Research Article Algebro-Geometric Solutions for a Discrete Integrable Equation

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1 Hidawi Discrete Dyamics i ature ad Society Volume 27 Article ID pages Research Article Algebro-Geometric Solutios for a Discrete Itegrable Equatio Megshuag Tao ad Huahe Dog College of Mathematics ad Systems Sciece Shadog Uiversity of Sciece ad Techology Qigdao Chia Correspodece should be addressed to Huahe Dog; mathsdog@26.com Received 25 April 27; Revised 4 August 27; Accepted 24 October 27; Published 4 ovember 27 Academic Editor: Chris Goodrich Copyright 27 Megshuag Tao ad Huahe Dog. This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited. With the assistace of a Lie algebra whose elemet is a matrix we itroduce a discrete spectral problem. By meas of discrete zero curvature equatio we obtai a discrete itegrable hierarchy. Accordig to decompositio of the discrete systems the ew differetial-differece itegrable systems with two-potetial fuctios are derived. By costructig the Abel-Jacobi coordiates to straighte the cotiuous ad discrete flows the Riema theta fuctios are proposed. Based o the Riema theta fuctios the algebro-geometric solutios for the discrete itegrable systems are obtaied.. Itroductio As we all kow the geeratio of itegrable system determiatio of exact solutio ad the properties of the coservatio laws are becomig more ad more rich [ 5]; i particular the discrete itegrable systems have may applicatios i statistical physics quatum physics ad mathematical physics [6 ]. It is worth discussig the properties of discrete itegrable systems such as Darboux trasformatios [2 3] Hamiltoia structures [4 6] exact solutios [7] ad the trasformed ratioal fuctio method [8]. I the past decades some methods have bee proposed to gai explicit solutios of the cotiuous solito equatios for istace the algebro-geometric method [9 2] the iverse scatterig trasformatio [2] the Backlud trasformatio [22] ad the sie-cosie method [23]. However it is very hard to obtai algebro-geometric solutios for discrete solito equatios due to the treatmet of discrete variables. I 975 Its ad Matveev first preseted the algebro-geometric approaches [24] which permitted us to seek out a class of exact solutios to the solito equatios. The elliptic fuctios ad multisolito solutios may be acquired by these degeerated solutios [25]. Recetly Qiao et al. further improved the algebrogeometric methods by makig use of the oliearizatio theory [26 29]. Trigoal curves are also systematically used to costruct algebro-geometric solutios [3 3]. But we ote that there is few research to focus o the algebro-geometric solutios of discrete solito equatios. I this paper we will geerate the algebro-geometric solutios of the discrete itegrable system by takig advatage of the Riema-Jacobi iversio theorem ad Abel coordiates. I Sectios 2 ad 3 we will costruct a ew discrete itegrable system by usig Lie algebra ad spectral problem. By itroducig Abel-Jacobi coordiates straighteig out of the cotiuous ad discrete flows will be give ad placed i Sectio 4. Sectio 5 will be devoted to derive the algebro-geometric solutios of the abovemetioed discrete itegrable equatio by utilizig the Riema theta fuctio. 2. The Discrete Itegrable Hierarchy We cosider the algebra h =( ) h 2 =( ) e=( ) f=( ) ()

2 2 Discrete Dyamics i ature ad Society which is the simple subalgebra of the Lie algebras A ad correspodig loop algebras ca be expressed as h i () =h i λ e () =eλ f () =fλ Z. (2) we get λ (EA A) +s EB r C= r (EA A) +( λ +λr s )EB λ B= λ EC s (EA + A) ( λ +λr s )C= r EC ( λ +λr s ) (EA A) s B=. (8) Accordig to the loop algebras we itroduce the followig discrete spectral problems Eψ =U ψ dψ dt =V ψ U =h ( ) +h 2 ( ) +r s h 2 () +r e () +s f () V = Thus = [a (h ( 2) h 2 ( 2))+b e ( 2 + ) +c f ( 2 + )]. r U =( λ ) s λ +λr s V =( A B C A ) A= B= C= = = = a λ 2 b λ 2+ c λ 2+. Accordig to the followig statioary discrete zero curvature equatio for V (3) (4) (5) (6) (EV )U =U V (7) Substitutig (6) ito (8) yields Δa +s Eb + r c + = r s Eb + +r (Ea +a )+Δb = s r c + s (Ea +a )+Δc = r s Δa + +r Ec + s b + Δa = Δ=E. We choose the iitial values a = /2 b = ad eed to select zero costats for the iverse operatio of the differece operator Δ i computig a.o this coditio recursio relatios (9) uiquely determie a b c.theweobtaithefirstfewquatities From a = s r b = s c = r a 2 = s 2 r 2 r 2 s s s 2 r r b 2 = s 2 r s 2 r c 2 = r 2 s r 2... s Eψ =U ψ dψ dt =V (m) ψ we have the discrete zero curvature equatio V (m) = (9) () () du (EV (m) )U dt +U V (m) = (2) m m = ( a λ 2m 2 a m b λ 2m 2+ c λ 2m 2+ a λ 2m 2 +a m ). (3)

3 Discrete Dyamics i ature ad Society 3 Thus we obtai the followig itegrable discrete hierarchy Ad r tm =Eb m b m s tm =Ec m c m. r t = s s s t = r + r with m=.equatio(5)cabereadas t l r = r ( s s ) t l s = s ( r + r ). It is easy to fid that the Lax pair of (5) is give by Eφ () =U φ () (4) (5) (6) φ t () =V () φ () (7) V () 2 λ2 λ s =( r λ ). (8) λ2 2 I the followig we express Leard's gradiet sequeces S ( Z) by the recursio equatio with two operators J S + () =K S () J S () = K =( J =( S () = (s () =... Δ s (E+) Δ r (E+) ) r Δ s Δ r s Δ r s r s E r Δ s Δ r s Δ s(2) s(3) )T. ) (9) (2) From the equatio J S () = ad (9) respectively S () =( ) 2 r S () = ( ) s ( s r ) S () r 2 ( + ) s s = ( s 2 ( + ) ). r r ( + + ) ( s r s r s r s r ) Equatio (9) implies that Δs (2) +r (E+) s (3) +r s Es (2) + = Δs () s (E+) s (3) r s s () + = r Δs () +s Δs (2) r s Δs (3) =. (2) (22) The discrete itegrable hierarchical (4) could be rewritte as geeratio of the followig so spectrum problem is V (m) ψ (+) =U ψ () A (m) = B (m) = C (m) = ψ () tm =( A(m) C (m) =V (m) ψ () B (m) A (m) ) m s (3) λ2(m ) s (3) m = m s (2) λ2(m )+ = m s () λ2(m )+. = (23) (24) From the compatibility coditios of the discrete Lax pair (23) we ca read that the hierarchical equatio is r tm =Es (2) m s(2) m s tm =Es () m s() m. (25)

4 4 Discrete Dyamics i ature ad Society Thus we also have ( r ) =X m () =( s t m Δs (2) m Δs () m 3. Decompositio of the Differetial-Differece Equatios ). (26) I this sectio we shall resolve the discrete systems (6) ito solvable ordiary differetial equatios. We assume that (23) hastwobasicsolutiosψ() = (ψ () () ψ () ()) T φ() = (φ () () φ () ()) T ad we defie a Lax matrix W as follows: W = 2 (φ () ψ ()T +ψ() φ () T )σ f () g () =( h () f () ) σ=( ) ad W should meet the followig equatios: W + U U W = W tm =[V (m) W ]. (27) (28) Substitutig (3) ito (29) yields J G + =K G () J G () = K G () = G () = (h () g () f ()) T. It is evidet that (3) G =α S () (32) α is a costat. Actig with J K ad K J respectively o (32) yields Thus G () =α S () +α S (). (33) G k () =α S k () +α S k () + +α k+ S () (34) α α...α k+ are costats. Substitutig (34) ito the K G () = gives the followig discrete statioary equatio: α X () +α X () + +α X () =. (35) Accordig to (3) we have (α =) It is easy to see that (28) ca be writte as λ Δf () +s Eg () r h () = r Ef () + λ Δg () +λr s Eg () +r f () = λ Δh () s Ef () s f () λr s h () = r Eh () λ Δf () λr s Δf () s g () = f () tm g () tm h () tm =B (m) h () B (m) g () =2A (m) g () 2B(m) f () =2C (m) f () 2A (m) h () f () = g () = h () = + = + = + = f () λ 2(+ ) g () λ 2(+ )+ h () λ 2(+ )+. (29) (3) h () = r g () = s f () = h () = r 2 r s 2 α ( s + s )+α r g () = s 2 ( + )+α r r. s (36) Defie the elliptic coordiates μ ad ] by expressig g() ad h(): g () = λ (λ 2 μ s () 2 ) = λ ( λ s μ ()) h () = r λ = = (λ 2 ] () 2 ) r λ = ( λ ] ()) (37) we deote λ 2 μ () 2 ad] () 2 with λ μ () ad ] ()respectively.

5 Discrete Dyamics i ature ad Society 5 By comparig coefficiets of the same power for λweget g () = s = μ () h () = ] r () g 2 () = = s i< μ i () μ () h 2 () = ] r i () ] (). i< Equatio(38)caberewritteas s ( r + r ) α = r ( s + s ) α = by makig use of (37). Thus (6) ca be writte as t l r = t l s = = = = = ] () +α μ (+) +α. μ () ] () (38) (39) (4) Cosider the fuctio det W which is a (4+2)th-order polyomial i λ: det W =f 2 () +g() h () = 2 4 λ2 (λ 2 λ 2 ) = = 4 λ 2 = ( λ λ )= 4 R( λ). (4) Substitutig (3) ito (4) ad comparig coefficiets of the same powers of λ read ad deduce that α = 2 2 = f () λ= μk () = 2 R( μ k ()) λ (42) Hece it follows that g () t λ= μk () = s μ k () R( μ k ()) = μ s k () ( t μ k ()) ( μ k () μ i ()) i=i=k h () t λ= ]k () = r ] k () R( ] k ()) = r ] k () ( t ] k ()) i=i=k Agai from (37) ad (44) we have Thus ( ] k () ] i ()). t μ k () = R( μ k ()) i=i=k ( μ k () μ i ()) t ] k () = R( ] k ()) Similarly whe m=2 g () t2 λ= μk () i=i=k ( ] k () ] i ()). = [ μ s k () 3 s 2 ( + )μ r r k ()] R( μ k ()) = s μ k () ( t2 μ k ()) i=i=k ( μ k () μ i ()) h () t2 λ= ]k () =[ r ] k () 3 r 2 R( ] k ()) = r ] k () ( t2 ] k ()) i=i=k ( ] k () ] i ()). ( s + s ) ] k ()] (44) (45) (46) by takig m=. f () λ= ]k () = 2 R( ] k ()) g () t =2A () h () t =2C () g () 2B() f () f () 2A() h () (43) t2 μ k () R( μ k ()) t2 ] k () R( ] k ()) = μ k () = μ () α i=i=k ( μ k () μ i ()) = ] k () = ] () α i=i=k ( ] k () ] i ()). (47)

6 6 Discrete Dyamics i ature ad Society 4. Straighteig out of the Cotiuous ad Discrete Flows I order to acquire the algebro-geometric solutios of systems (6) we first itroduce the Riema surface Γ of the hyperelliptic curve with geus : Γ:ξ 2 =R( λ) R ( λ) = 2 = ( λ λ ) (48) which has two ifiite poits ad 2 otbrachpoit of Γ.Wefixasetofregularcyclepaths:a...a ; b...b which are idepedet ad have the itersectio umbers: ad the Able-Jacobi coordiates are defied as ρ () () = A ( ρ (2) () = A ( k= k= p( μ k ())) = p( ] k ())) = k= k= p( μ k ()) ω p p( ] k ()) ω p (56) a k a =b k b = a k b =δ k k. (49) p( μ k )=( λ = μ k () ξ = R( μ k ())) Γ p( ] k )=( λ = ] k () ξ = R( ] k ())) Γ (57) We choose the holomorphic differetials o Γ ad defie ω l = λl dλ l (5) R (λ) ad p isachosebasepoitoγ. The compoets of the Abel-Jacobi coordiates i (56) are A k = a ω i B k = b ω i A=(A i ) B = (B i ). Thus we deote the matrices C ad τ by (5) ρ () () = ρ (2) () = k= k= p( μ k ) ω = p p( ] k ) ω = p k= l= k= l= μ k () c λl d λ l λ(p ) R( λ) ] k () c λl d λ l λ(p ) R( λ) (58) C=(c k )=A τ=a B (52) ad verify that τ is symmetric ad has positive defied imagiary part. By ormalizig ω ito the ew basis ω which meets ω = a k ω = b k ω = l= l= l= c l ω l =2... (53) c l ω l = a k c l ω l = b k l= l= The Abel map A(p) is itroduced as c l A lk =δ k c l B lk =τ k. (54) p A (p) = ω (55) p λ(p ) is the local coordiate of p. We ifer that t ρ () () = t ρ (2) () = μ c k () l t μ k () l k= l= R( ] k ()) c l k= l= = = c Ω () μ k () l i=i=k ( μ k () μ i ()) ] c k () l t ] k () l k= l= R( ] k ()) c l k= l= = Ω (). ] k () l i=i=k ( ] k () ] i ()) =c (59)

7 Discrete Dyamics i ature ad Society 7 Similarly we have t2 ρ () () = μ k () l ( μ k () = c μ () α ) l i=i=k ( μ k () μ i ()) l= k= =Ω (2) t2 ρ (2) () ] k () l ( ] k () = = c ] () α ) l l= k= i=i=k ( ] k () ] i ()) = Ω (2). (6) Let the fudametal solutio matrix of (3) be of the form Q =(φ() φ ()) =( φ() () φ () () φ (2) () φ (2) () ) Q =I. It is easy to obtai that from which we have (6) Q + =U U U (62) φ () () = λ φ (2) () =s φ () () =r φ (2) () = λ +λr s φ () (2) = λ 2 +r s φ (2) (2) =s λ +s ( λ +λr s ) φ () (2) =r λ +( λ +λr s )r φ (2) (2) =s r +( λ +λr s )( λ +λr s ). (63) Suppose that δ is eigevalue of the Lax matrix W i the solutio space of equatio ψ( + ) = U ψ() whichis ivariat uder the actio of W due to (EW )U =U W. The correspodig eigefuctio is ψ() that ca be called the Baker fuctio which satisfies that ψ (+) =U ψ () W ψ () =δψ(). (64) It is easy to check that det δi W =δ2 f 2 () g() h () = (65) which has two eigevalues δ ± =±δ δ= f 2 () g() h () = 2 R( λ). (66) The correspodig Baker fuctio ca be take as φ ± () =φ() +b ± φ () φ ± () = φ () +c ± φ () b ± = or b ± = c ± = or c ± = ±δ f () g () h () ±δ + f () ±δ + f () h () g () ±δ f (). (67) (68) Let p ± ( λ) q ± ( λ) be the compoets of the Baker fuctios φ ± () ad φ ± () respectively.actuallystartig from Q + =U Q W Q =W U Q = =U U 2 U W Q we ca ifer that =Q W Q =Q W p + ( λ) p ( λ) = s λ μ () λ μ () s = p + ( λ) p ( λ) = r λ ] () λ ] (). Similarly we have r = Δρ () () =ρ () (+) ρ () () =Ω () (mod J) Δρ (2) () =ρ (2) (+) ρ (2) () =Ω () (mod J) Ω () = 2 ω. 5. Algebro-Geometric Solutios The well-kow Riema theta fuctio of Γ is defied by θ (ξ τ) = z Z exp (πi τz z + 2πi ξ z ) ξ=(ξ...ξ ) T ξ z = = ξ z. ξ C (69) (7) (7) (72)

8 8 Discrete Dyamics i ature ad Society Accordig to the Riema theorem there exists a costat M (i) C so that (i) F =θ(a(p) ρ () () M () ) has exactly zeros at λ = μ ()... μ (); (ii) F 2 =θ(a(p) ρ (2) () M (2) ) has exactly zeros at λ = ] ()... ] (). We have the iversio formula = = μ () k =I k (Γ) ] () k =I k (Γ) 2 s= 2 s= Res λ= s λk d l F ( λ) Res λ= s λk d l F 2 ( λ) (73) with the costat I k (Γ) = = a λk ω. Through a stadard treatmet we arrive at = = μ () k =I k (Γ) + t l θ(ρ() () +M () +π (2) ) θ(ρ () () +M () +π () ) ] () k =I k (Γ) + t l θ(ρ(2) () +M (2) +π () ) θ(ρ (2) () +M (2) +π (2) ) π (s) = p s ω. Substitutig (74) ito (4) yields Thus r = exp [ l θ(ρ(2) () +M (2) +π () ) θ(ρ (2) () +M (2) +π (2) ) 2 2 t λ = [ +I (Γ) t] ] s = exp [ l θ(ρ() (+) +M () +π (2) ) θ(ρ () (+) +M () +π () ) [ 2 2 t λ +I (Γ) t]. = ] r = exp [ l θ(ω() Ω () t+ Υ () ) θ(ω [ () Ω () t+ Υ (2) ) 2 2 t λ = +I (Γ) t] ] s = exp [ l θ(ω() (+) +Ω () t+υ (2) ) θ(ω () (+) +Ω () t+υ () ) [ 2 2 t λ +I (Γ) t] = ] (74) (75) (76) Υ () =ρ () +M () +π () Υ (2) =ρ () +M () +π (2) Υ () =ρ (2) +M (2) +π () Υ (2) =ρ (2) +M (2) +π (2). which is the algebro-geometric solutio to (6). (77) Remark. We have cocluded the algebro-geometric solutios of the discrete system (6). It is sigificace of a maor work for ivestigatig umerical solutios of the discrete itegrable system (6) like the way preseted i [32]. Comparig the umerical solutios ad algebro-geometric solutios about the discrete itegrable system we ca get lots of useful properties. These problems will be studied i the future. Coflicts of Iterest The authors declare that there are o coflicts of iterest regardig the publicatio of this paper. Ackowledgmets This work was supported by atioal atural Sciece Foudatio of Chia (o ad o. 7334) Ope Fud of the Key Laboratory of Ocea Circulatio ad Waves Chiese Academy of Scieces (o. KLOCAW4) ad the SDUST Research Fud (24TDJH2). Refereces [] X. B. Hu A powerful approach to geerate ew itegrable systems Joural of Physics A: Mathematical ad Geeral vol. 27o.7pp [2] X. Li ad Q. Zhao A ew itegrable symplectic map by the biary oliearizatio to the super AKS system Joural of Geometry ad Physicsvol.2pp [3] X.-Y. Li Q.-L. Zhao Y.-X. Li ad H.-H. Dog Biary bargma symmetry costrait associated with 3 3 discrete matrix spectral problem Joural of oliear Scieces ad Applicatiosvol.8o.5pp [4] X.-Y. Li Y.-X. Li ad H.-X. Yag Two families of Liouville itegrable lattice equatios Applied Mathematics ad Computatiovol.27o.2pp [5] H. Dog K. Zhao H. Yag ad Y. Li Geeralised (2 + )- dimesioal super MKdV hierarchy for itegrable systems i solito theory East Asia Joural o Applied Mathematicsvol. 5 o. 3 pp [6]X.WagH.DogadY.Li SomereductiosfromaLax itegrable system ad their Hamiltoia structures Applied Mathematics ad Computatiovol.28o.2pp [7] H. Yag B. Yi Y. Shi ad Q. Wag Forced ILW-Burgers equatio as a model for Rossby solitary waves geerated by topography i fiite depth fluids Joural of Applied Mathematics vol. 22 Article ID

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