Lump Solutions for Two Mixed Calogero-Bogoyavlenskii-Schi ffand Bogoyavlensky-Konopelchenko Equations∗

Bo Ren(任博), Wen-Xiu Ma(马文秀), and Jun Yu(俞军)

1Institute of Nonlinear Science,Shaoxing University,Shaoxing 312000,China

2Department of Mathematics and Statistics,University of South Florida,Tampa,FL 33620-5700,USA

3Department of Mathematics,Zhejiang Normal University,Jinhua 321004,China

4College of Mathematics and Physics,Shanghai University of Electric Power,Shanghai 200090,China

5College of Mathematics and Systems Science,Shandong University of Science and Technology,Qingdao 266590,China

6International Institute for Symmetry Analysis and Mathematical Modelling,Department of Mathematical Sciences,North-West University,Ma fikeng Campus,Private Bag X2046,Mmabatho 2735,South Africa

AbstractBased on the Hirota bilinear operators and their generalized bilinear derivatives,we formulate two new(2+1)-dimensional nonlinear partial differential equations,which possess lumps. One of the new nonlinear differential equations includes the generalized Calogero-Bogoyavlenskii-Schi ffequation and the generalized Bogoyavlensky-Konopelchenko equation as particular examples,and the other has the same bilinear form with different Dp-operators.A class explicit lump solutions of the new nonlinear differential equation is constructed by using the Hirota bilinear approaches.A specific case of the presented lump solution is plotted to shed light on the charateristics of the lump.

Key words:generalized Calogero-Bogoyavlenskii-Schiffequation,generalized Bogoyavlensky-Konopelchenko equation,Hirota bilinear,Lump solution

1 Introduction

The investigation of exact solutions to nonlinear partial differential equations is one of the most important problems.Many kinds of soliton solutions are studied by a variety of methods including the inverse scattering transformation,[1]the Darboux transformation,[2−3]the Hirota bilinear method,[4]and symmetry reductions,[5]etc.[6−10]Recently,lump solutions which are rational,analytical and localized in all directions in the space,[11−20]have attracted much attention.As another kind of exact solutions,it exsits potential applications in physics,partically in atmospheric and oceanic sciences.[21]

The Hirota bilinear method in soliton theory provides a powerful approach to finding exact solutions.[4]A kind of lump solutions can be also obtained by means of the Hirota bilinear formuation.Recently,the generalized bilinear operators are proposed by exploring the linear superposition principle.[22]Many new nonlinear systems are constructed by using the generalized Hirota bilinear operators.[23−26]The lump solutions and integrable propertites for those new nonlinear systems are interesting topic in nonlinear science.

The paper is organized as follows.In Sec.2,a new nonlinear differential equation is constructed by means of the bilinear formulation.The new nonlinear equation includes a Calogero-Bogoyavlenskii-Schiff equation and a Bogoyavlensky-Konopelchenko(gCBS-BK)equation.A class of gCBS-BK-like equations can be obatined by using the generalized bilinear method.In Sec.3,a lump solution to the newly presented gCBS-BK systems is obtained bsaed on the Maple symbolic computations.Two figures are given theoretically and graphically.The last section is devoted to summary and discussions.

2 A Generalized gCBS-BK Equation

We consider a(2+1)-dimensioanl nonlinear partial differential equation

where δi,i=1,2,...,6 are arbitrary constants.While the constants satisfy δ3= δ4= δ5= δ6=0 and δ5= δ6=0,(1)becomes a generalized Calogero-Bogoyavlenskii-Schi ff(CBS)equation[18]and a generalized[18,27]Bogoyavlensky-Konopelchenko(BK)equation,[19]respectively.The CBS equation was constructed by the modified Lax formalism and the self-dual Yang-Mills equation respectively.[28−29]

The BK equation is described as the interaction of a Riemann wave propagating along y-axis and a long wave propagating along x-axis.[30]These two equations have been widely studied in different ways.[31−32]The(2+1)-dimensional nonlinear differential equation(1)is thus called gCBS-BK equation.The Hirota bilinear form of gCBS-BK equation(1)has

by the relationship between u,w,and f

Based on the generalized bilinear thoery,[22]the generalized bilinear operators read

where m,n ≥ 0 and αsp=(−1)rp(s)if s=rp(s)mod p.Here αpis a symbol.For a prime number p>2,we can not write the relationship

Taking the prime number p=3,we have

and then,we have the concrete operators

By the above analysis,the corresponding bilinear form of the gCBS-BK equation(1)in p=3 reads

Bell polynomial theories suggest a dependent variable transfomation

to transfrom bilinear equations to nonlinear equations.By selecting the variable transformation(9),a gCBS-BK-like equation is obtained from the generalized bilinear form(8)

By selecting the prime number p=3,we get a new gCBSBK-like equtaion(10).We can aslo select p=5,7,9,...to get new nonlinear partial differential equations.This provides a useful method to get new nonlinear systems that possess bilinear forms.In this paper,we shall focus on the gCBS-BK equation(1)and the gCBS-BK-like equation(10)for the prime number p=3.

3 A Search for Lump Solution

Based on the bilinear form,a quadratic function solution to the(2+1)-dimensional bilinear gCBS-BK equation(2)and bilinear gCBS-BK-like equation(8),is defined by

where ai,1≤i≤9 are constant parameters to be de-termined.Substituting the expression(11)into Eqs.(2)and(8)and vanishing the coefficients of different powers of x,y,and t,we can get the same relationship among parameters for Eqs.(2)and(8).The following set of solutions for the parameters a3,a7,and a9

Fig.1 (Color online)Profiles of the lump solution(13).(a)3D lump plot with the time t=0,(b)the corresponding density plot,(c)the curve by selecting different parameters y and t,(d)the curve by selecting different parameters x and t.

which need to satisfy the following conditions

(i)a5=0,to guarantee the well-posedness for f;

(iii)a1a6−a2a5=0,to ensure the localization of u,w in all directions in the space.

The parameters take a1=1,a2= −2,a4= −2,a5= −2,a6=2,a8=1,δ1=1,δ2=1,δ3=1,δ4=1,δ5=1,δ6=2.By substituting Eq.(11)into Eq.(9)and combining the relationship(12),we get the lump solution to have the positivity of f;

The 3D plot,density plot,and curve plot for this lump solution are depicted in Fig.1.The parameters take a1=1,a2=1,a4=1,a5= −2,a6=3,a8=1,δ1=1,δ2=1,δ3=1,δ4=1,δ5= −2,δ6=2.The lump solution has the following form

The 3D plot,density plot and curve plot for the lump solution are shown in Fig.2.

Fig.2 (Color online)Profiles of the lump solution(14).(a)3D lump plot with the time t=0,(b)the corresponding density plot,(c)the curve by selecting different parameters y and t,(d)the curve by selecting different parameters x and t.

4 Summary and Discussions

In summary,the gCBS-BK equation was derived in terms of Hirota bilinear forms.By selecting the prime number p=3,a gCBS-BK-like equation was formulated by the generalized Hirota operators.The lump solution of the gCBS-BK equation and the gCBS-BK-like equation was generated by their Hirota bilinear forms.The phenomena of lump solutions were presented by fi gures.The results provide a new example of(2+1)-dimensional nonlinear partial differential equations,which possess lump solutions.Other new nonlinear equations can be also obtained by seleting the prime numbers p=5,7,...It is demonstrated that the generalized Hirota operators are very useful in constructing new nonlinear differential equations,which possess nice math properties.In the meanwhile,lumpkink interaction solutions,[34−35]lump-soliton interaction solutions,[36]lump type solutions for the(3+1)-dimensional nonlinear differential equations[36−38]and solitons-cnoidal wave interaction solutions[39−41]are important and will be explored in the future.