theta }_{2}+{\theta }_{3}}+{a}_{12}{a}_{13}{a}_{23}{e}^{{\theta }_{1}+{\theta }_{2}+{\theta }_{3}},$$

where

$${\theta }_{1}={k}_{1}x+{r}_{1}y+{s}_{1}z-\left({k}_{1}^{5}+{c}_{1}{k}_{1}+{c}_{2}{r}_{1}+{c}_{3}{s}_{1}\right)t,$$
$${\theta }_{2}={k}_{2}x+{r}_{2}y+{s}_{2}z-\left({k}_{2}^{5}+{c}_{1}{k}_{2}+{c}_{2}{r}_{2}+{c}_{3}{s}_{2}\right)t,$$
$${\theta }_{3}={k}_{3}x+{r}_{3}y+{s}_{3}z-\left({k}_{3}^{5}+{c}_{1}{k}_{3}+{c}_{2}{r}_{3}+{c}_{3}{s}_{3}\right)t,$$
$${a}_{12}=\frac{{k}_{1}^{4}-3{k}_{2}{k}_{1}^{3}+4{k}_{1}^{2}{k}_{2}^{2}-3{k}_{1}{k}_{2}^{3}+{k}_{2}^{4}}{{k}_{1}^{4}+3{k}_{2}{k}_{1}^{3}+4{k}_{1}^{2}{k}_{2}^{2}+3{k}_{1}{k}_{2}^{3}+{k}_{2}^{4}},$$
$${a}_{13}=\frac{{k}_{1}^{4}-3{k}_{3}{k}_{1}^{3}+4{k}_{1}^{2}{k}_{3}^{2}-3{k}_{1}{k}_{3}^{3}+{k}_{3}^{4}}{{k}_{1}^{4}+3{k}_{3}{k}_{1}^{3}+4{k}_{1}^{2}{k}_{3}^{2}+3{k}_{1}{k}_{3}^{3}+{k}_{3}^{4}},$$
$${a}_{23}=\frac{{k}_{2}^{4}-3{k}_{3}{k}_{2}^{3}+4{k}_{2}^{2}{k}_{3}^{2}-3{k}_{2}{k}_{3}^{3}+{k}_{3}^{4}}{{k}_{2}^{4}+3{k}_{3}{k}_{2}^{3}+4{k}_{2}^{2}{k}_{3}^{2}+3{k}_{2}{k}_{3}^{3}+{k}_{3}^{4}}.$$

Figure 1 demonstrates double- and triple-soliton waves and their energy distributions for

$$\text{Set }1: \left\{{c}_{1}=1,{c}_{2}=1,{c}_{3}=1,{k}_{1}=1, {r}_{1}=-2, {s}_{1}=2, {k}_{2}=-0.5, {r}_{2}=0.75, {s}_{2}=-1.75, y=0.5, t=0\right\},$$
$$\text{Set }2: \left\{{c}_{1}=1,{c}_{2}=1,{c}_{3}=1,{k}_{1}=1, {r}_{1}=-2, {s}_{1}=2, {k}_{2}=-0.5, {r}_{2}=0.75, {s}_{2}=-1.75, y=0.5, t=2\right\},$$
$$\text{Set }3: \left\{{c}_{1}=1,{c}_{2}=1,{c}_{3}=1,{k}_{1}=1, {r}_{1}=-2, {s}_{1}=2, {k}_{2}=-2, {r}_{2}=1, {s}_{2}=3,{k}_{3}=3, {r}_{3}=2, {s}_{3}=1, y=0.5, t=0\right\},$$
$$\text{Set }4: \left\{{c}_{1}=1,{c}_{2}=1,{c}_{3}=1,{k}_{1}=1, {r}_{1}=-2, {s}_{1}=2, {k}_{2}=-2, {r}_{2}=1, {s}_{2}=3,{k}_{3}=3, {r}_{3}=2, {s}_{3}=1, y=0.5, t=0.03\right\},$$

respectively. These figures signify the collision of two and three bright waves.

Fig. 1
figure 1

The double-soliton wave for a Set 1, b Set 2; The triple-soliton wave for c Set 3, d Set 4

Full size image

Now, the authors show that the governing equation is integrable. For this purpose, let’s consider the following generalized Laurent expansion

$$u={\psi }^{\alpha }\sum_{j=0}^{\infty }{u}_{j}{\psi }^{j}, \psi =\psi \left(x,y,z,t\right),$$

as the solution of Eq. (1). The leading order analysis yields

$$\alpha =-2, {u}_{0}=-4{\psi }_{x}^{2},$$
$$\alpha =-2, {u}_{0}=-2{\psi }_{x}^{2}.$$

Now, we substitute

$$u={u}_{0}{\psi }^{-2}+\sum_{j=1}^{\infty }{u}_{j}{\psi }^{j-2}$$

into Eq. (1). Such an operation yields

$$\left(j+2\right)\left(j+1\right)\left(j-5\right)\left(j-6\right)\left(j-12\right){u}_{j}{\psi }_{x}^{5}=F\left({u}_{j-1}, {u}_{j-2}, \dots , {u}_{0}, {\psi }_{t},{\psi }_{x},\dots \right),$$
$$\left(j+1\right)\left(j-2\right)\left(j-3\right)\left(j-6\right)\left(j-10\right){u}_{j}{\psi }_{x}^{5}=F({u}_{j-1}, {u}_{j-2}, \dots , {u}_{0}, {\psi }_{t},{\psi }_{x},\dots ).$$

The resonances of the first branch are

$$j=-2, -1, 5, 6, 12,$$

while resonances of the second branch are

$$j=-1, 2, 3, 6, 10,$$

where the resonance at \(j=-1\) is related to the singular manifold \(\psi \left(x,y,z,t\right)=0.\) Based on Kruskal’s method with \(\psi \left(x, y,z, t\right)=x-\varphi (y,z, t)\), the coefficients of the first branch are found as

$$j=0: {u}_{0}=-4,$$
$$j=1: {u}_{1}=0,$$
$$j=2:{ u}_{2}=0,$$
$$j=3:{ u}_{3}=0,$$
$$j=4:{ u}_{4}=\frac{1}{60}({c}_{1}-{\varphi }_{t}-{c}_{2}{\varphi }_{y}-{c}_{3}{\varphi }_{z}),$$
$$j=5: {u}_{5}={u}_{5},$$
$$j=6:{ u}_{6}={u}_{6},$$
$$j=7:{ u}_{7}=0,$$
$$j=8: {u}_{8}=-\frac{1}{43200}\left({\varphi }_{t}^{2}+{c}_{2}^{2}{\varphi }_{y}^{2}+{c}_{3}^{2}{\varphi }_{z}^{2}+{c}_{1}^{2}\right)+\frac{1}{21600}\left({c}_{1}{\varphi }_{t}+{{c}_{1}c}_{2}{\varphi }_{y}+{{c}_{1}c}_{3}{\varphi }_{z}-({c}_{2}{c}_{3}{\varphi }_{y}{\varphi }_{z}+{c}_{2}{\varphi }_{t}{\varphi }_{y}+{c}_{3}{\varphi }_{t}{\varphi }_{z})\right),$$
$$j=9: {u}_{9}=\frac{1}{440}{u}_{5}\left({\varphi }_{t}+{c}_{2}{\varphi }_{y}+{c}_{3}{\varphi }_{z}-{c}_{1}\right)-\frac{1}{118800}\left({c}_{2}{\varphi }_{yt}+{c}_{3}{\varphi }_{zt}+{c}_{2}{c}_{3}{\varphi }_{yz}\right)-\frac{1}{237600}\left({\varphi }_{tt}+{c}_{2}^{2}{\varphi }_{yy}+{c}_{3}^{2}{\varphi }_{zz}\right),$$
$$j=10: {u}_{10}=\frac{1}{880}{u}_{6}\left({\varphi }_{t}+{c}_{2}{\varphi }_{y}+{c}_{3}{\varphi }_{z}-{c}_{1}\right)+\frac{1}{5280}\left({u}_{{5}_{t}}+{c}_{2}{u}_{{5}_{y}}+{c}_{3}{u}_{{5}_{z}}\right)-\frac{3}{44}{u}_{5}^{2},$$
$$j=11: {u}_{11}=-\frac{5}{52}{u}_{5}{u}_{6}+\frac{1}{4680}\left({u}_{{6}_{t}}+{c}_{2}{u}_{{6}_{y}}+{c}_{3}{u}_{{6}_{z}}\right),$$
$$j=12: {u}_{12}={u}_{12},$$

whereas the coefficients for the second branch are

$$j=0: {u}_{0}=-2,$$
$$j=1: {u}_{1}=0,$$
$$j=2:{ u}_{2}={u}_{2},$$
$$j=3:{ u}_{3}={u}_{3},$$
$$j=4:{ u}_{4}=-\frac{3}{2}{u}_{2}^{2}+\frac{1}{30}\left({\varphi }_{t}+{c}_{2}{\varphi }_{y}+{c}_{3}{\varphi }_{z}-{c}_{1}\right),$$
$$j=5: {u}_{5}={-u}_{2}{u}_{3},$$
$$j=6:{ u}_{6}={u}_{6},$$
$$j=7:{ u}_{7}=\frac{3}{4}{u}_{2}^{2}{u}_{3}-\frac{1}{80}{u}_{3}\left({\varphi }_{t}+{c}_{2}{\varphi }_{y}+{c}_{3}{\varphi }_{z}-{c}_{1}\right)+\frac{1}{480}\left({u}_{{2}_{t}}+{c}_{2}{u}_{{2}_{y}}+{c}_{3}{u}_{{2}_{z}}\right),$$
$$j=8: {u}_{8}=-\frac{3}{8}{u}_{2}^{4}+\frac{1}{60}{u}_{2}^{2}\left({\varphi }_{t}+{c}_{2}{\varphi }_{y}+{c}_{3}{\varphi }_{z}-{c}_{1}\right)+\frac{1}{4}{u}_{2}{u}_{3}^{2}-\frac{1}{5400}\left({\varphi }_{t}^{2}+{c}_{2}^{2}{\varphi }_{y}^{2}+{c}_{3}^{2}{\varphi }_{z}^{2}+{c}_{1}^{2}\right)-\frac{1}{2700}\left({c}_{2}{c}_{3}{\varphi }_{y}{\varphi }_{z}+{c}_{2}{\varphi }_{t}{\varphi }_{y}+{c}_{3}{\varphi }_{t}{\varphi }_{z}-\left({c}_{1}{\varphi }_{t}+{c}_{1}{c}_{2}{\varphi }_{y}+{{c}_{1}c}_{3}{\varphi }_{z}\right)\right)+\frac{1}{1080}\left({u}_{{3}_{t}}+{c}_{2}{u}_{{3}_{y}}+{c}_{3}{u}_{{3}_{z}}\right),$$
$$j=9: {u}_{9}=-\frac{3}{7}{u}_{2}^{3}{u}_{3}+\frac{1}{280}\left(3{u}_{2}{u}_{3}\left({\varphi }_{t}+{c}_{2}{\varphi }_{y}+{c}_{3}{\varphi }_{z}-{c}_{1}\right)\right)+\frac{1}{28}{u}_{3}^{3}-\frac{1}{560}{u}_{2}\left({u}_{{2}_{t}}+{c}_{2}{u}_{{2}_{y}}+{c}_{3}{u}_{{2}_{z}}\right)+\frac{1}{18900}\left({c}_{2}{\varphi }_{ty}+{c}_{3}{\varphi }_{tz}+{c}_{2}{c}_{3}{\varphi }_{yz}\right)+\frac{1}{37800}\left({\varphi }_{tt}+{c}_{2}^{2}{\varphi }_{yy}+{c}_{3}^{2}{\varphi }_{zz}\right),$$
$$j=10: {u}_{10}={u}_{10}.$$

According to the above relations, we see that \({u}_{5}\), \({u}_{6}\), and \({u}_{12}\) in the first branch and \({u}_{2}\), \({u}_{3}\), \({u}_{6}\), and \({u}_{10}\) in the second branch are arbitrary functions. Accordingly, Eq. (1) passes the Painlevè test.

4.2 Lump wave

To detect the lump wave of Eq. (1), the authors take an ansatz as follows

$$G={\left({a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}t+{a}_{5}\right)}^{2}+{\left({a}_{6}x+{a}_{7}y+{a}_{8}z+{a}_{9}t+{a}_{10}\right)}^{2}+{a}_{11},$$
(7)

in which \({a}_{i},i=\text{1,2},\dots ,11\) are unknowns. By considering Eqs. (6) and (7), collecting the expressions, and solving the resulting system, we have

$${a}_{4}=-\left({a}_{1}{c}_{1}+{a}_{2}{c}_{2}+{a}_{3}{c}_{3}\right), {a}_{9}=-\left({a}_{6}{c}_{1}+{a}_{7}{c}_{2}+{a}_{8}{c}_{3}\right).$$

Consequently, a lump wave to Eq. (1) is acquired as

$$u=2{\left(\text{ln}\left(G\right)\right)}_{xx},$$

where

$$G={\left({a}_{1}x+{a}_{2}y+{a}_{3}z-\left({a}_{1}{c}_{1}+{a}_{2}{c}_{2}+{a}_{3}{c}_{3}\right)t+{a}_{5}\right)}^{2}+{\left({a}_{6}x+{a}_{7}y+{a}_{8}z-\left({a}_{6}{c}_{1}+{a}_{7}{c}_{2}+{a}_{8}{c}_{3}\right)t+{a}_{10}\right)}^{2}+{a}_{11},$$

where \({a}_{11}>0\) to guarantee the function \(G\) is positive.

Energy distributions of such a lump wave for

$$\text{Set }1: \left\{{c}_{1}=1,{c}_{2}=1,{c}_{3}=1,{a}_{1}=0.1,{a}_{2}=0.5, {a}_{3}=-0.75, {a}_{5}=0.5, {a}_{6}=0.5,{a}_{7}=0.1, {a}_{8}=0.1, {a}_{10}=-0.35, {a}_{11}=0.25, y=0.5, t=0\right\},$$
$$\text{Set }2: \left\{{c}_{1}=1,{c}_{2}=1,{c}_{3}=1,{a}_{1}=0.1,{a}_{2}=0.5, {a}_{3}=-0.75, {a}_{5}=0.5, {a}_{6}=0.5,{a}_{7}=0.1, {a}_{8}=0.1, {a}_{10}=-0.35, {a}_{11}=0.25,x=0.5,z=0.5\right\},$$

have been represented in Fig. 2. It is figured out that by decreasing/increasing spatial and temporal variables, the lump wave tends to zero.

Fig. 2
figure 2

The lump wave for a Set 1, b Set 2

Full size image

4.3 Lump-soliton wave

To identify the lump-soliton wave of Eq. (1), the authors take an ansatz as follows

$$G={f}^{2}+{g}^{2}+k{e}^{h}+{a}_{11},$$
(8)

where

$$f={a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}t+{a}_{5},$$
$$g={a}_{6}x+{a}_{7}y+{a}_{8}z+{a}_{9}t+{a}_{10},$$
$$h={k}_{1}x+{k}_{2}y+{k}_{3}z+{k}_{4}t.$$

Here, \({a}_{i},i=\text{1,2},\dots ,11\), \({k}_{i},i=\text{1,2,3} ,4\), and \(k\) are unknowns. By considering Eqs. (6) and (8), collecting the expressions, and solving the resulting system, we find

$${a}_{4}=-\left({a}_{1}{c}_{1}+{a}_{2}{c}_{2}+{a}_{3}{c}_{3}\right), {a}_{9}=-\left({a}_{6}{c}_{1}+{a}_{7}{c}_{2}+{a}_{8}{c}_{3}\right), {a}_{10}=0, {a}_{11}={a}_{5}^{2}, {k}_{1}=0, {k}_{4}=-\left({c}_{2}{k}_{2}+{c}_{3}{k}_{3}\right).$$

Consequently, a lump-soliton wave to Eq. (1) is generated as

$$u=2{\left(\text{ln}\left(G\right)\right)}_{xx},$$

where

$$G={\left({a}_{1}x+{a}_{2}y+{a}_{3}z-\left({a}_{1}{c}_{1}+{a}_{2}{c}_{2}+{a}_{3}{c}_{3}\right)t+{a}_{5}\right)}^{2}+{\left({a}_{6}x+{a}_{7}y+{a}_{8}z-\left({a}_{6}{c}_{1}+{a}_{7}{c}_{2}+{a}_{8}{c}_{3}\right)t\right)}^{2}+k{e}^{{k}_{2}y+{k}_{3}z-\left({c}_{2}{k}_{2}+{c}_{3}{k}_{3}\right)t}+{a}_{5}^{2}.$$

The energy distribution of the above lump-soliton wave for

$$\text{Set }1: \left\{{c}_{1}=1,{c}_{2}=1,{c}_{3}=1,{a}_{1}=-0.5,{a}_{2}=1,{a}_{3}=-1.5,{a}_{5}=-1.5,{a}_{6}=-1,{a}_{7}=1,{a}_{8}=-0.1,{k}_{2}=3,{k}_{3}=10,k=1.5,x=0.5,t=-5\right\},$$
$$\text{Set }2: \left\{{c}_{1}=1,{c}_{2}=1,{c}_{3}=1,{a}_{1}=-0.5,{a}_{2}=1,{a}_{3}=-1.5,{a}_{5}=-1.5,{a}_{6}=-1,{a}_{7}=1,{a}_{8}=-0.1,{k}_{2}=3,{k}_{3}=10,k=1.5,x=0.5,t=-4\right\},$$

has been portrayed in Fig. 3.

Fig. 3
figure 3

The lump-soliton wave for a Set 1, b Set 2

Full size image

4.4 Breather wave

To discover the breather wave of Eq. (1), the authors start by considering the following ansatz

$$G={e}^{-kX}+{b}_{0}\text{cos}\left(hY\right)+{b}_{1}{e}^{kX},$$
(9)

and

$$X={a}_{1}x+{a}_{2}y+{a}_{3}z+{a}_{4}t+{a}_{5},$$
$$Y={a}_{6}x+{a}_{7}y+{a}_{8}z+{a}_{9}t+{a}_{10},$$

where \(k\), \({b}_{0}\), \(h\), \({b}_{1}\), and \({a}_{i}\), \(i=\text{1,2},\dots ,10\) are unknowns.

Due to Eqs. (6) and (9), the following system of algebraic type is constructed

$${h}^{6}{a}_{6}^{6}-15{a}_{6}^{4}{k}^{2}{a}_{1}^{2}{h}^{4}+{a}_{6}\left(15{k}^{4}{a}_{1}^{4}{a}_{6}+{a}_{6}{c}_{1}+{a}_{7}{c}_{2}+{a}_{8}{c}_{3}+{a}_{9}\right){h}^{2}-{k}^{2}{a}_{1}\left({k}^{4}{a}_{1}^{5}+{a}_{1}{c}_{1}+{a}_{2}{c}_{2}+{a}_{3}{c}_{3}+{a}_{4}\right)=0,$$
$${h}^{4}{a}_{6}^{5}{a}_{1}-\frac{10}{3}{h}^{2}{a}_{6}^{3}{k}^{2}{a}_{1}^{3}+{a}_{6}{k}^{4}{a}_{1}^{5}+\left(\frac{1}{3}{a}_{6}{c}_{1}+\frac{1}{6}{a}_{7}{c}_{2}+\frac{1}{6}{a}_{8}{c}_{3}+\frac{1}{6}{a}_{9}\right){a}_{1}+\frac{1}{6}{a}_{6}\left({a}_{2}{c}_{2}+{a}_{3}{c}_{3}+{a}_{4}\right)=0,$$
$$-32{a}_{6}{h}^{2}\left({h}^{4}{a}_{6}^{5}+\frac{1}{16}{a}_{6}{c}_{1}+\frac{1}{16}{a}_{7}{c}_{2}+\frac{1}{16}{a}_{8}{c}_{3}+\frac{1}{16}{a}_{9}\right){b}_{0}^{2}+8{b}_{1}{k}^{2}{a}_{1}\left(16{k}^{4}{a}_{1}^{5}+{a}_{1}{c}_{1}+{a}_{2}{c}_{2}+{a}_{3}{c}_{3}+{a}_{4}\right)=0,$$

whose solution gives

$${a}_{4}=-5{h}^{4}{a}_{1}{a}_{6}^{4}+10{h}^{2}{k}^{2}{a}_{1}^{3}{a}_{6}^{2}-{k}^{4}{a}_{1}^{5}-{a}_{1}{c}_{1}-{a}_{2}{c}_{2}-{a}_{3}{c}_{3},$$
$${a}_{9}=-{h}^{4}{a}_{6}^{5}+10{h}^{2}{k}^{2}{a}_{1}^{2}{a}_{6}^{3}-5{k}^{4}{a}_{1}^{4}{a}_{6}-{a}_{6}{c}_{1}-{a}_{7}{c}_{2}-{a}_{8}{c}_{3},$$
$${b}_{1}=-\frac{{h}^{2}{a}_{6}^{2}{b}_{0}^{2}\left(3{h}^{2}{a}_{6}^{2}-{k}^{2}{a}_{1}^{2}\right)}{4{k}^{2}{a}_{1}^{2}\left({h}^{2}{a}_{6}^{2}-3{k}^{2}{a}_{1}^{2}\right)}.$$

Subsequently, a breather wave to Eq. (1) is extracted as

$$u=2{\left(\text{ln}\left(G\right)\right)}_{xx},$$

where

$$G={e}^{-kX}+{b}_{0}\text{cos}\left(hY\right)-\frac{{h}^{2}{a}_{6}^{2}{b}_{0}^{2}\left(3{h}^{2}{a}_{6}^{2}-{k}^{2}{a}_{1}^{2}\right)}{4{k}^{2}{a}_{1}^{2}\left({h}^{2}{a}_{6}^{2}-3{k}^{2}{a}_{1}^{2}\right)}{e}^{kX},$$

and

$$X={a}_{1}x+{a}_{2}y+{a}_{3}z+\left(-5{h}^{4}{a}_{1}{a}_{6}^{4}+10{h}^{2}{k}^{2}{a}_{1}^{3}{a}_{6}^{2}-{k}^{4}{a}_{1}^{5}-{a}_{1}{c}_{1}-{a}_{2}{c}_{2}-{a}_{3}{c}_{3}\right)t+{a}_{5},$$
$$Y={a}_{6}x+{a}_{7}y+{a}_{8}z+\left(-{h}^{4}{a}_{6}^{5}+10{h}^{2}{k}^{2}{a}_{1}^{2}{a}_{6}^{3}-5{k}^{4}{a}_{1}^{4}{a}_{6}-{a}_{6}{c}_{1}-{a}_{7}{c}_{2}-{a}_{8}{c}_{3}\right)t+{a}_{10}.$$

The energy distribution of the above breather wave for \({c}_{1}=1\), \({c}_{2}=1\), \({c}_{3}=1\), \({a}_{1}=-0.1\), \({a}_{2}=0.25\), \({a}_{3}=0.2\), \({a}_{5}=0.35\), \({a}_{6}=0.3\), \({a}_{7}=0.3\), \({a}_{8}=0.5\), \({a}_{10}=0.1\), \({b}_{0}=0.1\), \(h=0.5\), \(k=1\), \(y=0.5\), and \(t=0\) has been portrayed in Fig. 4.

Fig. 4
figure 4

The breather wave for \({c}_{1}=1\), \({c}_{2}=1\), \({c}_{3}=1\), \({a}_{1}=-0.1\), \({a}_{2}=0.25\), \({a}_{3}=0.2\), \({a}_{5}=0.35\), \({a}_{6}=0.3\), \({a}_{7}=0.3\), \({a}_{8}=0.5\), \({a}_{10}=0.1\), \({b}_{0}=0.1\), \(h=0.5\), \(k=1\), \(y=0.5\), and \(t=0\)

Full size image

Theorem 5

For \({b}_{0}=-2\), \(h=k\), and \({a}_{1}={a}_{6}\), and assuming k → \(0\); a rational solution, from the breather wave, to the 3D generalized KdV equation can be retrieved as follows

$$u=\frac{8{a}_{6}^{2}}{{\overline{\overline{X}}}^{2}+{\overline{\overline{Y}}}^{2}}-\frac{{8{a}_{6}^{2}\left(\overline{\overline{X}}+\overline{\overline{Y}}\right)}^{2}}{{\left({\overline{\overline{X}}}^{2}+{\overline{\overline{Y}}}^{2}\right)}^{2}},$$

where

$$\overline{\overline{X}}={a}_{6}x+{a}_{2}y+{a}_{3}z-\left({a}_{6}{c}_{1}+{a}_{2}{c}_{2}+{a}_{3}{c}_{3}\right)t+{a}_{5},$$
$$\overline{\overline{Y}}={a}_{6}x+{a}_{7}y+{a}_{8}z-\left({a}_{6}{c}_{1}+{a}_{7}{c}_{2}+{a}_{8}{c}_{3}\right)t+{a}_{10}.$$

Proof

For \({b}_{0}=-2\), \(h=k\), and \({a}_{1}={a}_{6}\), we find that

$${a}_{4}=4{k}^{4}{a}_{6}^{5}-{a}_{2}{c}_{2}-{a}_{3}{c}_{3}-{a}_{6}{c}_{1},$$
$${a}_{7}=4{k}^{4}{a}_{6}^{5}-{a}_{6}{c}_{1}-{a}_{7}{c}_{2}-{a}_{8}{c}_{3},$$
$${b}_{1}=1.$$

So, \(G\) can be written as

$$G={e}^{-k\overline{X} }-2\text{cos}\left(k\overline{Y }\right)+{e}^{k\overline{X} }=2\text{cosh}\left(k\overline{X }\right)-2\text{cos}\left(k\overline{Y }\right),$$

where

$$\overline{X }={a}_{6}x+{a}_{2}y+{a}_{3}z+\left(4{k}^{4}{a}_{6}^{5}-{a}_{2}{c}_{2}-{a}_{3}{c}_{3}-{a}_{6}{c}_{1}\right)t+{a}_{5},$$
$$\overline{Y }={a}_{6}x+{a}_{7}y+{a}_{8}z+\left(4{k}^{4}{a}_{6}^{5}-{a}_{6}{c}_{1}-{a}_{7}{c}_{2}-{a}_{8}{c}_{3}\right)t+{a}_{10}.$$

By these findings, we have

$$u=2{\left(\text{ln}\left(G\right)\right)}_{xx}=\frac{4{k}^{2}{a}_{6}^{2}\text{cosh}\left(k\overline{X }\right)+4{k}^{2}{a}_{6}^{2}\text{cos}\left(k\overline{Y }\right)}{2\text{cosh}\left(k\overline{X }\right)-2\text{cos}\left(k\overline{Y }\right)}-\frac{2{\left(2k{a}_{6}\text{sinh}\left(k\overline{X }\right)+2k{a}_{6}\text{sin}\left(k\overline{Y }\right)\right)}^{2}}{{\left(2\text{cosh}\left(k\overline{X }\right)-2\text{cos}\left(k\overline{Y }\right)\right)}^{2}}.$$

Employing the Taylor expansion leads to

$$u=\frac{2{k}^{2}{a}_{6}^{2}\left(\left(1+\frac{{k}^{2}{\overline{X} }^{2}}{2!}+\dots \right)+\left(1-\frac{{k}^{2}{\overline{Y} }^{2}}{2!}+\dots \right)\right)}{\left(1+\frac{{k}^{2}{\overline{X} }^{2}}{2!}+\dots \right)-\left(1-\frac{{k}^{2}{\overline{Y} }^{2}}{2!}+\dots \right)}-\frac{{2\left(k{a}_{6}\left(\left(k\overline{X }+\frac{{k}^{3}{\overline{X} }^{3}}{3!}+\dots \right)+\left(k\overline{Y }-\frac{{k}^{3}{\overline{Y} }^{3}}{3!}+\dots \right)\right)\right)}^{2}}{{\left(\left(1+\frac{{k}^{2}{\overline{X} }^{2}}{2!}+\dots \right)-\left(1-\frac{{k}^{2}{\overline{Y} }^{2}}{2!}+\dots \right)\right)}^{2}}.$$

Now, when k → 0; a rational solution to Eq. (1) is derived as

$$u=\frac{8{a}_{6}^{2}}{{\overline{\overline{X}}}^{2}+{\overline{\overline{Y}}}^{2}}-\frac{{8{a}_{6}^{2}\left(\overline{\overline{X}}+\overline{\overline{Y}}\right)}^{2}}{{\left({\overline{\overline{X}}}^{2}+{\overline{\overline{Y}}}^{2}\right)}^{2}},$$

where

$$\overline{\overline{X}}={a}_{6}x+{a}_{2}y+{a}_{3}z-\left({a}_{6}{c}_{1}+{a}_{2}{c}_{2}+{a}_{3}{c}_{3}\right)t+{a}_{5},$$
$$\overline{\overline{Y}}={a}_{6}x+{a}_{7}y+{a}_{8}z-\left({a}_{6}{c}_{1}+{a}_{7}{c}_{2}+{a}_{8}{c}_{3}\right)t+{a}_{10}.$$

The energy distribution of the above rational solution for

$$\text{Set} 1: \left\{{c}_{1}=1, {c}_{2}=1, {c}_{3}=1,{a}_{2}=1.25,{a}_{3}=-1.5,{a}_{5}=0.2, {a}_{6}=-0.35, {a}_{7}=0.5, {a}_{8}=0.35, {a}_{10}=1, y=0.5, t=0\right\},$$
$$\text{Set} 2: \left\{{c}_{1}=1, {c}_{2}=1, {c}_{3}=1,{a}_{2}=1.25,{a}_{3}=-1.5,{a}_{5}=0.2, {a}_{6}=-0.35, {a}_{7}=0.5, {a}_{8}=0.35, {a}_{10}=1, y=0.5, t=0.05\right\},$$

has been depicted in Fig. 5.□

Fig. 5
figure 5

The rational solution for a Set 1, b Set 2

Full size image

5 Conclusion

An extensive investigation has been undertaken on a 3D generalized KdV equation describing long-wave behavior in shallow water. To this end, the authors used the Bell polynomial method to determine the Hirota bilinear form, bilinear Bäcklund transformation, and Lax pair of the governing equation. As a result of the Painlevè test (and the three-soliton condition), the integrability of the governing equation was examined, and its multi-soliton waves were successfully derived using the simplified Hirota method. Furthermore, using symbolic computations, lump, lump-soliton, and breather waves of the 3D-gKdV equation were constructed using various ansatz methods. For clarification of the physical characteristics of nonlinear waves, some graphical representations were presented in two- and three-dimensional postures using Maple. There is no doubt that the current paper’s findings will enrich studies regarding KdV-type equations and their nonlinear wave structures. In the future, the authors aim to consider some generalizations of well-known NLPDEs [52,53,54,55,56,57,58,59,60] and to derive their different wave structures.