LUMP AND INTERACTION SOLUTIONS TO LINEAR(4+1)-DIMENSIONAL PDES?

Wen-Xiu MA(馬文秀)

College of Mathematics and Physics,Shanghai University of Electric Power,Shanghai 200090,China;Department of Mathematics,King Abdulaziz University,Jeddah,Saudi Arabia;Department of Mathematics and Statistics,University of South Florida,Tampa,FL 33620,USA;Department of Mathematics,Zhejiang Normal University,Jinhua 321004,China;College of Mathematics and Systems Science,Shandong University of Science and Technology,Qingdao 266590,China;International Institute for Symmetry Analysis and Mathematical Modelling,Department of Mathematical Sciences,North-West University,Ma fikeng Campus,Private Bag X2046,Mmabatho 2735,South Africa E-mail:mawx@cas.usf.edu

Abstract Taking a class of linear(4+1)-dimensional partial di ff erential equations as examples,we would like to show that there exist lump solutions and interaction solutions in(4+1)-dimensions.We will compute abundant lump solutions and interaction solutions to the considered linear(4+1)-dimensional partial di ff erential equations via symbolic computations,and plot three speci fic solutions with Maple plot tools,which supplements the existing literature on lump,rogue wave and breather solutions and their interaction solutions in soliton theory.

Key words symbolic computation;lump solution;interaction solution

1 Introduction

Di ff erential equations played a prominent role in a bunch of disciplines including engineering,physics,chemistry,economics and biology;and they were studied from di ff erent perspectives,mostly concerned with their solutions–functions that satisfy the di ff erential equations[1,2].One of the fundamental problems in the theory of di ff erential equations,called the Cauchy problem,is to find a solution of a di ff erential equation satisfying what are known as initial data.Laplace’s method and the Fourier transform method are established for solving Cauchy problems for linear ordinary and partial di ff erential equations,respectively.Soliton scientists had thought about nonlinearity innovatively and developed novel solution techniques–the isomonodromic transform method and the inverse scattering transform method–for dealing with Cauchy problems for nonlinear ordinary and partial di ff erential equations,respectively[3,4].

Only the simplest di ff erential equations,often linear,are solvable by explicit formulas.However,soliton theory does bring many di ff erent approaches for finding explicit solutions to nonlinear di ff erential equations.Recently,some systematical studies were made on a kind of interesting explicit solutions called lumps,originated from the study on formulation of solitons[5–7].Mathematically,lumps are a kind of rational function solutions that are localized in all directions in space,historically found for nonlinear integrable equations,and solitons are analytic solutions exponentially localized in all directions in space and time.Particular lumps can be generated from solitons by taking long wave limits[8].There also exist positons and complexitons to nonlinear integrable equations,enriching the diversity of solitons[9,10].Furthermore,interaction solutions between two di ff erent kinds of solutions are found to exist in soliton theory[11],and they can explain various nonlinear phenomena in sciences.

Within the Hirota bilinear formulation,solitons can be usually generated as follows

where

with kiand ωisatisfying the so-called dispersion relation and ξi,0being arbitrary phase shifts.The polynomial P determines a Hirota bilinear form

where Dxand Dtare Hirota’s bilinear derivatives,for a partial di ff erential equation with the dependent variable u.As an example of lumps,we point out that the KPI equation

possesses a class of lump solutions[12]

where two wave frequencies and a positive position shift are given by

and four wave numbers and two translation shifts are arbitrary but need to satisfy0,which guarantees rational localization in all directions in the(x,y)-plane.Other integrable equations that possess lump solutions contain the three-dimensional three-wave resonant interaction[13],the BKP equation[14,15],the Davey-Stewartson equation II[8],the Ishimori-I equation[16]and many others(see e.g.,[5,7,17]).

It is recognized through symbolic computations that many nonintegrable equations possess lump solutions as well,which include(2+1)-dimensional generalized KP,BKP,KP-Boussinesq and Sawada-Kotera equations[18–21].Moreover,many recent works exhibited interaction solutions between lumps and other kinds of exact solutions to nonlinear integrable equations in(2+1)-dimensions,including lump-kink interaction solutions(see e.g.,[22–25])and lump-soliton interaction solutions(see e.g.,[26–29]).In the(3+1)-dimensional case,lump-type solutions,which are rationally localized in almost all directions in space,were worked out for the integrable Jimbo-Miwa equations.Abundant such solutions were generated for the(3+1)-dimensional Jimbo-Miwa equation[30–32]and the(3+1)-dimensional Jimbo-Miwa like equation[33].It is de finitely interesting to search for lump and interaction solutions to partial di ff erential equations in(4+1)-dimensions or higher dimensions.

This paper aims at exploring lump solutions and their interaction solutions to a class of linear partial di ff erential equations in(4+1)-dimensions.Concrete examples of(4+1)-dimensional linear equations will be presented to show lump solution phenomena.Both lump solutions and interaction solutions,including lump-periodic,lump-kink and lump-soliton solutions,will be computed explicitly through Maple symbolic computations.Sufficient conditions which guarantee the existence of lump and interaction solutions will be acquired,and three-dimensional plots and contour plots of speci fic examples of the presented solutions will be made via Maple plot tools.A few concluding remarks will be presented in the last section.

2 Abundant Lump and Interaction Solutions

Let u=u(x1,x2,x3,x4,t)be a real function of the variables x1,x2,x3,x4,t∈R.We consider a class of linear(4+1)-dimensional partial di ff erential equations(PDEs)

where the subscripts denote partial di ff erentiation and αi,1 ≤ i≤ 10,are real constants.

We look for a kind of exact solutions

where v is an arbitrary real function,and ξi,1 ≤ i≤ 5,are five linear functions of the dependent variables

in which aij,1≤i≤5 and 1≤j≤6,are real constants to be determined.Then,the above class of linear PDEs(2.1)becomes

where wij,1≤i≤j≤5,are quadratic functions of the parameters aij,1≤i,j≤5.By equating all coefficients of the second partial derivatives of v to zero,we get a system of fifteen equations on the parameters aij,1≤ i,j≤ 5,and the coefficients αi,1≤ i≤ 10,

The arbitrariness of the parameters ai6,1≤i≤5,is due to the translation invariance of the equations in(2.1).

Through Maple direct symbolic computations,we can determine many solutions to this system of cubic equations.We classify the solutions we obtain into the following three categories

and

where

In each set of the three solutions above,the constants not determined in the set are arbitrary provided that all expressions in the set will make sense.Those three categories of the constants will present lumps and their interaction solutions,since a sufficient condition for u to be a lump

can be achieved,though all the three categories of the constants satisfy a determinant equation

Speci fically,we can generate the corresponding three examples on lump and interaction solutions as follows.

Example 1Upon taking a45=?a44and a55=?a54,the first solution(2.6)shows that the following linear(4+1)-dimensional PDE

possesses a kind of exact and explicit solutions

where ξi,1 ≤ i≤ 5,are de fined by

and the function g is arbitrary.

Example 2Upon taking a35=?a33and a45=?(a43+a44),solution(2.7)tells that the following linear(4+1)-dimensional PDE

possesses a kind of exact and explicit solutions

where ξi,1 ≤ i≤ 5,are de fined by

and the function g is again arbitrary.

Example 3Upon taking a15=?a11,a35=?(a31+a33)and a45=?(a41+a43+a44),solution(2.8)implies that the following linear(4+1)-dimensional PDE

possesses a kind of exact and explicit solutions

where ξi,1 ≤ i≤ 5,are de fined by

and the function g is once again arbitrary.

Now,further taking

where βi,1 ≤ i≤ 5,are proper constants which need to ensure the positivity of the generating function f,we can obtain lump solutions,and interaction solutions:lump-periodic,lump-kink and lump-soliton solutions to the above three linear(4+1)-dimensional PDEs,(2.11),(2.14)and(2.17),as follows

All solutions obtained above provide supplements to the theories available on soliton solutions and dromion-type solutions,formulated through basic approaches such as the Hirota perturbation technique and symmetry constraints(see e.g.,[34–39]).

Particularly taking

we get the three speci fic solutions to(2.17)

and

where ξ5= ?4x1+2x2+x3? 3x4+6t+1.The first solution is a lump,and the second and third ones are lump-periodic and lump-soliton solutions,respectively.Three three-dimensional plots and contour plots of those three solutions are made,to shed light on the characteristics of lump and interaction solutions,in Figure 1,Figure 2,and Figure 3.

Figure 1 Pro files of u1when t=0,1,2 and x3=2,x4=1:3d plots(top)and contour plots(bottom)

Figure 2 Pro files of u2when t=0,3,5 and x3=1,x4=?2:3d plots(top)and contour plots(bottom)

Figure 3 Pro files of u3when t=0,1.5,3 and x3=1,x4=3:3d plots(top)and contour plots(bottom)

3 Concluding Remarks

We studied a class of linear(4+1)-dimensional partial di ff erential equations to exhibit abundant lump and interaction solutions,including lump-periodic,lump-kink and lump-soliton solutions,via Maple symbolic computations.The results amend the existing soliton theory on nonlinear integrable equations and the recent studies on lumps and interaction solutions to linear partial di ff erential equations in(2+1)-and(3+1)-dimensions(see e.g.,[40]).Three concrete examples which possess lump and interaction solutions were explicitly presented,and three-dimensional plots and contour plots of three specially chosen solutions were made via Maple plot tools.

We remark that the obtained lump and interaction solutions also provide supplements to exact solutions generated from di ff erent kinds of combinations[41–43].Moreover,it will be interesting to look for lump and interaction solutions to other generalized bilinear and tri-linear di ff erential equations involving generalized bilinear derivatives[44].The corresponding interaction solutions will generally not be resonant solutions generated through the linear superposition principle[41,43].Though lump solutions generated from quadratic functions remain the same as in the Hirota case,integrable equations determined by generalized bilinear derivatives[44]can possess di ff erent interaction solutions(see[6]for a detailed discussion).

It is direct to formulate models in(n+1)-dimensions and their lump and interaction solutions following the pattern in the examples presented above.Diverse interaction solutions also imply that there exist the corresponding Lie-B?cklund symmetries,thereby supplementing symmetry theories on partial di ff erential equations.It is known that the Wronskian technique can solve nonlinear integrable equations,and therefore,our study brings up a new question:how can we formulate novel Wronskian solutions by adopting matrix entries of new type?It is also absolutely important to establish a basic theory of lumps and interaction solutions to di ff erence-di ff erential equations,and to see if such solutions can be constructed via Riemann-Hilbert problems[45].All those problems deserve further studies.