Breather wave and lump-type solutions of new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in incompressible fluid
aa r X i v : . [ n li n . PS ] F e b Noname manuscript No. (will be inserted by the editor)
Breather wave and lump-type solutions of new(3+1)-dimensional Boiti-Leon-Manna-Pempinelliequation in incompressible fluid
Jian-Guo Liu ∗ , Abdul-Majid Wazwaz Received: date / Accepted: date
Abstract
Under investigation is a new (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation. The main results are listed as follows: (i) lump solutions;(ii) Interaction solutions between lump wave and solitary waves; (iii) Inter-action solutions between lump wave and periodic waves; (iv) Breather wavesolutions. Furthermore, graphical representation of all solutions is studied andshown in some 3D- and contour plots.
Keywords lump solutions, breather wave solutions, solitary waves, periodicwaves.
Mathematics Subject Classification (2000) · · The (3+1)-dimensional Boiti-Leon-Manna-Pempinelli (BLMP) equation canbe used to describe the wave propagation in incompressible fluid [1] and theinteraction of the Riemann wave [2]. Many results about BLMP equation havebeen obtained. Darvishi [2] presented the stair and step Soliton Solutions. Zuo[3] derived the bilinear form, Lax pairs and B¨acklund transformations. Tang [4]obtained the new periodic-wave solutions. Liu [5,6] given the new three-wavesolutions and new non-traveling wave solutions. Mabrouk [7] obtained somenew analytical solutions. Li [8] presented the multiple-lump waves solutions.
Jian-Guo Liu(*Corresponding author)College of Computer, Jiangxi University of Traditional Chinese Medicine, Jiangxi 330004,China Tel.: +8613970042436Fax: +86079187119019E-mail: [email protected] WazwazDepartment of Mathematics, Saint Xavier University, Chicago, IL 60655, USAE-mail: [email protected] Jian-Guo Liu ∗ , Abdul-Majid Wazwaz Osman [9] derived lump waves, breather waves, mixed waves, and multi-solitonwave solutions. Peng [10] studied the breather waves and rational solutions. Xu[11] given the Painlev´e analysis, lump-kink solutions and localized excitationsolutions.In this paper, a new (3+1)-dimensional BLMP equation is studied asfollows [12]( u x + u y + u z ) t + ( u x + u y + u z ) xxx + ( u x ( u x + u y + u z )) x = 0 , (1)where u = u ( x, y, z, t ). The new BLMP equation was first proposed by Wazwaz[12]. The integrability, compatibility conditions, multiple soliton solutions andmultiple complex soliton solutions were discussed by Painlev´e test and Hirota’sdirect method.Based on the transformation u = − lnξ ( x, y, z, t )] x , (2)the new BLMP equation has the bilinear form[ D x + D y D x + D z D x + D t D x + D t D y + D t D z ] ξ · ξ = 0 . (3)The organization of this paper is as follows. Section 2 gives the lump so-lutions; Section 3 obtains the interaction solutions between lump wave andsolitary waves; Section 4 derives the interaction solutions between lump waveand periodic waves; Section 5 presents the breather wave solutions. All resultshave been verified to be correct by using Mathematica software [13-25]. Section6 gives the conclusion. Generally speaking, the lump solutions of nonlinear integrable equations canbe assumed as follows ξ = γ + ( α t + α x + α y + α z ) + ( β t + β x + β y + β z ) , (4)where α i (1 ≤ i ≤ β i (1 ≤ i ≤
4) and γ are undetermined real parameters.Substituting Eq. (4) into Eq. (3) and making the coefficients of x , y , xy , xz et al. be zero, undetermined real parameters in Eq. (4) have the followingresults( I ) α = − α − α , β = − β − β . (5)( II ) β = − α α β , α = α β β , β = − α β α + α , β = ( α + α ) α β . (6)( III ) α = α β β , β = − α β α + α , β = − α β α + α . (7)( IV ) α = − β β α , β = α + α α β , β = − α ( α + α ) + β β . (8) itle Suppressed Due to Excessive Length 3 ( V ) α = − α , β = − β , α = β = 0 . (9)( V I ) α = β = 0 , β = − α ( α + α + α ) + β ( β + β ) β . (10)( V II ) α = − α − α , β = − β − β , β = 2 α α β α − β . (11)( V III ) α = − α − α , β = α β α ,β = − ( α + α ) β α , β = 2 α α β α − β . (12)Substituting Eqs. (5)-(12) into Eq. (2) and Eq. (4), eight different types oflump solutions are derived. As an example, substituting Eq. (5) into Eq. (2)and Eq. (4), we have u = − [2[2 α [ α t + α x + α y − ( α + α ) z ] + 2 β [ β t + β x − ( β + β ) y + β z ]]] / [ γ + [ α t + α x + α y − ( α + α ) z ] + [ β t + β x − ( β + β ) y + β z ] ] . (13)Graphical representation of lump solution (13) is shown in Fig. 1. It’s obviousthat there is a peak and a bottom in Fig. 1, and they are symmetric. In thepeak, solution (13) has a maximum 2 √ x = − / √ y = 0. In thebottom, solution (13) has a minimum − √ x = 1 / √ y = 0. ( a ) -10 -5 0 5 10x-10-50510 y H b L Fig. 1 . Lump solution (13) with α = β = − β = − α = 2, t = z = 0, α = β = γ = 1. In this section, we will discuss the interaction phenomenon between lumpwave and solitary waves. Considering the following mixed functions ξ = γ + ( α t + α x + α y + α z ) + ( β t + β x + β y + β z ) + γ e tϕ + ϕ x + ϕ y + ϕ z + γ e − tϕ − ϕ x − ϕ y − ϕ z , (14) Jian-Guo Liu ∗ , Abdul-Majid Wazwaz where ϕ i (1 ≤ i ≤
4) and γ i ( i = 1 ,
2) are undetermined real parameters. As anexample, substituting Eq. (14) and Eq. (5) into Eq. (3) and making the coeffi-cients of e tϕ + ϕ x + ϕ y + ϕ z x , e tϕ + ϕ x + ϕ y + ϕ z y et al. be zero, undeterminedreal parameters in Eq. (14) have the following results α = − α − α , β = − β − β , ϕ = − ϕ − ϕ . (15)Substituting Eq. (15) into Eq. (2) and Eq. (14), the interaction solutions be-tween lump wave and solitary waves can be read as u = − [2[ − γ ϕ exp[ − tϕ − ϕ x − ϕ y + ( ϕ + ϕ ) z ] + 2 α [ α t + α x + α y − ( α + α ) z ] + 2 β [ β t + β x − ( β + β ) y + β z ]+ γ ϕ e tϕ + ϕ x + ϕ y − ( ϕ + ϕ ) z ]] / [ γ + γ exp[ − tϕ − ϕ x − ϕ y + ( ϕ + ϕ ) z ] + [ α t + α x + α y − ( α + α ) z ] + [ β t + β x − ( β + β ) y + β z ] + γ e tϕ + ϕ x + ϕ y − ( ϕ + ϕ ) z ] . (16)Graphical representation of the interaction solutions (16) is shown in Fig.2, Fig. 3 and Fig. 4. Fig. 2 describes the interaction phenomenon betweenlump wave and one solitary wave. Fig. 3 and Fig. 4 represent the interactionphenomenon between lump wave and two solitary waves. ( a ) ( b ) ( c ) -10 -5 0 5 10x-10-50510 y H d L -10 -5 0 5 10x-10-50510 y H e L -10 -5 0 5 10x-10-50510 y H f L Fig. 2 . Solution (16) with α = β = γ = − β = ϕ = − α = ϕ = 2, γ = z = 0, α = β = γ = ϕ = 1, when t = − t = 0 in (b,e), t = 3 in (c,f). itle Suppressed Due to Excessive Length 5 In this section, we will investigate the interaction phenomenon between lumpwave and periodic waves. Assuming the following mixed functions ξ = γ + ( α t + α x + α y + α z ) + ( β t + β x + β y + β z ) + γ sin ( µ t + µ x + µ y + µ z ) + γ cos ( tϕ + ϕ x + ϕ y + ϕ z ) , (17)where µ i (1 ≤ i ≤
4) is undetermined real parameter. ( a ) ( b ) ( c )( d ) ( e ) Fig. 3 . Solution (16) with α = β = γ = γ = − β = ϕ = − z = 0, α = ϕ = 2, α = β = γ = ϕ = 1, when t = − t = − t = 0 in (c), t = 2 in (d), t = 4 in (e). -10 -5 0 5 10x-10-50510 y H a L -10 -5 0 5 10x-10-50510 y H b L -10 -5 0 5 10x-10-50510 y H c L -10 -5 0 5 10x-10-50510 y H d L -10 -5 0 5 10x-10-50510 y H e L Jian-Guo Liu ∗ , Abdul-Majid Wazwaz Fig. 4 . The corresponding contour plots of Fig. 3.Substituting Eq. (17) and Eq. (5) into Eq. (3) and making the coefficients ofcos ( tϕ + ϕ x + ϕ y + ϕ z ) x , sin ( µ t + µ x + µ y + µ z ) x et al. be zero,undetermined real parameters in Eq. (17) have the following results α = − α − α , β = − β − β , ϕ = − ϕ − ϕ , µ = − µ − µ . (18)Substituting Eq. (18) into Eq. (2) and Eq. (17), the interaction solutions be-tween lump wave and periodic waves can be written as u = − [2[2 α [ α t + α x + α y − ( α + α ) z ] + 2 β [ β t + β x − ( β + β ) y + β z ] + γ µ cos[ µ t + µ x + µ y − ( µ + µ ) z ] − γ ϕ sin[ tϕ + ϕ x + ϕ y − ( ϕ + ϕ ) z ]]] / [ γ + [ α t + α x + α y − ( α + α ) z ] + [ β t + β x − ( β + β ) y + β z ] + γ sin[ µ t + µ x + µ y − ( µ + µ ) z ]+ γ cos[ tϕ + ϕ x + ϕ y − ( ϕ + ϕ ) z ]] . (19) ( a ) -10 -5 0 5 10x-10-50510 y H b L Fig. 5 . Solution (19) with α = β = γ = − β = ϕ = − z = t = γ = 0, α = ϕ = µ = µ = 2, α = β = γ = ϕ = 1, µ = − ( a ) -10 -5 0 5 10x-10-50510 y H b L itle Suppressed Due to Excessive Length 7 Fig. 6 . Solution (19) with α = β = γ = γ = − β = − z = t = 0, α = µ = µ = ϕ = ϕ = 2, α = β = γ = 1, µ = ϕ = − In this section, we will study the breather wave solutions. Choosing thefollowing mixed functions ξ = k e tϕ + ϕ x + ϕ y + ϕ z + γ sin ( µ t + µ x + µ y + µ z )+ γ cos ( ν t + ν x + ν y + ν z ) + e − tϕ − ϕ x − ϕ y − ϕ z , (20)where ν i (1 ≤ i ≤
6) and k are unknown real parameters. Substituting Eq.(20) into Eq. (3), we have the following results ν = − ν − ν , ϕ = − ϕ − ϕ , µ = − µ − µ . (21)Substituting Eq. (21) into Eq. (2) and Eq. (20), the breather wave solutionscan be presented as u = − [2[ − ϕ exp[ − tϕ − ϕ x − ϕ y + ( ϕ + ϕ ) z ] + γ µ cos[ µ t + µ x + µ y − ( µ + µ ) z ] + k ϕ e tϕ + ϕ x + ϕ y − ( ϕ + ϕ ) z − γ ν sin[ ν t + ν x + ν y − ( ν + ν ) z ]]] / [exp[ − tϕ − ϕ x − ϕ y + ( ϕ + ϕ ) z ]+ k e tϕ + ϕ x + ϕ y − ( ϕ + ϕ ) z + γ sin[ µ t + µ x + µ y − ( µ + µ ) z ]+ γ cos[ ν t + ν x + ν y − ( ν + ν ) z ]] . (22)Graphical representation of the breather wave solutions (22) is displayed inFig. 7. ( a ) -10 -5 0 5 10x-10-50510 y H b L Fig. 7 . Breather wave solutions (22) with γ = − ϕ = ν = − z = t = 0, γ = µ = µ = ϕ = ν = 2, ν = ϕ = 1, µ = k = − Jian-Guo Liu ∗ , Abdul-Majid Wazwaz Recently, a new (3+1)-dimensional BLMP equation is introduced by Wazwaz.At present, there is no literature on the lump solution of this equation. In thispaper, breather wave and lump-type solutions of the new (3+1)-dimensionalBLMP equation are presented. Lump-type solutions contain the interactionsolutions between lump and solitary waves, and the interaction solutions be-tween lump and periodic waves, which have not been studied in any literature.Furthermore, graphical representation for lump solution is displayed in Fig. 1.Interaction phenomenon of lump-type solutions are demonstrated in Figs. 2-7.
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