Sigma 27, , 2009

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1 ornal of Engineering and Naral Sciences Mühendislik ve Fen Bilimleri ergisi Research Aricle / Araşırma Makalesi A NEW +-IMENSIONAL HAMILTONIAN INTEGRABLE SYSTEM Sigma 7, 8-8, 9 evrim YAZICI* Yıldız Teknik Üniversiesi, Fen-Edebia Fakülesi, Fizik Bölümü, Esenler-İSTANBUL Received/Geliş:..9 Revised/üzelme: Acceped/Kabl: ABSTRACT I is shown ha a new +-dimensional second-order parial differenial eaion, when wrien as a firs-order nonlinear evolionar ssem, admis bi-hamilonian srcre. Therefore, b Magri s heorem i is a compleel inegrable ssem. For his ssem a Lagrangian is inrodced and irac s heor is applied in order o obain firs Hamilonian srcre. Then recrsion operaor is consrced and finall he second Hamilonian srcre for his ssem is obained. acobi ideni for he Hamilonian srcre is proved b sing Olver s mehod. Ths, i is an eample of a compleel inegrable ssem in hree dimensions. Kewords: Inegrable ssems, Hamilonian inegrable ssems. PACS nmbers/nmaraları: 4..b,.4.K. + BOYUTTA YENİ İNTEGRE EİLEBİLİR HAMİLTONIAN SİSTEMLER ÖZET Yeni + bol ikinci merebeden kismi ürevli diferansiel denklem, birinci merebe lineer olmaan değişim sisemi olarak azıldığında, b eni sisemin bi-hamilonian apıa sahip oldğ göserilmişir. Bölece Magri eoremine göre amamen inegere edilebilir bir sisem elde edilmişir. B sisem için Lagrangian elde edilmiş, ve birinci Hamilonian apıı elde emek için irac eori glanmışır. Sisem için ekrarlama (recrsion operaorü krlmş ve son olarak ikinci Hamilonian apı elde edilmişir. Hamilonian apılar için acobi özdeşliği Olver in meod kllanılarak ispalanmışır. Bölece eni denklem üç boa amamen inegre edilebilir siemlere bir örnek eşkil emekedir. Anahar Sözcükler: İnegre edilebilir sisemler, Hamilonian ineger edilebilir sisemler.. INTROUCTION Inegrable Hamilonian ssems are sdied for more han hree decades and here are man eamples of +-dimensional ones in lierare. The well-known eample in his field is Koreweg- de Vries (KdV eaion. The firs discover, made b Gardner [], was ha he KdV eaion cold be wrien as a compleel inegrable Hamilonian ssem. This idea was frher developed b Zakharov and Fadeev []. The general concep of a Hamilonian ssem of evolion eaions firs appears in he works of Magri [3], Kpershmid [4] and Manin [5]. Frher developmens, inclding he simplified echnies for verifing he acobi ideni, appear in Gelfand and orfman [6], Olver [7] and Kosmann Schwarzbach [8]. The basic * /e-ilei: azici@ildiz.ed.r, el: (

2 . Yazıcı Sigma 7, 8-8, 9 heorem on bi-hamilonian ssems is de o Magri [3, 9], who was also he firs one o pblish he second Hamilonian srcre for he KdV and he oher eaions. For a long ime here were onl few eamples of +- and even no eamples of 3+- dimensional inegrable ssems. Ver recenl Nezi, Nk and Shefel [] discovered ha he second heavenl eaion of Plebanski, when being presened in a wo componen form, is a 3+- dimensional bi-hamilonian inegrable ssem. Laer, i was discovered ha he comple Monge- Ampere eaion in (3+ real dimensions is compleel inegrable in he sense of he Magri s heorem []. In [] we sdied smmer redcion of second heavenl eaion and we obained a +-dimensional bi-hamilonian ssem. In his paper I will presen a new +-dimensional bi- Hamilonian ssem. The Lagrangian of his ssem is, L ( and Eler Lagrange eaion gives he following non linear + dimensional parial differenial eaion in one-componen form. ( ( where is an arbirar consan. This ssem is obained b sing a linear combinaion of invarians given in []. In secion, I will give Lagrangians and consrc he firs Hamilonian srcre sing irac s heor [3] of consrains. In secion 3 I derive a recrsion operaor for a new ssem. In secion 4 I obain he second Hamilonian srcre and Hamilonian fncion b appling he recrsion operaor o he firs Hamilonian srcre. Finall in secion 5, he acobi ideni for he Hamilonian srcre will be checked in deail b sing Olver s mehod [4].. LAGRANGIAN AN FIRST HAMILTONIAN STRUCTURE In his par I se he mehod of [] for he calclaion of he firs Hamilonian srcre. The Lagrangian densi ( for he eaion in one-componen form (, b his ms be convered o a form siable for appling irac s heor of consrains. For his prpose, I inrodce an ailiar variable whereb ssem ( assmes he form [ + ( + ] (3 Q of a firs-order wo-componen ssem. Here sb indees, and sand for parial derivaive of, and respecivel and in all paper I will se he same noaion. Lagrangian densi for ssem (3 is given b, shold be degenerae, ha is, linear in he ime derivaive of he nknown and wih no : L ( ( This Lagrangian is degenerae [6], becase is Hessian ( (4 9

3 A New +-imensional Hamilonian Inegrable Sigma 7, 8-8, 9 L vanishes idenicall. Alernaivel, he canonical momenm given b; L Π ( can no be invered he veloci and we have degenerae Lagrangian. Afer sbsiing, coincides wih or original Lagrangian ( p o a oal divergence. We can easil check k ha Eler-Lagrange eaions for (4 give he ssem (3. The variaional derivaive for defined as following. δ k δ + + k k k k k k Here k, wih δl δ and δl δ + and ( [ + ( + ], hence we ge, π π L L Since he Lagrangian densi (4 is linear in and has no, he canonical momena ( (5 canno be invered for he velociies and and so he Lagrangian is degenerae. Therefore, according o he irac s heor [3], we impose (5 as consrains φ π φ π + ( where he canonical momena shold saisf canonical Possion brackes (6

4 . Yazıcı Sigma 7, 8-8, 9 k k [ π (, (, ] δ δ ( δ (,, i, k, i i and calclae he Passion brackes of he consrains K K K K [ φ (, φ ( ] i i,,. (7 If we organize hem ino a mari form, we find [ φ (,, φ(, ] ( δ ( δ( + ( δ ( δ( + δ ( δ( δ( δ ( + δ ( δ( δ( δ ( [ ( ( ] ( δ( φ,, φ, δ (,, φ (, K [ φ ], where he sbscrips rn from o wih and corresponding o and, respecivel. In all he coefficiens of K if we kill facor ( inverse of he Hamilonian operaor : K i + i ( (8 his ields he smplecic operaor K i ha is an, (9 which is an eplicil skew-smmeric local mari-differenial operaor. The firs Hamilonian operaor ( Ki is obained b invering K i in (9 as + ( which is eplicil skew-smmeric. Also i saisfies he acobi ideni, as will be shown in deail in secion 5. The Hamilonian densi is H π + π L, which resls in H ( + (. ( The new ssem (3 can now be wrien in a Hamilonian form wih he Hamilonian densi H defined b (

5 A New +-imensional Hamilonian Inegrable Sigma 7, 8-8, 9 δ Η δ Η [ ( ] + + δ δ ( where δ and δ are Eler-Lagrange operaors [4], defined as E δ δ ( H δ Η d Η δ d d d wih, and similarl for Hamilonian fncional 3. RECURSION OPERATOR (3 Η Hdd. δ Η, which correspond o he variaional derivaives of he We sar wih he eaion deermining smmeries of he wo-componen ssem (3. We inrodce he wo-componen smmer characerisic Φ b τ τ ϕ ψ (,,,,,,,, (,,,,,,,, A Q From he Freché derivaive of he flow we find + ϕ ψ, Φ. (4 ( So ha he eaion deermining smmeries of he new hree-dimensional evolion ssem is given b ( Φ A. (6 If we combine he firs deermining eaion wih he second eaion in (6, mliplied b he overall facor, we reprodce he deermining eaion for smmeries of original eaion (. The eaion for smmeries (6 can be se in a -erm divergence form ( ( ϕ ϕ + ψ ( ϕ + ψ ϕ ϕ (7 ha implies he local eisence of he poenial variable ϕ ~ defined b (5

6 . Yazıcı Sigma 7, 8-8, 9 ~ ϕ ϕ ~ ϕ ( ϕ ϕ ( ϕ + ψ ϕ + ψ which also saisfies he same deermining eaion for he smmeries of ( and herefore i is a parner smmer for ϕ. In he wo-componen form, we define he second componen of his new smmer, similar o he definiion of ψ, as ~ ~ ϕ Φ ~ ψ ~ ψ ~ ϕ. Then he wo-componen vecor saisfies he deermining eaion for smmeries in he form (6 and hence a smmer characerisic of he ssem (3, provided he vecor (4 is also a smmer characerisic. ~ Φ R Φ ( wih he recrsion operaor R given b R where f [ + ( ] ( Q + is he inverse of and defined as f ( ξ dξ (8 (9, (. ( For he properies of his operaor see [5]. Moreover, vanishing of he commaor[ R, A], comped wiho sing he eaions (3, reprodce he new ssem (3 and hence he operaor R and A form a La pair for he -componen ssem. The commaor reads, [ R, A] ( Q ( Q ( {( Q ( + ( ( ( Q } ( Q I can be easil see ha [, A] and A form a La pair for -componen ssem (3. R is eivalen o he ssem (3 and herefore R 3

7 A New +-imensional Hamilonian Inegrable Sigma 7, 8-8, 9 4. SECON HAMILTONIAN STRUCTURE AN HAMILTONIAN FUNCTION The second Hamilonian operaor is obained b appling he recrsion operaor ( o he firs Hamilonian operaor R wih he resl where, ( ( Q + ( Q ( + (. Operaor is obviosl skew-smmeric and he acobi ideni for his operaor will be checked in he ne secion in deail. The Hamilonian fncion for which generaes he ssem (3 is given b H + (. (3 The Hamilonian fncion H saisfies he recrsion relaion of Magri, δ Η δ Η δ Η δη which shows ha he new eaion ( in he -componen form (3 is a bi-hamilonian ssem. The second Hamilonian operaor is obained b acing wih he recrsion operaor R on he Hamilonian operaor. In order o have he higher flows we generalize his relaion as n n R (5 In he case of ( we have n. If we ake, for eample, n we can generae a new Hamilonian operaor R R. Here we sed he relaion R. B he repeaed applicaion of he recrsion operaor ( o he Hamilonian operaors, and so on, we cold obain mli-hamilonian represenaion of or new ssem. (4 4

8 . Yazıcı Sigma 7, 8-8, 9 5. ACOBI IENTITY In his secion, I will concenrae on checking of he acobi ideni for he Hamilonian operaors and. efiniion: A linear operaor : A A is called Hamilonian if is Possion bracke P, Q δp δq d saisfies skew-smmer proper { } { P, Q} { Q, P}, (6 and he acobi ideni {, Q}, R} + { R, P}, Q} + { Q, R}, P} P (7 for all fncionals P, Q and R B sing his definiion direcl, he verificaion of acobi ideni (7, even for simples skew-adoin operaors, appears a hopelessl complicaed compaional ask. For his reason we will se he Olver s mehod [4] b following he heorem below. Theorem: Le be a skew-adoin Θ θ d mari differenial operaor and ( be he corresponding bi-vecor. Then is Hamilonian if and onl if ( Θ Pr V θ. (8 We menioned ha if we can presen he ssem (3 in he form (4, he ssem is called bi-hamilonian ssem. We sa ha, form a Hamilonian pair if ever linear combinaion a + b where a and b are consans, shold saisf he acobi ideni. Therefore, if we direcl compe he acobi ideni for Γ a + b, hen we garanee ha and saisf he acobi ideni. Becase, if we choose, b a and a, b hen we will end p wih he acobi ideni for and respecivel. In his wa, we will prove ha, and independenl saisf he acobi ideni and also ha an linear combinaion a + b also saisfies acobi ideni. Therefore we sar wih, a + b b b a b a A + B + C Γ (9 where 5

9 A New +-imensional Hamilonian Inegrable Sigma 7, 8-8, 9 A b a + b + b a b ( b a B + C A B+ ( ab. ( B sing heorem (8, we define he wo-form bi-vecor i ( Γiω Θ ω d d (3 i, Where he ni-vecors correspond o (3 becomes Θ ω η and ω θ i (3 and,,. Hence b b a ( η η + θ η + η Aθ + Bθ dd If we sbsie Θ in (8 we obain b Γω ω b b + V ω Pr ( Θ ( ( b a ( b a ( + b ( + b ( and b sing he following relaion [4], Γω β ( βω, β, ( + ( b a d d η A θ θ (3 Γ,,,, (33 we can compe he erms given in (3 as given below Pr V b a ( + + η Aθ Bθ Cθ b ( a η Aθ + Bθ + Cθ 6

10 . Yazıcı Sigma 7, 8-8, 9 b a b a bη θ θ b a b a ω η θ θ 3 b ( b a b a Pr V bη θ θ. Afer we sbsie hese erms in (3 we ge a ver large eaion which is no siable o wrie here. Afer ha we do a ver lengh and cmbersome calclaion and finall we ge zero. This means, b vire of (8, ha he acobi ideni is saisfied boh for and and he form a Hamilonian pair. 6. CONCLUSION We discover new +-dimensional nonlinear evolion eaion and we wrie his eaion in a wo-componen form in order o obain is Hamilonian srcre. We sar wih he firs Hamilonian srcre. We se irac s heor of consrains o consrc he mari operaor K which is an inverse of he firs Hamilonian operaor. We obain a recrsion operaor for smmeries and we show ha he recrsion operaor R and he linear operaor A of he eaion deermining smmeries comme and, moreover, he form a La pair for he new wocomponen evolionar ssem. We have fond second Hamilonian srcre b acing wih he recrsion operaor R on he firs Hamilonian operaor. Finall, we prove ha boh Hamilonian operaors, and also heir linear combinaion a + b saisf he acobi ideni. Therefore, his new ssem is bi-hamilonian and, b Magri s heorem, he mli- Hamilonian srcre makes he new ssem o be a compleel inegrable ssem in +- dimensions. In he fre work we will presen he Lie algebra of all poin smmeries and inegrals of moion which generae all variaional poin smmeries of he new evolion ssem. Acknowledgemens I wold like o hank M B Shefel for frifl discssions and encoraging me o wrie his paper. Also I wold like o hank o he referees for heir imporan sggesion and correcions. REFERENCES / KAYNAKLAR [] Gardner C. S., Koreweg-de Vrise eaion and generalizaions. IV. The Koreweg-de Vrise eaion as Hamilonian ssem,. Mah. Phs., , 97. [] Zakharov V. E., Fadeev L.., Koreweg-de Vrise eaion: a compleel inegrable Hamilonian ssem, Fnc. Anal. Appl 5, 8-87, 97. [3] Magri F., A simple model of he inegrable Hamilonian eaion,. Mah. Phs. 9, 56-6,

11 A New +-imensional Hamilonian Inegrable Sigma 7, 8-8, 9 [4] Kpershmid B. A., Geomer of e bndles and srcre of Lagrangian and Hamilonian formalism, in Geomeric Mehods in Mahemaical Phsics, G. Kaiser and. E. Marsden, eds., Lecre Noes in Mah. No. 775, pp. 6-8, Springer-Verlag, Nework, 98 [5] Manin Y. I., Algebraic aspecs of nonlinear differenial eaions,. Sovie Mah., -, 979. [6] I. M. Gelfand and I. Ya. orfman, Hamilonian operaors and algebraic srcres relaed o hem, Fnc. Anal. Appl. 3, 48-6, 979. [7] P.. Olver, On he Hamilonian srcre of evalaion eaions, Mah. Proc. Camb. Phil. Soc. 88, 7-88, 98. [8] Y. Kosmann Schwarzbach, Hamilonian ssem on fibered manifolds, Le. Mah. Phs. 5, 9-37, 98. [9] Magri F., A geomerical approach o he nonlinear solvable eaions, in Nonlinear Evolion Eaions and namic Ssems, M. Boii, F. Pempinelli and G. Soliani, eds., Lecre Noes in Phsics, No. Springer-Verlag, Nework 98. [] F. Nezi, Y. Nk and M. B. Shefel, Mli Hamilonian srcre of Plebanski s second heavenl eaion,. Phs. A: Mah. Gen , 5. [] Y. Nk, M. B. Shefel,. Kalacı and. Yazıcı, Self-dal gravi is compleel inegrable,. Phs. A: Mah. Gen. 4, 3956, 3pp, 8. []. Yazıcı, M. B. Shefel, Smmer redcions of second heavenl eaion and +- dimensional Hamilonian inegrable ssem, ornal of Nonlinear Mahemaical Phsics, Volme 5, spplemen 3, (8. [3] irac P. A. M., Lecre Noes in Qanm Mechanics, Belfer Gradae School of Science Monographs series, New York, 964. [4] P.. Olver, Applicaion of Lie grops o differenial eaions, Springer, New York, 986. [5] P. M. Sanini and A. S. Fokas, Commn. Mah. Phsics., 5, 375, 988. [6] Y. Nk, Lagrangian approach o inegrable ssems ields new smplecic srcre for KdV, hep-h/5v, 8 Nov 8. 8