Commun. heo. Phys. 64 05 37 378 Vol. 64, No. 4, Ocobe, 05 Dual Hieachies of a Muli-Componen Camassa Holm Sysem LI Hong-Min, LI Yu-Qi, and CHEN Yong Shanghai Key Laboaoy of uswohy Compuing, Eas China Nomal Univesiy, Shanghai 0006, China Received Mach, 05 Absac In his pape, we deive he bi-hamilonian sucue of a muli-componen Camassa Holm sysem, which associaes wih he muli-componen AKNS hieachy and muli-componen KN hieachy via he i-hamilonian dualiy mehod. Fuhemoe, he specal poblems of he dual hieachies may be obained. PACS numbes: 0.30.Ik,.0.Ef Key wods: bi-hamilonian sucue, dual hieachies, Camasssa Holm sysem Inoducion In 993, he Camassa Holm CH equaion m + um x + u x m 0, m u u xx, was deived by Camassa and Holm fom an appoximaion o he incompessible Elue equaions. [] Like he KdV equaion, he CH equaion is inegable wih Lax pai and bi-hamilonian sucue, [] an unusual feaue is ha i admis peakon soluions. [3 4] I is ineesing ha he CH equaion is associaed wih he fis negaive flow of he KdV hieachy by ecipocal ansfomaion, [5] and he Hamilonian pai fo i can be consuced by eaanging ha of he KdV equaion. Via his connecion, he specal poblem fo he CH equaion can be obained fom ha of he KdV equaion. Moivaed by he emakable popey of he CH equaion, many ohe CH sysems have been consuced [6 0] and sudied. [ 5] Recenly, Xia and Qiao pesened a muli-componen CH sysem, [6] m s + [ m v + v x u u x + u u x v + v x m], n s + [ n u u x v + v x + v + v x u u x n], m u u xx, n v v xx, whee u u, u,..., u s, v v, v,, v s, m m, m,..., m s, n n, n,..., n s and is he anspose of a veco. hey found ha he sysem possessed a Lax pai and infiniely many consevaion laws, and discussed he peakon soluions as s. When s, he bi-hamilonian sucue of he sysem was consideed in Ref. [7]. he muli-componen CH sysem is bi- Hamilonian as a by-poduc of he esuls in his pape. Olve and Rosenau consuced CH sysems via he i- Hamilonian dualiy mehod ha eaanging he Hamilonian opeaos of he classical solion equaions in an algoihmic manne. [8] hey called he CH sysems he dual hieachies of he associaed solion equaions. Indeed, he mehod of eaanging he Hamilonian opeaos appeaed in he ealie wok of Fuchsseine and Fokas. [9] And vice vesa, a pope ecombinaion of Hamilonian opeaos of he CH sysems can also geneae he classical solion hieachies. he aim of his pape is o consuc he dual hieachies of he CH sysem. he pape is aanged as follows: In Sec., we deive he bi-hamilonian sucue of he s-componen CH sysem, and consuc he dual hieachies of i using he i-hamilonian dualiy mehod. [8] In Sec. 3, we sudy he dual vesions of a wo-componen s CH sysem and a fou-componen s CH sysem by ecombining hei Hamilonian opeaos. In Appendix, we pesen he deail poof of he Jacobi ideniy fo he opeao J 3 as well as he compaibiliy wih he Hamilonian opeao K by he iveco echnique of Olve. [0] Dual Hieachies of he Muli-Componen Camassa Holm Sysem In his secion, we deive he bi-hamilonian sucue of he muli-componen CH sysem and conside is dual hieachies using he i-hamilonian dualiy appoach. [8] Moeove, via his connecion, we ecove he specal poblems of he dual hieachies. In ode o bee undesand and display, we denoe m, n, u, v by M, N, U, V especively, i.e., M m, m,..., m s, N n, n,..., n s, Suppoed by he Naional Naual Science Foundaion of China unde Gan Nos. 7507 and 375090, Reseach Fund fo he Docoal Pogam of Highe Educaion of China unde No. 00076004, he Innovaive Reseach eam Pogam of he Naional Naual Science Foundaion of China unde Gan No. 63064, Shanghai Knowledge Sevice Plafom fo uswohy Inene of hings unde Gan No. ZF3, alen Fund and K.C. Wong Magna Fund in Ningbo Univesiy E-mail: ychen@sei.ecnu.edu.cn c 05 Chinese Physical Sociey and IOP Publishing Ld hp://www.iopscience.iop.og/cp hp://cp.ip.ac.cn
No. 4 Communicaions in heoeical Physics 373 U u, u,..., u s, V v, v,..., v s. And hen he sysem can be ewien as follows F s + M s + [ M, V + V x U U x + U U x, V + V x M], N s + [ N, U U x V + V x + V + V x, U U x N], s λm λn I s, G s + whee λ is he specal paamee and I s is he s s ideniy maix. In ode o obain he bi-hamilonian sucue of he sysem, we ewie he ime pa of he sysem 5 as follows V s + v A B C M U U xx, N V V xx, 3 whee, denoes he sandad inne poduc. As poined in Ref. [6], he muli-componen CH sysem 3 aises fom a zeo cuvaue equaion F G x + [F, G] 0, 4 his being he compaibiliy fo he linea sysem wih λ s + s + U U x, V + V x, 6 whee v is a funcion vaiable and A, B ae boh s dimension ow vecos depending on veco poenials M, N and λ, C is an s s maix depending on veco poenials M, N and λ. he compaible condiion yields M λ A + A x + s + M v s + M C, 7 ϕ x F ϕ, ϕ Gϕ, 5 λ U U x λ V + V x λ I s s + V + V xu U x, N λ B x B s + Nv + CN, s + 8 v λ s + M, B N, A, 9 C λ s + NA B M, 0 whee M M 0s s + I s K I s 0 s s M M + M M M N M NI s J N M N MI s N N + N N, B x B x. Subsiuing he equaliies 9 and 0 o Eqs. 7 and 8, we have M N λ K + λ B s + J A, wih,, 3 whee 0 s s is he s s zeo maix. In he following, we show ha he opeaos K, J ae Hamilonian opeaos and fom a bi-hamilonian pai. he Jacobi ideniy and compaibiliy condiions fo he opeaos K, J may be checked using he muliveco appoach o Hamilonian sysems in infinie dimensions, as descibed in he wok of Olve. [0] heoem he muli-componen CH sysem 3 can be wien in he bi-hamilonian fom M N K δh δm δh δn J δh 0 δm δh 0 δn using he opeaos K and J 3 and H 0 s + U xx U x, N dx, H Poof s +, 4 U U x, V + V x U U x, N dx. he equaliies and 3 imply ha he opeaos K, J ae skew-symmeic. Fuhemoe K is a Hamilonian opeao. Hence, we need o pove ha he Jacobi ideniy fo J and compaibiliy of J wih K. aking θ θ,..., θ s, θ θ,..., θ s as he basic uni-vecos coesponding o M, N especively, we know ha he opeao J is a Hamilonian opeao if whee PV J θ Θ J 0, 5 θ θ and Θ J is he associaed bi-veco of J. o check whehe K and J fom a bi-hamilonian pai, we only need o pove θ PV Kθ Θ J 0. 6 he poof of he heoem is ahe echnical and lenghy, so ae given in Appendix. In he following, we will sudy he dual hieachies of he muli-componen CH sysem 3 by ecombining he
374 Communicaions in heoeical Physics Vol. 64 0s s I s K, Ĵ I s 0 s s whee Q q, q,..., q s, R,,..., s. he above Hamilonian pai K, Ĵ is nohing bu he bi-hamilonian pai fo he muli-componen AKNS hieachy. [] In he equaliy, fo he Hamilonian opeaos K and J, we make he following ansfomaion x λx, M Q, N R. 7 s + Afe he above ansfomaion, he equaliy becomes Q R λ K + B, 8 s + Ĵ Hamilonian opeaos K and J in Eqs. and 3 accodingly. he dual Hamilonian opeaos of he opeaos K and J ae obained by he following pocedue. ansfeing he ems I s fom he opeao K o he coesponding elemens of he opeao J and eplacing M, N by Q, R especively in he opeao J, we ge wo Hamilonian opeaos Q Q + Q Q Q RI s Q R A which leads o he Hamilonian opeaos K and Ĵ. heewih, he spaial pa of he linea poblem 5 is 0s s I s K, J I s 0 s s R QI s R Q R R + R R, ansfomed ino s Q ϕ x F ϕ, F λ, 9 R λ I s which, by he ansfomaion λ /λ, may be efomulaed as λs Q ϕ x F ϕ, F. 0 R λi s he above specal poblem is jus he one fo he mulicomponen AKNS hieachy see. in Ref. [] fo deails. On he ohe hand, if we ansfe he ems I s and I s insead of he ems I s fom K o he coesponding elemens of he J, and eplace M, N by Q, R especively as well, we have Q Q + Q Q Q RI s Q R + R QI s R Q R R + R R, which ae jus he compaible Hamilonian opeaos of muli-componen KN hieachy. Fuhemoe, afe he ansfomaion x λx, M s + Q, N s + λ R, Eq. yields Q λ R s + K + s + J B A. /λ he specal poblem of he CH sysem 3 becomes s ϕ x F ϕ, F s + Q λ s + s + λ R, 3 λ I s which, by he ansfomaion leads o λ λs +, R s + R, λs Q ϕ x F ϕ, F λr λi s. 4 he specal poblem 4 is nohing bu he one of he muli-componen KN hieachy. 3 Dual Hieachies of he wo-componen CH Sysem and Fou-Componen Camassa Holm Sysem In his secion, we conside he dual hieachies of he wo-componen CH sysem and fou-componen CH sysem. 3. Dual Hieachies of he wo-componen Camassa Holm Sysem As s, he muli-componen CH sysem 3 is m muv u xv x + uv x u x v, n nuv u xv x + uv x u x v, m u u xx, n v v xx, 5 which appeas in he bi-hamilonian fom 4 wih 0 + m m m n K, J 0 n m n, n H 0 u xx u x ndx, H u u x v + v x ndx. 6 4 Fom he esuls in Sec., we know he dual Hamilonian pais of he opeao 6 ae especively 0 q K q q, Ĵ 0 q, 7 0 q K, J q q 0 q, 8 which ae he Hamilonian pais fo he AKNS hieachy
No. 4 Communicaions in heoeical Physics 375 and KN hieachy especively. he associaed specal poblems of he dual hieachies ae especively λ q ϕ x Uϕ, U, 9 λ λ q ϕ x Uϕ, U, 30 λ λ which can be deduced via he connecion. Remak In Ref. [3], Ma and Zhou sudied he Hamilonian opeao α q q α + α 3 α q M, 3 α + α 3 α q α whee α, α, α 3 ae abiay consans. hey gave he Hamilonian pais 7 and 8 fo he AKNS hieachy and KN hieachy saing fom Hamilonian opeao 3. Indeed he Hamilonian opeao 3 can lead o anohe Hamilonian pai 6 fo he wo-componen CH sysem 5. 3. Dual Hieachies of he Fou-Componen Camassa Holm Sysem When s, he muli-componen CH sysem 3 becomes m 9 {m [u u x v + v x + u u x v + v x ] + m u u x v + v x }, m 9 {m [u u x v + v x + u u x v + v x ] + m u u x v + v x }, n 9 {n [u u x v + v x + u u x v + v x ] + n u u x v + v x }, n 9 {n [u u x v + v x + u u x v + v x ] + n u u x v + v x }, m u u xx, m u u xx, n v v xx, n v v xx, 3 which can be wien as he bi-hamilonian fom 4, using he 0 0 + 0 m m J J 3 m n 0 0 0 + K 0 0 0, J J m m m n J 4 J3 n m n n J 34, whee 0 0 0 n m J4 J34 n n H 0 u xx u x n + u xx u x n dx, 9 H [u u x v + v x + u u x v + v x ][u u x n + u u x n ]dx, 9 J m m + m m, J 3 m n + m n, J 4 m n + m n, J 34 n n + n n. Afe applying he i-hamilonian dualiy mehod o he Hamilonian opeaos K and J, we ge he dualiy Hamilonian opeaos 0 0 0 q q Ĵ Ĵ 3 q 0 0 0 K 0 0 0, Ĵ Ĵ q q q Ĵ 4 Ĵ3 q Ĵ 34, 33 0 0 0 0 0 0 0 0 0 K 0 0 0, J 0 0 0 q Ĵ4 Ĵ34 q q J J3 q J q q q J4 J 3 q J34 q J 4 J 34, 34
376 Communicaions in heoeical Physics Vol. 64 whee Ĵ q q + q q, Ĵ 3 q + q, Ĵ 4 q + q, Ĵ 34 +, J q q + q q, J 3 q + q, J 4 q + q, J 34 +. he K, Ĵ 33 and K, J 34 ae jus he Hamilonian pais of he coupled AKNS hieachy, [4] and coupled KN hieachy, he specal poblems of hem ae especively ϕ x Uϕ, U λ q q λ 0, 35 0 λ λ q q ϕ x Uϕ, U λ λ 0. 36 λ 0 λ Remak he coupled nonlinea Schödinge equaion [5] is a educion of he coupled AKNS hieachy. In fac, i can be educed o he coupled MKdV equaion [6] and he Sasa Sasuma equaion [7] unde he consains q, q and q, q especively. [8] Fuhemoe, hough he Diac educions of he Hamilonian opeaos J and KJ K unde he coesponding consains, one can ge a Hamilonian sucue and a symplecic sucue fo he coupled MKdV equaion and he Sasa Sasuma equaion especively. A naual quesion is wha ae he educed sysems of he fou-componen CH sysem 3 and hei coesponding Hamilonian sucues. Besides, i is wohwhile o invesigae he ecipocal ansfomaions beween he CH sysems 3, 5, 3 and hei dual hieachies. Appendix Fis, we pove ha he opeao J is Hamilonian, namely o veify 5. o simplify he pesenaion and calculaions, we inoduce M i, N i i,,..., s as m m i θ i n i θ i + m i m θ i n i θ J θ M M s N m s m i θ i n i θ i + m i m s θ i n i θ s N s i n m i θ i n i θ i n i m i θ n θ i. A n s m i θ i n i θ i n i m i θ s n s θ i he associaed bi-veco of J is Θ J θ J θdx θ j M j + θ j N j dx j θ j m j m i θ i n i θ i + m i m j θ i n i θ j i,j + θ j n j m i θ i n i θ i n i m i θ j n j θ i dx m j θ j n j θ j m i θ i n i θ i + m j θ i n i θ j m i θ j n j θ i dx, i,j whee we have subsiued he expessions of M i, N i i,,..., s and used he skew-symmey of he opeao. By diec calculaion, we have he polongaion PV J θ Θ J θ j M j θ j N j m i θ i n i θ i + θ i M j θ j N i m i θ j n j θ i dx i,j i,j,k [θ j m j m k θ k n k θ k + m k m j θ k n k θ j + θ j n j m k θ k n k θ k + n k m k θ j n j θ k ] m i θ i n i θ i + [θ i m j m k θ k n k θ k + m k m j θ k n k θ j + θ j n i m k θ k n k θ k + n k m k θ i n i θ k ] m i θ j n j θ i dx
No. 4 Communicaions in heoeical Physics 377 i,j,k m j θ j + n j θ j m k θ k n k θ k m i θ i n i θ i + m j θ k + n k θ j m k θ j n j θ k m i θ i n i θ i + m j θ i + n i θ j m k θ k n k θ k m i θ j n j θ i + m k θ i m j θ k n k θ j m i θ j n j θ i + n k θ j m k θ i n i θ k m i θ j n j θ i dx m k θ i m j θ k n k θ j m i θ j n j θ i i,j,k + n k θ j m k θ i n i θ k m i θ j n j θ i dx, A whee we have used inegaion by pas and he skew-symmey of he opeao. Afewads, expanding he wo ems in Eq. A ino eigh ems, we ge PV J θ Θ J m k θ i m j θ k m i θ j m k θ i m j θ k n j θ i i,j,k m k θ i n k θ j m i θ j + m k θ i n k θ j n j θ i + n k θ j m k θ i m i θ j n k θ j m k θ i n j θ i n k θ j n i θ k m i θ j + n k θ j n i θ k n j θ i dx m k θ i m j θ k n j θ i m k θ i n k θ j m i θ j i,j,k + m k θ i n k θ j n j θ i + n k θ j m k θ i m i θ j n k θ j m k θ i n j θ i n k θ j n i θ k m i θ j dx m k θ i m j θ k n j θ i + m i θ j m k θ i n k θ j i,j,k + n j θ i n k θ j m k θ i n k θ j n i θ k m i θ j dx 0. A3 In he above, we have dopped he ems which only conain m i o n i using he inegaion by pas and he skewsymmey of he opeao, which ae also applied o he emaining ems. Fom Eq. 3, we know J is Hamilonian. Secondly, we will show he compaibiliy of K and J, i.e., he equaliy 6. Noice ha θx + θ Kθ, A4 θ x θ we calculae PV Kθ Θ J [θ j θ jx + θ j θ j θ jx θ j ] m i θ i n i θ i i,j + [θ i θ jx + θ j θ j θ ix θ i ] m i θ j n j θ i dx θ j θ j m i θ i n i θ i θ i θ j m i θ j n j θ i dx 0, i,j which implies he opeaos K and J ae compaible Hamilonian opeaos. hus, we complee he poof of he heoem. A5 Refeences [] R. Camassa and D.D. Holm, Phys. Rev. Le. 7 993 66. [] R. Camassa, D.D. Holm, and J.M. Hyman, Adv. Appl. Mech. 3 994. [3] R. Beals, D.H. Sainge, and J. Szmigielski, Adv. Mah. 54 000 9.
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