Diversity soliton solutions for the (2+1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation

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1 Jornal of Mathematics Research; Vol. 6, No. ; ISSN E-ISSN Pblished b Canadian Center of Science and Edcation Diversit soliton soltions for the (+)-Dimensional Boiti-Leon-Manna-Pempinelli Eqation Xichao Deng, Hanlin Chen & Zhenhi X 3 Applied Technolog College, Sothwest Universit of Science and Technolog, Mianang 6, PR China School of Science, Sothwest Universit of Science and Technolog, Mianang 6, PR China. 3 Applied Technolog College, Sothwest Universit of Science and Technolog, Mianang 6, PR China Correspondence: Zhenhi X, Applied Technolog College, Sothwest Universit of Science and Technolog, Mianang 6, PR China. Tel: xzhenhi9@63.com Received: September 5, Accepted: October 8, Online Pblished: October 8, doi:.5539/jmr.v6np85 URL: This work was spported b Chinese Natral Science Fondation Grant No.68, Sichan Edcational science Fondation Grant No.9zc8. Abstract Diversit soliton soltions, inclding breather-tpe kink two wave soltions, cross-kink two solitar soltions, breather-tpe kink three wave soltions, kink three soliton soltions, are obtained for the (+)-Dimensional Boiti-Leon-Manna-Pempinelli Eqation b sing Hirota s bilinear form and extended homoclinic test approach, respectivel. Moreover, the properties for some new soltions are shown with some figres. Kewords: Boiti-Leon-Manna-Pempinelli Eqation, extended homoclinic test approach, bilinear form, breathertpe soliton, cross-kink two-soliton. Introdction It is well known that exact soltion of nonlinear evoltion eqations pla an important role in nonlinear science fields, especiall in nonlinear phsical science since the can provide mch phsical information and more insight into the phsical aspects of the problem and ths lead to frther applications. Man effective and powerfl methods to seek exact soltions were proposed, sch as the inverse scattering method, the homogeneos balance method, Hirotas bilinear method, the Darbox transformation method, Wronskian techniqe and so on [-6]. Ver recentl, Dai et al.[7-9] proposed a new techniqe called the extended homoclinic test approach to seek solitar wave soltions for integrable eqations, and this method has been sed to investigate several eqations[-]. The extended homoclinic test techniqe is an extension of the homoclinic test method, the main difference between the two methods mentioned above is the test fnction of constrcting exact soltion. In this paper, we consider (+)-Dimensional Boiti-Leon-Manna-Pempinelli Eqation: φ t 3φ x φ x 3φ φ xx + φ xxx =, () where φ(x,, t) : Rx R Rt R is a real fnction. It is called potential Boiti-Leon-Manna-Pempinelli (BLMP) eqation. The Painlevé analsis, lax pairs and some exact soltions of Eq. () were given in Ref[-3]. In this work, we mainl appl extended homoclinic test approach to determine diversit soliton soltions for Eq.(). As a reslt, breather-tpe kink two wave soltions, cross-kink two solitar soltions and kink three soliton soltions are obtained. B sing Painlevé analsis, we assme φ = (ln f ) x, () where f = f (x,, t) is nknown real fnction. Sbstitting Eq.() into Eq.() and sing the bilinear form, we have (D D t + D 3 xd ) f f =, (3) 85

2 Jornal of Mathematics Research Vol. 6, No. ; where the bilinear operator D is defined as D m x D n D s t f (x,, t) g(x,, t) = ( x x ) m ( ) n ( t t ) s [ f (x,, t)g(x,, t )] x =x,, =,t =t.. Breather-tpe Kink and Kink Two Wave Soltions Now we choose extended homoclinic test fnction f (x,, t) = e θ + ξ cos(τ) + ξ e θ, () where θ = α x + β + γ t, τ = α x + β + γ t and α i, β i, γ i, ξ i (i =, ) are some constants to be determined later. Sbstitting Eq.() into Eq.(3) and eqating all the coefficients of different powers of e θ, e θ, sin(τ), cos(τ) and constant term to zero ields a set of algebraic eqations: ξ (β γ 3α α β 3α α β β γ + α 3 β + α 3 β ) = ξ ξ (β γ 3α α β 3α α β β γ + α 3 β + α 3 β ) = ξ ξ (3α α β α 3 β + α 3 β 3α α β + β γ + β γ ) = ξ (3α α β α 3 β + α 3 β 3α α β + β γ + β γ ) = 6α 3 β ξ + β γ ξ β γ ξ + α3 β ξ =. Solving the sstem of Eqs.(5), we obtain the following cases Case(I): α = ξ α β ξ β, γ = α3 (8ξ β ξ β ) ξ β (5), γ = ξ α 3 β (3ξ β 6ξ β ), (6) ξ 6β3 where α, β, β, ξ, ξ are free real constants. Sbstitting Eq. (6) into Eq. () and taking ξ >, ξ, β, we have f = ξ cosh(α x + β + E t + ln(ξ )) + ξ cos(i x β + G t), (7) where E = α3 (8ξ β ξ β ), I ξ = ξ α β, G β ξ β = ξ α 3 β (3ξ β 6ξ β ). Sbstitting Eq. (7) into Eq. (), we obtain the ξ 6β3 breather-tpe kink two-wave soltions for BLMP eqation as follows(see Fig (a)) φ = ξ β sin(τ) α (ξ β ξ sinh(θ+ ln(ξ ))) ξ β (, (8) ξ cosh(θ+ ln(ξ )))+ξ cos(τ) where θ = α x+β + E t, τ = I x β +G t. The soltion represented b Eq.(8) is a breather-tpe kink two-wave which has breather effect when wave along with straight line I x β + G t = d and it also is a kink solitar wave as it along with straight line I x β + G t = d, where d are constants. Case(II): β = β i, γ = 3α α i + 3α α α 3 + α3 i γ i, ξ = ξ (γ α 3 ) (3α α i+3α 3 i+3α α +γ α 3 ), (9) where α, α, β, γ, ξ are free real constants. Sbstitting Eq. (9) into Eq. () and taking α = A i, β = B i, γ = C i and M >, we have f = M cosh(α x B + E t + ln(m)) + ξ cosh(a x + B + C t), () where E = 3α A 3A α α 3 + A3 + C ξ, M = (α3 +γ ). Sbstitting Eq. () into Eq. (), we (3α A 3α A +3α3 +A3 +γ ) obtain the cross-kink two-solitar wave soltions for BLMP eqation as follows(see Fig (b)) φ = (α M sinh(θ+ where θ = α x B + E t, τ = A x + B + C t. ln(m))+ξ A sinh(τ)) M cosh(θ+ ln(m))+ξ cosh(τ), () 3. Breather-tpe Kink and Kink Three Wave Soltions If we choose extended homoclinic test fntion as f = e θ + ξ cos(τ) + ξ sinh(ω) + ξ 3 e θ, () 86

3 Jornal of Mathematics Research Vol. 6, No. ; where θ = α x + β + γ t, τ = α x + β + γ t, ω = α 3 x + β 3 + γ 3 t and α i, β i, γ i, ξ i (i =,, 3) are some constants to be determined later. Sbstitting Eq. () into Eqs. (3), and eqating all the coefficients of different powers of e θ, e θ, sin(τ), cos(τ), sinh(ω), cosh(ω) and constant term to zero ields a set of algebraic eqations: ξ (α 3 β α 3 β β γ 3α α β + β γ 3α α β ) = ξ (α 3 3 β + β γ 3 + β 3 γ + 3α α 3β + α 3 β 3 + 3α α 3 β 3) = ξ (β 3 γ 3 + 3α 3 α β + β γ + α 3 3 β 3 + 3α α 3β 3 + α 3 β ) = ξ ξ (α 3 β + α 3 3 β 3 3α α 3 β + β 3 γ 3 β γ 3α α 3β 3 ) = ξ ξ 3 ( α 3 β + α 3 β + β γ + β γ + 3α α β 3α α β ) = ξ ξ 3 (α 3 β + α 3 β β γ 3α α β + β γ 3α α β ) = ξ ξ 3 (β 3 γ 3 + 3α 3 α β + β γ + α 3 3 β 3 + 3α α 3β 3 + α 3 β ) = ξ ( α 3 β + α 3 β + β γ + β γ + 3α α β 3α α β ) = ξ ξ ( β 3 γ + 3α α 3β + α 3 β 3 3α 3 α β 3 α 3 3 β β γ 3 ) = ξ ξ 3 (α 3 3 β + β γ 3 + β 3 γ + 3α α 3β + α 3 β 3 + 3α α 3 β 3) = ξ α3 β ξ β γ + 6ξ 3 α 3 β + ξ 3 β γ + ξ α3 3 β 3 + ξ β 3γ 3 =, Solving the sstem Eqs.(3), we obtain the following cases Case(III): α 3 = α, β 3 = β, γ = α (3α α ), γ = α (α 3α ), γ 3 = α (α 3α ), ξ 3 = α β ξ α β ξ α β, () where α, α, β, β, ξ, ξ are free real constants. Sbstitting Eq.() into Eq.() and and taking M >, we have f 3 = M cosh(α x + β E 3 t + ln(m)) + ξ cosh(α x + β + I 3 t) ξ sinh(α x β E 3 t), (5) where E 3 = α (3α α ), I 3 = α (α 3α ), M = α β ξ α β ξ α β. Sbstitting Eq. (5) into Eq. (), we obtain the breather-tpe kink three wave soltions for BLMP eqation as follows(see Fig (c)) φ 3 = (α M sinh(θ+ ln(m)) ξ α sin(τ) ξ α cosh(ω)) M cosh(θ+ ln(m))+ξ cos(τ) ξ sinh(ω) (3), () where θ = α x + β E 3 t, τ = α x + β + I 3 t, ω = α x β E 3 t. The soltion represented b Eq.() is a breather-tpe kink three wave which has breather effect when wave along with straight line α x + β + I 3 t = d and also is a two-solitar wave as α x + β + I 3 t = d, where d is a constant. Case(IV): α = (α α 3 )i, β = β i, β 3 = β i, γ = α 3, γ = (5α 3 3α α 3 + 3α 3 α α 3 3 )i, γ 3 = 3α 3 α3 3 3α α 3 3α 3, ξ = ξ (α +α 3 ) α 3 3α. (5) where α, α 3, β, γ, ξ, ξ 3 are free real constants. Sbstitting Eq.(5) into Eq.() and taking δ 3 >, we have f = ξ 3 cosh(α x B α 3 t + ln(ξ 3)) + M cosh(e x + B + I t) + ξ sinh(α 3 x + B + G t), (6) where M = ξ (α +α 3 ) α 3 3α, E = α α 3, I = 3α α 3 + α 3 3 5α3 3α 3 α, G = 3α 3 α3 3 3α α 3 3α 3. Sbstitting Eq. (6) into Eq. (), we obtain the kink three soliton soltions for BLMP eqation as follows φ = (α ξ3 sinh(θ+ ln(ξ 3)) M(α 3 α ) sinh(τ)+ξ α 3 cosh(ω)), (7) ξ 3 cosh(θ+ ln(ξ 3))+M cosh(τ)+ξ sinh(ω) where θ = α x B α 3 t, τ = E x + B + I t, ω = α 3 x + B + G t.. Conclsion In smmar, sccessfll appling the extended homoclinic test method to the (+)-Dimensional Boiti-Leon- Manna-Pempinelli eqation, we obtain new breather soliton and cross-kink soliton soltions. With the aid of Maple, this method provides a powerfl mathematical tool to obtain exact soltions. The extended homoclinic test approach can also be applied to solve other tpes of higher dimensional integrable and non-integrable sstems. 87

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