Integrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem

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1 Commu. Theor. Phys. Beijig Chia pp c Chiese Physical Society ad IOP Publishig Ltd Vol. 52 No. 6 December Itegrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem LI Xi-Yue 1 ad WANG Xi-Zeg 2 1 College of Sciece Shadog Uiversity of Sciece ad Techology Qigdao Chia 2 Iformatio School Shadog Uiversity of Sciece ad Techology Qigdao Chia Received December ; Revised May Abstract Based o a ew discrete three-by-three matrix spectral problem a hierarchy of itegrable lattice equatios with three potetials is proposed through discrete zero-curvature represetatio ad the resultig itegrable lattice equatio reduces to the classical Toda lattice equatio. It is show that the hierarchy possesses a Hamiltoia structure ad a hereditary recursio operator. Fially ifiitely may coservatio laws of correspodig lattice systems are obtaied by a direct way. PACS umbers: Ik Key words: discrete Hamiltoia structure discrete zero-curvature represetatio coservatio laws 1 Itroductio I the past twety years the discrete itegrable systems treated as models of some physical pheomea has become the focus of commo cocer ad the most importat topic because of its applicatio i physics chemistry ad biology. May itegrable lattice equatios have bee preseted ad studied systematically ad they are almost all derived from a 2 2 discrete matrix spectral problem. [ ] It is kow to us that the discrete zero curvature represetatio is a effective method to geerate the lattice solito equatios. However the work of studyig the higher order discrete matrix spectral problem [6 7] i the literatures of costructig itegrable lattice equatios is very few ad wats more efforts because it is much more difficult ad complicated. Therefore searchig for ew lattice solito equatios which are derived from a higher order discrete matrix spectral problem is still a importat ad more complicated task. Furthermore the coservatio laws [8] play the importat role existece of ifiitely may coservatio laws for the lattice solito equatio may further cofirm their itegrability. [ ] There are however other examples talkig about Hamiltoia structures from Lax pairs of differet orders [16 18] ad eve oisopectal hierarchies from Lax pairs. [19] Also a much more geeral idea tha the trace idetity to yield Hamiltoia structures to oliear itegrable lattice equatios was the method of usig variatioal idetities. [20] This article is orgaized as follows. I Sec. 2 a discrete 3 3 matrix spectral problem is itroduced ad a hierarchy of lattice solito equatios is derived by the method of discrete zero curvature represetatio. A lattice system is proposed it is a typical lattice system i resultig hierarchy. The we establish the Hamiltoia structures of the resultig hierarchy by meas of the discrete trace idetity. [2] Ifiitely may commutig symmetries ad ifiitely may commutig coserved fuctioals for the obtaied hierarchy are give. I Sec. 3 ifiitely may coservatio laws for correspodig lattice solito equatio are give by a direct method. 2 A Family of Lattice Solito Equatios ad Their Liouville Itegrability I this sectio we would like to propose ad discuss the lattice equatio r t r w 1 w + s +1 s s t s w 2 w w t r +1 r 1 here r r t s s t ad w w t are real fuctios defied over Z R ad satisfy lim r t 0 lim s t 0 lim w t 0. The shift operator E the iverse of E are defied by ad Ef f +1 E 1 f f 1 Z 2 E E E 2m+1 2 m E 2m+1 1 E E E E 1. Supported by the Sciece ad Techology Pla Project of the Educatioal Departmet of Shadog Provice of Chia uder Grat No. J09LA54 ad the Research Project of SUST Sprig Bud of Shadog Uiversity of Sciece ad Techology of Chia uder Grat No. 2009AZZ071 Correspodig author xiyueli 2008@yahoo.com.c

2 982 LI Xi-Yue ad WANG Xi-Zeg Vol. 52 To obtai the Lax pairs of Eqs. 1 we itroduce the discrete 3 3 matrix spectral problem Eφ U u λφ φ 3 r s λ + w where eigefuctio vector φ φ 1 φ2 φ3 T ad the potetial vector U r s w T λ t 0 ad solve the statioary discrete zero curvature equatio with EΓ U U Γ 0 4 a b c Γ d e f g h a + e the equatio 4 becomes b +1 + r c +1 g 0 s c +1 h 0 a +1 + a + e + λc +1 + w c +1 0 e +1 a + r f +1 0 s f +1 b 0 λf +1 + w f +1 + d +1 c 0 h +1 r a +1 r e +1 r a λg +1 s d w g 0 s a +1 s e +1 s e λh r b w h 0 λa +1 + e +1 a e w a +1 + e +1 a e + g +1 r c s f 0. 5 Substitutig the followig expressios a a m λ m b c e c m λ m d e m λ m f λ m Γ + m i0 Based o 6 a direct computatio gives b m λ m d m λ m Eλ m Γ + U U λ m Γ + f m λ m a i λ m i d i λ m i g i λ m i c We must take a modificatio m m+1 b i λ m i e i λ m i h i λ m i g g m λ m h ito 5 we obtai the iitial data h m λ m c 0 +1 g0 f0 +1 h0 b0 +1 d0 0 a 0 +1 a0 e0 +1 e0 0. For λ m m 0 we get the recursio relatios b m +1 + r c m +1 gm 0 m 0 s c m +1 hm 0 m 0 a m +1 + am + e m + c m w c m +1 0 m 0 e m +1 am + r f m +1 0 s f m +1 bm 0 m 0 f m w f m +1 + dm +1 cm 0 m 0 h m +1 r a m +1 s e m +1 gm+1 s d m w g m 0 m 0 s a m +1 + em+am +1 + em+em +1 r b m w h m 0 m 0 h m+1 a m+1 +1 a m+1 + e m+1 +1 e m+1 w a m +1 am + e m +1 em r c m s f m g m +1 0 m 0. 6 We choose a 0 e 0 1/3. I order to obtai a m b m c m d m e m f m g m h m m 1 we have to ivert the differece operator D E 1 to solve for a m+1 ad a m+1 i 6. By selectig zero costat for the iverse operatio of the differece operator D i computig a m e m m 1 the recursio 6 uiquely determies a m b m c m d m e m f m g m h m m 1. Startig from the above iitial values ad usig the recursio relatio 6 we ca easily calculate the first few terms as follows: a b1 0 c 1 1 d 1 0 e f1 0 g 1 r h 1 s. Now we defie c i λ m i f i λ m i a i + e i λ m i 0 0 c m f m+1 +1 g m f m+1 c m V m h m+1 ad defie m 0. 7 a m e m+1 +1 a m+1 e m+1. λ m Γ + + m m 0. 8

3 No. 6 Itegrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem 983 It is easy to coclude that EV m U U V m g m+1 r c m+1 s f m+1 h m+1 s c m+1 1 E 1a m+1. + e m+1 Let the time evolutio of the eigefuctio of the spectral problem 3 obeys the differetial equatio φ tm V m φ. 9 The the compatibility coditios of 3 ad 9 are U tm EV m U U V m m 0 10 which give rise to the followig hierarchy of lattice solito equatios r tm s +1 f m+1 +2 s f m+1 s tm s c m+1 +1 c m+1 1 w tm E 1a m+1 + e m+1 m So Eq. 10 are discrete zero curvature represetatio of 11 the discrete spectral problem 3 ad 9 costitute the Lax pair of 11 ad 11 is a hierarchy of Lax itegrable oliear lattice equatios. It is easy to verify that the first oliear lattice equatio i 11 whe m 1 is r t r w 1 w + s +1 s s t s w 2 w w t r +1 r 12 whe s 0 12 becomes the Toda lattice equatio. [2 4] r t r w 1 w w t r +1 r which is Eq. 1 whe we set t 1 t. The time part of the Lax pairs of the system 1 is give by 1 V 1 3 λ w λ λ w λr λs 2 3 λ The variatioal derivative the Gateaux derivative ad the ier product are defied respectively by δh E m H δu u +m m Z J u [v ] ε Ju + εv ε0 14 f g Zf g R 2 15 where f g are required to be rapidly vaished at the ifiity ad f g R 2 deotes the stadard ier product of f ad g i the Euclidea space R 2. Operator J is defied by f J g Jf g; it is called adjoit operator of J with respect to 15. If a operator J has the property J J the J is called to be a skew-symmetric. A liear operator J is called a Hamiltoia operator if J is a skew-symmetric operator ad satisfies the Jacobi idetity i.e. it satisfies that f Jg Jf g J u [Jf]g h + Cyclef g h 0. Based o a give Hamiltoia operator J we ca defie a correspodig Poisso bracket δf {f g} J J δg δf J δg. 16 δu δu δu δu Z To establish the Hamiltoia structures for 11 we defie R Γ U 1 λb + w b s + c s a r b b s s λe + w e + f s d r e e s s s λh + w h s a e g r h h s s s ad A B TrAB where A ad B are the some order square matrices. We have U λ U r U U. s w Hece R U h λ s R U e s s R U b r s R U h. w s By virtue of the discrete trace idetity [2] δ R U λ ε δu λ λ we have Z i δ δu Z h λ ε s λ λ ε The substitutio of a a m λ m b c e g c m λ m d e m λ m f g m λ m h λ ε R U u i f +1 e s c +1 b m λ m d m λ m f m λ m. 17 h m λ m

4 984 LI Xi-Yue ad WANG Xi-Zeg Vol. 52 ito 17 ad comparig the coefficiets of λ m 1 i Eq. 17 we get f m +1 δ h m e m ε m δu s Z. 18 c m +1 Whe m 0 i the Eq. 18 through a direct calculatio here Let u tm r s w tm J δh m δu we fid that ε 0. So we have δ δu Z hm+2 m + 1s f m+1 +1 e m+1 s c m+1 +1 m 0. Now we ca rewrite the Eqs. 11 i the followig Hamiltoia forms J Es s E 1 0 r 1 E 1 J 0 0 s 1 E 2 E 1r E 2 1s 0 δh m+1 δu Φ δhm Φ δu Ask for help the recursio relatio 6 we have f m+1 +1 e m+1 s c m+1 +1 m 0 19 H m+1 Z 1 + Er 1 + E + E 2 s w hm+2 m + 1s. 20 Φ 11 Φ 12 Φ 13 Φ 21 Φ 22 Φ Φ 11 Es s E 1 1 Ew s w Es w Φ 12 Es s E 1 1 r 1 Es Φ 13 E 1 Es s E 1 1 Es E 1 1 Φ 21 1 s E s 1 E s E Ew r w Er + 1 s 1 + E 1 r Es s E 1 1 w Es Ew s + 1 Φ E 1 r Es s E 1 1 r E 1s + 1 E w 1 E 2 s s s Φ 23 1 E Er 1 E E 1 r Es s E 1 1 Es E 1 1. s s At the same time let k 11 k 12 k 13 K Jφ k 21 k 22 k k 31 k 32 k 33 where k 11 s E 1 w w Es + r E 1 Er k 12 r 1 E + E 1 E 2 s k 13 Es s E 2 r 1 E 1 w k 21 s 1 + E 1 E + E 2 r k 22 s E 1 E + E 2 E 2 s k 23 s E 2 1w k 31 w 1 Er + E 2 s s E 1 k 32 w 1 E 2 s k 33 Er r E 1. It is easy to verify that K is a skew-symmetric operator i this way ad the positive hierarchy 10 is derived. It is easy to verify that the positive hierarchy has the discrete zero-curvature represetatio 9. Ad every solito equatio i 11 or the discrete Hamiltoia system 19 is a discrete Liouville itegrable system. The 19 become u tm r s w JΦ δhm δu tm J δh m+1 δu m+1 δh0 JΦ. δu Especially the lattice system 1 possess the Hamiltoia

5 No. 6 Itegrable Properties Associated with a Discrete Three-by-Three Matrix Spectral Problem 985 structure where u t H 1 r s w t m Z m δh1 JΦ δu s + w s s. 23 Now we show the Liouville itegrability of the Hamiltoia systems 19. It is ot difficult to verify that amely Hece {H m Hl } J JΦ JΦ Φ J JΦ. m δh δu Similarly we ca fid that It implies that Therefore ad {H l Hm m 1 δh1 Φ J δhl δu δu JΦ m 1 δh1 Φ l 1 δh1 δu l 2 δh1 Φ JΦ δu Φ m δh1 l 2 δh1 JΦ δu δu {H m+1 H l 1 } J {H m+1 1 H 1 } J. } J {H m+1 1 H 1 } J. {H m Hl } J {H l Hm } J. δu {H l Hm } J 0 m l 1 24 m δh H m t l u tl δu δh m δu {H m H l } J 0 m l 1. J δhl δu Moreover it is kow that if J is a Hamiltoia operator the [ J δf δu δg ] J δ{f G } J δu δu where the commutator is defied by [X Y ] : ε Xu + εy Y u + εx ε0. From 24 we have ] [J δhm δu J δhl J δ{hm H l } J 0 m l 0. δu δu Based o above discussio we ca obtai the followig propositio ad theorem. Propositio 1 The lattice hierarchy 11 has ifiitely may commo commutig symmetries {JδH m / δu } m1 ad ifiitely may coserved fuctioals {H m }. Theorem 1 The oliear lattice solito equatios i hierarchy 19 are all discrete Liouville itegrable. I particular the oliear lattice system 1 r t r w 1 w + s +1 s s t s w 2 w w t r +1 r or the discrete Hamiltoia equatio 21 is a discrete itegrable system i Liouville sese. 3 The Coservatio Laws I this sectio we would like to derive the coservatio laws of the lattice solito equatio 1 by a simple ad direct way. [8] From the 3 ad 9 we have ad φ 3 +1 r φ s φ λ + w φ 3 25a φ 3 t λr φ λs φ λ φ 3. 25b Assume that θ φ3 φ a direct calculatio gives lθ t E 1 λr θ 1 + λs θ 2 θ λ. 27 I virtue of 1 ad 25 we have Assume 1 r θ 1 θ + s θ 2 θ 1 θ + λ + w θ. 28 θ j1 θ j λ j 29 the expadig θ i the power series of 1/λ ad substitutig 29 ito 28 we have θ 1 1 θ 2 1 w θ 3 1 w2 r... From 25b ad 27 we have lθ t E 1λr θ 1 + λs θ 2 θ Substitutig 29 ito 30 1 k θ j+1 k λ t j k1 k1 j1 E 1 r θ j 1 λ j 1 + s k1 θ j 2 λ j 1 k1 θ j 1 λ j 31 Equatig the power of 1/λ i 31 we ca get a ifiite umber of coservatio laws for the lattice solito equatio 1. The first two coservatio laws are poited out as follows w t E 1s r 1 2 w2 E 1s w 1 + s w 2... t

6 986 LI Xi-Yue ad WANG Xi-Zeg Vol. 52 Similarly we ca get the coservatio laws of other lattice equatio i the hierarchy 11. I this paper we discuss a 3 3 discrete iso-spectral problem ad derive the correspodig family of discrete Hamiltoia system. It is show that the resultig lax itegrable lattice equatios are all liouville itegrable discrete Hamiltoia systems. Moreover ifiitely may coservatio laws for correspodig lattice solito equatio are give. It should be poited out that further ivestigatio for the higher order matrix discrete spectral problems will be paid more ad more attetio. I subsequet papers we shall study Darboux trasformatio ad the biary o-liearizatio of Lax pairs for the itegrable lattice equatio 1. Refereces [1] M. Ablowitz ad J. Ladik J. Math. Phys [2] G.Z. Tu J. Phys. A: Math. Ge [3] M. Toda J. Phys. Soc. Jp [4] W. Oevel H.W. Zhag ad B. Fuchssteier Prog. Theor. Phys [5] Y. Ohta ad R. Hirota J. Phys. Soc. Jp [6] Y.T. Wu ad X.G. Geg J. Phys. Soc. Jp [7] W.M. Li ad X.G. Geg Chi. Phys. Lett [8] D.J. Zhag ad D.Y. Che Chaos Solitos ad Fractals [9] W.X. Ma X.X. Xu ad Y.F. Zhag Phys. Lett. A [10] K.M. Tamizhmai ad W.X. Ma J. Phys. Soc. Jp [11] X.X. Xu Noliear Aalysis [12] X.Y. Li X.X. Xu ad Q.L. Zhao Phys. Lett. A [13] W.X. Ma J. Math. Phys [14] T. Tsuchida ad M. Wadati J. Phys. Soc. Jp [15] Y.P. Su D.Y. Che ad X.X. Xu J. Noliear Aal [16] W.X. Ma ad B. Fuchssteier J. Math. Phys [17] W.X. Ma ad X.X. Xu J. Phys. A: Math. Ge [18] W.X. Ma X.X. Xu ad Y.F. Zhag J. Math. Phys [19] W.X. Ma Phys. Lett. A [20] W.X. Ma J. Phys. A: Math. Ge