The bi-Hamiltonian structure and new solutions of KdV6 equation

The bi-Hamiltonian stru cture and new solut i o ns of KdV6 equatio n Y uqin Y ao 1 and Y un b o Zeng 2 Dep ar tment of Mathematics, Tsingh ua U niversity, Beijing 1 00084 , PR China Abstract. W e sho w that the KdV6 equation recen tly studied in [1, 2] is equiv ale nt to the Roso c hatius deformation of KdV equation with se lf-consisten t sources (RD-K dVESCS) recen tly prese nted in [9]. The t -type bi-Hamiltonian fo rmalism of KdV6 equation (RD- KdVESCS) is constructed b y taking x as ev olution parameter. Some new solutions of KdV6 equation, such as soliton, p ositon and negaton solution, are presen ted. Mathematics Sub ject C lassifications (2000): 35Q5 3, 37 K35. Key w ords: KdV6 eq uation, Roso c hatious deformation of KdV equation with self-consisten t source, bi-Hamiltonian structure, p ositon solution, negaton solution. 1 In tro duction Recen tly , the 5 authors of [1] applied the P ainlev ´ e analysis to the class of sixth-order nonlinear wa v e equations, a nd found 4 cases that pass the P ainlev ´ e t est. Three of those cases corresp ond to previously known in tegrable equations, whereas the fourth one turns out to b e new: ( ∂ 3 x + 8 u x ∂ x + 4 u xx )( u t + u xxx + 6 u 2 x ) = 0 . (1) This equation, a s it stands, do es not b elong to any recognizable theory . In the v ariables v = u x , w = u t + u xxx + 6 u 2 x , (1) is con v erted to v t + v xxx + 12 v v x − w x = 0 , (2a) w xxx + 8 v w x + 4 w v x = 0 , (2b) whic h is referred as KdV6 equ atio n in [1] and regarded as a nonholonomic deformation of the KdV equation. The authors of [1] found Lax pair and an a ut o -B¨ a c klund tra nsforma- tion for (2). They claimed that (2) is differen t from the KdV eq uation w ith self-consis ten t sources (KdVESCS) and rep orted tha t they w ere una ble to find higher symmetries and ask ed if higher conserv ed densities and a Hamiltonian formalism exist for (2) . In [2], Kup ershmidt describ ed (2) as a nonholonomic p erturbations of bi-Hamiltonian systems . By rescaling v and t in (2), one g ets u t = 6 u u x + u xxx − w x , (3a) w xxx + 4 uw x + 2 w u x = 0 , (3b) 1 yqyao@math.tsinghua.edu.cn 2 yzeng@math.tsinghua.edu.cn 1 whic h can b e con v erted in to a nonholonomic p erturbations of bi-Hamiltonian systems [2] u t = B 1 ( δ H n +1 δ u ) − B 1 ( w ) = B 2 ( δ H n δ u ) − B 1 ( w ) , (4a) B 2 ( w ) = 0 , (4b) where B 1 = ∂ = ∂ x , B 2 = ∂ 3 + 2( u∂ + ∂ u ) (5) are the tw o standard Hamiltonian op erators of the KdV hierarc h y , n = 2, and H 1 = u, H 2 = u 2 2 , H 3 = u 3 3 − u 2 x 2 , · · · Then the author in [2 ] b eliev ed that he could prov e the in tegrability of KdV6 equation b y constructing the infinite comm uting hierarch y KdV n 6 (4) with a common infinite set of conserv ed densities. Some solutions for (2) were obtained in [1, 3]. The soliton equations with self-consisten t sources (SESCS) ha v e attra cted m uc h at- ten tion (see [4]- [7]) and ha v e imp ortant ph ysical applications, for example, the KdV equation with self-consisten t sources (KdVES CS) describ es the in teraction of lo ng a nd short capillary-gra vit y w a v es [4]. The Roso c hatius deformation of finite-dimensional in te- grable Hamiltonian syste m (FD IHS) also has imp ortant phy sical application, for example, the Garner-Roso c hatius sys tem can be used to solv e the m ulticomp onen t coupled nonlin- ear Sc hr ¨ o dinger equation [8]. W e g eneralized the Roso c hatius deformation from FDIHS to SESCS and presen ted man y Roso ch atius deformations of SESCS (RD-SESCS) in [9], suc h as RD -KdVESCS whic h stationary reduction giv es rise to the w ell-kno wn generalized Henon-Heiles system [10]. In this pap er, w e would lik e to answ er the questions men tioned in [1]. W e will first sho w that (3) is equiv alen t to the Rosochatius deformation of KdVESCS (RD-KdVESCS) presen ted in [9]. It is kno wn [11, 12] that some soliton equations ha v e b oth x − and t − t yp e Hamilto nian for mulation. Ho w ev er the Hamiltonian f orm ulation for KdV6 equation (RD-KdVESCS) can no t b e written in usual w a y . W e w ill formulate it as an infinite- dimensional in tegrable bi-Hamiltonian sy stem with a t − t ype Hamilto nia n operat o r b y taking t as the ’spatial’ v ariable and x as the ev olution parameter as in the case of KdVESCS [13, 1 4]. Since the KdV6 equation can b e regarded as the KdV equation with no-homogeneous term and w is related to the square o f eigenfunction, we may apply t he metho d of v aria nt of constan t to find some new solutions of KdV6 equation starting from the kno wn solutions of KdV equation. The pr esen t pap er is organized a s follows. W e will first con v ert the KdV6 equation into RD-KdVESCS, and presen t exten sion of KdV6 equation in s ection 2. In section 3, w e will describe RD- KdVESCS (KdV6 equation) and RD-mKdVESCS as a t − t yp e Hamiltonian system b y taking x as the ev olution parameter, resp ectiv ely . Then followin g the pro cedure giv en in [11]- [14 ] by means of the t − type Miura transformat io n relating these tw o Hamiltonian systems, w e will construct the sec ond t − ty p e Hamiltonian structure for KdV6 eq uation (RD-KdVESCS) from the first Hamiltonian structure of RD-mKdVESCS, 2 and presen t infinite c hain of local comm uting v ector fields for KdV6 equation. Finally in section 4, star t ing fr om the solutions o f KdV, w e obtain many new solutions of KdV6 equation, suc h as soliton, p o siton and nega t o n solution. 2 KdV6 equation is equi v alen t to RD-KdVESC S By r escaling u and t and using the Galilean inv ariance of KdV equation, KdV6 equation (3) can b e rewritten as u t = 1 4 ( u xxx + 6 uu x ) − w x , (6a) w xxx + 4( u − λ 1 ) w x + 2 w u x = 0 (6b) where λ 1 is a parameter. Set w = ϕ 2 , (7) then ( 6b) yields w xxx + 4( u − λ 1 ) w x + 2 w u x = 2 ϕ [ ϕ xx + ( u − λ 1 ) ϕ ] x + 6 ϕ x [ ϕ xx + ( u − λ 1 ) ϕ ] = 0 , whic h immediately giv es rise to ϕ xx + ( u − λ 1 ) ϕ = µ ϕ 3 , where µ is an in tegrable constan t. So KdV6 equation (6) is equiv alen t to u t = 1 4 ( u xxx + 6 uu x ) − ( ϕ 2 ) x , (8a) ϕ xx + ( u − λ 1 ) ϕ = µ ϕ 3 , (8b) whic h is just the RD- KdVESCS presen ted in [9]. The Lax pair for (8) reads [9]  ψ 1 ψ 2  x = U  ψ 1 ψ 2  , U =  0 1 λ − u 0  (9a)  ψ 1 ψ 2  t = N  ψ 1 ψ 2  , N =  − u x 4 λ + u 2 λ 2 − u 2 λ − u xx 4 − u 2 2 + 1 2 ϕ 2 u x 4  − 1 2 1 λ − λ 1  ϕϕ x − ϕ 2 ϕ 2 x + µ ϕ 2 − ϕϕ x  . (9b) More generally , the multi-component extension of KdV6 equation is given by u t = 1 4 ( u xxx + 6 uu x ) − N X j =1 w j x , (10a) w j xx + 4( u − λ j ) w j x + 2 u x w j = 0 , j = 1 , 2 , · · · , · · · N . (10b) 3 Under the transformatio n w j = ϕ 2 j , (10) can b e con v erted in to the follo wing RD-KdVESCS u t = 1 4 ( u xxx + 6 uu x ) − N X j =1 ( ϕ 2 j ) x , (11a) ϕ j xx + ( u − λ j ) ϕ j = µ j ϕ 3 j , j = 1 , 2 , · · · , N . (11b) The Lax pair for (11) is given by (9a) with N =   − u x 4 − λ + u 2 − λ 2 − u 2 λ − u xx 4 − u 2 2 + 1 2 N P j =1 ϕ 2 j u x 4   − 1 2 N X j =1 1 λ − λ j ϕ j ϕ j x − ϕ 2 j ϕ 2 j x + µ ϕ 2 j − ϕ j ϕ j x ! . (12) 3 Bi-Hamiltonian stru cture o f KdV6 equation In this section, we will follow the metho d in [11]- [14] to construct the bi-Hamiltonian formalism for KdV6 equation ( R D-KdVESCS). First w e will presen t the t − t yp e Hamil- tonian formalism f or RD- KdVESCS and RD-mKdVESCS. F o r the RD- KdVESCS (8), set 1 4 u xx + 3 4 u 2 − ϕ 2 = c, q t = c x , q = u , p = − 1 8 u x , Q = ϕ, P = ϕ x , R = ( Q, q , P , p, c ) T , (13) then (8) b ecomes x − ev olution equations and can b e written as a t − type Hamiltonian system R x =       P − 8 p ( λ 1 − q ) Q + µ Q 3 3 8 q 2 − 1 2 Q 2 − 1 2 c q t       = K 1 = Π 0 ∇ H 1 , (14a) w her e ∇ means v ar iational der iv ativ e, ∇ H = ( δ H δ Q , δ H δ q , δ H δ P , δ H δ p , δ H δ c ) T , and the t − ty pe P oisson oper ator Π 0 and conser v ed density H 1 ar e g iv en by Π 0 =       0 0 1 0 0 0 0 0 1 0 − 1 0 0 0 0 0 − 1 0 0 0 0 0 0 0 2 ∂ t       , (14b) H 1 = 1 2 P 2 − 4 p 2 − 1 2 λ 1 Q 2 + 1 2 q Q 2 − 1 8 q 3 + 1 2 cq + 1 2 µ Q 2 . (14c) 4 The R oso c hatius deformation of mKdV equation with self-consisten t source (RD-mKdVESCS) is defined as [9] v t = 1 4 ( v xxx − 6 v 2 v x ) + 1 2 ( ¯ ϕ 1 ¯ ϕ 2 ) x , (15a) ¯ ϕ 1 x = v ¯ ϕ 1 + λ 1 ¯ ϕ 2 , ¯ ϕ 2 x = ¯ ϕ 1 − v ¯ ϕ 2 + µ λ 1 ¯ ϕ 1 3 . (15b) Let 1 4 ( v xx − 2 v 3 ) + 1 2 ¯ ϕ 1 ¯ ϕ 2 = − ¯ c, v t = − ¯ c x , ¯ q = v , ¯ p = 1 2 v x , ¯ Q = ¯ ϕ 1 , ¯ P = ¯ ϕ 2 , ¯ R = ( ¯ Q, ¯ q , ¯ P , ¯ p, ¯ c ) T , (16) then R D-mKdVESCS (15) can b e written as a t − t yp e Hamiltonian system ¯ R x =       ¯ q ¯ Q + λ 1 ¯ P 2 ¯ p ¯ Q − ¯ q ¯ P + µ λ 1 ¯ Q 3 ¯ q 3 − ¯ Q ¯ P − 2 ¯ c − ¯ q t       = ¯ K 1 = ¯ Π 0 ∇ ¯ H 1 , (17a) w her e ∇ ¯ H = ( δ ¯ H δ ¯ Q , δ ¯ H δ ¯ q , δ ¯ H δ ¯ P , δ ¯ H δ ¯ p , δ ¯ H δ ¯ c ) T , and the t − ty pe P oisson oper ator ¯ Π 0 and conser v ed density ¯ H 1 ar e g iv en by ¯ Π 0 =       0 0 1 0 0 0 0 0 1 0 − 1 0 0 0 0 0 − 1 0 0 0 0 0 0 0 − 1 2 ∂ t       , (17b) ¯ H 1 = ¯ q ¯ P ¯ Q + 1 2 λ 1 ¯ P 2 + ¯ p 2 − 1 2 ¯ Q 2 − 1 4 ¯ q 4 + 2¯ c ¯ q + 1 2 µ λ 1 ¯ Q 2 . (17c) The Miura map relating systems (14) to (17 ), ie R = M ( ¯ R ), is giv en by M : Q = ¯ Q, q = − ¯ q 2 − 2 ¯ p, P = λ 1 ¯ P + ¯ q ¯ Q, p = 1 4 ¯ q 3 − 1 2 ¯ c − 1 4 ¯ Q ¯ P + 1 2 ¯ q ¯ p, c = ¯ H 1 − ¯ q t = − 1 2 ¯ Q 2 − 1 4 ¯ q 4 + ¯ p 2 + 2¯ c ¯ q + 1 2 λ 1 ¯ P 2 + ¯ q ¯ Q ¯ P + 1 2 µ λ 1 ¯ Q 2 − ¯ q t (18) whic h can b e pro v ed through direct calculations. Denote M ′ ≡ D R D ¯ R T 5 where D R D ¯ R T is the Jacobi matrix consisting of F rec het deriv ativ e of M , M ′ ∗ denotes adjoin t of M ′ . According to the standard pro cedure [11]- [14], applying the map M (18) to the first Hamiltonian structure of RD -mKdVESCS (17), w e can generate the second Hamilto nian structure of the RD- KdVESCS (14), Π 1 = M ′ ¯ Π 0 M ′ ∗ =       0 0 λ 1 − 1 4 Q P 0 0 2 Q − 1 2 q − 8 p + 2 ∂ t − λ 1 − 2 Q 0 1 4 P ( λ 1 − q ) Q + µ Q 3 1 4 Q 1 2 q − 1 4 P − 1 8 ∂ t 3 8 q 2 − 1 2 c − 1 2 Q 2 − P 8 p + 2 ∂ t ( q − λ 1 ) Q − µ Q 3 − 3 8 q 2 + 1 2 c + 1 2 Q 2 q ∂ t + ∂ t q       (19) and the bi-Hamiltonian structure for KdV6 equation or RD-KdVESCS (8) under the transformation (7) and (13) is giv en b y R x = Π 0 δ H 1 δ R = Π 1 δ H 0 δ R , H 0 = c. (20) Since the t − type P ossion op era t or Π 0 is inv e rtible, w e can immediately construct a recursion op erator Φ = Π 1 (Π 0 ) − 1 =       λ 1 − 1 4 Q 0 0 1 2 P ∂ − 1 t 2 Q − 1 2 q 0 0 − 4 p∂ − 1 t + 1 0 1 4 P λ 1 2 Q 1 2 [( λ 1 − q ) Q + µ Q 3 ] ∂ − 1 t − 1 4 P − 1 8 ∂ − 1 t − 1 4 Q − 1 2 q 1 2 ( 3 8 q 2 − 1 2 c − 1 2 Q 2 ) ∂ − 1 t ( − λ 1 + q ) Q − µ Q 3 − 3 8 q 2 + 1 2 c + 1 2 Q 2 P − 8 p − 2 ∂ t 1 2 q + 1 2 ∂ t q ∂ − 1 t       (21) whic h has the hereditary prop ert y [11]. Applying Φ to the vector field K 1 (14), we can generate the hierarch y of Hamilto nian comm uting v ector fields (symmetries ) K n = Φ n − 1 K 1 , (22) and obtain a hierarc h y of infinite-dimensional in tegrable bi-Hamiltonian systems R x = Φ n − 1 K 1 = K n = Π 0 ∇ H n = Π 1 ∇ H n − 1 , (23) for example K 2 =       λ 1 P + 2 pQ + 1 2 q P 2 Q ¯ P + q t − 2 p ¯ P + λ 2 1 Q − 1 2 λ 1 Qq + 1 4 q 2 Q − Q 3 − cQ + λ 1 µ Q 3 + q µ 2 Q 3 − 1 4 P 2 + p t + 1 4 q Q 2 − 1 4 λ 1 Q 2 − 1 4 µ Q 2 2 QQ t + c t       with t he conserv ed functional densities H 2 giv en b y H 2 = 1 4 Q 4 +2 pP Q − 1 8 q 2 Q 2 + 1 4 P 2 q + 1 4 λ 1 q Q 2 − 1 2 λ 2 1 Q 2 + 1 2 λ 1 P 2 + µq 4 Q 2 + µλ 1 2 Q 2 − q p t + 1 2 cQ 2 + 1 4 c 2 . 6 4 The new solu t ions of KdV6 The KdV equation is u t = 1 4 ( u xxx + 6 uu x ) . (24) The lax pair is ψ xx + uψ = λψ , (25a) ψ t = − u x 4 ψ + ( u 2 + λ ) ψ x . (25b) Let the functions φ 1 , φ 2 , · · · , φ n b e n differen t solutions of the system (25 ) with the corresp onding λ = λ 1 , λ 2 , · · · , λ n . W e construct t w o W ronskian determinants from these functions: W 1 = W ( φ 1 , · · · , φ ( m 1 ) 1 , φ 2 , · · · , φ ( m 2 ) 2 , · · · , φ n , · · · , φ ( m n ) n ) , (26a) W 2 = W ( φ 1 , · · · , φ ( m 1 ) 1 , φ 2 , · · · , φ ( m 2 ) 2 , · · · , φ n , · · · , φ ( m n ) n , ψ ) , (26b) where m i ≥ 0 are giv en num bers and φ ( n ) j := ∂ n λ φ j ( x, λ ) | λ = λ j . The generalized Darb oux transformation of equation ( 2 4) a nd system (25) is given by [15] ¯ u = u + 2 ∂ 2 x lnW 1 , (27a) ¯ ψ = W 2 W 1 , (27b) namely , system (25) is co v arian t with resp ect to the action o f ( 27). F or any initia l solution of (24), ¯ u and ¯ ψ are new solution of (24) and (25). No w w e will tak e u = 0 in what follo ws. 4.1 soliton solution In (26), let n = 1 , m 1 = 0 , λ = k 2 4 , λ 1 = k 2 1 4 and take φ 1 ( x, t, k ) = cosh Θ , ψ 1 ( x, t, k ) = sinh Θ , (28a) Θ = k 2 ( x + 1 4 k 2 t ) + α, Θ 1 = k 1 2 ( x + 1 4 k 2 1 t ) + α (28b) where α is an a rbitrary constan t. By using (26) and (27), we obtain the single-soliton solution and the corresp onding eigenfunction with k = k 1 for t he KdV equation (24) ¯ u = k 2 1 2 sech 2 Θ 1 , (29) ¯ ψ 1 ( x, t, k 1 ) = β k 1 2 sech Θ 1 , (30) where β is an arbitrary constan t as well. Since KdV6 equation (6) can b e considered to b e KdV equation ( 2 4) with non- homogeneous terms and w is related to the square of eigenfunction b y (7), w e ma y apply 7 the metho d of v ariat ion of constant to find the solution of Eq.(6) by using the solution ¯ u of Eq.(24) and cor r esp o nding eigenfunc tion ¯ ψ 1 . T aking α and β in (28b) and (30) to b e time-dep enden t functions α ( t ) and β ( t ) and using (7), and requiring that u = k 2 1 2 sech 2 ¯ Θ 1 , (31a) w = ¯ ψ 2 1 ( x, t, k 1 ) = β ( t ) 2 k 2 1 4 sech 2 ¯ Θ 1 , (31b) ¯ Θ 1 = k 1 2 ( x + 1 4 k 2 1 t ) + α ( t ) , (31c) satisfy the Eq.(6). W e find that α ( t ) can b e a n arbitrary function of t and β ( t ) 2 = − 4 α ′ ( t ) k 1 . (32) So the single-soliton solution of KdV6 equation (6) is give n by u = k 2 1 2 sech 2 ¯ Θ 1 , (33a) w = − α ′ ( t ) k 1 sech 2 ¯ Θ 1 . (33b) Its shap e is sho wn in figure 1. Notice tha t ¯ Θ 1 con tains an arbitrary t-function α ( t ). This implies that the insertion of sources into KdV equation may cause the v a r ia tion of the sp eed of the soliton solution. So the dynamics of solution of KdV6 equation turns out to b e m uc h ric her than that of solution o f KdV equation. -5 5 10 15 20 25 x 0.1 0.2 0.3 0.4 0.5 u -5 5 10 15 20 x 0.5 1 1.5 2 w (a) (b) Figure 1. The shap e of single soliton solution for u and w when α ( t ) = − 2 t, k 1 = 1 , t = 3 . 4.2 The first and second order of p ositon solution In (26), set n = 1 , m 1 = 1 , λ = − k 2 4 , λ 1 = − k 2 1 4 and tak e φ 1 ( x, t, k ) = sin Θ , ψ 1 ( x, t, k ) = cos Θ , (34a) Θ = k 2 ( x + x 1 ( k ) − 1 4 k 2 t ) − 1 8 ( k − k 1 ) α, (34b) 8 where x 1 ( k ) is a f unction that is analytic in the vicinit y of the p oin t k and has real T a ylor expansion co efficien ts. By using (26) , (27) and (34), w e o bt a in first order of the one- p ositon solution and the corresp onding eigenfunction with k = k 1 for the KdV equation (24) [15] ¯ u = − 16 k 2 1 sin Θ 1 (8 sin Θ 1 + k 1 γ cos Θ 1 ) (4 sin 2Θ 1 + k 1 γ ) 2 , (35a) ¯ ψ 1 ( x, t, k 1 ) = − 4 β k 2 1 sin Θ 1 4 sin 2Θ 1 + k 1 γ , (35b) Θ 1 = k 1 2 ( x + x 1 ( k 1 ) − 1 4 k 2 1 t ) , (35c) γ = − 8 ∂ k Θ | k = k 1 = 3 k 2 1 t − 4( x + x 2 ( k 1 )) + α , x 2 ( k 1 ) = [ x 1 + 4 k ∂ k x 1 ( k )] k = k 1 (35d) where α , β are a rbitrary constants. Similarly , by using (7) and the metho d of v aria tion of constant w e presen t first order of t he one-p ositon solution fo r the KdV6 equation (6) u = − 16 k 2 1 sin Θ 1 (8 sin Θ 1 + k 1 ¯ γ cos Θ 1 ) (4 sin 2Θ 1 + k 1 ¯ γ ) 2 , (36a) w = − 16 k 2 1 α ′ ( t ) sin 2 Θ 1 (4 sin 2Θ 1 + k 1 ¯ γ ) 2 , (36b) ¯ γ = 3 k 2 1 t − 4( x + x 2 ( k 1 )) + α ( t ) . (36c) (36) implies tha t for fixed t a nd x → ±∞ , w e ha v e the asymptotic estimate u = 2 k 1 x sin 2Θ 1 [1 + O ( x − 1 )] , (37a) w = − k 1 α ′ ( t ) x 2 sin 2 Θ 1 [1 + O ( x − 1 )] . (37b) If x is fixed and t → ±∞ , the solution has the asymptotic b eha vior u = − 8 sin 2Θ 1 3 k 1 t [1 + O ( t − 1 )] , (38a) w = − 16 α ′ ( t ) 9 k 4 1 t 2 sin 2 Θ 1 [1 + O ( t − 1 )] . (38b) A p ositon solution as a function of x, u, and w hav e a sec ond-o r der p o le. This p o le is situated at the p o in t x = x 0 ( t ) which o scillates aro und the p oint x as ( t ) with the amplitude 4 k 1 , where x as ( t ) = − 3 4 k 2 1 t − x 2 ( k 1 ) − α ( t ) 4 . The exact p osition of this p ole can b e determined b y solving the f o llo wing equation with δ = k 1 ¯ γ , δ = − 4 sin 1 8 [ δ − 4 k 3 1 t + 4 k 1 ( x 1 − x 2 ) − k 1 α ( t )] . So the p ositon solution of KdV6 equation (6) is lo ng -range analogue of soliton and is slo wly decreas ing, oscillating solution. The shap e and motion of the single p ositon is sho wn in figure 2. 9 -15 -10 -5 5 10 x -5 -4 -3 -2 -1 u -10 -5 5 10 x -8 -6 -4 -2 u -10 -5 5 10 15 x -8 -6 -4 -2 u t=-5 t=3 t=10 -20 -10 10 20 x 0.1 0.2 0.3 0.4 0.5 w -30 -20 -10 10 20 30 x 0.025 0.05 0.075 0.1 0.125 0.15 w -20 -10 10 20 x 0.1 0.2 0.3 0.4 w t=-10 t=3 t=25 Figure 2 . The shap e a nd motion of o ne-p ositon so lution for u and w whe n α ( t ) = − 2 t, x 1 ( k 1 ) = 2 k 1 , k 1 = 1 . In order to find the second o rder of one-p ositon solution, w e take Θ in (34 a ) to b e Θ = k 2 ( x + x 1 ( k ) − 1 4 k 2 t ) − 1 8 ( k − k 1 ) 2 α, (39) w e ha v e W 1 = W ( φ 1 , ∂ k φ 1 , ∂ 2 k φ 1 ) | k = k 1 = 1 128 {− 32 sin 2 Θ 1 cos Θ 1 + k 2 1 γ 2 cos Θ 1 + [12 k 2 1 ν − 4 k 1 (4 x + 4 x 1 ( k 1 ) + k 1 α − 2 k 2 1 x ′′ 1 ( k 1 ))] sin Θ 1 } , (40a) W 2 = W ( φ 1 , ∂ k φ 1 , ∂ 2 k φ 1 , ψ 1 ) | k = k 1 = − 1 64 k 3 1 (4 sin 2Θ 1 + k 1 γ ) , (40b) Θ 1 = k 1 2 ( x + x 1 ( k 1 ) − 1 4 k 2 1 t ) , γ = − 8 ∂ k Θ | k = k 1 = 3 k 2 1 t − 4( x + x 2 ( k 1 )) , (40c) ν = − 4 ∂ 2 k Θ | k = k 1 = 3 k 1 t − 4 ∂ k x 2 ( k ) | k = k 1 + α. (40d) By the similar pro cess , by taking α = α ( t ) in (40 d) w e obtain the second-order of the one-p ositon solution from (7) and (27) u = 2 ∂ 2 x lnW 1 , (41a) w = − 2 α ′ ( t ) k 3 1 ( W 2 W 1 ) 2 . (41b) Its shap e is sho wn in figure 3. -20 -15 -10 -5 5 10 15 x -60 -50 -40 -30 -20 -10 u -20 -10 10 20 x 2 4 6 8 10 w (a) (b) Figure 3. The shap e of second-order posito n solution for u and w when x 1 ( k 1 ) = 2 k 1 , α ( t ) = − 2 t 2 , k 1 = 1 , t = 2 . 10 4.3 The first and second order of negaton solution In (26), set n = 1 , m 1 = 1 , λ = k 2 4 , λ 1 = k 2 1 4 and tak e φ 1 ( x, t, k ) = sinh Θ , ψ 1 ( x, t, k ) = cosh Θ , (42a) Θ = k 2 ( x + x 1 ( k ) + 1 4 k 2 t ) + 1 8 ( k − k 1 ) α. (42b) W e obtain the first order of one-negaton solutio n and the corresp onding eigenfunction with k = k 1 for t he KdV equation (24) ¯ u = − 16 k 2 1 sinh Θ 1 (8 sinh Θ 1 − k 1 γ cosh Θ 1 ) (4 sinh 2Θ 1 − k 1 γ ) 2 , (43a) ¯ ψ 1 ( x, t, k 1 ) = − 4 β k 2 1 sinh Θ 1 4 sinh 2Θ 1 − k 1 γ , (43b) Θ 1 = k 1 2 ( x + x 1 ( k 1 ) + 1 4 k 2 1 t ) , (43c) γ = 8 ∂ k Θ | k = k 1 = 3 k 2 1 t + 4 ( x + x 2 ( k 1 )) + α , (43d) where α, β are arbitrary constan ts. -30 -20 -10 10 20 x -2 -1 1 u -30 -20 -10 10 20 x -2.5 -2 -1.5 -1 -0.5 0.5 1 u -30 -20 -10 10 20 x -2 -1.5 -1 -0.5 0.5 1 u t=-15 t=2 t=20 -20 -10 10 20 x 0.25 0.5 0.75 1 1.25 1.5 w -20 -10 10 20 x 0.2 0.4 0.6 0.8 1 w -20 -10 10 20 x 0.2 0.4 0.6 0.8 1 w t=-10 t=2 t=10 Figure 4. The shap e and motion of one- negaton solution fo r u and w when α ( t ) = − 2 t, x 1 ( k 1 ) = 2 k 1 , k 1 = 1 . Similarly , by using (7) and the method of v aria t io n of constan t w e presen t the first order o f one-negaton solution for the KdV6 equation (6) 11 u = − 16 k 2 1 sinh Θ 1 (8 sinh Θ 1 − k 1 ¯ γ cosh Θ 1 ) (4 sinh 2Θ 1 − k 1 ¯ γ ) 2 (44a) w = − 16 k 2 1 α ′ ( t ) sinh 2 Θ 1 (4 sinh 2Θ 1 − k 1 ¯ γ ) 2 , (44b) ¯ γ = 3 k 2 1 t + 4 ( x + x 2 ( k 1 )) + α ( t ) . (44c) Similarly , nega t on solution of (6) hav e second-order p ole. The shap e and motion of the negaton is shown in figure 4. No w w e tak e Θ in ( 42a) to b e Θ = k 2 ( x + x 1 ( k ) + 1 4 k 2 t ) + 1 8 ( k − k 1 ) 2 α, ( 4 5) w e ha v e W 1 = W ( φ 1 , ∂ k φ 1 , ∂ 2 k φ 1 ) | k = k 1 = 1 128 { 32 sinh 2 Θ 1 cosh Θ 1 − k 2 1 γ 2 cosh Θ 1 + [12 k 2 1 ν − 4 k 1 ( − 4 x − 4 x 1 ( k 1 ) + k 1 α + 2 k 2 1 x ′′ 1 ( k 1 ))] sinh Θ 1 } , (46a) W 2 = W ( φ 1 , ∂ k φ 1 , ∂ 2 k φ 1 , ψ 1 ) | k = k 1 = 1 64 k 3 1 ( − 4 sin 2Θ 1 + k 1 γ ) , (46b) Θ 1 = k 1 2 ( x + x 1 ( k 1 ) + 1 4 k 2 1 t ) , γ = 8 ∂ k Θ | k = k 1 = 3 k 2 1 t + 4( x + x 2 ( k 1 )) , (46c) ν = 4 ∂ 2 k Θ | k = k 1 = 3 k 1 t + 4 ∂ k x 2 ( k ) | k = k 1 + α. (46d) Similarly , w e obtain the second-order of negaton solution from ( 7 ) and (27) u = 2 ∂ 2 x lnW 1 , (47a) w = 2 α ′ ( t ) k 3 1 ( W 2 W 1 ) 2 . (47b) Its shap e is sho wn in figure 5. -20 -10 10 20 x -12.5 -10 -7.5 -5 -2.5 u -20 -10 10 20 x 1 2 3 w (a) (b) Figure 5. The shape of second-order negaton solution for u and w when x 1 ( k 1 ) = 2 k 1 , α ( t ) = 2 t 2 , k 1 = 1 , t = 5 . 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