## BRIEF RESEARCH REPORT article

Front. Phys., 06 October 2022
Sec. Interdisciplinary Physics
Volume 10 - 2022 | https://doi.org/10.3389/fphy.2022.1041100

# Multiple lump solutions of the (2+1)-dimensional sawada-kotera-like equation

Feng-Hua Qi1 Shuang Li1* Zhenhuan Li1 Pan Wang2
• 1School of Information, Beijing Wuzi University, Beijing, China
• 2Sports Business School, Beijing Sport University, Beijing, China

In this paper, 1-lump solution and 2-lump solution of a (2 + 1)-dimensional Sawada-Kotera-like equation are obtained by means of the Hirota’s bilinear method and long wave limit method. The propagation orbits, velocities and the collisions among waves are analyzed. By setting the parameter values, the dynamic characteristics of the obtained solutions are shown in 3D and density plots. These conclusions enrich the dynamical theory of higher-dimensional nonlinear dispersive wave equations.

## 1 Introduction

Nonlinear evolution equations can be used to simulate various nonlinear phenomena in the real world, which appear in fluid mechanics [13], optical fibers[4], applied mathematics[57], chemistry and biology[810], etc. In recent years, searching for exact solutions of nonlinear evolution equations has attracted considerable attention, such as lump solutions[1116], soliton solutions[1721] and breather solutions[2225].

The (2 + 1)-dimensional Sawada–Kotera equation:

$5uxu2+5uuy+5uxxxu+5ux∂x−1uy+5uxxux−ut+5uxxy+uxxxxx−5∂x−1uyy=0,(1)$

has important and wide applications in conformal field theory, quantum gravity field theory and conserved current of Liouville equation[2628]. Soliton solutions[2931], lump solutions[32,33], travelling wave solutions[34] and some other exact solutions[35] of Eq. 1 have been detailed. In this paper, we mainly consider the (2 + 1)-dimensional Sawada-Kotera-like equation[36]:

$−180∂x−1uyy−36ut+180uxxy+30uxxuw+15uxxxw2+180uxv+180uuy+135uxu2+90uxxxu−30uxxux+15u3w+35uxuw2+5u2w3+536uw5+203uxxw3+54uxw4=0,(2)$

in which −1 represents the partial integration operator. Eq. 2 is gained from Eq. 1 by the generalized bilinear method[36]. When vx = uy and ωx = u, Eq. 2 can be reduced to Eq. 1. And Eq. 2 is different from the Sawada-Kotera-like equations which have been mentioned by [32,37].As far as we known, multiple lump solutions of Eq. 2 have not been presented in any existing articles. Classic lump, generalized lump solutions and new rogue wave solutions of Eq. 2 have been obtained by [36]. In this paper, we will study multiple lump solutions of Eq. 2. In Section 2, we construct 1-lump solution and 2-lump solution of Eq. 2 by employing the Hirota’s bilinear method and long wave limit method. The dynamical behaviors of the solutions are analyzed in Section 3. Section 4 is our conclusions.

## 2 1-lump solution and 2-lump solution

The long wave limit method is an effective method to generate M-lump solutions from N-soliton solutions[3844]. In this section, we will construct the 1-lump solution and 2-lump solution of Eq. 2. As a preparation for constructing 1-lump solution and 2-lump solution of Eq. 2, we first study the N-soliton solutions[45]. With the aid of the variable transformation

$u=6ln⁡fxx,ω=6ln⁡fx,ν=6ln⁡fxy,(3)$
Eq. 2 can be transformed into a bilinear form[36]:
$D5,x6+5D5,yD5,x3−5D5,y2−D5,xD5,tf⋅f=−2ftxf−10fyyf+10fxxxyf+2fxft−30fxxyfx+10fy2−10fyfxxx+30fxyfxx+30fxxxxfxx−20fxxx2=0,(4)$

where Dt, Dx, and Dy are the bilinear derivative operators, which can be defined by generalized D operator[46]:

$Dp,x1n1…Dp,xMnMf⋅f=∏i=1M∂∂xi+α∂∂xi′nifx1,…,xMfx1′,…,xM′x1′=x1,…,xM′=xM.(5)$

It means that Eq. 3 are solutions of Eq. 2 if and only if f is a solution of Eq. 4. Based on the Hirota’s bilinear method, the N-soliton solutions of Eq. 4 have been obtained[45]:

$f=fN=∑μ=0,1exp∑i=1Nμiηi+∑i

where

$ηi=kix+piy+ωit+ηi0,ωi=ki4+5ki2pi−5pi2,$
$expAij=ki4−3ki3kj+kj4+pi−pj2−3kikjkj2+pi+pj+ki24kj2+2pi+pj+kj2pi+2pjki4+3ki3kj+kj4+pi−pj2+3kikjkj2+pi+pj+ki24kj2+2pi+pj+kj2pi+2pj,(7)$

with ki, pi and $ηi0$ are arbitrary constants,$∑μ=0,1$is the summation of possible combinations of μi = 0, 1(i = 1, 2⋯N).By taking a limit of $ηi0=−1,ki→0(i=1,2,⋯N)$ and considering all the ki in the same asymptotic order in Eq. 6, we have

$fN=∏i=1Nθi+12∑i,jNBij∏s≠i,jNθs+⋯+1M!2M∑i,j,…,p,qNBijBkl…Bpq︷M∏m≠i,j,k,l,…,p,qNθm+⋯,(8)$

where

$θi=x+ypi−5tpi2,Bij=−6pi+pjpi−pj2,(9)$

and $∑i,j,⋯NN$ represents the summation over all possible compositions of i, j, ⋯p, q, which are taken different values from 1, 2, ⋯N. Taking N = 2M in Eq. 8 and $pM+i=pi*i=1,2,…,M$ in Eq. 9, where “*” denotes the complex conjugation, the M-lump solutions to Eq. 4 can be obtained.In the case of N = 2, Eq. 8 is changed into

$f2=θ1θ2+B12,(10)$

where $θ1=x+yp1−5tp12$, $θ2=x+yp2−5tp22$, $B12=−6p1+p2p1−p22$. If we take $p2=p1∗=a−b∗I$, where a and b are real constants, I is an imaginary number unit, Eq. 10 is changed into

$f2=3ab2+25a4t2+50a2b2t2+25b4t2−10a2tx+10b2tx+x2−10a3ty−10ab2ty+2axy+a2y2+b2y2,(11)$

then we can obtain the 1-lump solution of Eq. 2:

$u=6lnf2xx=−12b225a4b2t2+25b6t2+b2x2−10a3b2ty+a−3+30b4ty+2b2xy+b410tx−y2+a2b2−150b2t2−10tx+y225a4b2t2+25b6t2+b2x2−10a3b2ty+a3−10b4ty+2b2xy+a2b250b2t2−10tx+y2+b410tx+y2−2.(12)$

This wave keeps moving on the line $y=−2axa2+b2$, the velocity along the x-axis is vx = x + 5b2t + 5a2t, and the velocity along the y-axis is vy = y − 10 at.In the case of N = 4, Eq. 8 is changed into

$f4=θ1θ2θ3θ4+B12θ3θ4+B13θ2θ4+B14θ2θ3+B23θ1θ4+B24θ1θ3+B34θ1θ2+B12B34+B13B24+B14B23,(13)$

where $θi=x+ypi−5tpi2$, $Bij=−6pi+pjpi−pj2,(1≤i and $p1=p2*=a1+b1Ip3=p4*=a2+b2I$. Substituting Eq. 13 into Eq. 3, we can obtain 2-lump solution of Eq. 2:

$u=6⁡lnf4xx=−6β1+β2+β3+β4+β5+β6+β72+12β8+β9+β10+β11β12+β13+β14+β15+β16×−β12−β13−β14−β15−β16−2,(14)$

where

$β1=6p1+p3α2p1−p3−2+6p2+p3α1p2−p3−2,$
$β2=6p1+p2α3p1−p2−2−α1α2α3,β3=6p1+p4α2+α3p1−p4−2,$
$β4=6p2+p4α1+6p2+p4α3p2−p4−2,β5=6p3+p4α1+α2p3−p4−2,$
$β6=6p1+p2α4p1−p2−2−α1α2α4,β7=6p1+p3α4p1−p3−2+6p2+p3α4p2−p3−2,$
$β8=α1α3α4,β9=α2α3α4,β10=α1α3+α2α3,$
$β11=α1α2−6p1+p2p1−p2−2−6p1+p3p1−p3−2−6p2+p3p2−p3−2,$
$β12=α1α4−6p1+p4p1−p4−2−6p2+p4p2−p4−2−6p3+p4p3−p4−2,$
$β13=α2α4+α3α4,$
$β14=36p2+p3p1+p4p2−p3p1−p4−2−6p1+p4α2α3p1−p4−2,$
$β15=36p1+p3p2+p4p1−p3p2−p4−2−6p2+p4α1α3p2−p4−2,$
$β16=36p1+p2p3+p4p1−p2p3−p4−2−6p3+p4α1α2p3−p4−2,$
$β17=−6p1+p3α2α4p1−p3−2,β18=−6p2+p3α1α4p2−p3−2,$
$β19=−6p1+p2α3α4p1−p2−2,β20=α1α2α3α4,$

in which $αi=x+ypi−5tpi2,i=1,2,3,4$. The wave keeps moving along the line $y1=−2a1xa12+b12$, $y2=−2a2xa22+b22$.

Figures 12, show the evolution of the 1-lump solution Eq. 12 and 2-lump solution Eq. 14 with the time variation. Figure 1 show the 1-lump waves for Eq. 2 under a = 1, b = 1 but with the different values of (a) and (d) t = −1, (b) and (e) t = 0, (c) and (f) t = 1. Figure 2 are the 2-lump waves for Eq. 2 with parameters a1 = 1, $a2=13$, b1 = 1, $b2=12$ with the different values of (a) and (d) t = −1, (b) and (e) t = 0, (c) and (f) t = 1. Figures 1D,E,F are the density plot of Figures 1A,B,C separately and Figures 2D,E,F are the density plot of Figures 2A,B,C respectively.

FIGURE 1

FIGURE 1. 1-lump solution Eq. 12 for Eq. 2 with a = 1, b = 1: (A) t = −1; (B) t = 0; (C) t = 1; (D), (E), (F) are the density plot of (A), (B), (C) respectively.

FIGURE 2

FIGURE 2. 2-lump solution Eq. 14 for Eq. 2 with a1 = 1, $a2=13$, b1 = 1, $b2=12$: (A) t = −1; (B) t = 0; (C) t = 1; (D), (E), (F) are the density plot of (A), (B), (C) respectively.

## 3 Conclusion

In this paper, we have presented the 1-lump solution Eq. 12 and 2-lump solution Eq. 14 of the (2 + 1)-dimensional Sawada-Kotera-like Eq. 2 by using a variable transformation. Dynamical features and density distributions of the presented solutions have been depicted through plots. It is expected that these results can be useful to understand the dynamical behavior of relevant fields in physics.

## Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

## Author contributions

FQ: work out the whole idea of this paper, including method and writing. SL: some calculations and writing of the paper. ZL: polish the whole paper. PW: check the English gramma.

## Funding

The study is supported by the key project of Beijing Social Science Foundation “strategic research on improving the service quality of capital logistics based on big data technology (18GLA009)”.

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

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Keywords: multiple lump solution, long wave limit, sawada-kotera-like equation, hirota bilinear, partial differential equations

Citation: Qi F-H, Li S, Li Z and Wang P (2022) Multiple lump solutions of the (2+1)-dimensional sawada-kotera-like equation. Front. Phys. 10:1041100. doi: 10.3389/fphy.2022.1041100

Received: 10 September 2022; Accepted: 15 September 2022;
Published: 06 October 2022.

Edited by:

Yunqing Yang, Zhejiang Ocean University, China

Reviewed by:

Xing Lue, Beijing Jiaotong University, China
Xiazhi Hao, Zhejiang University of Technology, China

Copyright © 2022 Qi, Li, Li and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Shuang Li, liangsh222@gmail.com