Welcome to visit Communications in Theoretical Physics,
Mathematical Physics

Dynamical rational solutions and their interaction phenomena for an extended nonlinear equation

  • Karmina K Ali , 1 ,
  • Abdullahi Yusuf 2, 3 ,
  • Wen-Xiu Ma , 4, 5, 6, 7, *
Expand
  • 1Department of Mathematics, University of Zakho, Zakho, Iraq
  • 2Department of Computer Engineering, Biruni University, Istanbul, Turkey
  • 3Department of Computer Science and Mathematics, Lebanese, American University, Beirut, Lebanon
  • 4Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620, United States of America
  • 5Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
  • 6Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
  • 7School of Mathematical and Statistical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa

*Author to whom all correspondence should be addressed.

Received date: 2022-10-29

  Revised date: 2023-01-03

  Accepted date: 2023-01-11

  Online published: 2023-03-17

Copyright

© 2023 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing

Abstract

In this paper, we analyze the extended Bogoyavlenskii-Kadomtsev-Petviashvili (eBKP) equation utilizing the condensed Hirota's approach. In accordance with a logarithmic derivative transform, we produce solutions for single, double, and triple M-lump waves. Additionally, we investigate the interaction solutions of a single M-lump with a single soliton, a single M-lump with a double soliton, and a double M-lump with a single soliton. Furthermore, we create sophisticated single, double, and triple complex soliton wave solutions. The extended Bogoyavlenskii-Kadomtsev-Petviashvili equation describes nonlinear wave phenomena in fluid mechanics, plasma, and shallow water theory. By selecting appropriate values for the related free parameters we also create three-dimensional surfaces and associated counter plots to simulate the dynamical characteristics of the solutions offered.

Cite this article

Karmina K Ali , Abdullahi Yusuf , Wen-Xiu Ma . Dynamical rational solutions and their interaction phenomena for an extended nonlinear equation[J]. Communications in Theoretical Physics, 2023 , 75(3) : 035001 . DOI: 10.1088/1572-9494/acb205

1. Introduction

Nonlinear partial differential equations (NPDEs) are used to explain diverse significant nonlinear phenomena in nature that demonstrate significant properties, such as the existence of numerous conservation laws, soliton solutions, bi-Hamiltonian structures, and various symmetries [1]. The quest for analytical solutions to nonlinear partial differential equations is essential in scientific and engineering applications since it provides a wealth of information on the mechanisms of complicated physical phenomena. Numerous effective methods have been devised to seek exact solutions for NPDEs in mathematical physics, such as the Burgan et al method [2], the similarity transformations [3], the parabolic equation method [4], the new modified unified auxiliary equation method [5], the $(1/G^{\prime} )-$ expansion method [6, 7], Jacobi elliptic function expansion (JEFE) method [8], the simplified Hirota's method [9], the Kudrayshov approach and its modified version [10, 11], the modified auxiliary expansion method [12], and the generalized exponential rational function method and its modified version [13, 14], as well as some numerical methods [15-18]. Each of these methods has its characteristics, and the simplified Hirota method is commonly used owing to its efficiency and directness. In [19-28], the authors have constructed multiple solitons, complexiton solutions, fusions, breather solutions, lump solutions, and mixed kink-lump and periodic lump solutions of some NPDEs by using the simplified Hirota's method. The Bogoyavlenskii-Kadomtsev-Petviashvili (BKP) equation is given by [29]
$\begin{eqnarray}\begin{array}{l}{u}_{{xxt}}+{u}_{{xxxy}}+12{u}_{{xx}}{u}_{{xy}}\\ \,\,+8{u}_{x}{u}_{{xxy}}+4{u}_{{xxx}}{u}_{y}=\alpha {u}_{{yyy}},\end{array}\end{eqnarray}$
Equation (1) represents nonlinear wave phenomena in fluid mechanics, plasma physics, and shallow water theory. The solution function $u\left(x,y,t\right)$ stands for the wave amplitude. When α = 0, equation (1) reduces to the Calogero-Bogoyavlenskii-Schiff (CBS) equation. When α ≠ 0, equation (1) represents a modification of the CBS equation, often known as a modification to the Kadomtsev-Petviashvili (KP) equation [30]. Equation (1) was actually extracted in [31-33] by reducing the renowned (3 + 1)-dimensional KP equation. Equation (1) represents the spread of non-linear waves in several scientific fields, like fluid mechanics, plasma, and shallow waves.
Wazwaz developed an extended form of the BKP equation (1). The Painlevé analysis has been used to explore the integrability of the eBKP equation. The eBKP has the following form [34]:
$\begin{eqnarray}\begin{array}{l}{u}_{{xxt}}+{u}_{{xxxxy}}+12{u}_{{xx}}{u}_{{xy}}+8{u}_{x}{u}_{{xxy}}\\ \,\,+4{u}_{{xxx}}{u}_{y}=\alpha {u}_{{yyy}}+\beta {u}_{{xxx}}+\gamma {u}_{{xxy}}.\end{array}\end{eqnarray}$
Equivalently, it reads
$\begin{eqnarray}\begin{array}{l}{\left({u}_{{xt}}+{u}_{{xxxy}}+8{u}_{x}{u}_{{xy}}+4{u}_{{xx}}{u}_{y}\right)}_{x}\\ \,=\alpha {u}_{{yyy}}+{\left(\beta {u}_{x}+\gamma {u}_{y}\right)}_{{xx}}.\end{array}\end{eqnarray}$
It is clear that the extended form equation (3) acknowledges two extra weak terms of dispersion, namely uxxx and uxxy. When α = β = γ = 0, equation (3) is simplified to the CBS equation [34]
$\begin{eqnarray}{u}_{{xt}}+{u}_{{xxxy}}+8{u}_{x}{u}_{{xy}}+4{u}_{{xx}}{u}_{y}=0.\end{eqnarray}$
Nevertheless, for β = γ = 0, equation (3) is reduced to the BKP equation (1). More specifically, for α = 0 the eBKP equation (3) can be simplified to [34]:
$\begin{eqnarray}{u}_{{xt}}+{u}_{{xxxy}}+8{u}_{x}{u}_{{xy}}+4{u}_{{xx}}{u}_{y}={\left(\beta {u}_{x}+\gamma {u}_{y}\right)}_{x}.\end{eqnarray}$
In [34], multiple soliton solutions of equation (2) have been extracted via the Hirota approach. In this paper, we will study multiple M-lump waves and their interactions with single, and double soliton solutions. Also, we will derive complex single, double, and triple soliton solutions. Lump waves are rational solutions that locate in all directions of space [35-39]. For the first time, Hirota used a direct method to obtain a soliton solution [40, 41]. Manakov et al were the first to discover the lump wave solutions [42]. Then, Satsuma and Ablowitz et al developed the long-wave limit method to construct the multiple lump (M-lump) waves solutions [43]. Zhang et al, developed the extended long-wave limit method to study the high-order M-lump solutions [44]. Subsequently, the interaction of lump waves and solitons has developed and so have many interactive solutions, including lump-kink and lump-strip solutions [45, 46]. Ma used the positive quadratic function to construct the single lump wave of the KP equation [47]. Thereupon, several functions are used to construct different types of rational solutions [48, 49].

2. Rational solutions to the eBKP equation

In this part, we propose to provide single, double, and triple M-lump solutions and also generate single, double, and triple complex soliton solutions to the eBKP equation. We also investigate the mixed single M-lump wave with single and double soliton solutions from the perspective of the suggested equation.
Consider the following transformation:
$\begin{eqnarray}u\left(x,y,t\right)={\left(\mathrm{ln}f\left(x,y,t\right)\right)}_{x}.\end{eqnarray}$
Plugging equation (6) into equation (2), we obtain
$\begin{eqnarray}\begin{array}{l}-6\alpha {f}_{y}^{3}{f}_{x}+6\alpha {ff}{{}_{y}}_{y}^{2}{f}_{{xy}}+6{f}_{x}^{2}\left(-2{f}_{{xy}}{f}_{{xx}}+{f}_{x}\left({f}_{t}-\beta {f}_{x}+2{f}_{{xxy}}\right)\right)\\ \,\,+6f\left(-{f}_{t}{f}_{x}{f}_{{xx}}+{f}_{{xy}}{f}_{{xx}}^{2}+{f}_{x}^{2}\left(-{f}_{{xt}}+\gamma {f}_{{xy}}+2\beta {f}_{{xx}}-2{f}_{{xxxy}}\right)\right)\\ \,\,+{f}^{2}(-\alpha {f}_{{yyy}}{f}_{x}-3\alpha {f}_{{yy}}{f}_{{xy}}+3{f}_{{xt}}{f}_{{xx}}-3\gamma {f}_{{xy}}{f}_{{xx}}-3\beta {f}_{{xx}}^{2}\\ \,\,+3{f}_{x}{f}_{{xxt}}-3\gamma {f}_{x}{f}_{{xxy}}+{f}_{t}{f}_{{xxx}}-4\beta {f}_{x}{f}_{{xxx}}-2{f}_{{xxy}}{f}_{{xxx}}\\ \,\,+2{f}_{{xx}}{f}_{{xxxy}}+{f}_{{xy}}{f}_{{xxxx}}+5{f}_{x}{f}_{{xxxxy}})\\ \,\,+{f}_{y}(-6{f}_{x}\left(\gamma {f}_{x}^{2}+{f}_{2}^{2}-2{f}_{x}{f}_{{xxx}}\right)+6{{ff}}_{x}\left(\alpha {f}_{{yy}}+\gamma {f}_{{xx}}-{f}_{{xxxx}}\right)\\ \,\,+{f}^{2}\left(-3\alpha {f}_{{xyy}}-\gamma {f}_{{xxx}}+{f}_{{xxxxx}}\right))\\ \,\,+{f}^{3}\left(\alpha {f}_{{xyyy}}-{f}_{{xxxt}}+\gamma {f}_{{xxxy}}+\beta {f}_{{xxxx}}-{f}_{{xxxxy}}\right)=0.\end{array}\end{eqnarray}$
Motivated by Hirota's bilinear method, the N-soliton solution in general form is given by [50]
$\begin{eqnarray}f={f}_{N}=\displaystyle \sum _{\mu =0,1}\,\exp \left(\displaystyle \sum _{1\leqslant i\lt j}^{N}{\mu }_{i}{\mu }_{j}\,{A}_{{ij}}+\displaystyle \sum _{i=1}^{N}{\mu }_{i}{v}_{i}\right)\end{eqnarray}$
where
$\begin{eqnarray}{v}_{i}={q}_{i}\left(x+{r}_{i}y+{s}_{i}t\right)+{\alpha }_{i},\end{eqnarray}$
$\begin{eqnarray}{e}^{{A}_{{ij}}}=1-\displaystyle \frac{4{q}_{i}{q}_{j}}{{\left({q}_{i}+{q}_{j}\right)}^{2}+{\left({r}_{i}-{r}_{j}\right)}^{2}\alpha }.\end{eqnarray}$
The ∑μ=0,1 notation indicates summing over all possible combinations of μ1 = 0, 1, μ2 = 0, 1,…,μN = 0, 1; the ${\sum }_{i\lt j}^{(N)}$ summation is over all possible combinations of the N elements with the specific condition i < j.
The first three terms of equation (8) have the following forms:
$\begin{eqnarray}{f}_{1}=1+{e}_{{v}_{1}},\end{eqnarray}$
$\begin{eqnarray}{f}_{2}=1+{e}_{{v}_{1}}+{e}_{{v}_{2}}+{e}_{{v}_{1}+{v}_{2}+{A}_{12}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{f}_{3} & = & 1+{e}_{{v}_{1}}+{e}_{{v}_{2}}+{e}_{{v}_{3}}+{e}_{{v}_{1}+{v}_{2}+{A}_{12}}+{e}_{{v}_{1}+{v}_{3}+{A}_{13}}\\ & & +{e}_{{v}_{2}+{v}_{3}+{A}_{23}}+c\,{e}_{{v}_{1}+{v}_{2}+{v}_{3}+{A}_{123}}.\end{array}\end{eqnarray}$

2.1. M-lump solutions

We explore a single, double, and triple M-lump solution of equation (2) in this portion. To obtain one M-lump solution, the long-wave limit method will be used, and taking ${q}_{i}\to 0,\,\tfrac{{q}_{1}}{{q}_{2}}=O\left(1\right),$ and ${e}^{{\alpha }_{i}}=-1$, $\left(i=1,2\right),$ so equation (11) reduces to
$\begin{eqnarray}{f}_{2}={\phi }_{1}{\phi }_{2}+{B}_{12},\end{eqnarray}$
where $\phi$1 = x + r1y + s1t, $\phi$2 = x + r2y + s2t, ${B}_{12}=-\tfrac{4}{{\left({r}_{1}-{r}_{2}\right)}^{2}\alpha }$, ${s}_{1}={r}_{1}^{3}\alpha +\beta +{r}_{1}\gamma $, and ${s}_{2}={r}_{2}^{3}\alpha +\beta +{r}_{2}\gamma $,
Plugging equation (14) in equation (6), one M-lump solution will be obtained,
$\begin{eqnarray}u\left(x,y,t\right)=\displaystyle \frac{2x+{r}_{1}y+{r}_{2}y+t\left({r}_{1}^{3}\alpha +\beta +{r}_{1}\gamma \right)+t\left({r}_{2}^{3}\alpha +\beta +{r}_{2}\gamma \right)}{-\tfrac{4}{{\left({r}_{1}-{r}_{2}\right)}^{2}\alpha }+\left(x+{r}_{1}y+t\left({r}_{1}^{3}\alpha +\beta +{r}_{1}\gamma \right)\right)\left(x+{r}_{2}y+t\left({r}_{2}^{3}\alpha +\beta +{r}_{2}\gamma \right)\right)},\end{eqnarray}$
where r1 = a + ib, and r2 = aib. equation (15) represents a one M-lump solution of equation (2) as shown in figure 1.
Figure 1. 3D profile and counter plot of equation (15) using t = 1, b = 1.2 a = 0.1, β = 2, α = 2 and γ = 2.
To obtain a 2-M-lump solution of equation (2), we take ${q}_{m}\to 0,\,{\alpha }_{i}=-1\left(i=1,2,3,4\right)$ in equation (7), and we get
$\begin{eqnarray}\begin{array}{rcl}{f}_{4} & = & {\phi }_{1}{\phi }_{2}{\phi }_{3}{\phi }_{4}+{B}_{12}{\phi }_{3}{\phi }_{4}+{B}_{13}{\phi }_{2}{\phi }_{4}+{B}_{14}{\phi }_{2}{\phi }_{3}\\ & & +{B}_{23}{\phi }_{1}{\phi }_{4}+{B}_{24}{\phi }_{1}{\phi }_{3}+{B}_{34}{\phi }_{1}{\phi }_{2}+{B}_{12}{\phi }_{34}\\ & & +{B}_{13}{B}_{24}+{B}_{14}{B}_{23},\end{array}\end{eqnarray}$
$\begin{eqnarray}{\phi }_{i}=x+{r}_{i}y+{s}_{i}t,\end{eqnarray}$
$\begin{eqnarray}{B}_{{ij}}=-\displaystyle \frac{4}{{\left({r}_{i}-{r}_{j}\right)}^{2}\alpha },\end{eqnarray}$
and
$\begin{eqnarray}{s}_{i}={r}_{i}^{3}\alpha +\beta +{r}_{i}\gamma .\end{eqnarray}$
Using equation (16) into equation (6), we have a two M-lump solution of equation (2) as given in figure 2.
Figure 2. 3D profile and counter plot of equation (2) using t = 2, a1 = 0.1, b1 = 1.2, β = 1.5, a2 = 0.2, b2 = 3, γ = 0.5 and α = 1.
To generate a 3-M-lump solution of equation (2), we take ${q}_{m}\to 0,\,{\alpha }_{i}=-1\left(i=1,2,3,4,5,6\right)$ in equation (7), which give
$\begin{eqnarray}\begin{array}{rcl}{f}_{6} & = & {\phi }_{1}{\phi }_{2}{\phi }_{3}{\phi }_{4}{\phi }_{5}{\phi }_{6}+{B}_{12}{B}_{34}{B}_{56}+{B}_{12}{B}_{35}{B}_{46}\\ & & +{B}_{12}{B}_{45}{B}_{36}+{B}_{13}{B}_{24}{B}_{56}+{B}_{13}{B}_{25}{B}_{46}+{B}_{13}{B}_{45}{B}_{26}\\ & & +{B}_{23}{B}_{14}{B}_{56}+{B}_{14}{B}_{25}{B}_{36}+{B}_{14}{B}_{35}{B}_{26}+{B}_{24}{B}_{15}{B}_{36}\\ & & +{B}_{34}{B}_{15}{B}_{26}+{B}_{23}{B}_{15}{B}_{46}+{B}_{23}{B}_{45}{B}_{16}+{B}_{24}{B}_{35}{B}_{16}\\ & & +{B}_{34}{B}_{25}{B}_{16}+{\phi }_{2}{\phi }_{3}{\phi }_{4}{\phi }_{5}{B}_{16}+{\phi }_{2}{\phi }_{3}{\phi }_{5}{\phi }_{6}{B}_{14}+{\phi }_{2}{\phi }_{3}{\phi }_{4}{\phi }_{6}{B}_{15}\\ & & +{\phi }_{3}{\phi }_{4}{\phi }_{5}{\phi }_{6}{B}_{12}+{\phi }_{2}{\phi }_{4}{\phi }_{5}{\phi }_{6}{B}_{13}+{\phi }_{1}{\phi }_{2}{\phi }_{4}{\phi }_{6}{B}_{35}\\ & & +{\phi }_{1}{\phi }_{2}{\phi }_{4}{\phi }_{5}{B}_{36}+{\phi }_{1}{\phi }_{4}{\phi }_{5}{\phi }_{6}{B}_{23}+{\phi }_{1}{\phi }_{3}{\phi }_{5}{\phi }_{6}{B}_{24}+{\phi }_{1}{\phi }_{3}{\phi }_{4}{\phi }_{6}{B}_{25}\\ & & +{\phi }_{1}{\phi }_{3}{\phi }_{4}{\phi }_{5}{B}_{26}+{\phi }_{1}{\phi }_{2}{\phi }_{3}{\phi }_{4}{B}_{56}+{\phi }_{1}{\phi }_{2}{\phi }_{3}{\phi }_{6}{B}_{45}\\ & & +{\phi }_{1}{\phi }_{2}{\phi }_{3}{\phi }_{5}{B}_{46}+{\phi }_{1}{\phi }_{2}{\phi }_{5}{\phi }_{6}{B}_{34}+{\phi }_{1}{\phi }_{2}{B}_{34}{B}_{56}+{\phi }_{1}{\phi }_{2}{B}_{35}{B}_{46}\\ & & +{\phi }_{1}{\phi }_{2}{B}_{45}{B}_{36}+{\phi }_{1}{B}_{23}{\phi }_{4}{B}_{56}+{\phi }_{1}{B}_{23}{B}_{45}{\phi }_{6}\\ & & +{\phi }_{1}{B}_{23}{\phi }_{5}{{B}_{4}}_{6}+{\phi }_{1}{\phi }_{3}{B}_{24}{B}_{56}+{\phi }_{1}{\phi }_{6}{B}_{24}{B}_{35}+{\phi }_{1}{\phi }_{5}{B}_{24}{B}_{36}\\ & & +{\phi }_{1}{\phi }_{3}{B}_{25}{B}_{46}+{\phi }_{1}{\phi }_{6}{B}_{34}{B}_{25}+{\phi }_{1}{\phi }_{4}{B}_{25}{B}_{36}\\ & & +{\phi }_{1}{\phi }_{3}{B}_{45}{B}_{26}+{\phi }_{1}{\phi }_{5}{B}_{34}{B}_{26}+{\phi }_{1}{\phi }_{4}{B}_{35}{B}_{26}+{\phi }_{4}{\phi }_{5}{B}_{12}{B}_{36}\\ & & +{\phi }_{3}{\phi }_{4}{B}_{12}{B}_{56}+{\phi }_{3}{\phi }_{6}{B}_{12}{B}_{45}+{\phi }_{3}{\phi }_{5}{B}_{12}{B}_{46}\\ & & +{\phi }_{5}{\phi }_{6}{B}_{12}{B}_{34}+{\phi }_{4}{\phi }_{6}{B}_{12}{B}_{35}+{\phi }_{5}{\phi }_{6}{B}_{13}{B}_{24}+{\phi }_{4}{\phi }_{6}{B}_{13}{B}_{25}\\ & & +{\phi }_{4}{\phi }_{5}{B}_{13}{B}_{26}+{\phi }_{2}{\phi }_{4}{B}_{13}{B}_{56}+{\phi }_{2}{\phi }_{6}{B}_{13}{B}_{45}\\ & & +{\phi }_{2}{\phi }_{5}{B}_{13}{B}_{46}+{\phi }_{2}{\phi }_{3}{B}_{14}{B}_{56}+{\phi }_{2}{\phi }_{6}{B}_{14}{B}_{35}+{\phi }_{2}{\phi }_{5}{B}_{14}{B}_{36}\\ & & +{\phi }_{5}{\phi }_{6}{B}_{23}{B}_{14}+{\phi }_{3}{\phi }_{6}{B}_{14}{B}_{25}+{\phi }_{3}{\phi }_{5}{B}_{14}{B}_{26}\\ & & +{\phi }_{4}{\phi }_{6}{B}_{23}{B}_{15}+{\phi }_{3}{\phi }_{6}{B}_{24}{B}_{15}+{\phi }_{3}{\phi }_{4}{B}_{15}{B}_{26}+{\phi }_{2}{\phi }_{3}{B}_{15}{B}_{46}\\ & & +{\phi }_{2}{\phi }_{6}{B}_{34}{B}_{15}+{\phi }_{2}{\phi }_{4}{B}_{15}{B}_{36}+{\phi }_{2}{\phi }_{4}{B}_{35}{B}_{16}\\ & & +{\phi }_{4}{\phi }_{5}{B}_{23}{B}_{16}+{\phi }_{3}{\phi }_{5}{B}_{24}{B}_{16}+{\phi }_{3}{\phi }_{4}{B}_{25}{B}_{16}\\ & & +{\phi }_{2}{\phi }_{3}{B}_{45}{B}_{16}+{\phi }_{2}{\phi }_{5}{B}_{34}{B}_{16},\end{array}\end{eqnarray}$
where Bij, $\phi$i, and si are given in the earlier steps.
Plugging equation (20) into equation (6), we have a three M-lump solution of equation (2) as seen in figure 3.
Figure 3. 3D profiles and counter plot of equation (2), using t = 2, a1 = 0.1, b1 = 1.52, a2 = 0.2, b2 = 31.9, γ = 2, β = 1.5, α = 3, a3 = 0.5 and b3 = 1.

2.2. Mixed between one M-lump solution and one soliton solution

We generate the mixed single M-lump solution with a single soliton solution in this part. To do that, we will take equation (12) into account and use the limit ${q}_{\delta }\to 0,\left(\delta =1,2\right)$ and $\tfrac{{q}_{1}}{{q}_{2}}=O\left(1\right)$. Then f3 can be rewritten as follows:
$\begin{eqnarray}\begin{array}{rcl}{f}_{3} & = & {\phi }_{1}{\phi }_{2}+{B}_{12}+\left({\phi }_{1}{\phi }_{2}+{B}_{12}+{C}_{23}{\phi }_{1}\right.\\ & & \left.+{C}_{13}{\phi }_{2}+{C}_{13}\right){e}_{{v}_{3}},\end{array}\end{eqnarray}$
where B12 is given in the previous subsection, and v3 is given in equation (8). The constants C13 and C23 are given as follows
$\begin{eqnarray}\begin{array}{l}{C}_{13}=-\displaystyle \frac{4{q}_{3}}{{q}_{3}^{2}+{r}_{1}^{2}\alpha -2{r}_{1}{r}_{3}\alpha +{r}_{3}^{2}\alpha },\\ {C}_{23}=-\displaystyle \frac{4{q}_{3}}{{q}_{3}^{2}+{r}_{2}^{2}\alpha -2{r}_{2}{r}_{3}\alpha +{r}_{3}^{2}\alpha }.\end{array}\end{eqnarray}$
Using equation (21) into equation (6), we get mixed of single M-lump solution with a single soliton solution as seen in figure 4.
Figure 4. 3D profile of mixed of single M-lump solution with single soliton solution of equation (2) for the values of t = 2, $a=\tfrac{1}{2}$, b = 2, q3 = 2, r3 = 2, γ = 3, β = 1 and $\alpha =-\tfrac{1}{2}$.

2.3. Mixed solutions between single M-lump solution and double soliton solution

We provide the mixed single M-lump solution and double soliton solution in this part. For this purpose, we take equation (7) into account and take qn → 0, (n = 1, 2, and $\tfrac{{q}_{1}}{{q}_{2}}=O\left(1\right)).$ Then f4 may be rewritten as follows
$\begin{eqnarray}\begin{array}{rcl}{f}_{4} & = & {\phi }_{1}{\phi }_{2}+{B}_{12}+{{\rm{\Omega }}}_{1}{e}_{{v}_{3}}+{{\rm{\Omega }}}_{2}{e}_{{v}_{4}}\\ & & +{A}_{34}{e}_{{v}_{3}+{v}_{4}}\left({{\rm{\Omega }}}_{1}+{{\rm{\Omega }}}_{2}-{\phi }_{1}{\phi }_{2}+{B}_{12}+{C}_{13}{C}_{24}+{C}_{14}{C}_{23}\right),\end{array}\end{eqnarray}$
where ${{\rm{\Omega }}}_{2}=\left({\phi }_{1}{\phi }_{2}+{B}_{12}+{C}_{24}{\phi }_{1}+{C}_{14}{\phi }_{2}+{C}_{14}{C}_{24}\right)$, $\phi$1, $\phi$2, and B12 are given in the previous section, v3, v4 are given in equation (8), and Cj4, (j = 1, 2) is given as follows
$\begin{eqnarray}{C}_{j4}=-\displaystyle \frac{4{q}_{4}}{{q}_{4}^{2}+{r}_{j}^{2}\alpha -2{r}_{j}{r}_{4}\alpha +{r}_{4}^{2}\alpha }.\end{eqnarray}$
The interaction between a single M-lump solution and a double soliton solution is presented in figure 5.
Figure 5. 3D profile and their counter plot of the mixed of single M-lump wave and double soliton solution, for the values of $t=1,a=\tfrac{1}{2}$, b = 3, r3 = 1, q3 = 4, r4 = 1, q4 = 3, γ = 1, β = 1 and $\alpha =\tfrac{1}{2}$.

2.4. Mixed between a two M-lump solution and a one soliton solution

Here, we extract the mixed two M-lump solution and one soliton solution motivated by [23], we use
$\begin{eqnarray}\begin{array}{rcl}{f}_{5} & = & {\phi }_{1}{\phi }_{2}{\phi }_{3}{\phi }_{4}+{B}_{34}{\phi }_{1}{\phi }_{2}+{B}_{24}{\phi }_{1}{\phi }_{3}\\ & & +{B}_{23}{\phi }_{1}{\phi }_{4}+{B}_{14}{\phi }_{2}{\phi }_{3}+{B}_{13}{\phi }_{2}{\phi }_{4}\\ & & +{B}_{12}{\phi }_{3}{\phi }_{4}+Q{{\rm{e}}}^{{q}_{5}\left(x+{r}_{5}y+{s}_{5}t\right)+{\alpha }_{5}}+{B}_{14}{B}_{23}\\ & & +{B}_{13}{B}_{24}+{B}_{12}{B}_{34},\end{array}\end{eqnarray}$
where r3 = e + if, r4 = eif, and Q = $\phi$1$\phi$2$\phi$3$\phi$4 + C45$\phi$1$\phi$2$\phi$3 + C15$\phi$2$\phi$3$\phi$4 + C25$\phi$1$\phi$3$\phi$4 + C35$\phi$1$\phi$2$\phi$4 + $\left({B}_{34}+{C}_{35}{C}_{45}\right){\phi }_{1}{\phi }_{2}$ + $\left({B}_{24}+{C}_{25}{C}_{45}\right){\phi }_{1}{\phi }_{3}$ + $\left({B}_{14}+{C}_{15}{C}_{45}\right){\phi }_{2}{\phi }_{3}$ + $\left({B}_{23}+{C}_{25}{C}_{35}\right){\phi }_{1}{\phi }_{4}$ + $\left({B}_{13}+{C}_{15}{C}_{35}\right){\phi }_{2}{\phi }_{4}$ + $\left({B}_{12}+{C}_{15}{C}_{25}\right){\phi }_{3}{\phi }_{4}$ + $\left({B}_{34}{C}_{25}+{B}_{24}{C}_{35}+{B}_{23}{C}_{45}+{C}_{25}{C}_{35}{C}_{45}\right){\phi }_{1}$ + $\left({B}_{34}{C}_{15}+{B}_{14}{C}_{35}+{B}_{13}{C}_{45}+{C}_{15}{C}_{35}{C}_{45}\right){\phi }_{2}$ + $\left({B}_{24}{C}_{15}+{B}_{14}{C}_{25}+{B}_{12}{C}_{45}+{C}_{15}{C}_{25}{C}_{45}\right){\phi }_{3}$ + $\left({B}_{23}{C}_{15}+{B}_{13}{C}_{25}+{B}_{12}{C}_{35}+{C}_{15}{C}_{25}{C}_{35}\right){\phi }_{4}+{B}_{14}{B}_{23}$ + B13B24 + B12B34 + B34C15C25 + B24C15C35 + B14C25C35 + B23C15C45 + B13C25C45 + B12C35C45 + C15C25C35C45.
Plugging equation (25) into equation (6), we get the mix between the two M-lump solution and the one soliton wave solution as presented in figure 6.
Figure 6. 3D profile and counter plot of the mixed double M-lump solution and single soliton solution, when ${a}_{1}=\tfrac{1}{2}$, b1 = 2, q5 = 2, r5 = 2, ${a}_{2}=\tfrac{1}{4}$, b2 = 2, γ = 1, $\beta =\tfrac{1}{2}$, t = 1 and α = 1.

2.5. Complex soliton solutions

In this subsection, we generate complex multiple soliton solutions of equation (2). Rotschild et al illustrated experimentally that the lone range of nonlocality enables the formation of several scalar solitons possessing complex features from dipole tripoles to quadrupoles, to necklaces [51]. Wazwaz introduced a complex algorithm of Hirota's simple method in order to determine multiple complex solutions [52].
To derive mentioned solutions, one may use
$\begin{eqnarray}{s}_{n}=-{q}_{n}^{2}{r}_{n}+{r}_{n}^{3}\alpha +\beta +{r}_{n}\gamma ,\,\,n=1,2,3.\end{eqnarray}$
To identify complex single, double, and triple soliton solutions, one uses
$\begin{eqnarray}{f}_{1}={\rm{i}}+{e}^{{q}_{1}\left(x+{r}_{1}y+{s}_{1}t+{\alpha }_{1}\right)},\end{eqnarray}$
$\begin{eqnarray}{f}_{2}={\rm{i}}+{e}^{{v}_{1}}+{e}^{{v}_{2}}+{e}^{{v}_{1}+{v}_{2}+{A}_{12}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{f}_{3} & = & {\rm{i}}+{e}^{{v}_{1}}+{e}^{{v}_{2}}+{e}^{{v}_{3}}+{e}^{{v}_{1}+{v}_{2}+{A}_{12}}+{e}^{{v}_{1}+{v}_{3}+{A}_{13}}\\ & & +{e}^{{v}_{2}+{v}_{3}+{A}_{23}}+{\rm{i}}{e}^{{v}_{1}+{v}_{2}+{v}_{3}+{A}_{123}},\end{array}\end{eqnarray}$
where
$\begin{eqnarray}{e}^{{A}_{{ij}}}=\displaystyle \frac{4{\rm{i}}{q}_{i}{q}_{j}}{{\left({q}_{i}+{q}_{j}\right)}^{2}+{\left({r}_{i}-{r}_{j}\right)}^{2}\alpha }-{\rm{i}},\,\,\mathrm{and}\,\,{\rm{i}}=\sqrt{-}1.\end{eqnarray}$
Inserting equation (27) in equation (6), we have a complex 1-soliton solution as shown in figure 7.
$\begin{eqnarray}u\left(x,y,t\right)=\displaystyle \frac{{{\rm{e}}}^{{q}_{1}\left(x+{r}_{1}y+t\left(-{q}_{1}^{2}{r}_{1}+{r}_{1}^{3}\alpha +\beta +{r}_{1}\gamma \right)\right)}{q}_{1}}{{\rm{i}}+{{\rm{e}}}^{{q}_{1}\left(x+{r}_{1}y+t\left(-{q}_{1}^{2}{r}_{1}+{r}_{1}^{3}\alpha +\beta +{r}_{1}\gamma \right)\right)}}.\end{eqnarray}$
Figure 7. 3D profile and counter plot of complex one soliton solution of equation (31) when t = 2, γ = −2, β = 2, r1 = 1, q1 = 0.5, and α = 1.
Inserting equation (28) into equation (6), we have a complex two-soliton solution, as seen in figure 8.
Figure 8. 3D profile and their counter plot of complex two soliton solution of equation (2) of the values of t = 2, q1 = − 2, q2 = 2, r1 = 1, r2 = 0.5, α1 = −1.5, α2 = 2, ε = 0.5, γ = 2, β = 3 and α = 0.5.
Using equation (29) in equation (6), we have a complex three-soliton solution, as given in figure 9.
Figure 9. 3D profile of complex three soliton solution of equation (2) for t = 2, q1 = −2, q2 = 2, q3 = −1, r1 = 1, r2 = 0.5, r3 = −2, α1 = −1.5, α2 = 1, α3 = 2, ε = 0.5, γ = 2, β = 1.5 and α = 0.5.

3. Conclusion

In this study, we analyze the extended Bogoyavlenskii-Kadomtsev-Petviashvili equation using the streamlined Hirota's approach. First, we turn the suggested equation into a quadratic form using a logarithmic derivative transformation. Then, we produce M-lump solutions with one, two, and three lumps. The interactions between a single M-lump wave and a single soliton solution, a single M-lump wave, and a double soliton solution, and a double M-lump wave and a single soliton solution are also investigated. Furthermore, we have developed sophisticated single, double, and triple solitons. The created 3D profiles and associated contour plots help to better comprehend the dynamical properties of the built solutions. All built-in solutions, to the best of our knowledge, satisfy equation (2). In future work, some new features and physical patterns for the governing equation will be considered from a different point of view such as fractional calculus and some extended classical derivatives which include M-truncated, conformable, and beta derivatives among others.

Ethical approval

Not applicable.

Competing interests

It is affirmed that there are no financial or other conflicts of interest.

Authors' contributions

Karmina K. Ali: Conceptualization, Formal analysis, Writing—original draft; Abdullahi Yusuf: Formal analysis, Writing—original draft, Writing—review and editing; Wen-Xiu Ma: Conceptualization, Writing—original draft, Writing—review and editing.

Funding

Not applicable.
1
Liu Y Wen X Y 2019 Soliton, breather, lump and their interaction solutions of the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation Advances in Difference Equations 2019 332

DOI

2
Ray J R 1982 Exact solutions to the time-dependent Schrödinger equation Phys. Rev. A 26 729

DOI

3
Yan Z Konotop V V 2009 Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potential and nonlinearities Phys. Rev. E 80 036607

DOI

4
Kiselev A P 2007 Localized light waves: Paraxial and exact solutions of the wave equation (a review) Opt. Spectrosc. 102 603 622

DOI

5
Tarla S Ali K K Yilmazer R 2022 Newly modified unified auxiliary equation method and its applications Optik 269 169880

DOI

6
Yokus A Durur H Ahmad H Thounthong P Zhang Y F 2020 Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G,1/G)-expansion and (1/G′)-expansion techniques Results in Physics 19 103409

DOI

Yokus A Durur H Ahmad H Thounthong P Zhang Y F 2020 Construction of exact traveling wave solutions of the Bogoyavlenskii equation by (G′/G,1/G)-expansion and (1/G′)-expansion techniques Int. J. Mod. Phys. B 34 2050227

DOI

7
Ali K K Yilmazer R Bulut H Yokus A 2022 New wave behaviours of the generalized Kadomtsev-Petviashvili modified equal Width-Burgers equation Appl. Math. 16 249 258

8
Tarla S Ali K K Yilmazer R Yusuf A 2022 Investigation of the dynamical behavior of the Hirota-Maccari system in single-mode fibers Opt. Quantum Electron. 54 1 2

DOI

9
Manafian J Ilhan O A Ali K K Abid S 2020 Cross-Kink Wave Solutions and Semi-Inverse Variational Method for (3+1)-Dimensional Potential-YTSF Equation East Asian Journal on Applied Mathematics 10 549 565

DOI

10
Alharthi M S Baleanu D Ali K K Nuruddeen R I Muhammad L Aljohani A F Osman M S 2022 The dynamical behavior for a famous class of evolution equations with double exponential nonlinearities Journal of Ocean Engineering and Science

DOI

11
Tariq K U Rezazadeh H Zubair M Osman M S Akinyemi L 2022 New exact and solitary wave solutions of nonlinear Schamel-KdV equation Int. J. Appl. Comput. Math. 8 114

DOI

12
Ismael H F Bulut H Baskonus H M Gao W 2021 Dynamical behaviors to the coupled Schrödinger-Boussinesq system with the beta derivative AIMS Mathematics 6 7909 7928

DOI

13
Tarla S Ali K K Yilmazer R Osman M S 2022 On dynamical behavior for optical solitons sustained by the perturbed Chen-Lee-Liu model Commun. Theor. Phys. 74 075005

DOI

14
Ali K K Yokus A Seadawy A R Yilmazer R 2022 The ion sound and Langmuir waves dynamical system via computational modified generalized exponential rational function Chaos, Solitons Fractals 161 112381

DOI

15
Tassaddiq A Tanveer M Azhar M Nazeer W Qureshi S 2022 Time-efficient reformulation of the Lobatto III family of order eight Journal of Computational Science 63 101792

DOI

16
Qureshi S Ramos H Soomro A E Hincal Four Step A 2022 A four step feedback iteration and its applications in fractals Fractal and Fractional 6 662

DOI

17
Qureshi S Abro K A Gómez-Aguilar J F 2022 On the numerical study of fractional and non-fractional model of nonlinear duffing oscillator: a comparison of integer and non-integer order approaches Int. J. Model. Simul. 1 14

DOI

18
Soomro A Naseem A Qureshi S Al Din Ide N 2022 Development of a new multi-step iteration scheme for solving non-linear models with complex polynomiography Complexity 2022 2596924

DOI

19
Ismael H F Seadawy A Bulut H 2021 Multiple soliton, fusion, breather, lump, mixed kink-lump and periodic solutions to the extended shallow water wave model in (2+1)-dimensions Mod. Phys. Lett. B 35 2150138

DOI

20
Ismael H F Bulut H Park C Osman M S 2020 M-lump, N-soliton solutions, and the collision phenomena for the (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation Results in Physics 19 103329

DOI

21
Ma W X Zhang L 2020 Lump solutions with higher-order rational dispersion relations Pramana 94 43

DOI

22
Ali K K Yilmazer R Osman M S 2022 Dynamic behavior of the (3+ 1)-dimensional KdV-Calogero-Bogoyavlenskii-Schiff equation Opt. Quantum Electron. 54 1 5

DOI

23
Ismael H F Bulut H 2021 Multi soliton solutions, M-lump waves and mixed soliton-lump solutions to the Sawada-Kotera equation in (2+1)-dimensions Chin. J. Phys. 71 54 61

24
Chen S T Ma W X 2018 Lump solutions of a generalized Calogero-Bogoyavlenskii-Schiff equation Comput. Math. Appl. 76 1680 1685

DOI

25
Ismael H F Ma W X Bulut H 2021 Dynamics of soliton and mixed lump-soliton waves to a generalized Bogoyavlensky-Konopelchenko equation Phys. Scr. 96 035225

DOI

26
Ali K K Yilmazer R 2022 M-lump solutions and interactions phenomena for the (2+ 1)-dimensional KdV equation with constant and time-dependent coefficients Chin. J. Phys. 77 2189 2200

DOI

27
Ma W X 2021 N-soliton solutions and the Hirota conditions in (1+1)-dimensions International Journal of Nonlinear Sciences and Numerical Simulation 22

DOI

28
Ali K K Yilmazer R Osman M S 2021 Extended Calogero-Bogoyavlenskii-Schiff equation and its dynamical behaviors Phys. Scr. 96 125249

DOI

29
Rui W Zhang Y 2020 Soliton and lump-soliton solutions in the Grammian form for the Bogoyavlenskii-Kadomtsev-Petviashvili equation Advances in Difference Equations 2020 195

DOI

30
Wazwaz A M 2017 Negative-order forms for the Calogero-Bogoyavlenskii-Schiff equation and the modified Calogero-Bogoyavlenskii-Schiff equation Proceedings of the Romanian Academy. Series A 18 337 344

31
Ma W X Zhu Z 2012 Solving the (3+ 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm Appl. Math. Comput. 218 11871 11879

DOI

32
Chen Y Yan Z Zhang H 2003 New explicit solitary wave solutions for (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation Phys. Lett. A 307 107 113

DOI

33
Khalfallah M 2009 New exact traveling wave solutions of the (3+1) dimensional Kadomtsev-Petviashvili (KP) equation Commun. Nonlinear Sci. Numer. Simul. 14 1169 1175

DOI

34
Wazwaz A M 2021 On integrability of an extended Bogoyavlenskii-Kadomtsev-Petviashvili equation: multiple soliton solutions Int. J. Numer. Modell. Electron. Networks Devices Fields 34 e2817

DOI

35
Chang W Soto-Crespo J M Ankiewicz A Akhmediev N 2009 Dissipative soliton resonances in the anomalous dispersion regime Phys. Rev. A 79 033840

DOI

36
Gilson C R Nimmo J J 1990 Lump solutions of the BKP equation Phys. Lett. A 147 472 476

DOI

37
Ma W X Zhou Y 2018 Lump solutions to nonlinear partial differential equations via Hirota bilinear forms J. Differ. Equ. 264 2633 2659

DOI

38
Xu G Q Wazwaz A M 2019 Characteristics of integrability, bidirectional solitons and localized solutions for a (3+1)-dimensional generalized breaking soliton equation Nonlinear Dyn. 96 1989 2000

DOI

39
Xu G Q Liu Y P Cui W Y 2022 Painlevé analysis, integrability property and multiwave interaction solutions for a new (4+ 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation Appl. Math. Lett. 108184

DOI

Hirota R 1971 Exact solution of the Kortewegde Vries equation for multiple collisions of solitons Phys. Rev. Lett. 27 1192

DOI

40
Hirota R 1972 Exact solution of the sine-Gordon equation for multiple collisions of solitons J. Phys. Soc. Jpn. 33 1459 1463

DOI

41
Hirota R 1973 Exact three-soliton solution of the two-dimensional sine-Gordon equation J. Phys. Soc. Jpn. 35 1566

DOI

42
Manakov S V Zakharov V E Bordag L A Its A R Matveev V B 1977 Two-dimensional solitons of the Kadomtsev-Petviashvili equation and their interaction Phys. Lett. A 63 205 206

DOI

43
Satsuma J Ablowitz M J 1979 Two-dimensional lumps in nonlinear dispersive systems J. Math. Phys. 20 1496 1503

DOI

44
Zhang Y Liu Y Tang X 2018 M-lump and interactive solutions to a (3+1)-dimensional nonlinear system Nonlinear Dyn. 93 2533 2541

DOI

45
Zhao H Q Ma W X 2017 Mixed lump-kink solutions to the KP equation Comput. Math. Appl. 74 1399 1405

DOI

46
Zhang X Chen Y 2017 Deformation rogue wave to the (2+1)-dimensional KdV equation Nonlinear Dyn. 90 755 763

DOI

47
Ma W X 2015 Lump solutions to the Kadomtsev-Petviashvili equation Phys. Lett. A 379 1975 1978

DOI

48
X Ma W X Zhou Y Khalique C M 2016 Rational solutions to an extended Kadomtsev-Petviashvili-like equation with symbolic computation Comput. Math. Appl. 71 1560 1567

DOI

49
Zhao H Q Ma W X 2017 Mixed lump-kink solutions to the KP equation Comput. Math. Appl. 74 1399 1405

DOI

50
Satsuma J 1976 N-soliton solution of the two-dimensional Korteweg-deVries equation J. Phys. Soc. Jpn. 40 286 290

DOI

51
Rotschild C Segev M Xu Z Kartashov Y V Torner L Cohen O 2006 Two-dimensional multipole solitons in nonlocal nonlinear media Opt. Lett. 31 3312 3314

DOI

52
Wazwaz A M 2018 Multiple complex soliton solutions for the integrable Sinh-Gordon and the modified KdV-Sinh-Gordon equation Appl. Math. Inf. Sci. 12 899 905

DOI

Outlines

/