Nonlinear Dynamic Behaviors of the (3+1)-Dimensional B-Type Kadomtsev—Petviashvili Equation in Fluid Mechanics

Abstract

This paper provides an investigation on nonlinear dynamic behaviors of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation, which is used to model the propagation of weakly dispersive waves in a fluid. With the help of the Cole—Hopf transform, the Hirota bilinear equation is established, then the symbolic computation with the ansatz function schemes is employed to search for the diverse exact solutions. Some new results such as the multi-wave complexiton, multi-wave, and periodic lump solutions are found. Furthermore, the abundant traveling wave solutions such as the dark wave, bright-dark wave, and singular periodic wave solutions are also constructed by applying the sub-equation method. Finally, the nonlinear dynamic behaviors of the solutions are presented through the 3-D plots, 2-D contours, and 2-D curves and their corresponding physical characteristics are also elaborated. To our knowledge, the obtained solutions in this work are all new, which are not reported elsewhere. The methods applied in this study can be used to investigate the other PDEs arising in physics.

1. Introduction

Nonlinear partial differential equations (NPDEs) are powerful tools to model various complex phenomena in nature [1,2,3,4,5,6]. How to obtain the exact solution of the NPDEs has always been the purpose of researchers. To this day, some effective methods have been proposed to seek for the exact solution of the NPDEs such as the Bäcklund transformation method [7,8,9,10], Cole—Hopf transformation method [11,12,13,14], variational method [15,16], extended rational sine—cosine and sinh—cosh methods [17,18], Sardar subequation method [19,20], the Kudryashov approach [21,22], extended sinh—Gordon equation expansion method [23,24], trial equation method [25,26], sub-equation method [27,28,29], Wang’s direct mapping method [30], and so on [31,32,33,34,35]. In this work, we aim to study the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation that reads as [36]:

$${\psi }_{ty}-{\psi }_{xxxy}-3{\left({\psi }_x{\psi }_y\right)}_x+3{\psi }_{xz}=0,$$

(1)

Equation (1) is a subclass of the Kadomtsev—Petviashvili (KP) hierarchy and is used to describe the weakly dispersive waves in a fluid. $\psi =\psi \left(x,y,z,t\right)$ represents the amplitude or elevation of the relevant wave and is a real function of space x, y, z, and time t. Up to now, many effective methods have been proposed to deal with Equation (1) and some results have been obtained. In [37], two methods, namely the simplest equation method and multiple exp-function method, are used to construct the exact solutions of Equation (1). In [38], one- and two-soliton solutions of Equation (1) are obtained by using the Bäcklund transformations. In [39], the extended homoclinic test technique is employed to study Equation (1) and some new rational breather solutions are found. In [40], the direct algebraic method is adopted and the interaction solutions among solitons, rational waves, and periodic waves are obtained. In [41], the soliton molecules, asymmetric solitons, and some novel hybrid solutions are attained by using the velocity resonance mechanism and the long wave limit method. In [42], the exp-function method aided with symbolic computation is utilized to seek for the anti-kink solutions. However, to our knowledge, the multi-wave complexiton, multi-wave, and periodic lump have not been studied. In addition, the sub-equation method has not been employed to study Equation (1) either. Thus, the objective of this work is to search for the abundant solutions of Equation (1) by means of the Hirota bilinear method and sub-equation method. The results obtained in this work are expected to extend the solutions of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation. The structure of this article is arranged as follows. In Section 2, the Hirota bilinear equation is developed via the Cole—Hopf transform and the symbolic computation with the ansatz function schemes is utilized to find the multi-wave complexiton solutions, multi-wave solutions, and periodic lump solutions, respectively. In Section 3, the sub-equation method is applied to seek for the traveling wave solutions. In Section 4, the physical characteristics and numerical results of the solutions are presented via the 3-D plots, 2-D contours, and 2-D curves. Finally, the conclusion is provided in Section 5.

2. Cole—Hopf Transform and the Exact Solutions

For constructing the exact solutions of Equation (1), we use the Cole—Hopf transform as:

$$\psi =2\ln {\left(g\right)}_x.$$

(2)

Applying this transform, we can obtain the bilinear form of Equation (1) as:

$$\left(D_y{D}_t-D_x^3{D}_y+3D_x{D}_z\right)g\cdot g=0.$$

(3)

Here, the definition of the operators $D_x^m{D}_t^n$ is [43,44,45,46]:

$$D_x^m{D}_t^{n}f\cdot g={\left(\frac{\partial }{\partial x}-\frac{\partial }{\partial x^{\prime }}\right)}^m{\left(\frac{\partial }{\partial t}-\frac{\partial }{\partial t^{\prime }}\right)}^{n}f\left(x,t\right)g\left(x^{\prime },t^{\prime }\right){|}_{x=x^{\prime },t=t^{\prime }}.$$

(4)

Additionally, there are:

$D_x\left(f\cdot g\right)=f_{x}g-f{g}_x$,

$D_x^2\left(f\cdot g\right)=f_{xx}g-2f_x{g}_x+f{g}_{xx}$,

$D_x^2\left(f\cdot f\right)=2\left(f_{xx}f-f_x^2\right)$,

$D_t{D}_x\left(f\cdot g\right)=f_{tx}g-f_t{g}_x-f_x{g}_t+g_{tx}f$,

……

2.1. The Multi-Waves Complexiton Solutions

In order to find the multi-wave complexiton solution, it is assumed that the test function g is:

$$\varphi =x+{\epsilon }_{1}y+{\epsilon }_{2}z+{\epsilon }_{3}t,$$

$$\phi =x+{\epsilon }_{4}y+{\epsilon }_{5}z+{\epsilon }_{6}t,$$

$$\gamma =x+{\epsilon }_{7}y+{\epsilon }_{8}z+{\epsilon }_{9}t,$$

$$g={\Delta }_1{e}^{\phi }+{\Delta }_2{e}^{-\phi }+{\Delta }_3\sin \left(\phi \right)+{\Delta }_4\sin \mathrm{h}\left(\gamma \right),$$

(5)

where ${\epsilon }_i$($i=1,2,\dots ,9$) and ${\Delta }_i$($i=1,2,3,4$) are real parameters to be determined later. Substituting Equation (5) into Equation (3) and performing the corresponding adjustments, we have:

• Case 1:

$$\begin{array}{l}{\epsilon }_1=\frac{3{\epsilon }_2}{4-{\epsilon }_9}, {\epsilon }_2={\epsilon }_2, {\epsilon }_3={\epsilon }_9, {\epsilon }_4=\frac{3{\epsilon }_2}{{\epsilon }_9-4}, {\epsilon }_5=\frac{{\epsilon }_2\left(2+{\epsilon }_9\right)}{4-{\epsilon }_9}, {\epsilon }_6={\epsilon }_9-2, {\epsilon }_7=\frac{3{\epsilon }_2}{4-{\epsilon }_9}, {\epsilon }_8={\epsilon }_2, {\epsilon }_9={\epsilon }_9, \\ {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2, {\Delta }_3={\Delta }_3, {\Delta }_4={\Delta }_4.\end{array}$$

where $4-{\epsilon }_9\ne 0$. Applying the above results into Equation (5), we can obtain the multi-wave complexiton solution through the transformation provided by Equation (2) as:

$$\psi (x,y,z,t)=\frac{2\left[\begin{array}{c}{\Delta }_1{e}^{x+\frac{3{\epsilon }_2}{4-{\epsilon }_9}y+{\epsilon }_{2}z+{\epsilon }_{9}t}-{\Delta }_2{e}^{-(x+\frac{3{\epsilon }_2}{4-{\epsilon }_9}y+{\epsilon }_{2}z+{\epsilon }_{9}t)}+{\Delta }_3\cos (x+\frac{3{\epsilon }_2}{{\epsilon }_9-4}y+\frac{{\epsilon }_2(2+{\epsilon }_9)}{4-{\epsilon }_9}z+({\epsilon }_9-2)t)\\ +{\Delta }_4\cosh (x+\frac{3{\epsilon }_2}{4-{\epsilon }_9}y+{\epsilon }_{2}z+{\epsilon }_{9}t)\end{array}\right]}{\left[\begin{array}{c}{\Delta }_1{e}^{x+\frac{3{\epsilon }_2}{4-{\epsilon }_9}y+{\epsilon }_{2}z+{\epsilon }_{9}t}+{\Delta }_2{e}^{-(x+\frac{3{\epsilon }_2}{4-{\epsilon }_9}y+{\epsilon }_{2}z+{\epsilon }_{9}t)}+{\Delta }_3\sin (x+\frac{3{\epsilon }_2}{{\epsilon }_9-4}y+\frac{{\epsilon }_2(2+{\epsilon }_9)}{4-{\epsilon }_9}z+({\epsilon }_9-2)t)\\ +{\Delta }_4\sinh (x+\frac{3{\epsilon }_2}{4-{\epsilon }_9}y+{\epsilon }_{2}z+{\epsilon }_{9}t)\end{array}\right]}$$

(6)

• Case 2:

$$\begin{array}{l}{\epsilon }_1={\epsilon }_7, {\epsilon }_2={\epsilon }_2, {\epsilon }_3=\frac{4{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}, {\epsilon }_4=-{\epsilon }_7, {\epsilon }_5=2{\epsilon }_7-{\epsilon }_2, {\epsilon }_6=\frac{2{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}, {\epsilon }_7={\epsilon }_7, {\epsilon }_8={\epsilon }_2, \\ {\epsilon }_9=\frac{4{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}, {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2, {\Delta }_3={\Delta }_3, {\Delta }_4={\Delta }_4.\end{array}$$

Inserting the above results into Equation (5) and using the transformation provided by Equation (2), we can obtain the multi-wave complexiton solution as:

$$\psi (x,y,z,t)=\frac{2\left[\begin{array}{c}{\Delta }_1{e}^{x+{\epsilon }_{7}y+{\epsilon }_{2}z+\frac{4{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}t}-{\Delta }_2{e}^{-\left(x+{\epsilon }_{7}y+{\epsilon }_{2}z+\frac{4{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}t\right)}+{\Delta }_3\cos \left(x-{\epsilon }_{7}y+\left(2{\epsilon }_7-{\epsilon }_2\right)z+\frac{2{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}t\right)\hfill \\ +{\Delta }_4\cosh \left(x+{\epsilon }_{7}y+{\epsilon }_{2}z+\frac{4{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}t\right)\hfill \end{array}\right]}{\left[\begin{array}{c}{\Delta }_1{e}^{x+{\epsilon }_{7}y+{\epsilon }_{2}z+\frac{4{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}t}+{\Delta }_2{e}^{-\left(x+{\epsilon }_{7}y+{\epsilon }_{2}z+\frac{4{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}t\right)}+{\Delta }_3\sin \left(x-{\epsilon }_{7}y+\left(2{\epsilon }_7-{\epsilon }_2\right)z+\frac{2{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}t\right)\hfill \\ +{\Delta }_4\sinh \left(x+{\epsilon }_{7}y+{\epsilon }_{2}z+\frac{4{\epsilon }_7-3{\epsilon }_2}{{\epsilon }_7}t\right)\hfill \end{array}\right]}$$

(7)

• Case 3:

$$\begin{array}{l}{\epsilon }_1={\epsilon }_7, {\epsilon }_2=\frac{4{\epsilon }_7-3{\epsilon }_3{\epsilon }_7}{3}, {\epsilon }_3={\epsilon }_3, {\epsilon }_4=-{\epsilon }_7, {\epsilon }_5=\frac{\left({\epsilon }_3+2\right){\epsilon }_7}{3}, {\epsilon }_6={\epsilon }_3-2, {\epsilon }_7={\epsilon }_7, {\epsilon }_8=\frac{4{\epsilon }_7-3{\epsilon }_3{\epsilon }_7}{3}, \\ {\epsilon }_9={\epsilon }_3, {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2, {\Delta }_3={\Delta }_3, {\Delta }_4={\Delta }_4.\end{array}$$

In the same way, we can obtain the multi-wave complexiton solution as:

$$\psi (x,y,z,t)=\frac{2\left[\begin{array}{c}{\Delta }_1{e}^{x+{\epsilon }_{7}y+\frac{4{\epsilon }_7-3{\epsilon }_3{\epsilon }_7}{3}z+{\epsilon }_{3}t}-{\Delta }_2{e}^{-\left(x+{\epsilon }_{7}y+\frac{4{\epsilon }_7-3{\epsilon }_3{\epsilon }_7}{3}z+{\epsilon }_{3}t\right)}+{\Delta }_3\cos \left(x-{\epsilon }_{7}y+\frac{\left({\epsilon }_3+2\right){\epsilon }_7}{3}z+\left({\epsilon }_3-2\right)t\right)\hfill \\ +{\Delta }_4\cosh \left(x+{\epsilon }_{7}y+\frac{4{\epsilon }_7-3{\epsilon }_3{\epsilon }_7}{3}z+{\epsilon }_{3}t\right)\hfill \end{array}\right]}{\left[\begin{array}{c}{\Delta }_1{e}^{x+e\eta y+\frac{4{\epsilon }_r-3{\epsilon }_2{\eta }_3}{3}+\widehat{s}t}+{\Delta }_2{e}^{-\left(x+{\epsilon }_{7}y+\frac{4{\epsilon }_7-3{\epsilon }_3{\epsilon }_7}{3}z+{\epsilon }_{3}t\right)}+{\Delta }_3\sin \left(x-{\epsilon }_{7}y+\frac{\left({\epsilon }_3+2\right){\epsilon }_7}{3}z+\left({\epsilon }_3-2\right)t\right)\hfill \\ +{\Delta }_4\sinh \left(x+{\epsilon }_{7}y+\frac{4{\epsilon }_7-3{\epsilon }_3{\epsilon }_7}{3}z+{\epsilon }_{3}t\right)\hfill \end{array}\right]}$$

(8)

• Case 4:

$$\begin{array}{l}{\epsilon }_1={\epsilon }_7, {\epsilon }_2={\epsilon }_8, {\epsilon }_3=\frac{4{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}, {\epsilon }_4=-{\epsilon }_7, {\epsilon }_5=2{\epsilon }_7-{\epsilon }_8, {\epsilon }_6=\frac{2{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}, {\epsilon }_7={\epsilon }_7, {\epsilon }_8={\epsilon }_8, \\ {\epsilon }_9=\frac{4{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}, {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2, {\Delta }_3={\Delta }_3, {\Delta }_4={\Delta }_4.\end{array}$$

Accordingly, the multi-wave complexiton solution of Equation (1) is attained as:

$$\psi (x,y,z,t)=\frac{2\left[\begin{array}{c}{\Delta }_1{e}^{x+{\epsilon }_{7}y+{\epsilon }_{8}z+\frac{4{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}t}-{\Delta }_2{e}^{-\left(x+{\epsilon }_{7}y+{\epsilon }_{8}z+\frac{4{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}t\right)}+{\Delta }_3\cos \left(x-{\epsilon }_{7}y+\left(2{\epsilon }_7-{\epsilon }_8\right)z+\frac{2{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}t\right)\hfill \\ +{\Delta }_4\cosh \left(x+{\epsilon }_{7}y+{\epsilon }_{8}z+\frac{4{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}t\right)\hfill \end{array}\right]}{\left[\begin{array}{c}{\Delta }_1{e}^{x+{\epsilon }_{7}y+{\epsilon }_{8}z+\frac{4{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}t}+{\Delta }_2{e}^{-\left(x+{\epsilon }_{7}y+{\epsilon }_{8}z+\frac{4{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}t\right)}+{\Delta }_3\sin \left(x-{\epsilon }_{7}y+\left(2{\epsilon }_7-{\epsilon }_8\right)z+\frac{2{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}t\right)\hfill \\ +{\Delta }_4\sinh \left(x+{\epsilon }_{7}y+{\epsilon }_{8}z+\frac{4{\epsilon }_7-3{\epsilon }_8}{{\epsilon }_7}t\right)\hfill \end{array}\right]}$$

(9)

• Case 5:

$$\begin{array}{l}{\epsilon }_1={\epsilon }_1, {\epsilon }_2={\epsilon }_8, {\epsilon }_3=\frac{4{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}, {\epsilon }_4=-{\epsilon }_1, {\epsilon }_5=2{\epsilon }_1-{\epsilon }_8, {\epsilon }_6=\frac{2{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}, {\epsilon }_7={\epsilon }_1, {\epsilon }_8={\epsilon }_8, \\ {\epsilon }_9=\frac{4{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}, {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2, {\Delta }_3={\Delta }_3, {\Delta }_4={\Delta }_4.\end{array}$$

Using the above results, we can obtain the multi-wave complexiton solution of Equation (1) via Equation (2) and Equation (5) as:

$$\psi (x,y,z,t)=\frac{2\left[\begin{array}{c}{\Delta }_1{e}^{x+{\epsilon }_{1}y+{\epsilon }_{8}z+\frac{4{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}t}-{\Delta }_2{e}^{-\left(x+{\epsilon }_{1}y+{\epsilon }_{8}z+\frac{4{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}t\right)}+{\Delta }_3\cos \left(x-{\epsilon }_{1}y+{\epsilon }_{5}z+\frac{2{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}t\right)\hfill \\ +{\Delta }_4\cosh \left(x+{\epsilon }_{1}y+{\epsilon }_{8}z+\frac{4{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}t\right)\hfill \end{array}\right]}{\left[\begin{array}{c}{\Delta }_1{e}^{x+{\epsilon }_{1}y+{\epsilon }_{8}z+\frac{4{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}}+{\Delta }_2{e}^{-\left(x+{\epsilon }_{1}y+{\epsilon }_{8}z+\frac{4{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}t\right)}+{\Delta }_3\sin \left(x-{\epsilon }_{1}y+{\epsilon }_{5}z+\frac{2{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}t\right)\hfill \\ +{\Delta }_4\sinh \left(x+{\epsilon }_{1}y+{\epsilon }_{8}z+\frac{4{\epsilon }_1-3{\epsilon }_8}{{\epsilon }_1}t\right)\hfill \end{array}\right]}$$

(10)

• Case 6:

$$\begin{array}{l}{\epsilon }_1=-{\epsilon }_4, {\epsilon }_2=\frac{{\epsilon }_4\left({\epsilon }_9-4\right)}{3}, {\epsilon }_3={\epsilon }_9, {\epsilon }_4={\epsilon }_4, {\epsilon }_5=-\frac{2{\epsilon }_4+{\epsilon }_4{\epsilon }_9}{3}, {\epsilon }_6={\epsilon }_9-2, {\epsilon }_7=-{\epsilon }_4, {\epsilon }_8=\frac{{\epsilon }_4\left({\epsilon }_9-4\right)}{3}, \\ {\epsilon }_9={\epsilon }_9, {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2, {\Delta }_3={\Delta }_3, {\Delta }_4={\Delta }_4.\end{array}$$

Putting the above results into Equation (5) and using the transformation of Equation (2), we can obtain the multi-wave complexiton solution as:

$$\psi (x,y,z,t)=\frac{2\left[\begin{array}{c}{\Delta }_1{e}^{x-{\epsilon }_{4}y+\frac{{\epsilon }_4\left({\epsilon }_9-4\right)}{3}z+{\epsilon }_{9}t}-{\Delta }_2{e}^{-\left(x-{\epsilon }_{4}y+\frac{{\epsilon }_4\left({\epsilon }_9-4\right)}{3}z+{\epsilon }_{9}t\right)}+{\Delta }_3\cos \left(x+{\epsilon }_{4}y-\frac{2{\epsilon }_4+{\epsilon }_4{\epsilon }_9}{3}z+\left({\epsilon }_9-2\right)t\right)\hfill \\ +{\Delta }_4\cosh \left(x-{\epsilon }_{4}y+\frac{{\epsilon }_4\left({\epsilon }_9-4\right)}{3}z+{\epsilon }_{9}t\right)\hfill \end{array}\right]}{\left[\begin{array}{c}{\Delta }_1{e}^{x-{\epsilon }_{4}y+\frac{{\epsilon }_4\left({\epsilon }_9-4\right)}{3}z+{\epsilon }_{9}t}+{\Delta }_2{e}^{-\left(x-{\epsilon }_{4}y+\frac{{\epsilon }_4\left({\epsilon }_9-4\right)}{3}z+{\epsilon }_{9}t\right)}+{\Delta }_3\sin \left(x+{\epsilon }_{4}y-\frac{2{\epsilon }_4+{\epsilon }_4{\epsilon }_9}{3}z+\left({\epsilon }_9-2\right)t\right)\hfill \\ +{\Delta }_4\sinh \left(x-{\epsilon }_{4}y+\frac{{\epsilon }_4\left({\epsilon }_9-4\right)}{3}z+{\epsilon }_{9}t\right)\hfill \end{array}\right]}$$

(11)

2.2. The Multi-Wave Solutions

To seek for the multi-wave solutions of Equation (1), we assume that $g$ has the following form:

$$\varphi ={\epsilon }_{1}x+{\epsilon }_{2}y+{\epsilon }_{3}z+{\epsilon }_{4}t+{\epsilon }_5,$$

$$\phi ={\epsilon }_{6}x+{\epsilon }_{7}y+{\epsilon }_{8}z+{\epsilon }_{9}t+{\epsilon }_{10},$$

$$\gamma ={\epsilon }_{11}x+{\epsilon }_{12}y+{\epsilon }_{13}z+{\epsilon }_{14}t+{\epsilon }_{15},$$

$$g={\Delta }_1\cos \left(\varphi \right)+{\Delta }_2\cos \mathrm{h}\left(\phi \right)+{\Delta }_3\cos \mathrm{h}\left(\gamma \right)+{\epsilon }_{16},$$

(12)

where ${\epsilon }_i$($i=1,2,\dots ,16$) and ${\Delta }_i$($i=1,2,3$) are real parameters to be determined later. Incorporating Equation (12) into Equation (3) yields a set of algebraic equations. Solving them, we have:

• Case 1:

$$\begin{array}{l}{\epsilon }_1={\epsilon }_1, {\epsilon }_2=0, {\epsilon }_3=0, {\epsilon }_4={\epsilon }_4, {\epsilon }_5={\epsilon }_5, {\epsilon }_6={\epsilon }_6, {\epsilon }_7=0, {\epsilon }_8=0, {\epsilon }_9=\frac{{\epsilon }_6\left({\epsilon }_1^3+{\epsilon }_4+{\epsilon }_1{\epsilon }_6^2\right)}{{\epsilon }_1}, {\epsilon }_{10}={\epsilon }_{10}, \\ {\epsilon }_{11}=0, {\epsilon }_{12}={\epsilon }_{12}, {\epsilon }_{13}=-\frac{{\epsilon }_{12}\left({\epsilon }_1^3+{\epsilon }_4\right)}{3{\epsilon }_1}, {\epsilon }_{14}=0,\\ {\epsilon }_{15}={\epsilon }_{15}, {\epsilon }_{16}={\epsilon }_{16}, {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2, {\Delta }_3={\Delta }_3.\end{array}$$

Substituting the above results into Equation (12) and adopting the transformation provided by Equation (2), we can obtain the multi-wave solution as:

$$\psi \left(x,y,z,t\right)=\frac{2\left[-{\epsilon }_1{\Delta }_1\sin \left({\epsilon }_{1}x+{\epsilon }_{4}t+{\epsilon }_5\right)+{\epsilon }_6{\Delta }_2\sin \mathrm{h}\left({\epsilon }_{6}x+\frac{{\epsilon }_6\left({\epsilon }_1^3+{\epsilon }_4+{\epsilon }_1{\epsilon }_6^2\right)}{{\epsilon }_1}t+{\epsilon }_{10}\right)\right]}{\begin{array}{l}{\Delta }_1\cos \left({\epsilon }_{1}x+{\epsilon }_{4}t+{\epsilon }_5\right)+{\Delta }_2\cos \mathrm{h}\left({\epsilon }_{6}x+\frac{{\epsilon }_6\left({\epsilon }_1^3+{\epsilon }_4+{\epsilon }_1{\epsilon }_6^2\right)}{{\epsilon }_1}t+{\epsilon }_{10}\right)\\ +{\Delta }_3\cos \mathrm{h}\left({\epsilon }_{12}y-\frac{{\epsilon }_{12}\left({\epsilon }_1^3+{\epsilon }_4\right)}{3{\epsilon }_1}z+{\epsilon }_{15}\right)+{\epsilon }_{16}\end{array}}.$$

(13)

• Case 2:

$$\begin{gathered} {\epsilon }_1={\epsilon }_1, {\epsilon }_2=-\frac{{\epsilon }_{11}{\epsilon }_{12}}{{\epsilon }_1}, {\epsilon }_3=\frac{{\epsilon }_{11}{\epsilon }_{12}\left(4{\epsilon }_1^3+{\epsilon }_4\right)}{3{\epsilon }_1^2}, {\epsilon }_4={\epsilon }_4, {\epsilon }_5={\epsilon }_5, {\epsilon }_6={\epsilon }_6, {\epsilon }_7=\frac{{\epsilon }_{11}{\epsilon }_{12}}{{\epsilon }_6}, \\ {\epsilon }_8=-\frac{{\epsilon }_{11}{\epsilon }_{12}\left({\epsilon }_1^3+{\epsilon }_4-3{\epsilon }_1{\epsilon }_6^2\right)}{3{\epsilon }_1{\epsilon }_6}, {\epsilon }_9=\frac{{\epsilon }_6\left({\epsilon }_1^3+{\epsilon }_4+{\epsilon }_1{\epsilon }_6^2\right)}{{\epsilon }_1}, {\epsilon }_{10}={\epsilon }_{10}, {\epsilon }_{11}={\epsilon }_{11}, {\epsilon }_{12}={\epsilon }_{12}, \\ {\epsilon }_{13}=-\frac{{\epsilon }_{12}\left({\epsilon }_1^3-3{\epsilon }_1{\epsilon }_{11}^2+{\epsilon }_4\right)}{3{\epsilon }_1}, {\epsilon }_{14}=\frac{{\epsilon }_{11}\left({\epsilon }_1^3+{\epsilon }_1{\epsilon }_{11}^2+{\epsilon }_4\right)}{{\epsilon }_1}, {\epsilon }_{15}={\epsilon }_{15}, {\epsilon }_{16}=0,{\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2, {\Delta }_3={\Delta }_3 \end{gathered}$$

Using the obtained results, we can obtain the multi-wave solution of Equation (1) by Equation (12) and Equation (2) as:

$$\psi (x,y,z,t)=\frac{2\left[\begin{array}{c}-{\epsilon }_1{\Delta }_1\sin \left({\epsilon }_{1}x-\frac{{\epsilon }_{11}{\epsilon }_{12}}{{\epsilon }_1}y+\frac{{\epsilon }_{11}{\epsilon }_{12}\left(4{\epsilon }_1^3+{\epsilon }_4\right)}{3{\epsilon }_1^2}z+{\epsilon }_{4}t+{\epsilon }_5\right)\hfill \\ +{\epsilon }_6{\Delta }_2\sinh \left({\epsilon }_{6}x+\frac{{\epsilon }_{11}{\epsilon }_{12}}{{\epsilon }_6}y-\frac{{\epsilon }_{11}{\epsilon }_{12}\left({\epsilon }_1^3+{\epsilon }_4-3{\epsilon }_1{\epsilon }_6^2\right)}{3{\epsilon }_1{\epsilon }_6}z+\frac{{\epsilon }_6\left({\epsilon }_1^3+{\epsilon }_4+{\epsilon }_1{\epsilon }_6^2\right)}{{\epsilon }_1}t+{\epsilon }_{10}\right)\hfill \\ +{\epsilon }_{11}{\Delta }_3\sinh \left({\epsilon }_{11}x+{\epsilon }_{12}y-\frac{{\epsilon }_{12}\left({\epsilon }_1^3-3{\epsilon }_1{\epsilon }_{11}^2+{\epsilon }_4\right)}{3{\epsilon }_1}z+\frac{{\epsilon }_{11}\left({\epsilon }_1^3+{\epsilon }_1{\epsilon }_{11}^2+{\epsilon }_4\right)}{{\epsilon }_1}t+{\epsilon }_{15}\right)\hfill \end{array}\right]}{\begin{array}{c}{\Delta }_1\cos \left({\epsilon }_{1}x-\frac{{\epsilon }_{11}{\epsilon }_{12}}{{\epsilon }_1}y+\frac{{\epsilon }_{11}{\epsilon }_{12}\left(4{\epsilon }_1^3+{\epsilon }_4\right)}{3{\epsilon }_1^2}z+{\epsilon }_{4}t+{\epsilon }_5\right)\hfill \\ +{\Delta }_2\cosh \left({\epsilon }_{6}x+\frac{{\epsilon }_{11}{\epsilon }_{12}}{{\epsilon }_6}y-\frac{{\epsilon }_{11}{\epsilon }_{12}\left({\epsilon }_1^3+{\epsilon }_4-3{\epsilon }_1{\epsilon }_6^2\right)}{3{\epsilon }_1{\epsilon }_6}z+\frac{{\epsilon }_6\left({\epsilon }_1^3+{\epsilon }_4+{\epsilon }_1{\epsilon }_6^2\right)}{{\epsilon }_1}t+{\epsilon }_{10}\right)\hfill \\ +{\Delta }_3\cosh \left({\epsilon }_{11}x+{\epsilon }_{12}y-\frac{{\epsilon }_{12}\left({\epsilon }_1^3-3{\epsilon }_1{\epsilon }_{11}^2+{\epsilon }_4\right)}{3{\epsilon }_1}z+\frac{{\epsilon }_{11}\left({\epsilon }_1^3+{\epsilon }_1{\epsilon }_{11}^2+{\epsilon }_4\right)}{{\epsilon }_1}t+{\epsilon }_{15}\right)\hfill \end{array}}$$

(14)

2.3. The Periodic Lump Solutions

To search for the periodic lump solutions of Equation (1). Here, the ansatz function $g$ is supposed as the following form:

$$\varphi =x+{\epsilon }_{1}y+{\epsilon }_{2}z+{\epsilon }_{3}t,$$

$$\phi =x+{\epsilon }_{4}y+{\epsilon }_{5}z+{\epsilon }_{6}t,$$

$$g={\Delta }_1\sin \left(\varphi \right)+{\Delta }_2\cos \mathrm{h}\left(\phi \right)+{\epsilon }_7,$$

(15)

where ${\epsilon }_i$($i=1,2,\dots ,7$) and ${\Delta }_i$($i=1,2$) are real parameters that can be determined later. In the same manner, substituting Equation (15) into Equation (3) and performing the corresponding adjustments, we have:

• Case 1:

  $${\epsilon }_1=-{\epsilon }_4, {\epsilon }_2=\frac{{\epsilon }_3+4}{3}, {\epsilon }_3={\epsilon }_3, {\epsilon }_4={\epsilon }_4, {\epsilon }_5=\frac{2{\epsilon }_4-{\epsilon }_3{\epsilon }_4}{3}, {\epsilon }_6={\epsilon }_3+2, {\epsilon }_7=0, {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2.$$

Substituting the above results into Equation (15) and applying the transformation provided by Equation (2), we can obtain the periodic lump solution of Equation (1) as:

$$\psi \left(x,y,z,t\right)=\frac{2\left[{\Delta }_1\cos \left(x-{\epsilon }_{4}y+\frac{{\epsilon }_3+4}{3}z+{\epsilon }_{3}t\right)+{\Delta }_2\sin \mathrm{h}\left(x+{\epsilon }_{4}y+\frac{2{\epsilon }_4-{\epsilon }_3{\epsilon }_4}{3}z+\left({\epsilon }_3+2\right)t\right)\right]}{{\Delta }_1\sin \left(x-{\epsilon }_{4}y+\frac{{\epsilon }_3+4}{3}z+{\epsilon }_{3}t\right)+{\Delta }_2\cos \mathrm{h}\left(x+{\epsilon }_{4}y+\frac{2{\epsilon }_4-{\epsilon }_3{\epsilon }_4}{3}z+\left({\epsilon }_3+2\right)t\right)}.$$

(16)

• Case 2:

$${\epsilon }_1={\epsilon }_1, {\epsilon }_2=-\frac{2{\epsilon }_1+{\epsilon }_1{\epsilon }_6}{3}, {\epsilon }_3={\epsilon }_6-2, {\epsilon }_4=-{\epsilon }_1, {\epsilon }_5=\frac{{\epsilon }_1{\epsilon }_6-4{\epsilon }_1}{3}, {\epsilon }_6={\epsilon }_6, {\epsilon }_7=0, {\Delta }_1={\Delta }_1, {\Delta }_2={\Delta }_2.$$

Bringing the obtained results into Equation (15), we can obtain the periodic lump solution of Equation (1) with the help of the transformation provided by Equation (2) as:

$$\psi \left(x,y,z,t\right)=\frac{2\left[{\Delta }_1\cos \left(x+{\epsilon }_{1}y-\frac{2{\epsilon }_1+{\epsilon }_1{\epsilon }_6}{3}z+\left({\epsilon }_6-2\right)t\right)+{\Delta }_2\sin \mathrm{h}\left(x-{\epsilon }_{1}y+\frac{{\epsilon }_1{\epsilon }_6-4{\epsilon }_1}{3}z+{\epsilon }_{6}t\right)\right]}{{\Delta }_1\sin \left(x+{\epsilon }_{1}y-\frac{2{\epsilon }_1+{\epsilon }_1{\epsilon }_6}{3}z+\left({\epsilon }_6-2\right)t\right)+{\Delta }_2\cos \mathrm{h}\left(x-{\epsilon }_{1}y+\frac{{\epsilon }_1{\epsilon }_6-4{\epsilon }_1}{3}z+{\epsilon }_{6}t\right)}.$$

(17)

3. The Traveling Wave Solutions

In this section, we will inquire into the traveling wave solutions of Equation (1) with the aid of the sub-equation method [47,48].

To obtain the traveling wave solutions, we first introduce the following variable transformation:

$$\psi \left(x,y,z,t\right)=\mathsf{\Psi }\left(\zeta \right), \zeta =mx+ny+kz+st,$$

(18)

where m, n, k, s are non-zero constants. Putting it into Equation (1) yields:

$$m^{3}n{\mathsf{\Psi }}^{\left(4\right)}+3m^{2}n{\left[{\left({\mathsf{\Psi }}^{\prime }\right)}^2\right]}^{\prime }-\left(ns+3mk\right){\mathsf{\Psi }}^{″}=0,$$

(19)

Integrating Equation (19) with respect to $\zeta$ once and ignoring the integral constant, we have:

$$m^{3}n{\mathsf{\Psi }}^{‴}+3m^{2}n{\left({\mathsf{\Psi }}^{\prime }\right)}^2-\left(ns+3mk\right){\mathsf{\Psi }}^{\prime }=0,$$

(20)

Using the sub-equation method, we can assume the solution of Equation (20) is:

$$\mathsf{\Psi }\left(\zeta \right)={\displaystyle \sum _{i=0}^c{\alpha }_i{\Re }^i}\left(\zeta \right).$$

(21)

where ${\alpha }_i\left(i=0,1,2.\dots ,\hspace{1em}c\right)$ are constants that can be determined later. Additionally, there is:

$${\Re }^{\prime }\left(\zeta \right)=\mu +{\Re }^2\left(\gamma \right).$$

(22)

Here, $\mu$ is a constant. The solutions of Equation (22) are:

$$\Re \left(\zeta \right)=\left\{\begin{array}{l}-\sqrt{-\mu }\tan \mathrm{h}\left(\sqrt{-\mu }\zeta \right),\begin{array}{cc}& \mu <0\end{array}\\ -\sqrt{-\mu }\cot \mathrm{h}\left(\sqrt{-\mu }\zeta \right),\begin{array}{cc}& \mu <0\end{array}\\ \sqrt{\mu }\tan \left(\sqrt{\mu }\zeta \right),\begin{array}{cc}& \mu >0\end{array}\\ -\sqrt{\mu }\cot \left(\sqrt{\mu }\zeta \right),\begin{array}{cc}& \mu >0\end{array}\\ -\frac{1}{\zeta +\Lambda },\begin{array}{cccc}\Lambda & is& \mathrm{a}& \mathrm{constant},\begin{array}{cc}& \mu =0\end{array}\end{array}\end{array}\right..$$

(23)

By balancing ${\mathsf{\Psi }}^{‴}$ and ${\left({\mathsf{\Psi }}^{\prime }\right)}^2$ in Equation (20), the value of c in Equation (21) can be determined as:

$$c=1.$$

(24)

Thus, we have:

$$\mathsf{\Psi }\left(\zeta \right)={\alpha }_0+{\alpha }_1\Re \left(\zeta \right).$$

(25)

Now, substituting Equation (25) with Equation (22) into Equation (20) and setting their coefficients of the different powers of $\Re \left(\zeta \right)$ to be zero yields:

${\Re }^0\left(\zeta \right)$: $-3km\mu {\alpha }_1-ns\mu {\alpha }_1+2m^{3}n{\mu }^2{\alpha }_1+3m^{2}n{\mu }^2{\alpha }_1^2=0$,

${\Re }^2\left(\zeta \right)$: $-3km{\alpha }_1-ns{\alpha }_1+8m^{3}n\mu {\alpha }_1+6m^{2}n\mu {\alpha }_1^2=0$,

${\Re }^4\left(\zeta \right)$: $6m^{3}n{\alpha }_1+3m^{2}n{\alpha }_1^2=0$.

On solving them, we have:

• Case 1:

$${\alpha }_0={\alpha }_0, {\alpha }_1={\alpha }_1, m=-\frac{{\alpha }_1}{2}, n=n, k=k, s=\frac{{\alpha }_1\left(3k+n\mu {\alpha }_1^2\right)}{2n}, \mu =\mu .$$

So, the traveling wave solutions of Equation (1) can be obtained as:

$$\psi \left(x,y,z,t\right)={\alpha }_0-{\alpha }_1\sqrt{-\mu }\tan \mathrm{h}\left[\sqrt{-\mu }\left(-\frac{{\alpha }_1}{2}x+ny+kz+\frac{{\alpha }_1\left(3k+n\mu {\alpha }_1^2\right)}{2n}t\right)\right], \mu <0$$

(26)

$$\psi \left(x,y,z,t\right)={\alpha }_0-{\alpha }_1\sqrt{-\mu }\cot \mathrm{h}\left[\sqrt{-\mu }\left(-\frac{{\alpha }_1}{2}x+ny+kz+\frac{{\alpha }_1\left(3k+n\mu {\alpha }_1^2\right)}{2n}t\right)\right], \mu <0$$

(27)

$$\psi \left(x,y,z,t\right)={\alpha }_0+{\alpha }_1\sqrt{\mu }\tan \left[\sqrt{\mu }\left(-\frac{{\alpha }_1}{2}x+ny+kz+\frac{{\alpha }_1\left(3k+n\mu {\alpha }_1^2\right)}{2n}t\right)\right], \mu >0$$

(28)

$$\psi \left(x,y,z,t\right)={\alpha }_0-{\alpha }_1\sqrt{\mu }\cot \left[\sqrt{\mu }\left(-\frac{{\alpha }_1}{2}x+ny+kz+\frac{{\alpha }_1\left(3k+n\mu {\alpha }_1^2\right)}{2n}t\right)\right], \mu >0$$

(29)

• Case 2:

$${\alpha }_0={\alpha }_0, {\alpha }_1=-2m, m=m, n=n, k=-\frac{ns+4m^{3}n\mu }{3m}, s=s, \mu =\mu .$$

So, the traveling wave solutions of Equation (1) can be obtained as:

$$\psi \left(x,y,z,t\right)={\alpha }_0+2m\sqrt{-\mu }\tan \mathrm{h}\left[\sqrt{-\mu }\left(mx+ny-\frac{ns+4m^{3}n\mu }{3m}z+st\right)\right]$$

(30)

$$\psi \left(x,y,z,t\right)={\alpha }_0+2m\sqrt{-\mu }\cot \mathrm{h}\left[\sqrt{-\mu }\left(mx+ny-\frac{ns+4m^{3}n\mu }{3m}z+st\right)\right], \mu <0.$$

(31)

$$\psi \left(x,y,z,t\right)={\alpha }_0-2m\sqrt{\mu }\tan \left[\sqrt{\mu }\left(mx+ny-\frac{ns+4m^{3}n\mu }{3m}z+st\right)\right], \mu >0.$$

(32)

$$\psi \left(x,y,z,t\right)={\alpha }_0+2m\sqrt{\mu }\cot \left[\sqrt{\mu }\left(mx+ny-\frac{ns+4m^{3}n\mu }{3m}z+st\right)\right]$$

(33)

4. The Numerical Results and Physical Interpretations

The numerical results will be presented by the 3-D plots, 2-D contours, and 2-D curves in this section by assigning the reasonable parameters with the help of the Mathematica software.

Figure 1 presents the behavior of the multi-waves complexiton solution provided by Equation (6) for using ${\epsilon }_2=2$, ${\epsilon }_9=3$, ${\Delta }_1=1$, ${\Delta }_2=-1$, ${\Delta }_3=1$, ${\Delta }_4=1$ for $z=0$ at $t=0$, $t=4$ and $t=8$ in the form of a 3-D plot and 2-D contour. It can be observed that there were collision phenomena between the lump and singular periodic waves arising in the outline. As the time goes on, the wave is propagating along the negative direction of the x-axis and y-axis.

By assigning the parameters as ${\epsilon }_1=1$, ${\epsilon }_4=1$, ${\epsilon }_5=1$, ${\epsilon }_6=1$, ${\epsilon }_{10}=1$, ${\epsilon }_{12}=1$, ${\epsilon }_{15}=1$, ${\epsilon }_{16}=1$, ${\Delta }_1=1$, ${\Delta }_2=1$, and ${\Delta }_3=1$, we present the performance of Equation (13) in Figure 2 for $z=0$ at $t=0$, $t=2$, and $t=4$ through the 3-D plots and 2-D contours. We can find that the multi-wave is formed by the interleaving and superposition of two waveforms. As time goes on, one wave is propagating along the negative direction of x-axis and positive direction of y-axis, on the other hand, the other wave is propagating along the negative direction of the x-axis and y-axis.

Figure 3 presents the behaviors of Equation (16) by selecting the parameters as ${\epsilon }_3=1$, ${\epsilon }_4=1$, ${\Delta }_1=1$, and ${\Delta }_2=1$ at $z=1$ for $t=0$, $t=1$, and $t=2$ via the 3-D plots and 2-D contours. It is easy to see the collision phenomena between the lump and singular periodic waves arising in the outline. With the increase in $t$, the waveform propagates along the negative direction of the x-axis and y-axis.

Assigning the parameters as ${\alpha }_0=1$, ${\alpha }_1=1$, $n=1$, $k=1$, and $\mu =-1$, the behaviors of Equation (26) and Equation (27) are described in Figure 4 and Figure 5, respectively. We can find the waveform of Equation (26) is the dark wave and of Equation (27) is the bright-dark wave.

If we choose the parameters as ${\alpha }_0=1$, $m=1$, $n=1$, $s=1$, and $\mu =1$, we present the performances of Equation (28) and Equation (29) in Figure 6 and Figure 7, respectively. It can be observed that the profiles are both the singular periodic waves.

5. Conclusions and Future Recommendations

With the aid of the Cole—Hopf transform, we successfully constructed the multi-wave complexiton, multi-wave, and periodic-lump solutions of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation via applying the symbolic computation and the ansatz function schemes. Furthermore, we also find its abundant exact traveling solutions by means of the sub-equation method, which includes the dark wave, bright-dark wave, and singular periodic wave solutions. Finally, the numerical simulation of the solutions are presented through the 3-D plots, 2-D contours, and 2-D curves. Compared with the solutions reported in [35,36,37,38,39,40], it can be found that the diverse solutions in this work are all new, which can be used to extend the solutions of the (3+1)-dimensional B-type Kadomtsev—Petviashvili equation in fluid mechanics. The methods in this paper can be applied to study the exact solutions of the other PDEs in physics.

Recently, the fractal and fractional calculus have a wide range of applications in different fields [49,50,51,52,53,54,55,56,57]; how to apply them to Equation (1) is the direction of the future research.