The Degasperis-Procesi equation with self-consistent sources

The Degasp eris-Pro cesi equatio n with self-consi sten t sources Y eh ui Huang 1 ∗ , Y un b o Zeng 1 † and Orlando Ragnisco 2 , 3 ‡ 1 Departmen t of Mathematical Sciences, Tsingh ua Univ ersit y , Beijing, 100084 , P .R. China 2 Departmen t of Phy sics, Univ ersit` a Roma TRE, Rome, 001 46, Italy 3 I.N.F.N., sezione of Roma TRE, Rome, 00146, Italy Abstract The Degasp eris- P ro cesi equation with self-consistent sources(DPE SCS) is der ived. The Lax representation of the DPESCS is presented. The co nserv ation laws for DPE SCS are constructed. The p eakon solution of DPESCS is obtained by using the metho d o f v ariation of consta nt s. P A CS:02.30.IK KEYW ORDS: Degasp eris-Pr o cesi equation with self-consisten t sour ces; L ax represent ation; conserv ati on la ws; p eak on; the metho d of v ariation of constan ts ∗ Corresponding author: Y eh ui Huang, T el: +39-3318082062 , e- m ail:huangyh@mails.tsingh ua.edu.cn † yzeng@math.tsinghua.edu.cn ‡ ragnisco@fis.uniroma3.i t 1 1 In tro duc tion Soliton equatio ns with self-c onsistent sources (SESCS ) ha v e attracted m uch atte ntion in recent y ears. They are imp ortan t inte grable mo d els in many fields of physic s, suc h as h ydro dyn amics, state physic s, plasma physics, etc [1]-[18]. F or example, the KdV equation with self-consisten t sources describ es th e interac tion of long and short capillary-gra vit y w av es [5]. Th e KP equation with self- consisten t s ources d escrib es the interact ion of a long-w a v e with a short wa v e pac k et propagating on the x − y plane at some an gle to eac h other [2]. The nonlinear S c hr ¨ o dinger equ ation with self- consisten t sources represent s the nonlinear in teraction of an electrostatic high-frequency wa v e with the ion acoustic w a v e in a t w o comp onen t homogeneous plasma [6]. T he constrained flo ws of soliton hierarc hy ma y b e regarded as the stationary systems of the corresp ond ing in tegrable hierarch y with self-consisten t sources [15]-[18 ]. Since the Lax r epresen tation for constrained flo ws can b e d educed from the a djoint representa tion of the Lax r epresent ation of soliton equation [ 14], there is a naturally w a y to fi nd the zero-curv ature representati on for SES CS [15]-[18 ]. F rom this observ ation, the soliton equations with self-consisten t sources ma y b e view ed as integrable generalizations of the original soliton equations. A systematic w a y to construct the soliton equations with self-consisten t sources is prop osed in [15]-[18]. The Camassa-Holm equ ation, w hic h w as implicitly con tained in th e class of multi-H amiltonian systems introduced by F uchssteiner and F ok as in [20] and explicitly deriv ed as a sh allo w wa ter w a ve equation by Camassa and Holm in [21], has the form u t + 2 wu x − u xxt + 3 uu x = 2 u x u xx + uu xxx . (1.1) Since th e wo rks of Camassa and Holm, this equation has b ecome a well-kno wn example of integ rable systems and has b een studied f rom man y different p oin ts of view. 2 It is a natural qu estion to ask wh ether there are other thir d order disp ersive PDEs sh arin g the in tegrabilit y pr op erties of Camassa-Holm equation. An answe r has b een given in [19] wh ere the m etho d of asymptotic in tegrabilit y w as applied to a family of third -order d isp ersive P DE conserv ati on la ws u t + c 0 u x + γ u xxx − α 2 u xxt = ( c 1 u 2 + c 2 u 2 x + c 3 uu xx ) x , (1.2) where α , c 0 , c 1 , c 2 , c 3 and γ are some arbitrary constan ts. Only three equations in this family satisfy the asymptotic integrabilit y conditions. T hey are the KdV equ ation, the Camassa-Holm equ ation and the f ollo win g n ew equation u t + u x + 6 uu x + u xxx − α 2 ( u xxt + 9 2 u x u xx + 3 2 uu xxx ) = 0 . (1.3) By a co ordin ate trans formation, this new equation could b e written as [22] u t − u xxt + 4 uu x = 3 u x u xx + uu xxx . (1.4) This encourages us to study the family of equ ations [22] u t − u xxt + ( b + 1) uu x = bu x u xx + uu xxx . (1.5) All the equations in this family ha ve p eak on solutions of the form u = N X j =1 p j ( t ) e −| x − q j ( t ) | , (1.6) where p j and q j satisfy the d ynamical system ˙ p j = − ( b − 1) ∂ G N ∂ q j , ˙ q j = ∂ G N ∂ p j , (1.7) the generating function G N reading G N = 1 2 N X j,k =1 p j p k e −| q j − q k | . (1.8) 3 Let m = u − u xx . The equation (1.4) could b e written as m t + m x u + 3 mu x = 0 , (1.9) whic h is called the Degasp eris-Pro cesi (DP) equation. It is show ed in [23] that b oth the C amassa- Holm and the DP equation are derived as members of a one-parameter family of asymptotic s h allo w w ater appro ximations to th e Euler equations. They d escrib e the un idirectional propagat ion of nonlinear shallo w-wate r wa v es. Th e DP equation has a third ord er Lax pair and a bi-Hamiltonian structure. The existence of its global solutions are considered in [24]. A new integrable hierarc hy w as extended from the DP equ ation in [25]. In [26], the N-p eak on solutions of the DP equation h a v e b een deriv ed. The N-soliton solutions of th e DP equation are obtained in [27]. Interesting r esults on the solutions of th is equation ha ve also b een obtained in [28]-[32]. The soliton equation w ith self-consisten t sources was firstly studied by Melniko v in [1]-[3]. The problem of fi nding soliton solutions or other sp ecific solutions for equations with self-consisten t sources has b een considered in the past by sev eral auth ors [4]-[18]. T he present p ap er falls in that line of researc h, aiming at deriving Dega sp eris-Pro cesi equation with self-consisten t sour ces (DPESCS) and to fin d for its sp ecial explicit solutions. W e fi rst constru ct the high-order constrained fl o w of the DP equ ation. Based on it we estab- lish the DPESCS by regarding the constrained flow of DP equation as the stationary equation of DPESCS in the s ame wa y as in [15]-[18]. T h e Lax pair of the DPESC S is obtained, which means that th e DPESC S is Lax inte grable. In [33]-[34] we p oint ed out that soliton equations with self-consisten t sources can b e regarded as soliton equations with non-h omogeneous terms, and ac- cordingly prop osed to look for explicit solutions by u s ing the metho d s of v a riation of constan ts. Applying this tec hn ique to DPESCS w e ha v e b een able to fi nd its p eak on solutions and p eako n- 4 an tip eako n s olution. This p ap er is organized as f ollo ws. In section 2, w e extend DP equation including self-consisten t sources and constr u ct its Lax represen tation. In section 3 we derive its conserv ation la ws. In sectio n 4, the p eak on and the p eak on-antipeak on solution are obtained. In s ection 5, we menti on some op en problems. 2 The DPESCS and its Lax pair 2.1 The DPESCS First w e constru ct the high-order constrained flows of th e DP equation, then establish the DPESCS and describ e how to deriv e the Lax represent ation for the DPESCS. It is kno wn that the Lax pair f or DP equ ation (1.9) is [22] ψ xxx = ψ x − mλψ , (2.1a) ψ t = − 1 λ ψ xx − uψ x + ( u x + 2 3 λ ) ψ . (2.1b) Consider the f ollo wing equations obtained f rom the sp ectral pr ob lem and its formal adjoin t problem for n distinct real λ j . q j,xxx = q j,x − mλ j q j , j = 1 , · · · , n, (2.2a) r j,xxx = r j,x + mλ j r j , j = 1 , · · · , n. (2.2b) It is not difficu lt to find that δ λ j δ m = − λ j q j r j , j = 1 , · · · , n. (2.3) 5 It is kno wn that the DP equation p ossesses a bi-hamiltonian str ucture [22], namely: m t = B 1 δ H 1 δ m = B 0 δ H 0 δ m , (2.4) where B 0 = m 2 / 3 ∂ x m 1 / 3 ( ∂ x − ∂ 3 x ) − 1 m 1 / 3 ∂ x m 2 / 3 , (2.5) B 1 = ∂ x (1 − ∂ 2 x )(4 − ∂ 2 x ) , (2.6) H 0 = − 2 9 Z mdx, (2.7) H 1 = − 1 6 Z u 3 dx. (2.8) The high-ord er constrained flo w of the DP equation is obtained from (2.2) for n distinct λ j , requiring that the ”p otenti al” m ob eys the follo wing constraint B 1 ( δ H 1 δ m − n X j =1 α j δ λ j δ m ) = 0 , (2.9a) q j,xxx = q j,x − mλ j q j , (2.9b) r j,xxx = r j,x + mλ j r j j = 1 , · · · , n, (2.9 c) where α j , j = 1 , . . . , n are arbitrary constants. According to the app roac h prop osed in [15]-[18 ], the DPESCS consists of the follo wing equation m t = B 1 ( δ H 1 δ m − n X j =1 α j δ λ j δ m ) and the equ ations (2.2), whic h b y using (2.3) and taking α j = − 1 6 leads to the DPESC S m t = − um x − 3 u x m − 1 6 n X j =1 ∂ (1 − ∂ 2 )(4 − ∂ 2 )( λ j q j r j ) , (2.10a ) q j,xxx = q j,x − mλ j q j , (2.10b) r j,xxx = r j,x + mλ j r j , j = 1 , · · · , n. (2.10 c) 6 2.2 The Lax represen tation of the DPE SCS Comparing the DPESCS to the DP equation, we ma y assume that the Lax presenta tion of th e DPESCS has th e form ψ xxx = ψ x − mλψ , (2.11a ) ψ t = − 1 λ ψ xx − uψ x + ( u x + 2 3 λ ) ψ + aψ + bψ x + cψ xx , (2.11b) where a , b and c are some functions of q j and r j to b e d etermined. Requ iring th at under (2.10b) and (2.10c ) the compatibilit y condition of (2.11 a) and (2.11b), namely ψ xxxt = ψ txxx , leads to DPESCS (2.10a ) enables us to fin d th at a x − a xxx + 3 b x mλ + bm x λ + 3 c xx mλ + 3 c x m x λ + cm xx λ = λ n X j =1 α j ∂ (1 − ∂ 2 )(4 − ∂ 2 )( q j r j ) , (2.12a ) − 3 a xx − b xxx − 2 b x − 3 c xx + 3 c x mλ + 2 cm x λ = 0 , (2.12b) 3 a x + 3 b xx + 2 c x + c xxx = 0 . (2.12c ) F rom (2.10b) and (2.10c) we obtain tw o id en tities ∂ (1 − ∂ 2 )(4 − ∂ 2 )( q j r j ) = − 3 λ j ( m x W ( q j , r j ) + 3 mW ( q j , r j ) x ) , (2.13a ) W ( q j , r j ) xxx − W ( q j , r j ) x = λ j (2 m x q j r j + 3 m ( q j r j ) x ) , (2.13b) where W ( q j , r j ) = q j r j,x − q j,x r j is the usual W ronskian determinan t. F rom (2.12c) we obtain that 3 a + 3 b x + 2 c + c xx = 0 . (2.14) 7 Then (2.14) together with (2.12b) lead to a xxx − a x = 1 6 ∂ (1 − ∂ 2 )(4 − ∂ 2 ) c + ( cm x λ + 3 2 c x mλ ) x , (2.15a ) 2(( b + 1 2 c x ) x − ( b + 1 2 c x ) xxx ) = 2 cm x λ + 3 c x mλ. (2.15b) With the r elations ab o v e w e can r ewr ite (2.12a) as − 1 6 ∂ (1 − ∂ 2 )(4 − ∂ 2 ) c + λ (3 m ( b + 1 2 c x ) x + m x ( b + 1 2 c x )) = λ n X j =1 α j ∂ (1 − ∂ 2 )(4 − ∂ 2 )( q j r j ) . (2.16) Here we s uggest that c = n X j =1 A j q j r j , (2.17a ) b + 1 2 c x = n X j =1 B j W ( q j , r j ) , (2.17b) where A j , B j , j = 1 , . . . , n are some u n determined constan ts. Finally with some calculations to d etermine A j and B j w e solv e (2.12) for a , b , c yielding: a = n X j =1 1 6 λλ 2 j λ 2 j − λ 2 (3 λ ( q j r j,xx − q j,xx r j ) − 4 λ j q j r j − 2 λ j ( q j r j ) xx ) , ( 2.18a) b = n X j =1 − 1 2 λλ 2 j λ 2 j − λ 2 ( λ ( q j r j,x − q j,x r j ) + λ j ( q j r j ) x ) , (2.18 b ) c = n X j =1 λλ 3 j λ 2 j − λ 2 q j r j . (2.18c ) 8 In this w a y , we obtain the Lax pair for (2.10a) u nder(2.10b) and (2.10c) as follo ws ψ xxx = ψ x − mλψ , (2.19a ) ψ t = − 1 λ ψ xx − uψ x + ( u x + 2 3 λ ) ψ +( n X j =1 1 6 λλ 2 j λ 2 j − λ 2 (3 λ ( q j r j,xx − q j,xx r j ) − 4 λ j q j r j − 2 λ j ( q j r j ) xx )) ψ +( n X j =1 − 1 2 λλ 2 j λ 2 j − λ 2 ( λ ( q j r j,x − q j,x r j ) + λ j ( q j r j ) x )) ψ x +( n X j =1 λλ 3 j λ 2 j − λ 2 q j r j ) ψ xx , (2.19b) whic h means that th e DPESC S is Lax in tegrable. 3 The infin ite set of conserv ation la ws for the DPESCS With the h elp of the Lax represen tation of the DPESCS , w e could find the conserv ation la ws for the DPESCS b y a w ell-kno wn metho d. First we assume that m , u and its deriv ativ es tend to 0 when | x | → ∞ , and assu me that q j , r j and its deriv ativ es tends to 0 w hen x → −∞ . Set Γ = ψ x ψ , (3.1) then the iden tit y ∂ ∂ t ( ∂ ln ψ ∂ x ) = ∂ ∂ x ( ∂ ln ψ ∂ t ) together with (2.11) imp lies that DPESC S has the follo wing conserv ation la w: ∂ ∂ t (Γ) = ∂ ∂ x ( ψ t ψ ) = ∂ ∂ x ( u x + a − ( u + b )Γ − ( 1 λ + c )(Γ x + Γ 2 )) , (3.2) where a , b and c are giv en by (2.14 ). Here we define that Ω = u x + a − ( u + b )Γ − ( 1 λ + c )(Γ x + Γ 2 ). Using (2.11a) giv es r ise to Γ − Γ xx = mλ + 3ΓΓ x + Γ 3 . (3.3) 9 W e can hav e tw o kinds of exp ansions for Γ in the p o wer series of λ . The fi rst expansion is in p ositiv e p ow ers of λ [22], Γ = ∞ X k =0 h k λ k +1 , (3.4a) Ω = ∞ X k =0 g k λ k +1 . (3.4b) W e n ote that the od d densities h 2 k + 1 are exact deriv ativ es. The first t w o densities are h 0 = u , h 2 = u 3 , whic h yield the conserv ed qu an tities H 0 and H 1 in (2.7) and (2.8). The follo wing densities are nonlo cal b ecause that w ould b e an in verse of the op erator (1 − ∂ 2 x ) in the sequence. The second exp ansion could b e Γ = ∞ X k =0 µ k λ 1 − k 3 , (3.5a) Ω = ∞ X k =0 ω k λ 1 − k 3 . (3.5b) With the r elations of (3.2) and (3.3), w e can ob tain the infinite den s ities and fluxes of the conserv ati on la ws. F or brevity , w e omit the recursion r elations here. After some calculations, we can find that µ k with o dd sub scripts are deriv ativ es of some f unc- tions. S o we define H − s = R µ 2 s − 2 dx , w hic h are tak en as th e conserved quantitie s. W e could find the fi r st f ew conserv ed quantiti es give n by µ 0 , µ 2 are as follo ws H − 1 = Z m 1 / 3 dx, (3.6a) H − 2 = 1 27 Z ( m 2 x m − 7 / 3 + 9 m − 1 / 3 ) dx. (3.6b) The corresp onding flux of the conserv ation la ws are G − 1 = m 1 / 3 ( − u + 1 2 n X j =1 λ 2 j ( q j r j,x − q j,x r j )) , (3.7a) G − 2 = 1 2 u x n X j =1 λ 2 j ( q j r j,xx − q j,xx r j )) − 1 3 m x m − 1 ( − u + 1 2 n X j =1 λ 2 j ( q j r j,x − q j,x r j )) . (3.7b) 10 As the space part of the Lax p air is the same as that of DP equation, the densities of the conserv ati on la ws are also the same. Of course, as the time part is differen t, the fl uxes will b e also differen t. 4 Solution of the DPESCS 4.1 One p eak on solution of the DPESCS As mentioned in the int ro duction, we will construct p eak on solutions for DPESCS by the metho d of v ariation of constan ts. The DP equation has p eak on solutions [22]. The one p eako n is u = ce −| x − ct + α | , (4.1) where α is an arb itrary constan t. Th e corresp ond in g eigenfunction of (4.1) is q = β [ sg n ( x − ct + α )( e −| x − ct + α | − 1) − 1] , (4.2a) r = β [ sg n ( x − ct + α )( e −| x − ct + α | − 1) + 1] , (4.2b) where β is arbitrary constan t as wel l. T aking α and β in (4.1) and (4.2) to b e time-dep endent α ( t ) and β ( t ) and requir in g that u = ce −| x − ct + α ( t ) | , (4.3a) q = β ( t )[ sg n ( x − ct + α ( t ))( e −| x − ct + α ( t ) | − 1) − 1] , (4.3b) r = β ( t )[ sg n ( x − ct + α ( t ))( e −| x − ct + α ( t ) | − 1) + 1] (4.3c) satisfies the DPESCS (2.10) for n = 1. W e fin d that c = 1 λ 1 , α ( t ) can b e an arb itrary fun ction of t 11 and β ( t ) = p α ′ ( t ) c . S o w e ha ve the one p eak on solution for (2.9) w ith n = 1, λ 1 = λ = 1 c u = ce −| x − ct + α ( t ) | , (4.4a) q = c p α ′ ( t )[ sg n ( x − ct + α ( t ))( e −| x − ct + α ( t ) | − 1) − 1] , (4.4b) r = c p α ′ ( t )[ sg n ( x − ct + α ( t ))( e −| x − ct + α ( t ) | − 1) + 1] . (4 .4c) The one p eako n of th e DPESCS also has a cusp at its p eak, lo cated at x = ct − α ( t ). W e note that, w hile for the one p eak on solution of the DP equation trav els with sp eed c and has a cusp at its p eak of h eigh t c , for the DPES CS, the cusp is s till at its p eak of heigh t c , bu t the sp eed of th e w a ve is no longer a constan t. 4.2 The p eak on-an tip eak on solution of the DP ESCS The DP equation has p eak on-anti p eak on solutions [22] u = − sg n ( t ) c 1 − e − 2 c | t | ( e −| x + c | t | + α | − e −| x − c | t | + α | ) , (4.5) where α is an arb itrary constan t. Th e corresp ond in g eigenfunctions of (4.5) is q = β r c (1 − e − 2 c | t | ) ( sg n ( x + c | t | + α )( e −| x + c | t | + α | − 1) − sg n ( x − c | t | + α )( e −| x − c | t | + α | − 1)) , (4.6a) r = β r c (1 − e − 2 c | t | ) ( sg n ( x + c | t | + α )( e −| x + c | t | + α | − 1) + sg n ( x − c | t | + α )( e −| x − c | t | + α | − 1)) . (4.6b) T aking α and β in (4.5) and (4.6) to b e time-dep endent α ( t ) and β ( t ), and with the metho d of the v ariatio n of constan ts, the p eak on-an tip eako n solution of the DPESCS w ith n = 1 and λ 1 = 1 c 12 is u = c 1 − e − 2 c | t | ( e −| x + c | t | + α ( t ) | − e −| x − c | t | + α ( t ) | ) , (4.7a) q = c s − sg n ( t ) α ′ ( t ) (1 − e − 2 c | t | ) ( sg n ( x + c | t | + α ( t ))( e −| x + c | t | + α ( t ) | − 1) − sg n ( x − c | t | + α ( t ))( e −| x − c | t | + α ( t ) | − 1)) , (4.7b) r = c s − sg n ( t ) α ′ ( t ) (1 − e − 2 c | t | ) ( sg n ( x + c | t | + α ( t ))( e −| x + c | t | + α ( t ) | − 1) + sg n ( x − c | t | + α ( t ))( e −| x − c | t | + α ( t ) | − 1)) , (4.7c ) where α ( t ) is an arbitrary fu nction of t . 5 Conclusion As a concludin g remark, we w ant to stress that the construction of N p eak on solutions for the DPESCS is still an op en p roblem, as w ell as for the case of the C amassa-Holm equation with self- consisten t sources. In fact, in the case of the Camassa-Holm equation and DP equation a recipro cal transformation is u sed, bu t so far we ha v e not b een able to extend it in the case of those equations with self-consisten t sources. Ac kno wledgemen ts This w ork was s upp orted b y the National Basic Researc h Program of Chin a (973 program) (2007 CB814800) and the National Science F oundation of Ch in a (Gran t no 106010 28). Y eh ui Huang would like to thank ”Ant enn a Lazio” for ha ving a on e-yea r fello wship to stay at the Physics department of Roma T re Universit y . 13 References [1] V. K. Mel’niko v, Commun. Math. Phys. 120 (1989) 451. [2] V. K. Mel’niko v, Commun. Math, Phys. 126 (1989) 201. [3] V. K. Mel’niko v, Inv erse Pr oblems 6 (1990) 233. [4] D. J. Kaup , Phys. Rev. Lett. 59 (1987) 2063. [5] J. Leon and A. Latifi, J. Phys. A: Math. Gen. 23 (1990) 1385. [6] C. Claude, A. Latifi and J. Leon, J. Math. Ph ys. 32 (1991) 3321. [7] M. Nak az aw a, E. Y omada and H. Kub ota, Ph ys. Rev. Lett. 66 (1991) 2625. [8] E. V. Doktoro v and V. S. Shc hesn o vic h, Phys. L ett. A 207 (1995) 153. [9] V. S. S hc hesnovic h and E. V. Doktoro v, Phys. Lett. A 213 (1996) 23. [10] V. K. Mel’nik o v, J. Math. P hys. 31 (1990) 1106. [11] J. L eon, Phys. Lett. A 144 (1990) 444. [12] V. K. Mel’nik o v, Ph ys. Lett. A 133 (1988) 493. [13] Y. B. Zeng, P hys. Lett. A 160 (1991) 541. [14] Y. B. Zeng and Y. S. Li, J. Ph ys. A 26 (1993 ) L273. [15] Y. B. Zeng, P hysica D 73 (1994) 171. [16] Y. B. Zeng, W. X. Ma and R. L. Lin, J. Math. Ph ys. 41 (2000) 5453. [17] Y. B. Zeng, Y. J. S hao, W. M. Xue, J. Ph ys. A: Math. Gen. 36 (2003) 5035. 14 [18] T. Xiao and Y. B. Zeng, J. Phys. A: Math. Gen. 37 (2004 ) 7143. [19] A. Degasp eris and M. P r o cesi, ”Asymptotic in tegrabilit y”, in Symmetry and P erturbation Theory (A. Degasper is and G. Gaeta, eds) W orld Scienti fi c, Singap ore (1999) 23. [20] B. F uc hssteiner and A. S. F ok as, Ph ysica D 4 (1981) 47. [21] R. Camassa and D. Holm, Phys. Rev. Lett. 71 (1993) 1661. [22] A. Degasp eris, D. Holm and A.Hone, Theo. Math. Phys. 133(2 ) (2002) 1463. [23] H. R. Dullin, G.A. Gott w ald and D.D. Holm, Fluid Dyn. Res. 33 (2003) 7395. [24] Z. Y. Yin, J. F unct. Anal. 212 (1) (2004) 182. [25] Z. J . Qiao, Acta. App l. Math. 83 (2004) 199. [26] H. Lundm ark and J.Szmigielski, Inv erse P r oblems 19 (2003) 1241. [27] Y. Matsuno, Inv erse Problems 21 (2005) 1553. [28] Y. Matsuno, Physics letters A 359 (2006) 451. [29] H.Lundmark, J.Nonlinear S ci. 17 (2007) 169. [30] J.Lenells, J.Math.Anal.Appl. 306 (2005) 72. [31] V.O.V akhnenk o E.J.Park es, C haos,Solitons and F ractals 20 (2004) 1059. [32] B.L. Guo and Z .R. Liu, c haos S olitons F ractals 23 (2005) 1451. [33] H. X. W u , X. J. L iu , Y. H. Hu an g and Y. B. Zeng, Solving soliton equations with self-consisten t sources by constan t v ariation metho d, in submission. [34] H. X. W u, Y. B. Zeng and T. Y. F an, Inv erse Problems 24 (2008) 1. 15