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ORIGINAL RESEARCH article

Front. Phys., 15 November 2022
Sec. Interdisciplinary Physics
Volume 10 - 2022 | https://doi.org/10.3389/fphy.2022.1067405

Lax integrability and soliton solutions of the (2 + 1)- dimensional Kadomtsev– Petviashvili– Sawada–Kotera– Ramani equation

  • College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, China

In this paper, a new (2 + 1)-dimensional nonlinear evolution equation is investigated. This equation is called the Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation, which can be seen as the two-dimensional extension of the Korteweg–de Vries–Sawada–Kotera–Ramani equation. By means of Hirota’s bilinear operator and the binary Bell polynomials, the bilinear form and the bilinear Bäcklund transformation are obtained. Furthermore, by application of the Hopf-Cole transformation, the Lax pair is also derived. By introducing the new potential function, infinitely many conservation laws are constructed. Therefore, the Lax integrability of the equation is revealed for the first time. Finally, as the analytical solutions, the N-soliton solutions are presented.

1 Introduction

In recent years, the study of nonlinear evolution equations (NLEEs) has become more and more popular. The NLEEs are research hotspots not only in the field of mathematics but also in other scientific fields, for example, mathematical physics, fluid mechanics, nonlinear optics, marine science, electrical engineering, and atmospheric science. This is attributed to their role in explaining nonlinear phenomena (Feng et al. [1]; Shen et al. [2]; Kumar et al. [3]; Zhao et al. [4]; Liu et al. [5]; Manafian and Lakestani [6]; Osman [7]; Lan [8]).

Seeking exact solutions for NLEEs is one of the most vital research practices in the mathematics field. The solutions of the NLEEs can reveal many natural phenomena and properties. Researchers have proposed many approaches to find exact solutions for NLEEs, for instance, the Darboux transformation (Ma and Zhang [9]; Ling et al. [10]; Yang et al. [11]), Bäcklund transformation (Yin et al. [12]; Dong et al. [13]; Bershtein and Shchechkin [14]), inverse scattering transform (Ablowitz and Musslimani [15]; Ablowitz et al. [16]; Wang et al. [17]), Fourier transformation (Fokas and Gelfand [18]; Chekhovskoy et al. [19]; Segur and Ablowitz [20]), Riemann–Hilbert method Ai and Xu [21]; Ma [22]; Xu et al. [23], and Hirota’s bilinear method (Ma [24]; Ma and Zhou [25]; Cheng et al. [26]). In addition, with the development of computer science and technology, many numerical methods have been put forward. A number of numerical solutions have been obtained using this method.

The mathematical or physical properties of NLEEs are another indispensable research content. Integrability is one of the most important properties for NLEEs. So far, there is no strict unified definition of integrability. The integrability that includes Lax, Liouville, Painlevé, and symmetry types has been extensively studied.

In this paper, we aim to investigate the Lax integrability of the Kadomtsev–Petviashvili–Sawada–Kotera–Ramani (KPSKR) equation. The KPSKR equation is of the form

uxt+3u2+uxxxx+15u3+15uuxx+uxxxxxx+σuyy=0,(1)

where σ is a constant. When σ = 0, Eq. 1 reduces to the Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation:

ut+3u2+uxxx+15u3+15uuxx+uxxxxx=0.(2)

In Xiong et al. [27], the soliton molecules and symmetry groups of the KdVSKR equation have been studied. In Ma et al. [28], the Lie symmetries, exact solutions, and integrability of the KdVSKR equation have been investigated.

This paper is organized as follows. In Section 2, the Hirota bilinear form for the KPSKR equation is obtained. In Section 3, utilizing binary Bell polynomials, the bilinear Bäcklund transformation is derived. At the same time, the Lax pair is constructed. In Section 4, the infinitely many conservation laws are presented by introducing the potential function. In Section 5, the N-soliton solutions for the KPSKR equation are presented.

2 Hirota bilinear form of the (2 + 1)-dimensional Kadomtsev–Petviashvili –Sawada–Kotera–Ramani equation

The binary Bell polynomials establish the connection between the nonlinear evolution equation and the corresponding bilinear equation (Cheng et al. [29]). Therefore, the (2 + 1)-dimensional KPSKR equation can be transformed into the Hirota bilinear form (Ma [30,31]).

For Eq. 1, taking u = μwxx, μ = μ(t) and w = w(x, y, t) are functions to be determined. Taking it into Eq. 1, one obtains

μw3x,t+3μ2w2x2+μw4x2x+15μ3w2x3+15μ2w2xw4x+μw6x2x+σμw2x,2y=0.(3)

Integrating Eq. 3 with respect to x twice, one acquires

wx,t+3μw2x2+w4x+15μ2w2x3+15μw2xw4x+w6x+σw2y=0,(4)

where the integration constants are taken as zeros.

Taking μ(t) = 1, Eq. 4 is converted to P-polynomial form in the following:

Px,tw+P4xw+P6xw+σP2yw=0.(5)

According to the theory of Hirota’s bilinear operator D-operator and multivariate binary Bell polynomials, when w = 2(lnf), Eq. 1 has the Hirota bilinear form:

DxDt+Dx4+Dx6+σDy2ff=0(6)

under the dependent transformation u = 2(lnf)xx.

3 Bilinear Bäcklund transformation and the Lax pair of the (2 + 1)-dimensional Kadomtsev –Petviashvili –Sawada– Kotera– Ramani equation

The Bäcklund transformation is an effective method to seek the exact solution for NLEEs. The new solutions can be obtained from the known solutions using this method. In this section, the bilinear Bäcklund transformation of Eq. 1 can be obtained.

Assuming that p = p(x, y, t) and q = q(x, y, t) are two solutions of Eq. 4, i.e.,

p=2lnF,q=2lnG.(7)

Taking

Fw=wx,t+3w2x2+w4x+15w2x3+15w2xw4x+w6x+σw2y.(8)

According to the two-field condition, one has

FpFq=pqx,t+pq4x+3p2x2q2x2+pq6x+15p2x3q2x3+p2xp4xq2xq4x+σpq2y=0.(9)

To obtain the bilinear Bäcklund transformation, some additional constraints should be added. For this purpose, by introducing two new dependent variables,

ν=pq2=lnFG,ω=p+q2=lnFG.(10)

Therefore, Eq. 9 can be rewritten as

FpFq=Fω+νFων=2νx,t+ν4x+6ν2xω2x+ν6x+15ν2xω4x+ν4xω2x+3ν2xω2x2+ν2x3+σν2y=x2Ytν,ω+2Y3xν,ω3Y5xν,ω+ϕν,ω=0,(11)

where

ϕν,ω=5ν6x+12ν4xω2x+6ν3xω3x+9ν2xω4x+3νxω5x+12νxν2xν3x+6νx2ν4x+27ν2xω2x2+18νxω2xω3x+18νx2ν2xω2x+6νx3ω3x+3νx4ν2x+6ν2x3+6ν2xω2xνxω3xνx2ν2x+2σν2y.(12)

If ϕ(ν, ω) can be expressed as the derivative with respect to x of a linear combination of Y polynomials, then Eq. 11 can be expressed as the similar expression. For this purpose, the following constraint can be introduced:

Y3xν,ω=λ,(13)

where λ is an arbitrary parameter.

Therefore, Eq. 12 can be rewritten as

ϕν,ω=152νxν2xν3x+6νx2ν2xω2x+νx3ω3x+2νx4ν2x+ν3xω3x+3νxω2xω3x+6ν2xω2xνxω3xνx2ν2x+2σν2y.(14)

In addition, the following constraints can also be introduced:

Y2xν,ω+αYyν,ω=β,(15)

where β is an arbitrary parameter and α is a parameter to be determined later.

According to Eq. 15, one obtains

152νxν2xν3x+6νx2ν2xω2x+νx3ω3x+2νx4ν2x+ν3xω3x+3νxω2xω3x=15λανxy(16)

and

6ν2xω2xνxω3xνx2ν2x+2σν2y=2σαω2x,y+6α4σανxνxy6αν2xνy+6βν2x.(17)

Let

2σα=6α4σα=6α,(18)

then

α=3σ3.(19)

Therefore, Eq. 14 can be rewritten as

ϕν,ω=53σλνxy23σω2x,y+νxνxy+ν2xνy+6βν2x=x53σλYyν,ω23σYx,yν,ω+6βYxν,ω.(20)

Finally, Eq. 11 can be rewritten as

FpFq=x2Ytν,ω+2Y3xν,ω3Y5xν,ω+53σλYyν,ω23σYx,yν,ω+6βYxν,ω=0.(21)

From Eqs 13, 15, and 21, a coupled linear system of Y-polynomials can be derived as follows:

Y3xν,ω=λ,Y2xν,ω+3σ3Yyν,ω=β,2Ytν,ω+2Y3xν,ω3Y5xν,ω+53σλYyν,ω23σYx,yν,ω+6βYxν,ω+γ=0,(22)

where γ is an arbitrary parameter.

Accordingly, the bilinear Bäcklund transformation of Eq. 1 is as follows:

Dx3λFG=0,Dx2+3σ3DyβFG=0,2Dt+2Dx33Dx5+53σλDy23σDxDy+6βDx+γFG=0.(23)

By application of the Hopf-Cole transformation ν=lnψψ=FG, one has

Yxν,ω=νx=ψxψ,Yyν,ω=νy=ψyψ,Ytν,ω=νt=ψtψ,Y2xν,ω=ω2x+νx2=ν2x+q2x+νx2=q2x+ψ2xψ,Yx,yν,ω=ωx,y+νxνy=νx,y+qx,y+νxνy=qx,y+ψx,yψ,Y3xν,ω=ν3x+3νxω2x+νx3=ν3x+3νxν2x+3νxq2x+νx3=3q2xψxψ+ψ3xψ,Y5xν,ω=ν5x+5νxω4x+10ν3xω2x+10νx2ν3x+15νxω2x2+10νx3ω2x+νx5=ν5x+5νxν4x+5νxq4x+10ν3xν2x+10ν3xq2x+10νx2ν3x+15νxν2x2+30νxν2xq2x+15νxq2x2+10νx3ν2x+10νx3q2x+νx5=5q4xψxψ+10q2xψ3xψ+15q2x2ψxψ+ψ5xψ.(24)

Therefore, the Lax pair of Eq. 1 is of the form

ψ3x+3uψxλψ=0,ψ2x+uψ+3σ3ψxβψ=0,2ψt+2ψ3x+3uψx3ψ5x+10uψ3x+5u2xψx+15u2ψx+53σλψy23σuydxψ+ψx,y+6βψx=0.(25)

4 Infinitely many conservation laws of the (2 + 1)-dimensional Kadomtsev– Petviashvili–Sawada–Kotera– Ramani equation

The conservation law refers to the law that the value of a physical quantity is constant in nature. The conservation law is closely related to the Lax integrability of the system. Nonlinear systems with infinitely many conservation laws are often Lax integrable. The purpose of this section is to present conservation laws.

At first, in Eq. 20, ϕ(ν, ω) needs to be rewritten in the following form.

ϕν,ω=53σλνxy23σω2x,y+νxνxy+ν2xνy+6βν2x=x23σYx,yν,ω+6βYxν,ω+y53σλYxν,ω.(26)

Therefore, Eq. 21 can be rewritten as follows:

FpFq=x2Y3xν,ω3Y5xν,ω23σYx,yν,ω+6βYxν,ω+y53σλYxν,ω+t2Yxν,ω=0.(27)

Introducing a new potential function

ξ=pxqx2,(28)

and according to Eq. 10,

ξ=νx,ωx=νx+qx=ξ+qx.(29)

Taking λ = ς3 and β = ς2 into Eqs 13 and 15, respectively, one obtains two Riccati-type equations

ν3x+3νxω2x+νx3=ξ2x+3ξξx+q2x+ξ3=ς3,ω2x+νx2+3σ3νy=ξ2+ξx+q2x+3σ3ξydx=ς2,(30)

and a divergence-type equation

x2ς3+6ς2ξ3ξ4x+10ξ2xς2ξ23σ3ξydx+5ξ2ξx22ξξ2x3σ3ξxy+10ξ2ξ2x+15ξς42ς2ξ223σ3ς2ξydx+ξ4+23σ3ξ2ξydx+σ3ξydx2+10ξ3ς2ξ23σ3ξydx+ξ523σ2ξξydx3σ3ξ2ydxdx+ξξydx+y53σλξ+t2ξ=0.(31)

Taking

ξ=ς+n=1Tnq,qx,qy,q2x,qxy,q2y,ςn(32)

into the linear relation formula

ξ2x+3ξξx+q2x+ξ3ς3+ϵξ2+ξx+q2x+3σ3ξydxς2=0,(33)

where ϵ ≠ 0 and equating the coefficients of ς, then the conserved densities Tn’s are

T1=u,T2=ux+ϵ3u,.(34)

Substituting Eq. 32 into Eq. 31, one obtains

x2ς3+6ς2ς+n=1Tnςn3n=1Tn,4xςn+10ς2n=1Tn,2xςn10ς+n=1Tnςn2n=1Tn,2xςn103σ3n=1Tn,2xςnn=1Tn,ydxςn10ς+n=1Tnςnn=1Tn,xςn210ς+n=1Tnςn2n=1Tn,2xςn53σ3ς+n=1Tnςnn=1Tn,xyςn+10ς+n=1Tnςn2n=1Tn,2xςn+15ς4ς+n=1Tnςn30ς2×ς+n=1Tnςn3103σς2ς+n=1Tnςnn=1Tn,ydxςn+15ς+n=1Tnςn5+103σς+n=1Tnςn3n=1Tn,ydxςn+5σς+n=1Tnςnn=1Tn,ydxςn2+10ς2ς+n=1Tnςn310ς+n=1Tnςn5103σ3ς+n=1Tnςn3n=1Tn,ydxςn+ς+n=1Tnςn5+43σς+n=1Tnςnn=1Tn,yςndx+2σn=1Tn,2ydxdxςn23σς+n=1Tnςnn=1Tn,ydxςn+y53σλς+n=1Tnςn+t2ς+n=1Tnςn=0.(35)

For the conservation law equation,

Tn,t+In,x+Jn,y=0,n=0,1,2,,(36)

the fluxes In,x’s are as follows:

I1=6T33T1,4x+60T1T1,2x+30T1,x2+53σT2,xy+45T5+60T13+103σT4,ydx+303σT1T2,y+303σT2T1,y270T22540T1T3603σT1T2,ydx603σT2T1,ydx15σT1,ydx2+23σT2,ydx+2σT1,2ydxdx,I2=6T43T2,4x+60T1T2,2x+60T2T1,2x+103σT1,2xT1,y+60T1,xT2,x+53σT3,xy+53σT1T1,xy+45T6360T12T2+103σT5,ydx303σT1T3,ydx303σT2T2,ydx303σT3T1,ydx603σT12T1,ydx30σT1,ydxT2,ydx+43σT3,ydx+43σT1T1,ydx+2σT2,2ydxdx23σT3,ydx23σT1T1,ydx,,(37)

and the fluxes Jn,x’s are as follows:

J1=53σT4,J2=53σT5,.(38)

It is verified that Eqs. 34, 37, and 38 all satisfy Eq. 36.

5 N-soliton solutions of the (2 + 1)-dimensional Kadomtsev– Petviashvili –Sawada–Kotera– Ramani equation

According to the Hirota bilinear form of the (2 + 1)-dimensional KPSKR equation, the soliton solutions can be derived.

For one-soliton solutions, taking

F=1+eξ,ξ=px+qy+rt,p0,(39)

where the parameters p, q, and t need to satisfy the dispersion relation,

pr+p4+p6+σq2=0,(40)

then the one-soliton solutions of the (2 + 1)-dimensional KPSKR equation can be obtained as

u=2lnFxx=2ln1+epx+qyp3+p5+σq2ptxx.(41)

For two-soliton solutions, taking

F=1+eξ1+eξ2+a12eξ1+ξ2,ξi=pix+qiypi3+pi5+σqi2pit,pi0,(42)

where

a12=p1p2r1r2+p1p24+p1p26+σq1q22p1+p2r1+r2+p1+p24+p1+p26+σq1+q22,ri=pi3+pi5+σqi2pi,i=1,2,(43)

then the two-soliton solutions of the (2 + 1)-dimensional KPSKR equation can be obtained as

u=2lnFxx=2ln1+eξ1+eξ2+a12eξ1+ξ2xx.(44)

For three-soliton solutions, taking

F=1+eξ1+eξ2+eξ3+a12eξ1+ξ2+a13eξ1+ξ3+a23eξ2+ξ3+a123eξ1+ξ2+ξ3,ξi=pix+qiypi3+pi5+σqi2pit,pi0,a123=a12a13a23,(45)

where

aij=pipjrirj+pipj4+pipj6+σqiqj2pi+pjri+rj+pi+pj4+pi+pj6+σqi+qj2,ri=pi3+pi5+σqi2pi,1i<j3,(46)

then the three-soliton solutions of the (2 + 1)-dimensional KPSKR equation can be shown as

u=2lnFxx=2ln1+eξ1+eξ2+eξ3+a12eξ1+ξ2+a13eξ1+ξ3+a23eξ2+ξ3+a12a13a23eξ1+ξ2+ξ3xx.(47)

Accordingly, for N-soliton solutions, taking

F=η=0,1expi=1Nηiξi+1i<jNNηiηjAij,ξi=pix+qiypi3+pi5+σqi2pit,pi0,(48)

where

aij=eAij=pipjrirj+pipj4+pipj6+σqiqj2pi+pjri+rj+pi+pj4+pi+pj6+σqi+qj2,ri=pi3+pi5+σqi2pi,1i<jN,(49)

and the notation 1i<jNN is denoted as the summation of all pairs (i, j) that satisfy the condition 1 ≤ i < jN, the notation η=0,1 is denoted as the summation of all of the cases ηi, ηj = 0 or 1 that satisfy the condition 1 ≤ i < jN.

Therefore, the N-soliton solutions of the (2 + 1)-dimensional KPSKR equation can be shown as

u=2lnfxx=2lnη=0,1expi=1Nηiξi+1i<jNNηiηjAijxx.(50)

6 Conclusion

In this paper, the Lax integrability of the (2 + 1)-dimensional KPSKR equation is investigated. As a nonlinear evolution equation, the KPSKR equation is a high-dimensional extension of the KdVSKR equation. By applying Hirota’s bilinear operator and the binary Bell polynomials, the original equation is converted to the Hirota bilinear form. By reasonably assuming constraints on the parameters, the bilinear Bäcklund transformation and the Lax pair are obtained. By introducing a new potential function, infinitely many conservation laws are derived. Finally, the N-soliton solutions are provided. These results reveal that the KPSKR equations are completely integrable. At the same time, these results also show that Hirota’s bilinear method and the binary Bell polynomials are effective and practical for discussing the Lax integrability of nonlinear evolution equations. As a generalization, it is worth studying whether these methods can be applied to discrete equations.

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

BG contributed to the research and writing of the manuscript. BG: methodology and writing—original manuscript.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11975143 and 61602188).

Acknowledgments

The authors would like to thank all editors and reviewers for their comments toward the improvement of our paper.

Conflict of interest

The author declares 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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Keywords: (2 + 1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation, Hirota’s bilinear operator, binary Bell polynomials, Hopf-Cole transformation, Lax integrability, soliton solutions

Citation: Guo B (2022) Lax integrability and soliton solutions of the (2 + 1)- dimensional Kadomtsev– Petviashvili– Sawada–Kotera– Ramani equation. Front. Phys. 10:1067405. doi: 10.3389/fphy.2022.1067405

Received: 11 October 2022; Accepted: 26 October 2022;
Published: 15 November 2022.

Edited by:

Bo Ren, Zhejiang University of Technology, China

Reviewed by:

Biao Li, Ningbo University, China
Junchao Chen, Lishui University, China

Copyright © 2022 Guo. 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: Baoyong Guo, byguo1989@163.com

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