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

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 [1–3], optical fibers[4], applied mathematics[5–7], chemistry and biology[8–10], etc. In recent years, searching for exact solutions of nonlinear evolution equations has attracted considerable attention, such as lump solutions[11–16], soliton solutions[17–21] and breather solutions[22–25].

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

$5u_x{u}^2+5u{u}_y+5u_{\mathit{xxx}}u+5u_x{\partial }_x^{-1}{u}_y+5u_{xx}{u}_x-u_t+5u_{\mathit{xxy}}+u_{\mathit{xxxxx}}-5{\partial }_x^{-1}{u}_{yy}=0,(1)$

has important and wide applications in conformal field theory, quantum gravity field theory and conserved current of Liouville equation[26–28]. Soliton solutions[29–31], 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]:

$\begin{array}{ll}\hfill & -180{\partial }_x^{-1}{u}_{yy}-36u_t+180u_{\mathit{xxy}}+30u_{xx}uw+15u_{\mathit{xxx}}{w}^2+180u_{x}v+180u{u}_y+135u_x{u}^2\hfill \\ \hfill & +90u_{\mathit{xxx}}u-30u_{xx}{u}_x+15u^{3}w+35u_{x}u{w}^2+5u^2{w}^3+\frac{5}{36}u{w}^5+\frac{20}{3}{u}_{xx}{w}^3+\frac{5}{4}{u}_x{w}^4=0,\hfill \end{array}(2)$

in which ∂⁻¹ represents the partial integration operator. Eq. 2 is gained from Eq. 1 by the generalized bilinear method[36]. When v_x = u_y and ω_x = 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[38–44]. 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

$u=6{\left(\ln f\right)}_{xx},\omega =6{\left(\ln f\right)}_x,\nu =6{\left(\ln f\right)}_{xy},(3)$

Eq. 2 can be transformed into a bilinear form[36]:

$\begin{array}{ll}\hfill \left(D_{5,x}^6+5D_{5,y}{D}_{5,x}^3-5D_{5,y}^2-D_{5,x}{D}_{5,t}\right)f\cdot f& =-2f_{tx}f-10f_{yy}f+10f_{\mathit{xxxy}}f+2f_x{f}_t-30f_{\mathit{xxy}}{f}_x+10f_y^2-10f_y{f}_{\mathit{xxx}}+30f_{xy}{f}_{xx}+30f_{\mathit{xxxx}}{f}_{xx}-20f_{\mathit{xxx}}^2=0,\hfill \end{array}(4)$

where D_t, D_x, and D_y are the bilinear derivative operators, which can be defined by generalized D operator[46]:

$D_{p,x_1}^{n_1}\dots D_{p,x_M}^{n_M}\left(f\cdot f\right)={\left.\prod _{i=1}^M{\left(\frac{\partial }{\partial x_i}+\alpha \frac{\partial }{\partial x_i^{\prime }}\right)}^{n_i}f\left(x_1,\dots ,x_M\right)f\left(x_1^{\prime },\dots ,x_M^{\prime }\right)\right|}_{x_1^{\prime }=x_1,\dots ,x_M^{\prime }=x_M}.(5)$

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]:

$f=f_N=\sum _{\mu =\mathrm{0,1}}\exp \left(\sum _{i=1}^N{\mu }_i{\eta }_i+\sum _{i<j}^N{\mu }_i{\mu }_j{A}_{ij}\right),(6)$

where

${\eta }_i=k_i\left(x+p_{i}y+{\omega }_{i}t\right)+{\eta }_i^0,{\omega }_i=k_i^4+5k_i^2{p}_i-5p_i^2,$

$\exp A_{ij}=\frac{k_i^4-3k_i^3{k}_j+k_j^4+{\left(p_i-p_j\right)}^2-3k_i{k}_j\left(k_j^2+p_i+p_j\right)+k_i^2\left(4k_j^2+2p_i+p_j\right)+k_j^2\left(p_i+2p_j\right)}{k_i^4+3k_i^3{k}_j+k_j^4+{\left(p_i-p_j\right)}^2+3k_i{k}_j\left(k_j^2+p_i+p_j\right)+k_i^2\left(4k_j^2+2p_i+p_j\right)+k_j^2\left(p_i+2p_j\right)},(7)$

with k_i, p_i and ${\eta }_i^0$ are arbitrary constants,$\sum _{\mu =\mathrm{0,1}}$is the summation of possible combinations of μ_i = 0, 1(i = 1, 2⋯N).By taking a limit of ${\eta }_i^0=-1,k_i\to 0(i=\mathrm{1,2},\cdots N)$ and considering all the k_i in the same asymptotic order in Eq. 6, we have

$f_N=\prod _{i=1}^N{\theta }_i+\frac{1}{2}\sum _{i,j}^N{B}_{ij}\prod _{s\ne i,j}^N{\theta }_s+\cdots +\frac{1}{M!2^M}\sum _{i,j,\dots ,p,q}^N\stackrel{M}{\overbrace{B_{ij}{B}_{kl}\dots B_{pq}}}\prod _{m\ne i,j,k,l,\dots ,p,q}^N{\theta }_m+\cdots ,(8)$

where

${\theta }_i=x+y{p}_i-5t{p}_i^2,B_{ij}=-\frac{6\left(p_i+p_j\right)}{{\left(p_i-p_j\right)}^2},(9)$

and ${\sum }_{i,j,\cdots N}^N$ 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 $p_{M+i}=p_i^{*}\left(i=\mathrm{1,2},\dots ,M\right)$ 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

$f_2={\theta }_1{\theta }_2+B_{12},(10)$

where ${\theta }_1=x+y{p}_1-5t{p}_1^2$, ${\theta }_2=x+y{p}_2-5t{p}_2^2$, $B_{12}=-\frac{6\left(p_1+p_2\right)}{{\left(p_1-p_2\right)}^2}$. If we take $p_2=p_1^{\ast }=a-b\ast I$, where a and b are real constants, I is an imaginary number unit, Eq. 10 is changed into

$f_2=\frac{3a}{b^2}+25a^4{t}^2+50a^2{b}^2{t}^2+25b^4{t}^2-10a^{2}tx+10b^{2}tx+x^2-10a^{3}ty-10a{b}^{2}ty+2axy+a^2{y}^2+b^2{y}^2,(11)$

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

$\begin{array}{ll}\hfill u& =6{\left(\ln f_2\right)}_{xx}\hfill \\ \hfill & =-\left\{12b^2\left[25a^4{b}^2{t}^2+25b^6{t}^2+b^2{x}^2-10a^3{b}^{2}ty+a\left(-3+30b^{4}ty+2b^{2}xy\right)\right.\right.\hfill \\ \hfill & \left.\left.+b^4\left(10tx-y^2\right)+a^2{b}^2\left(-150b^2{t}^2-10tx+y^2\right)\right]\right\}\left[25a^4{b}^2{t}^2+25b^6{t}^2+b^2{x}^2-10a^3{b}^{2}ty\right.\hfill \\ \hfill & {\left.+a\left(3-10b^{4}ty+2b^{2}xy\right)+a^2{b}^2\left(50b^2{t}^2-10tx+y^2\right)+b^4\left(10tx+y^2\right)\right]}^{-2}.\hfill \end{array}(12)$

This wave keeps moving on the line $y=-\frac{2ax}{a^2+b^2}$, the velocity along the x-axis is v_x = x + 5b²t + 5a²t, and the velocity along the y-axis is v_y = y − 10 at.In the case of N = 4, Eq. 8 is changed into

$f4={\theta }_1{\theta }_2{\theta }_3{\theta }_4+B_{12}{\theta }_3{\theta }_4+B_{13}{\theta }_2{\theta }_4+B_{14}{\theta }_2{\theta }_3+B_{23}{\theta }_1{\theta }_4+B_{24}{\theta }_1{\theta }_3+B_{34}{\theta }_1{\theta }_2+B_{12}{B}_{34}+B_{13}{B}_{24}+B_{14}{B}_{23},(13)$

where ${\theta }_i=x+y{p}_i-5t{p}_i^2$, $B_{ij}=-\frac{6\left(p_i+p_j\right)}{{\left(p_i-p_j\right)}^2},(1\le i<j\le 4)$ and $p_1=p_2^{*}=a_1+b_{1}I{p}_3=p_4^{*}=a_2+b_{2}I$. Substituting Eq. 13 into Eq. 3, we can obtain 2-lump solution of Eq. 2:

$\begin{array}{ll}\hfill u& =6\ln {\left(f_4\right)}_{xx}\hfill \\ \hfill & =\left[-6{\left({\beta }_1+{\beta }_2+{\beta }_3+{\beta }_4+{\beta }_5+{\beta }_6+{\beta }_7\right)}^2+12\left({\beta }_8+{\beta }_9+{\beta }_{10}+{\beta }_{11}\right)\left({\beta }_{12}+{\beta }_{13}+{\beta }_{14}+{\beta }_{15}+{\beta }_{16}\right)\right]\hfill \\ \hfill & \times {\left(-{\beta }_{12}-{\beta }_{13}-{\beta }_{14}-{\beta }_{15}-{\beta }_{16}\right)}^{-2},\hfill \end{array}(14)$

where

${\beta }_1=6\left(p_1+p_3\right){\alpha }_2{\left(p_1-p_3\right)}^{-2}+6\left(p_2+p_3\right){\alpha }_1{\left(p_2-p_3\right)}^{-2},$

${\beta }_2=6\left(p_1+p_2\right){\alpha }_3{\left(p_1-p_2\right)}^{-2}-{\alpha }_1{\alpha }_2{\alpha }_3,{\beta }_3=6\left(p_1+p_4\right)\left({\alpha }_2+{\alpha }_3\right){\left(p_1-p_4\right)}^{-2},$

${\beta }_4=6\left(p_2+p_4\right){\alpha }_1+6\left(p_2+p_4\right){\alpha }_3{\left(p_2-p_4\right)}^{-2},{\beta }_5=6\left(p_3+p_4\right)\left({\alpha }_1+{\alpha }_2\right){\left(p_3-p_4\right)}^{-2},$

${\beta }_6=6\left(p_1+p_2\right){\alpha }_4{\left(p_1-p_2\right)}^{-2}-{\alpha }_1{\alpha }_2{\alpha }_4,{\beta }_7=6\left(p_1+p_3\right){\alpha }_4{\left(p_1-p_3\right)}^{-2}+6\left(p_2+p_3\right){\alpha }_4{\left(p_2-p_3\right)}^{-2},$

${\beta }_8={\alpha }_1{\alpha }_3{\alpha }_4,{\beta }_9={\alpha }_2{\alpha }_3{\alpha }_4,{\beta }_{10}={\alpha }_1{\alpha }_3+{\alpha }_2{\alpha }_3,$

${\beta }_{11}={\alpha }_1{\alpha }_2-6\left(p_1+p_2\right){\left(p_1-p_2\right)}^{-2}-6\left(p_1+p_3\right){\left(p_1-p_3\right)}^{-2}-6\left(p_2+p_3\right){\left(p_2-p_3\right)}^{-2},$

${\beta }_{12}={\alpha }_1{\alpha }_4-6\left(p_1+p_4\right){\left(p_1-p_4\right)}^{-2}-6\left(p_2+p_4\right){\left(p_2-p_4\right)}^{-2}-6\left(p_3+p_4\right){\left(p_3-p_4\right)}^{-2},$

${\beta }_{13}={\alpha }_2{\alpha }_4+{\alpha }_3{\alpha }_4,$

${\beta }_{14}=36\left(p_2+p_3\right)\left(p_1+p_4\right){\left[\left(p_2-p_3\right)\left(p_1-p_4\right)\right]}^{-2}-6\left(p_1+p_4\right){\alpha }_2{\alpha }_3{\left(p_1-p_4\right)}^{-2},$

${\beta }_{15}=36\left(p_1+p_3\right)\left(p_2+p_4\right){\left[\left(p_1-p_3\right)\left(p_2-p_4\right)\right]}^{-2}-6\left(p_2+p_4\right){\alpha }_1{\alpha }_3{\left(p_2-p_4\right)}^{-2},$

${\beta }_{16}=36\left(p_1+p_2\right)\left(p_3+p_4\right){\left[\left(p_1-p_2\right)\left(p_3-p_4\right)\right]}^{-2}-6\left(p_3+p_4\right){\alpha }_1{\alpha }_2{\left(p_3-p_4\right)}^{-2},$

${\beta }_{17}=-6\left(p_1+p_3\right){\alpha }_2{\alpha }_4{\left(p_1-p_3\right)}^{-2},{\beta }_{18}=-6\left(p_2+p_3\right){\alpha }_1{\alpha }_4{\left(p_2-p_3\right)}^{-2},$

${\beta }_{19}=-6\left(p_1+p_2\right){\alpha }_3{\alpha }_4{\left(p_1-p_2\right)}^{-2},{\beta }_{20}={\alpha }_1{\alpha }_2{\alpha }_3{\alpha }_4,$

in which ${\alpha }_i=x+y{p}_i-5t{p}_i^2,\left(i=\mathrm{1,2,3,4}\right)$. The wave keeps moving along the line $y_1=-\frac{2a_{1}x}{{a_1}^2+{b_1}^2}$, $y_2=-\frac{2a_{2}x}{{a_2}^2+{b_2}^2}$.

Figures 1–2, 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, $a_2=\frac{1}{3}$, b₁ = 1, $b_2=\frac{1}{2}$ 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, $a_2=\frac{1}{3}$, b₁ = 1, $b_2=\frac{1}{2}$: (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

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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