Bäcklund transformation, multiple wave solutions and lump solutions to a (3 + 1)-dimensional nonlinear evolution equation

Transcription

1 Nonlinear Dyn 7 89:33 4 DOI.7/s ORIGINAL PAPER Bäcklnd transformation, mltiple wave soltions and lmp soltions to a 3 + -dimensional nonlinear evoltion eqation Li-Na Gao Yao-Yao Zi Y-Hang Yin Wen-Xi Ma Xing Lü Received: 3 October 6 / Accepted: May 7 / Pblished online: 9 Jne 7 Springer Science+Bsiness Media Dordrecht 7 Abstract In this paper, a 3 + -dimensional nonlinear evoltion eqation is cast into Hirota bilinear form with a dependent variable transformation. A bilinear Bäcklnd transformation is then presented, which consists of si bilinear eqations and involves nine arbitrary parameters. With mltiple eponential fnction method and symbolic comptation, nonresonant-typed one-, two-, and three-wave soltions are obtained. Frthermore, two classes of lmp soltions to the dimensionally redced cases with y = and y = are both derived. Finally, some figres are given to reveal the propagation of mltiple wave soltions and lmp soltions. L.-N. Gao Y.-Y. Zi Y.-H. Yin X. Lü B Department of Mathematics, Beijing Jiao Tong University, Beijing 44, People s Repblic of China XLV@bjt.ed.cn inglv6@aliyn.com W.-X. Ma Department of Mathematics and Statistics, University of Soth Florida, Tampa, FL 336-7, USA W.-X. Ma College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 669, Shandong, People s Repblic of China W.-X. Ma International Institte for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Camps, Private Bag X 46, Mmabatho 73, Soth Africa Keywords Bäcklnd transformation Nonresonant mltiple wave soltions Lmp soltion Symbolic comptation Mathematics Sbject Classification 3Q 3Q 37K4 Introdction Nonlinear evoltion eqations NLEEs inclding soliton eqations play an important role in the areas of mathematical physics [ 7]. Generally speaking, it is very difficlt to find eact soltions to NLEEs [8 4]. The transformed rational fnction method and mltiple eponential fnction method provide two effective pathways to constrct mltiple wave soltions [3,4]. If we can get the Hirota bilinear form for a NLEE, then we can derive the eact soltions with mltiple eponential fnction algorithm. Frthermore, the Bäcklnd transformation BT can also be sed in soltion aspects []. Based on a known soltion, we can obtain another soltion by sing BT. In this paper, we will stdy the following dimensional NLEE [7,]as yt y 3 y =, which was proposed firstly in Ref. [7,], and the resonant behavior of mltiple wave soltions has been investigated []. With symbolic comptation, two classes of lmp soltions have been derived to the dimensionally redced eqations in +-dimensions 3

2 34 L.-N. Gao et al. with = y and = t, respectively, by searching for positive qadratic fnction soltions to associated bilinear eqation [7]. It is important to stdy other properties for Eq. sch as BT, nonresonant mltiple wave soltions, and lmp dynamics with novel dimensional redctions. The strctre of this paper is as follows: In Sect., we will constrct a BT for Eq. based on its bilinear form, and as an application, we will derive some eact soltions via this BT. Note that the BT consists of si bilinear eqations and involves nine arbitrary parameters. Nonresonant-typed mltiple wave soltions will be solved in Sect. 3 by se of mltiple eponential fnction method. In Sect. 4, we will give two classes of lmp soltions to the dimensionally redced eqations in +-dimensions with y = and y =, respectively. Finally, Sect. presents discssions and conclsions, and we will plot some figres to describe the characteristics of mltiple wave soltions and lmp soltions. Bilinear BT. Constrction of BT Sbstittion of the dependent variable transformation = ln f with f = f, y,, t into Eq. yields the bilinear representation for Eq. as D t D y D 3 D y 3D + 3 D f f =, where D t D y, D 3 D y, D, and D are all the bilinear derivative operators [] defined by D α Dβ y Dγ t ρ ϱ = α y β y t γ t ρ, y, tϱ, y, t. =, y =y, t =t 3 To constrct a bilinear BT by means of Eq., we consider P f D t D y D 3 D y 3D + 3D g g g D t D y D 3 D y 3D + 3D f f, 4 in which g = g, y,, t is another soltion to Eq.. By sing echange formlas, symbolic comptation on Eq. 4 leads to P = [ D t D y D 3 D y 3D + ] 3D g g f + g [ D t D y D 3 D y 3D + ] 3D f f = [ D t D y g g f g D t D y ] + [ D 3 D yg g f g D 3 D y f f ] + 6 [ D g g f g D f f ] 6 [ D g g f g D f f ] = D 3D D y g f fg+ D 3D g f D y f g + D 6D D y g f D f g + D D g f fg + D y D 3 g f fg+ D y 3D g f D f g + D t 4Dy g f fg D D g f fg [ = D 3D D y + λ D y + λ + D + λ 8 D g f ] fg [ ] + D y D 3 + λ 3 λ D 4D t λ 9 D g f fg [ ] + D 3D + λ 4 D y + λ 6 g f Dy f g [ ] + D y 3D + λ D λ 6 g f D f g [ ] + D 6D D y + 6λ 7 D g f D f g [ ] D D + λ 8 D + λ 9 D y g f fg, where we have introdced nine arbitrary coefficients of λ i i =,, 3, 4,, 6, 7, 8, 9. To this stage, eqation decopling of Eq. gives rise to an alternative BT for Eq. as B g f = 3D D y +λ D y +λ + D +λ 8 D g f =, B g f = D 3 + λ 3 λ D 4D t λ 9 D g f =, B 3 g f = 3D + λ 4 D y + λ 6 g f =, B 4 g f = 3D + λ D λ 6 g f =, B g f = 6D D y + 6λ 7 D g f =, B 6 g f = D + λ 8 D + λ 9 D y g f =, 6 which consists of si bilinear eqations and involves nine arbitrary parameters.. Application of BT We take f = as a soltion to Eq., which corresponds to the soltion = ln f = toeq.. Solving BT of Eq. 6, we obtain si linear partial differential eqations as 3g y + g + λ g y + λ 8 g + λ g =, g 4g t λ g λ 9 g + λ 3 g =, 3g + λ g λ 6 g =, 7 g y + λ 7 g =, g + λ 8 g + λ 9 g =, 3g + λ 4 g y + λ 6 g =. 3

3 Bäcklnd transformation, mltiple wave soltions and lmp soltions 3 In the following, we will derive two classes of eact soltions to Eq. by solving Eq. 7 with symbolic comptation... Eponential fnction soltion to Eq. 7 Firstly, we consider eponential fnction soltions to Eq. 7 by taking g = + εe θ with θ = k + ly + m wt, where ε, k, l, m, and w are all constants. Selecting λ = λ 3 = λ 6 =, and solving Eq. 7, we have 3k l + k + λ l + λ 8 m =, k 3 + 4w λ k λ 9 m =, 3k + λ k =, 8 3k + λ 4 l =, kl + λ 7 k =, m + λ 8 k + λ 9 l =. One choice of soltions to Eq. 8 isasfollows { λ = λ 8 k + λ 8λ 9 l 3k l k l, λ 4 = 3k, l λ = 3k, λ 7 = l, m = λ 8 k + λ 9 l, } ω = 3λ 8 k 3λ 9 l k 3 l 3k, 9 l then, = ln g 3λ = kεek+ly λ 8k+λ 9 l 8 k 3λ 9 l k 3 l 3k l t + εe k+ly λ 8k+λ 9 l 3λ 8 k 3λ 9 l k 3 l 3k l t is a soltion to Eq.... First-order polynomial soltion to Eq. 7, Secondly, we take a first-order polynomial soltion into consideration as g = k + ly + m wt, where ε, k, l, m, and w are all constants. Selecting λ i = i 7 and ptting Eq. into 7, we obtain the following algebraic eqations k + λ 8 m =, 4w λ 9 m =, m + λ 8 k + λ 9 l =, which reslt in 3k lw 3m =. To this stage, = ln g k = with 3k lw 3m =, k + ly + m wt 3 is a soltion to Eq.. 3 Mltiple wave soltions 3. One-wave soltion Sbstitting f = + εe θ, θ = k + ly + m wt, with ε, k, l, m, and w as constants into Eq., we can obtain w = k 3 + 3m 3k, 4 l which is called the dispersion relation. As a reslt, the one-wave soltion to Eq. can be written as εkek+ly+m wt =. + εek+ly+m wt 3. Two-wave soltion Sbstitting f = + ε e θ + ε e θ + ε ε a e θ +θ, θ i = k i + l i y + m i w i t, with ε,ε, k i, l i, m i, and w i i =, as constants into Eq., we get 3

4 36 L.-N. Gao et al. w i = k 3 i + 3m i 3k i l i, a = b c, 6 where b = k 3 l + k 3 l k 3 l k 3 l 3k k l 3k k l + 3k k l + 3k k l + l w + l w w l w l + 3k + 3k 6k k 3m 3m + 6m m, 7 c = k + k 3 l + l l + l w + w + 3m + m 3k + k. 8 where b ij = ki 3 l i + k 3 j l j ki 3 l j k 3 j l i 3ki k jl i 3k i k j l j +3ki k jl j +3k i k j l i +l i w i + l j w j w i l j w j l i + 3ki + 3k j 6k ik j 3mi 3m j + 6m im j, c ij = k i + k j 3 l i + l j l i + l j w i + w j + 3m i + m j 3k i + k j. Finally, the three-wave soltion to Eq. is = [ 3i= k i ε i e θ i + ] i< j 3 k i + k j ε i ε j a ij e θ i +θ j + k + k + k3ε ε ε 3 a 3 e θ +θ +θ i= ε i e θ i + i< j 3 ε iε j a ij e θ, i +θ j + ε ε ε 3 a 3 e θ +θ +θ 3 As a reslt, the two-wave soltion to Eq. can be written as = [ k ε e θ + k ε e θ + a k + k ε ε e θ ] +θ + ε e θ + ε e θ + a ε ε e θ, +θ 9 where w i and a are determined by Eq Three-wave soltion Following the derivation of one- and two-wave soltions, we assme f = + ε e θ + ε e θ + ε 3 e θ 3 + ε ε a e θ +θ + ε ε 3 a 3 e θ +θ 3 + ε ε 3 a 3 e θ +θ 3 + ε ε ε 3 a 3 e θ +θ +θ 3, where θ i = k i + l i y + m i w i t, i =,, 3, a 3 = a a 3 a 3, and ε,ε,ε 3, k i, l i, m i,w i are constants. With symbolic comptation, we get where ε,ε, k i, l i, m i are arbitrary constants, and w i and a ij are determined by Eq.. 4 Lmp soltions In this section, we will search for positive qadratic fnction soltions to dimensionally redced bilinear Eq. via taking y = or y =, correspondingly to constrct lmp soltions to dimensionally redced forms of Eq.. We begin with the assmption f = g + h + a 9, 3 and g = a + a + a 3 t + a 4, h = a + a 6 + a 7 t + a 8, where a i i 9 are all real parameters to be determined. To constrct the lmp soltions, we note that the conditions garanteeing the well definedness of f, positiveness of f and localiation of in all directions in the space need to be satisfied. w i = k 3 i + 3m i 3k i l i, a ij = b ij c ij, i =,, 3, 4. Lmp soltions to redction with y = With y =, the dimensionally redced form of the bilinear Eq. trns ot to be 3

5 Bäcklnd transformation, mltiple wave soltions and lmp soltions 37 D t D D 4 3D + 3 D f f =, 4 that is, f t f f t f 3 f f f which is transformed into f f 4 f f + 3 f + 3 f f f =, t =. 6 throgh the link between f and : [ ] = ln f,, t = f,, t f,, t. 7 Sbmitting Eq. 3 into, we obtain the following set of constraining eqations for the parameters: { a = a, a = a, a 3 = 3a a a + a + a 6 6a a a 6 a +, a a 4 = a 4, a = a, a 6 = a 6, a 7 = 3a a + a + a a 6 6a a a 6, a + a a a 8 = a 8, a 9 = + a 3 a a 6 a a }, 8 which needs to satisfy a 6 > a and a a 6 a a =. The positive qadratic fnction soltion to Eq. is f = a + a + 3a a a + a + a 6 6a a a 6 a + t + a 4 a + a + a 6 + 3a a + a + a a 6 6a a a 6 a + t + a 8 a a + a 3 + a a 6 a a, 9 which, in trn, generates a class of lmp soltions to dimensionally redced Eq. throgh the transformation = ln f as I = 4a g + a h, 3 f where the fnction f is defined by Eq. 9, and the fnctions g and h are given as follows: g = a + a + 3a a a + a + a 6 6a a a 6 a + t + a 4, a h = a + a 6 3a a + a + a a 6 6a a a 6 a + t + a 8. a 4. Lmp soltions to redction with y = With y =, the dimensionally redced form of the bilinear Eq. trns ot to be D t D D 3 D 3D + 3D f f =, 3 that is, f t f f t f f f f f 3 f f + 3 f f 3 f f f + 3 which is transformed into f f f =, 3 t =, 33 throgh the link of Eq. 7 between f and. Sbstitting f = g + h + a 9 into Eq. 3, we obtain the following set of constraining eqations for the parameters: { a = a, a = a, 3a a a a a 6 + 6a a a 6 a 3 = a +, a 6 a 4 = a 4, a = a, a 6 = a 6, 3a 6 a + a a + a 6 + 6a a a a 7 = a +, a 8 = a 8, a 6 a + a 6 a + a a a + a a 6 } a 9 = a a 6 a a, 34 3

6 38 L.-N. Gao et al. a a.. b 6 4 t t Fig. a Characteristics of the one-wave soltion via Eq. with ε =, k =, l =, m =, w = 4, and y = = ; b characteristics of the two-wave soltion via Eq. 9 with ε =, ε =, k =, k =, l = 3, l = 3, m =, m =, w =, w =, and y = = which needs to satisfy a a 6 a a = and a a + a a 6 <. The positive qadratic fnction soltion to Eq. 3 is 3a a f = a a a a 6 + 6a a a 6 + a + a + t + a 4 a 6 + a +a 6 + 3a 6a +a a + a 6 + 6a a a a + t + a 8 a 6 a + a 6 a + a a a + a a 6 a a 6 a a, 3 which, in trn, generates a class of lmp soltions to dimensionally redced Eq. throgh the transformation = ln f as II = 4a g + a h, 36 f b Fig. a Characteristics of lmp soltion I via Eq. 3 with a =, a =, a 4 =, a = 3, a 6 = 4, a 8 =, and t = ; b characteristics of lmp soltion II via Eq. 36 witha =, a =, a 4 =, a = 3, a 6 = 4, a 8 =, and t = where the fnction f is defined by Eq. 3, and the fnctions g and h are shown as follows: g = a + a + 3a a a a a 6 + 6a a a 6 a + t + a 4, a 6 h = a + a 6 + 3a 6 a + a a + a 6 + 6a a a a + t + a 8. a 6 Discssions and conclsion High-dimensional problems in soliton theory attract mch more attention in recent research. For eample, by sing mltiple ep-fnction method and symbolic comptation, one-wave, two-wave, and three-wave soltions have been presented to 3 + -dimensional generalied KP and BKP eqations [4,3,4]. Resonant behavior of mltiple wave soltions and lmp 3

7 Bäcklnd transformation, mltiple wave soltions and lmp soltions 39 a b a b = = = = = =. Fig. 3 a Contor of lmp soltion I ; b contor of lmp soltion II Fig. 4 a Plot of crves of lmp soltion I with =, =, and = 3; b plot of crves of lmp soltion II with =, =, and = dynamics has been stdied for a 3 + -dimensional NLEE [7,9]. In this paper, we have firstly transformed the 3+- dimensional nonlinear partial differential eqation, that is, Eq. into Hirota bilinear form by a dependent transformation. Then, a bilinear BT has been constrcted [see Eq. 6], which consists of si bilinear eqations and incldes nine arbitrary parameters. As an application, we have derived two classes of eact soltions [see Eqs. and 3] to Eq. by sing this BT. Moreover, with mltiple ep-fnction method and symbolic comptation, nonresonant-typed mltiple wave soltions have been given to Eq. inclding one-wave, two-wave, and three-wave soltions [see Eqs., 9, and ]. Characteristics of the onewave and two-wave soltions are shown in Fig.. Finally, two classes of lmp soltions have been investigated to the dimensionally redced forms of Eq. with y = and y =, respectively, i.e., Eqs. 6 and 33. We fond no lmp soltion in the form of f = g + h + a 9 to redced Eq. via taking y = t. To reveal the lmp dynamics, 3-dimensional plots, density plots and -dimensional crves with particlar choices of the involved parameters in the potential fnction are given in Figs., 3, and 4, respectively. Acknowledgements This work is spported by the Open Fnd of IPOC BUPT nder Grant No. IPOC6B8. Y. H. Yin is spported by the Project of National Innovation and Entreprenership Training Program for College Stdents nder Grant No W. X. Ma is spported in part by the National Natral Science Fondation of China nder Grant Nos and 78, Natral Science Fondation of Shanghai nder Grant No. ZR44, Zhejiang Innovation Project of China nder Grant No. T9, the First-class Discipline of Universities in Shanghai and the Shanghai University Leading Academic Discipline Project No. A3---4, and the Distingished Professorship at Shanghai University of Electric Power. References. Zhang, Y., Dong, H., Zhang, X., Yang, H: Rational soltions and lmp soltions to the generalied 3 + -dimensional shallow water-like eqation. Compt. Math. App. 73,

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