The New (2+1)-Dimensional Integrable Coupling of the KdV Equation: Auto-Bäcklund Transformation and Non-Travelling Wave Profiles

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1 MM Research Preprints, 36 3 MMRC, AMSS, Academia Sinica No. 3, December The New (+)-Dimensional Integrable Copling of the KdV Eqation: Ato-Bäcklnd Transformation and Non-Traelling Wae Profiles Zhena Yan Mathematics Mechanization Ke Laborator Academ of Mathematics and Sstems Science Chinese Academ of Sciences Beijing 8, China zan@mmrc.iss.ac.cn Abstract. The (+)-dimensional integrable copling of the KdV eqation, which was first presented b Ma and Fssteiner from the celebrated KdV eqation sing the pertrbation method of mltiple scales = + ɛ, = ɛ, is inestigated. With the aid of smbolic comptation, an ato-bäcklnd transformation is gained to seek new tpes of non-traelling wae soltions inoling an arbitrar fnction of. Moreoer a nontraelling wae similarit ariable transformation redces this sstem to a sstem of nonlinear ordinar differential eqations with constant coefficient, which is soled to get non-traelling wae Jacobi elliptic fnction soltions and Weierstrass elliptic fnction soltions inoling an arbitrar smooth fnction of. When the arbitrar fnction are taken as some special fnctions, these obtained soltions possess abndant strctres. The figres corresponding to these soltions are illstrated to show the rles of the wae propagation related to (+)-dimensional integrable copling of the KdV eqation.. Introdction PASC nmbers:.3ik; 3.6.Ge;..Y;.7.Wz. As is well known, the prototpical Korteweg-de Vries (KdV) eqation describes wae motion on the srface of shallow water and the nidirectional propagation of long wae of small amplitde and eists in man phsical branches. There are man etensions sch as the KdV-Brgers eqation, the KdV-mKdV eqation, (+)-dimensional KP eqation, (+)- dimensional generalized KdV eqation, the copled KdV eqations and KdV eqation hierarch(see Refs.[- and therein). Recentl, the inestigation of the integrable coplings has been paid more attention to[-7. The integrable coplings mean that for a gien integrable sstem (e.g. Painleé integrable, C-integrable, S-integrable, La integrable and Lioille integrable, etc.) t = K(), one needs to constrct a new bigger, non-triial integrable sstem, for eample { t = K(), t = C(, ), (.)

2 The New (+)-Dimensional Integrable Copling of the KdV Eqation 37 where C/ [ and [ denotes a ector consisting of all deriaties of with respect to the space ariables. Recentl, starting from the KdV hierarch t = Φ n (), Φ() = + +, n, (.) where =, = =. Ma and Fchssteiner[ made the pertrbation epansion û as depending on two space ariables and = ɛ[8 û = N φ i (,, t, t,...)ɛ i, = ɛ. (.3) i= The sbstittion of (.3) into (.) gies rise to the following (+)-dimensional pertrbation sstem φ t = φ + 6φ φ, φ t = φ + 3φ + 6(φ φ ) + 6φ φ, φ t = φ + 3φ + 3φ + 6(φ φ ) + 6(φ φ ) + 6φ φ, j φ jt = φ j + 3φ j, + 3φ j, + φ j 3, + 6( i= φ (.) iφ j i, + j i= φ iφ j i, ), 3 j N In the case N =, sstem (.) redces to the representatie sstem t = + 6, (.a) t = () + 6. (.b) In fact, this is a copling eqation of KdV eqation. In addition, we also deried a new integrable copling of the TC hierarch inclding KdV eqation from another wa[7. Sakoich[9 implemented the Painleé singlarit strctre analsis[ to (.a,b) sch that it is shown that (.a,b) possesses the Painlee integrabilit. Moreoer the La pair of (.) was obtained in terms of the relationship of independent ariable = ɛ and the known La pair of KdV eqation. Therefore it is said that (.) is a integrable copling of KdV eqation in + dimensions. Thogh Sakoich[9 applied the Weiss-Krskal algorithm related to the Painlee analsis to (.), i.e., (,, t) = i (, t)ψ i (,, t), (,, t) = j (, t)ψ j 3 (,, t), (.6) i= j= where ψ = ψ(,, t), ψ =, sch that it was shown that (.) passed the Painlee test. Therefore it is said that sstem (.a,b) is a (+)-dimensional (Painlee) integrable copling of the well-known KdV eqation (.a) Bt it was said that the eplicit epression for trncated epansions were too blk sch that Bäcklnd transformation of (.t) were not eplicitl fond. In addition, some traelling wae soltions in the form, (,, t) = (ξ), ξ = k + l + λt were obtained for sstem (.) sch as the Jacobi elliptic fnction soltion, the Weierstrass elliptic fnction soltion and the solitar wae soltions[.

3 38 Zhena Yan As far as we know, ato-bäcklnd transformation and eact non-traelling wae soltions (i.e., the relationship among these independent ariables,, t is not alwas linear) of (.a,b) were not reported et. In this paper we will trncate the special branch of Painlee analsis at the constant term leel sch that an ato-bäcklnd transformation is obtained. The obtained Bäcklnd transformation is sed to seek man tpes of non-traelling wae soltions of (.a,b). Moreoer sing some other transformations we also gain other tpes of non-traelling wae soltions of (.a,b). These soltions contains solitar-like wae soltions, singlarit solitar-like wae soltions, trigonometric fnctions soltions, dobl-periodic wae soltions and rational soltions, which ma be sefl to eplain some wae phenomena generating from (+)-dimensional integrable copling of the KdV eqation.. The deriation of Bäcklnd transformation for sstem (.a,b) As is well known the KdV eqation (.a) possesses the Painleé integrabilit and the following Bäcklnd transformation (,, t) = [log w(,, t) + (,, t), (.) where is a soltion of (.a) and w(,, t) satisfies w w t = w w 3w + 6w, (.) w t = 6w +. (.3) Therefore the transformation (.) makes balance linear term and nonlinear term 6. Bt it is eas to see that this transformation (.) does not make balance linear term 3 and nonlinear term 6 which are two terms onl inoling the fnction and its deriaties in (.b). Therefore the need balance with other terms in (.b), that is to sa and 6(). So the degree and 6() ma be greater than or eqal to ones of 3 and 6. There eist two cases to be considered: Case : If the degree of or 6() is greater than one of 3 or 6, then b leading order analsis we hae the epression for as (,, t) = p (,, t)w 3 (,, t) + p j (,, t)w j 3 (,, t), (.) j= where p i (,, t) and w(,, t) are both analtic fnctions of,, t near the manifold {(,, t) w(,, t) = }. This case with p (,, t) had been considered[9. If we trncate (.) at the constant leel O(w ) to get (,, t) = p (,, t) w 3 (,, t) + p (,, t) w (,, t) + p (,, t) w(,, t) + p 3(,, t), (.) then the sbstittion of (.) and (.) into (.b) can determine these fnctions p i (i =,,, 3) and the sstem of eqation that, p and w satisf. The determining sstem is

4 The New (+)-Dimensional Integrable Copling of the KdV Eqation 39 er complicated. We do not also consider this case here. Case : When the degree of or 6() is eqal to one of 3 or 6, we hae b the leading order analsis (,, t) = (,, t)w (,, t) + j (,, t)w j (,, t), (,, t) = w w (.6) j= and get the relationship = w b frther comparing the coefficient of w. Therefore with the aid of smbolic comptation, we trncate the Painleé epansion at the constant leel O(w ) to gain the ato-bäcklnd transformation (,, t) = (,, t) w (,, t) + (,, t) w(,, t) + (,, t) = log w(,, t) + (,, t). (.7) where the conditions in,, w are gien b w w w t + w w + 8w w w 3w w w w w +6 w w w =. (.8) (w w t + w w t + w w t ) + 6w w + w w w w w w + w w 6 w 8 (w w + w w ) 6 w 8 w w =. (.9) (w w t + 6 w + w ) + 6 ( w ) =. (.) Moreoer (, ) satisf sstem (.a,b). In fact, (.7) is eqialent to (.) in the first case with the conditions p, p = w w, p = w. To or knowledge, the Bäcklnd transformation (.7) is first obtained here. 3. Non-traelling wae soltions In the following we will se the aboe-obtained Bäcklnd transformations (.) and (.7) with the conditions (.),(.3) and (.8)-(.) to seek new non-traelling wae soltions of (.a,b). 3.. Solitar-like wae soltions We take the initial soltion of (.a,b) as { (,, t) = f(), (,, t) = g()+[6f()f ()+ 6f()g()t + h()}, where f(), g() and h() are all arbitrar smooth fnction of [-8 and assme that w is of the generalized form of -linear[-9 w(,, t) = P (, t) + ep[θ(, t) + Ψ(, t). (3.) where P (, t), Θ(, t) and Ψ(, t) are differentiable fnctions of and t onl to be determined. With the aid of smbolic comptation, sbstitting (3.) into (.) and (.3) and eqating coefficients of e (Θ+Ψ), e (Θ+Ψ) and e (Θ+Ψ) gies rise to these conditions P t =, Θ t =, Ψ t = Θ Θ. (3.)

5 33 Zhena Yan B sing (3.), the sbstittion of (3.) into (.8)-(.) leads to (Θ + f)p =, (Θ + f)θ + fθ =, (Θ + f)ψ + Θ[(6ff + 6fg)t + h + ΘΘ =, 3fΘ + fθ Θf =, fθ + fθ Θf =. (3.3) From (3.) and (3.3), we know that two cases need to be considered: Famil A: In the case Θ + f, we hae from (3.3) P (, t) = p = const, Θ(, t) = θ = const, f() = f = const θ, g() =, Ψ(, t) = (θ 3 θ + 6f θ)t θ h( )d. + f (3.) Therefore when p >, we get the bell-shaped solitar-like wae soltions inoling an arbitrar fnction h() from (.),(.7),(3.) and (3.) = θ sech [ θ + (θ 3 θh( ) + 6f θ)t θ d ln p + f, + f (3.a) = θ h() θ sech [ θ + (θ 3 θh( ) + 6f θ)t + f θ d ln p + h(), + f (3.b) while in the case p <, we get the singlar solitar-like wae soltion inoling an arbitrar fnction h() = θ csch [ θ + (θ 3 θh( ) + 6f θ)t θ d ln p + f, (3.6a) + f = θ h() θ csch [ θ + (θ 3 θh( ) + 6f θ)t + f θ d ln p + h(), (3.6b) + f Remark : When () = = const, p = in (3.a,b), these bell-shaped solitar-like wae soltion (non-traelling wae soltions) redce to the solitar wae soltions of (.a,b) which are the same as soltions fond [. When () = = const, p = in (3.6a,b), we get singlar solitar wae soltions of (.a,b). Remark : Since soltions (3.) and (3.6) inclde an arbitrar fnction h() of, the possesses abndant strctres. To illstrate the role of the arbitrar fnction h() in the soltion profiles (3.a,b) in the following we will take the different forms of the fnction h() to show the rles of the wae propagation related to the soliton model (.a,b). Tpe A: In the case h() = a + b (a, b, const.), the soltion (3.) redces to the form = θ sech [ θ + (θ 3 + 6f θ)t θ(a + b + c) θ ln p + f, (3.7a) + f

6 The New (+)-Dimensional Integrable Copling of the KdV Eqation 33 = θ (a + b) θ sech [ θ + (θ 3 + 6f θ)t θ(a + b + c) + f θ ln p + a + b, (3.7b) + f Figres and are the plots of soltions (3.7a,b) at time t = and show the semibarrel-shaped soliton-like soltion (3.7a) and the combination of solitar wae and plane wae (3.7b) in two dimensions with θ =, f = p = a = b =, c =, respectiel, since the soltion inoles the nonlinear term. Figre Figre Figre, (3.7a) Figre, (3.7b) Tpe A: In the case h() =, the soltion (3.) redces to the form = θ sech [ θ + (θ 3 θ log + 6f θ)t θ ln p + f, + f (3.8a) θ = [ (θ + f ) sech θ + (θ 3 θ log + 6f θ)t θ ln p + + f, (3.8b) Figres 3 and are the plots of soltions (3.8a,b) in two dimensions with θ =, f = p = a = b =, c = at time t =, respectiel, since the soltion inoles the nonlinear terms log and. Figre 3 Figre Figre 3, (3.8a) Figre, (3.8b) Tpe A3: In the case h() = e, the soltion (3.) redces to the form = θ sech [θ + (θ 3 + 6f θ)t θe θ ln p + f, + f (3.9a) = θ e θ sech [θ + (θ 3 + 6f θ)t θe + f θ ln p + e, + f (3.9b) Figres and 6 are the plots of soltions (3.9a,b) at time t = and show the qarterbarrel-shaped soliton-like soltion (3.9a) and another formal soliton-like wae (3.9b) in two

7 33 Zhena Yan dimensions with θ =, f = p = a = b =, c =, respectiel, since the soltion inoles the nonlinear term e. Figre Figre Figre, (3.9a) Figre 6, (3.9b) Tpe A: In the case h() = sin, the soltion (3.) redces to the form = θ sech [ θ + (θ 3 θ cos + 6f θ)t + θ ln p + f, (3.a) + f = θ sin θ sech [ θ + (θ 3 θ cos + 6f θ)t + + f θ ln p + sin, (3.b) + f Figres 7 and 8 are the plots of soltions (3.a,b) at time t = and show the rle of wae propagation related to (3.9a) and (3.9b) in two dimensions with θ =, f = p = a = b =, c =, respectiel, since the effect of the trigonometric fnction sin and cos. Figre 7 Figre Figre 7, (3.a) Figre 8, (3.b) Tpe A: In the case h() = sech, the soltion (3.) redces to the form = θ sech [ θ + (θ 3 θ tanh + 6f θ)t θ ln p + f, (3.a) + f = θ sech θ sech [ θ + (θ 3 θ tanh + 6f θ)t + f θ ln p + sech, (3.b) + f Figres 9 and are the plots of soltions (3.a,b) in two dimensions with θ =, f = p = a = b =, c = at time t =

8 The New (+)-Dimensional Integrable Copling of the KdV Eqation 333 Figre 9 Figre Figre 9, (3.a) Figre, (3.b) In addition, we can also take other fnction of sch that abndant strctres related to (3.a,b) are fond. Similarl the singlar soliton-like soltions (3.6a,b) hae also similar properties. We omit them here. Famil B: In the case Θ + f, we hae P (, t) = p(), Θ(, t) = θ = const, f() = θ, h() = g() =, Ψ(, t) = θ 3 t + ψ(). (3.) where p() and ψ() are arbitrar smooth fnctions of onl. Therefore when p() >, we get another famil of the bell-shaped solitar-like wae soltions from (.),(.7),(3.) and (3.) = θ sech [ θ θ 3 t + ψ() log p() θ, (3.3a) ( ) = θ ψ () p () sech [ θ θ 3 t + ψ() log p(). (3.3b) p() Similarl when p() <, we get the singlar solitar-like wae soltion = θ csch [ θ θ 3 t + ψ() log p() θ, (3.a) ( ) = θ ψ () p () csch [ θ θ 3 t + ψ() log p(). (3.b) p() Remark 3: The soltions (3.3a) and (3.a) are of similar properties to the soltions (3.a) and (3.6a), respectiel. Bt the soltions (3.3b) and (3.b) are different from the soltions (3.b) and (3.6b). For eample, let p() = θ =, the following figres - are the plots of the soltion (3.3b) with ψ() = ( + ),, sin, sech at t =. B comparing these figres related to (3.3b) and (3.b) we know that the are different. Figre Figre

9 33 Zhena Yan Figre, (3.3b) [ ψ() = ( + ) Figre, (3.3b) [ψ() = Figre 3 Figre Figre 3, (3.3b) [ψ() = sin Figre, (3.3b) [ψ() = sech 3.. Non-traelling wae rational soltions We take the initial soltion of (.a,b) as { (,, t) = f(), (,, t) = g()+[6f()f ()+ 6f()g()t + h()}, where f(), g() and h() are all arbitrar smooth fnction of and assme that w is of the -linear form[-8 w(,, t) = Σ(, t) + Ω(, t). (3.) where Σ(, t), Ω(, t) are differentiable fnctions of and t onl to be determined. With the aid of smbolic comptation, sbstitting (3.9) into (.), (.3), (.8)-(.9) and eqating coefficients of and reads Σ t =, Ω t = 6fΣ, Σ Ω t + 3gΣ + 9fΣΣ =, (3.6) Ω Ω t + 3Σ [(6ff + 6fg)t + h + 9fΣΩ =, ΣΩ t + Σ Ω t 3f Σ 3gΣ 8fΣΣ =, which leads to f() = f = const, Σ(, t) = σ = const, g() =, Ω(, t) = 6σf t σ f h( )d. (3.7) Therefore we hae the non-traelling wae rational soltion of (.a,b) from (.), (.7), (3.) and (3.7) σ (,, t) = ( σ + 6f t σ/f h( )d ) + f. (3.8a) (,, t) = σ /f h() ( σ + 6f t σ/f h( )d ) + h(). (3.8b) Remark : The non-traelling wae soltion (3.8a,b) inoles an arbitrar smooth fnction of which possesses abndant strctres. For eample, the figres and 6 are the plots of the soltions (3.8a,b) with h() = sech at t = with σ = f =

10 The New (+)-Dimensional Integrable Copling of the KdV Eqation Figre, (3.8a) [h() = sech Figre 6, (3.8b) [h() = sech and the figres 7 and 8 are the plots of the soltions (3.8a,b) with h() = sechtanh at t = with σ = f = Figre 7, (3.8a) [h() = sechtanh Figre 8, (3.8b) [ h() = sechtanh Remark : For the gien initial soltion (, ), one ma seek other transformations for w(,, t) sch that other tpes of soltions for (+)-dimensional integrable copling of KdV eqation (.a,b) are obtained with the aid of Bäcklnd transformation (.) and (.7).. Similarit redction and non-traelling dobl periodic soltions In the following we will consider other tpes of non-traelling wae soltions of (.a,b) from another wa. B inspection, we make the non-traelling wae transformation in the -linear form (,, t) = Γ (, t)(η) + Γ (, t), (,, t) = Λ (, t)(η) + Λ (, t), (.) η = α(, t) + β(, t). where Γ (, t), Γ (, t), Λ (, t), Λ (, t), α(, t) and β(, t) are the fnctions of, t to be determined later. The sbstittion of (.) into (.a,b) leads to Γ t + Γ t + Γ (α t + β t ) d dη Γ α 3 d3 dη 3 6Γ α d dη 6Γ Γ α d dη =, (.a) Λ t + Λ t + Λ (α t + β t ) d dη Λ α 3 d3 dη 3 3(Γ α d ) dη 6Γ Λ α d() dη 3Γ α (α + β ) d3 dη 3 6Γ Λ α d() dη 6Γ Λ α d() dη 6Γ Γ 6Γ Γ

11 336 Zhena Yan 6Γ (α + β ) d dη 6Γ Γ 6Γ Γ 6Γ Γ (α + β ) d dη =, (.b) where the sbscript denotes the partial deriaties. To make (.a,b) become the sstem of ODEs for (η) and (η) with constant coefficient. We hae the constraints for Γ (, t), Γ (, t), Λ (, t), Λ (, t), α(, t) and β(, t): α t (, t) = α (, t) =, β t (, t) = const, Γ t (, t) = Γ (, t) =, Γ t (, t) = Γ (, t) =, (.3) Λ t = Λ t =, Λ t = const β (, t), Λ t = const β (, t). From (.3) we hae α(, t) = k = const, β(, t) = λt + Ξ(), Γ (, t) = const, Γ (, t) = const, Λ (, t) = const Ξ (), Λ (, t) = const Ξ (), (.) where λ is an arbitrar constant, Ξ() an an arbitrar smooth fnctions of and the prime denotes the deriatie. Withot loss of generalit, we frther hae the transformations from (.) and (.) (,, t) = (η), (,, t) = Ξ ()(η), η = k λt + Ξ(), (.) nder which, sstem (.a,b) redces to the sstem of ODEs k 3 d dη + 3k + λ = c, where c, c are both integration constants. k 3 d dη + 3k d dη + 6k λ = c. (.6a) (.6b) Following or idea[-, we assme that (.6a,b) has the soltion (η) = A sn (η; m) + B sn(η; m)cn(η; m) + A sn(ξ; m) + B cn(ξ; m) + A, (.7a) (η) = a sn (η; m) + b sn(η; m)cn(η; m) + a sn(η; m) + b cn(η; m) + a. (.7b) where A i s, B i s, a i s, b is are constants to be determined, m( < m < ) is the modls of the Jacobian ellitic fnctions. With the aid of smbolic comptation, sbstitting (.7a,b) into (.6a,b) and sing the identities cn (η; m) = sn (η; m), dn (η; m) = m sn (η; m), (.8)

12 The New (+)-Dimensional Integrable Copling of the KdV Eqation 337 gies rise to a sstem of polnomials in sn i (η; m)cn(η; m)dn(η; m) and setting to zero their coefficients to leads to a set of algebraic eqations. We get the following soltions in real field b soling the sstem a = b = b = B = B = A =, λ = 6k a 8k 3 8k 3 m, A = k m, A = ka + k + k m, a = km, c = k m 3k 3 a + 8k a + 8k a m k k m, c = 3k a 8k 3 a 8k 3 m a + k m + k + k m. (.9) Therefore according to (.), (.7a,b) and (.9) we hae non-traelling wae Jacobi elliptic fnction (dobl periodic) soltions of (.a,b) = k m sn (k + Ξ() (6k a 8k 3 8k 3 m )t; m) ka + k + k m, (.a) = km Ξ ()sn (k + Ξ() (6k a 8k 3 8k 3 m )t; m) + a Ξ (). (.b) Remark 7: In the following we take Ξ() as some special fnction of in the soltions (.a,b) and (.a,b) sch that some phenomena are displaed. The figres 9 and are the plots of the soltions (.a) with Ξ() = at t = with k = a = Fig.9, (.a) [Ξ() =, m =. Fig., (.a) [Ξ() =, m =.99 The figres and are the plots of the soltions (.a) with Ξ() = sech at t = with k = a = Fig., (.a) [Ξ() = sech, m =. Fig., (.a) [Ξ() = sech, m =.99 The figres 3 and are the plots of the soltions (.b) with Ξ() = at t = with k = a =

13 338 Zhena Yan 3 3 Fig.3, (.b) [Ξ() =, m =. Fig., (.b) [Ξ() =, m =.99 The figres and 6 are the plots of the soltions (.b) with Ξ() = sech at t = with k = a = Fig., (.b) [Ξ() = sech, m =. Fig. 6, (.b) [Ξ() = sech, m =.99 In addition, if we assme that (.6a,b) has the soltion (η) = A ns (η; m) + B ns(η; m)cs(η; m) + A ns(ξ; m) + B cs(ξ; m) + A, (.a) (η) = a ns (η; m) + b ns(η; m)cs(η; m) + a ns(η; m) + b cs(η; m) + a. (.b) where A i s, B i s, a i s, b is are constants to be determined. With the aid of smbolic comptation, sbstitting (.a,b) into (.6a,b) we know that (.a,b) hae another famil of dobl periodic soltions = k ns (k + Ξ() (6k a 8k 3 8k 3 m )t; m) ka + k + k m, (.a) = kξ ()ns (k + Ξ() (6k a 8k 3 8k 3 m )t; m) + a Ξ (). (.b) Similarl, we also obtain non-traelling wae Weierstrass elliptic fnction soltions of (.a,b) = k (k + Ξ() 6k a t; g, g 3 ) ka, (.3a) = kξ () (k + Ξ() 6k a t; g, g 3 ) + a Ξ (), (.3a) where a, k, g, g 3 are arbitrar constants. The figres 7 and 8 are the plots of the soltions (.3a) with Ξ() =, sin at t = with k = a =, g =, g 3 =

14 The New (+)-Dimensional Integrable Copling of the KdV Eqation Fig.7, (.a) [Ξ() = Fig.8, (.a) [Ξ() = sin The figres 9 and 3 are the plots of the soltions (.3b) with Ξ() =, sin at t = with k = a =, g =, g 3 = e+6 3 e+6 Fig.9, (.b) [Ξ() = Fig.3, (.b) [Ξ() = sin In smmar, we hae fond an new ato-backlnd transformation of the new (+)- dimensional integrable copling of the KdV eqation (.a,b). B sing this transformation and some ansatze we hae arried at some families of eact solitar-like wae soltions and rational soltions, as well as dobl periodic soltions. Or reslt inclde both the known traelling wae solitar wae soltions, dobl periodic soltions and new non-traelling wae soltions inoling an arbitrar smooth fnction of. Therefore these soltions possess abndant strctres. We se figres to illstrate the motion rles of waes de to (.a,b). A frther std is needed to know whether there eist other tpes of eact soltions for (+)-dimensional integrable copling of the KdV eqation (.a,b). References [ M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Eoltion Eqations and Inerse Scattering (Cambridge: Cambridge Uniersit Press, 99). [ C.H.G, et. al., Soliton Theor and its Applications, Berlin: Spring-Verlag,99). [3 Z.Y. Yan, Commn. Theor. Phs., 36() 3. [ Z. Y. Yan and H. Q. Zhang, Proc. of th Asia Smp. Compt. Math., World Science, Singapore,, p93. [ W.X. Ma and B. Fchssteiner, Phs. Lett. A, 3(996) 9. [6 W.X.Ma, Methods Appl. Anal., 7(); J. Math. Phs., 3()8. [7 Z.Y.Yan and H.Q.Zhang, J. Math. Phs., 3()978. [8 A.H. Nafeh, Pertrbation Methods (Wile, New York, 973).

15 3 Zhena Yan [9 S. Y. Sakoich, J. Nonlinear Math. Phs., (998)3. [ J. Weiss, et al., J. Math. Phs., (983). [ E. G. Fan, Chaos, Solitons and Fractals, (3)67. [ Z.Y. Yan, Compt. Phs. Commn., 3(3). [3 Z.Y. Yan, Compt.Phs.Commn., 8() 3 [ Z.Y. Yan, Chaos, Solitons and Fractals, (3) 7. [ Z. Y. Yan and H. Q. Zhang, J. Phs. A, 3() 78. [6 Z. Y. Yan, J. Phs. A, 3()993. [7 Z. Y. Yan and H. Q. Zhang, Compt. Math. Appl. () 39. [8 Z.Y.Yan, Phs. Lett. A, 3(3)78. [9 B. Tian and Y. Gao, J. Phs. A, 9(996) 89.