Bidifferential graded algebras and integrable systems

Bidif ferential graded algebras and inte gra ble systems A R I S T O P H A N E S D I M A K I S Department of Financial and Management Engineering, Univ ersity of the Ae gean, 31 F ostini Str ., GR-82100 Chios, Greece E-mail: dimakis@ae gean.gr F O L K E RT M ¨ U L L E R - H O I S S E N Max-Planck-Institute for Dynamics and Self-Or ganization Bunsenstrasse 10, D-37073 G ¨ ottingen, Germany E-mail: folkert.mueller-hoissen@ds.mpg.de Abstract In the framework of bidifferential graded algebras, we present uni versal solution generating tech- niques for a wide class of integrable systems. 1 Introd uction Let A be a unital associati v e algebr a (ov er R or C ) with identity element I , and Ω( A ) = L r ≥ 0 Ω r ( A ) with Ω 0 ( A ) = A and A -bimodule s Ω r ( A ) , r = 1 , 2 , . . . . W e call (Ω( A ) , d , ¯ d) a bidif fer ential gra ded alg ebr a (BDGA ), or bidi f fer ential calcu lus , if (Ω( A ) , d) and (Ω( A ) , ¯ d) are both dif ferenti al graded algebr as, which means that Ω ( A ) is a graded algebra and the linear maps d , ¯ d : Ω r ( A ) → Ω r +1 ( A ) satisfy the graded Leibniz rule (anti deri v ati on prope rty) and d 2 = ¯ d 2 = 0 , d ¯ d + ¯ d d = 0 . (1.1) These conditions can be combined into d 2 z = 0 , where d z := ¯ d − z d with an in determin ate z . 1 In sectio n 2 we connect this structure with ‘inte grabl e’ partial dif ferent ial (or dif ferenc e) equatio ns [1–21]. Although this framew ork may not be a ble to cove r all po ssible (in some sens e) int egra ble e quatio ns, it has the adv antage o f admitting un iver sal tec hnique s for construct ing exa ct solu tions. Whereas previo us w ork concen trated on conserv ation laws and B ¨ acklun d transformat ions, the present work addresses Darboux transfo rmations and presents a ver y effecti ve ‘linea rizatio n ap proach ’, gene ralizin g res ults in [22] (see also [23, 24] for related ideas). After colle cting some basics in section 2, sectio n 3 addres ses uni v ersal soluti on ge neratin g techn iques. Section 4 then presents some exa mples. Section 5 cont ains fina l remarks. 2 Dr essing bidiffer ential graded algebras In the follo wing, (Ω( A ) , d , ¯ d) denotes a BDGA. Introdu cing ¯ D := ¯ d − A (2.1) 1 A generalization to an N -differential graded algebra is then obtained if d 2 z = 0 with d z = P N n =0 z n d n . But this will not be considered in this work. 1 with a 1-for m A , d and ¯ D satisfy again the BDGA relat ions iff ¯ d A − A A ≡ − ¯ D 2 = 0 and d A ≡ − (d ¯ D + ¯ D d) = 0 . (2.2) W e are interested in cases where these equations are equi v alent to a partial diff erentia l or di f ference equati on (or a fa mily of such equ ations) , which req uires th at A depends o n a set of independen t var iables and the diffe rential maps d , ¯ d in v olv e diffe rentia l or dif fer ence operators . As depicte d in the follo wing diagra m, w e can solv e either the first or the second equatio n. A = ( ¯ d g ) g − 1 d  ( ¯ d g ) g − 1  = 0 ¯ d A − A A = 0 d A = 0 ¯ d d φ = d φ d φ A = d φ ‘pseud odual ity’ This resul ts in two dif ferent equati ons that are related by a ‘Miura transfo rmation’ ( ¯ d g ) g − 1 = d φ , (2.3) and this relatio nship is sometimes referre d to as ‘pseud odual ity’. The condi tions (2.2) can be combined into D 2 z = 0 where D z = ¯ D − z d = d z − A . (2.4) Such a zero curvat ure cond ition is at the roots of the theory of integra ble systems. It is the integra bility condit ion of the linear equation D z W ( z ) = 0 . (2.5) T o g et some more informati on about this equatio n, let us deri ve it from ¯ d d φ = d φ d φ , (2.6) the equatio n in the lo wer left corner of the diagr am. Using (1.1), we write it as d[ ¯ d φ − (d φ ) φ ] = 0 . (2.7) W e shall assume that the first d -cohomolog y class vanis hes, so that d -closed 1-forms are d -exact. Then there is an element ψ ∈ A such that ¯ d φ − (d φ ) φ = d ψ . (2.8) Applying ¯ d , using (1.1), (2.6) and (2.8), we obtain d[ ¯ d ψ − (d φ ) ψ ] = 0 , ( 2.9) which in turn can be inte grated by introdu ction of a new po tential χ ∈ A , ¯ d ψ − (d φ ) ψ = d χ . (2.10) This proce dure can be iterated and yields the linear equatio n (2.5) w ith W ( z ) = I + X n ≥ 1 W n z − n , W 1 = − φ . (2.11) 2 Remark. A gauge tran sformati on of a BDGA (Ω ( A ) , d , ¯ d) is gi ven by d 7→ G d G − 1 = d ′ , ¯ d 7→ G ¯ d G − 1 = ¯ d ′ with an in ver tible map G : A → A . In case of (Ω( A ) , d , ¯ D) , choo sing G as multip licatio n by g − 1 , w e obtain the equiv alent BDGA (Ω( A ) , d ′ , ¯ D ′ ) with d ′ = d + g − 1 d g and ¯ D ′ = ¯ d , by use of A = ( ¯ d g ) g − 1 . Remark. W e can conside r a simultane ous dressing of d and ¯ d by introd ucing D = d − B in addition to ¯ D = ¯ d − A . Then (Ω( A ) , D , ¯ D) is a BDGA if f d B = B B , ¯ d A = A A , d A + ¯ d B = A B + B A . (2.12) Solving the first two condit ions by setting A = ( ¯ d g ) g − 1 , B = (d h ) h − 1 , the third (multiplied from the left by h − 1 , fro m the righ t by h ) becomes d[( ¯ d J ) J − 1 ] = 0 with J = h − 1 g . An equi v alen t form is ¯ d[(d J − 1 ) J ] = 0 . T his genera lizes Y ang’ s gauge in the case of the (anti-) self-dua l Y ang-Mills equati on (see also [25]). 3 Solution generating techniques 3.1 B ¨ acklund transformation (B T) An elementar y B T is gi ven by D ′ z = G ( z ) D z G ( z ) − 1 , D ′ z := d z − A ′ , (3.1) where G ( z ) = I + F z − 1 [6]. This is equi v alent to d F = A − A ′ , ¯ d F = A ′ F − F A . (3.2) Using A = d φ , the first equation can be integ rated, F = φ − φ ′ − C where d C = 0 , (3.3) and from the secon d equation w e obtain the elementar y B T ¯ d( φ ′ − φ + C ) = (d φ ′ ) ( φ ′ − φ + C ) − ( φ ′ − φ + C ) d φ . (3.4) Alternati vely , using A = ( ¯ d g ) g − 1 , the second of equati ons (3.2) is solve d by F = g ′ K g − 1 where ¯ d K = 0 , (3.5) and the first of equa tions (3.2) becomes d( g ′ K g − 1 ) = ( ¯ d g ′ ) g ′ − 1 − ( ¯ d g ) g − 1 , (3.6) which is the elementary BT for the pseudod ual equation . Using the M iura transf ormation (2.3), this can be inte grate d and yields φ ′ − φ + C = g ′ K g − 1 . (3.7) This equat ion connects the two elemen tary BTs. 3 3.2 Darboux transf ormation (DT) The linear system 2 ¯ d ψ = (d φ ) ψ + (d ψ ) ∆ , (3.8) has the follo wing inte grabili ty conditio n,  ¯ dd φ − (d φ ) 2  ψ − d  ψ ( ¯ d∆ − (d ∆) ∆)  = 0 , (3.9) which reduce s to (2.6) if ¯ d∆ = (d∆) ∆ . (3.10) Let θ be an in vertible solution of (3.8) with a solution ∆ ′ of (3.10), henc e ¯ d θ = (d φ ) θ + (d θ ) ∆ ′ . (3.11) As a conseq uence, ¯ d( θ ∆ ′ θ − 1 ) = (d φ ′ ) θ ∆ ′ θ − 1 − θ ∆ ′ θ − 1 d φ , (3.12) where φ ′ := φ + θ ∆ ′ θ − 1 − C ′ with d C ′ = 0 . (3.13) This is in accordance with (3.4 ), i.e. φ ′ is related to φ by an elementary BT . Hence, an y solution φ of (2.6) and any in vertible solut ion θ of the linear equation (3.11) determin e a ne w solution φ ′ of (2.6) via (3.13). This is an abstracti on of what is kno wn as a Darboux transformat ion (see e.g. [26]). Introducing ψ ′ = ( ψ ∆ − θ ∆ ′ θ − 1 ψ ) M , (3.14) where M satisfies ¯ d M = (d M ) ∆ , [∆ , M ] = 0 , (3.15) it follo ws that ψ ′ satisfies (3.8) with φ replaced by φ ′ , i.e. ¯ d ψ ′ = (d φ ′ ) ψ ′ + (d ψ ′ ) ∆ . (3.16) No w we can iterate this procedure . Let θ k , k = 1 , . . . , n , be in v ertibl e solu tions of ¯ d θ k = (d φ ) θ k + (d θ k ) ∆ k , and M k satisfy (3.15) with ∆ k . Set ψ [1] = ψ , θ [1] = θ 1 , ψ [ k +1] = ( ψ [ k ] ∆ − θ [ k ] ∆ k θ − 1 [ k ] ψ [ k ] ) M with θ [ k ] = ψ [ k ]    ψ → θ k , ∆ → ∆ k , M→M k (3.17) Then ψ [ n +1] satisfies ¯ d ψ [ n +1] = (d φ [ n +1] ) ψ [ n +1] + (d ψ [ n +1] ) ∆ with the fol lo wing solution of (2.6), φ [ n +1] = φ + n X k =1 ( θ [ k ] ∆ k θ − 1 [ k ] − C k ) . (3.18 ) If ψ [ n +1] is in vertible , then it solves (3.20) belo w (see the next subsec tion). 2 Instead of (3.8), we may consider ¯ d ψ = (d φ ) ψ + d( ψ ∆) , which results from (2.10) by setting χ = ψ ∆ . In this case we have to impose ¯ d∆ = ∆ d∆ in order to obtain (2.6) as integrab ility con dition. Some of the follo wing formulae, also in section 3.4, then hav e to be modified accordingly . One can prov e that the two possibilities are in fact equi v alent. 4 3.3 Modified Miura transf ormation If ψ in (3.8) is in vertible, w e ha v e [ ¯ d g − (d g ) ∆ ] g − 1 = d φ , (3.19) writing g instead of ψ . The integ rabilit y condition is d([ ¯ d g − (d g ) ∆ ] g − 1 ) = 0 , (3.20) a m odified pseudod ual of (2.6), related b y the modifi ed Miu ra tran sformatio n (3.19). 3 (3.20) correspond s to A = [ ¯ d g − (d g ) ∆ ] g − 1 , (3.21) which reduces the two equat ions (2.2) to a singl e one since ¯ d A − A A = (d A ) g ∆ g − 1 . W e note that (3.19) is equi v ale nt to ¯ d g − 1 + g − 1 d φ ′ = d(∆ g − 1 ) where φ ′ = φ + g ∆ g − 1 . (3.22) 3.4 Binary Darboux transf ormation (bDT) (2.6) is also inte grab ility conditio n of ¯ d ˜ ψ = − ˜ ψ d φ + ˜ ∆ d ˜ ψ where ¯ d ˜ ∆ = ˜ ∆ d ˜ ∆ . (3.23) Combining this with (3.8), we get ¯ d( ˜ ψ ψ ) = ˜ ∆ (d ˜ ψ ) ψ + ˜ ψ (d ψ ) ∆ . (3.24) Introd ucing Ω( ˜ ψ , ψ ) via ˜ ∆ Ω( ˜ ψ , ψ ) − Ω( ˜ ψ , ψ ) ∆ = ˜ ψ ψ , (3.25) the pre viou s equation is satisfied if ¯ dΩ( ˜ ψ , ψ ) = (dΩ( ˜ ψ , ψ )) ∆ − (d ˜ ∆) Ω( ˜ ψ , ψ ) + (d ˜ ψ ) ψ . (3.26) No w let Θ = ( θ 1 , . . . , θ N ) and ˜ Θ = ( ˜ θ 1 , . . . , ˜ θ N ) T be solutio ns of ¯ dΘ = (d φ ) Θ + (dΘ) ∆ , ¯ d ˜ Θ = − ˜ Θ d φ + ˜ ∆ d ˜ Θ , (3.27) with matrices ∆ , ˜ ∆ where ¯ d ∆ = (d ∆ ) ∆ , ¯ d ˜ ∆ = ˜ ∆ d ˜ ∆ , and Ω a matrix such that ˜ ∆ Ω − Ω ∆ = ˜ Θ Θ and ¯ d Ω = (d Ω ) ∆ − (d ˜ ∆ ) Ω + (d ˜ Θ) Θ (3.28) (or equi v ale ntly ¯ d Ω = ˜ ∆ d Ω − Ω d ∆ − ˜ Θ dΘ ) holds. 4 It follo ws that Θ ′ = Θ Ω − 1 ˜ N − 1 , ˜ Θ ′ = N − 1 Ω − 1 ˜ Θ , (3.29) satisfy (3.27) with ∆ and ˜ ∆ exchang ed and φ replaced by φ ′ = φ − Θ Ω − 1 ˜ Θ = φ − Θ ′ ˜ N ˜ Θ = φ − Θ N ˜ Θ ′ , (3.30) 3 If ∆ = λ I with λ ∈ C , the mod ification can be absorbed by a redefinition ¯ d ′ := ¯ d − λ d of ¯ d . 4 W e note that the first of equations (3.28) is a rank one condition. 5 if the matrices N , ˜ N are in v ertible and s atisfy [ ∆ , N ] = [ ˜ ∆ , ˜ N ] = 0 , ¯ d N = d( ∆ N ) and ¯ d ˜ N = d( ˜ N ˜ ∆ ) . In particu lar , φ ′ is again a sol ution of (2.6). If ψ and ˜ ψ are solutions of (3.8) and (3.23), respecti v ely , and if ω , ˜ ω satisfy ˜ ∆ ω − ω ∆ = ˜ ψ Θ , ˜ ∆ ˜ ω − ˜ ω ∆ = ˜ Θ ψ , (3.31) ¯ d ω = (d ω ) ∆ − (d ˜ ∆) ω + (d ˜ ψ ) Θ = ˜ ∆ d ω − ω d ∆ − ˜ ψ dΘ , (3.32) ¯ d ˜ ω = (d ˜ ω ) ∆ − (d ˜ ∆ ) ˜ ω + (d ˜ Θ) ψ = ˜ ∆ d ˜ ω − ˜ ω d∆ − ˜ Θ d ψ , (3.33) then one ve rifies by direct calcula tion that ψ ′ = ψ − Θ ′ ˜ N ˜ ω = ψ − Θ Ω − 1 ˜ ω , ˜ ψ ′ = ˜ ψ − ω N ˜ Θ ′ = ˜ ψ − ω Ω − 1 ˜ Θ , (3.34) satisfy again (3.8), res pecti vely (3.23), with φ rep laced by φ ′ defined abov e. If ψ is in vertib le, it is a soluti on of (3.20) and then also ψ ′ , if in ve rtible. Corres pondin gly , ˜ ψ − 1 and then als o ˜ ψ ′− 1 solv es d([ ¯ d g − d( g ˜ ∆)] g − 1 ) = 0 . 3.5 A linearization appr oach Let us cons ider (2.8 ) in the form ¯ dΦ = (dΦ ) Q Φ + dΨ , (3.35) where d Q = 0 . The reason for th e introductio n of Q w ill be giv en belo w . Setting Ψ = Φ R w ith a d -constan t R , this becomes ¯ dΦ = (dΦ)( Q Φ + R ) . (3.36) Next we e xpres s Φ as Φ = Y X − 1 , (3 .37) and impose the const raint RX + QY = X P (3.38 ) with some P . Multipl ying (3.36) by X from the right, leads to ¯ d Y − Φ ¯ d X = (d Y ) P − Φ (d X ) P , (3.39) which is a conse quenc e of the two linea r equations ¯ d Y = (d Y ) P , ¯ d X = (d X ) P . (3.40) The follo w ing theorem is now eas ily verified. 5 Theor em 3.1 L et X , Y solv e the linear equatio ns (3.40) and the constrain t (3.38) with d -constant R and some P , and let X be in vertible . Then Φ = Y X − 1 solves ¯ d d Φ = dΦ Q d Φ . (3.41) 5 The proof of the theorem does not use ¯ d 2 = 0 . 6 Let Φ take valu es in the algebra of M × N matr ices over A . The other object s abov e are then also matrices with appropri ate dimens ions. If Q has r ank one ov er A , i.e. Q = V U T with ( d - and ¯ d -) consta nt vec tors U, V havi ng entries in A , then φ = U T Φ V (3.42) solv es (2 .6) if Φ solv es ( 3.41). T he abo v e theo rem pro vid es us with a method to construc t exact solutions of the nonlinear equ ation (3.41 ) from so lution s of linear equations, and the last arg ument sho ws ho w these ge nerate e xact sol utions of (2.6). Since M and N can be cho sen freely , in this way we o btain ex act soluti ons of (2.6) in v olv ing an arbitrarily larg e number of parameters. This partly explain s the existe nce of infinite familie s of solutions like multi-sol itons. More gen erally , if Q = V U T with con stant M × m matrix U and N × m matrix V , then φ = U T Φ V solv es (2.6) in the algebra of m × m matrices (with entrie s in A ) if Φ solves (3.41). A some what weak er version of the theore m is obtain ed by ext ending (3.38) to H Z = Z P where Z =  X Y  , H =  R Q S L  , (3.43) with constant matrices L and S . This imposes the additi onal equation S X + LY = Y P on X and Y . T ogether with (3.38) it implies the algebraic Riccati equation S + L Φ − Φ R − Φ Q Φ = 0 . (3.44) The two equ ations (3.40) combine to ¯ d Z = d Z P . (3.45) The equati ons for Z are form-in varian t (with the same P ) under a transfor mation Z = Γ Z ′ , H = Γ H ′ Γ − 1 , (3 .46) with a ( d - and ¯ d -) cons tant matrix Γ . Such a trans formation relate s solutions of two versio ns of (3.41) corres pondin g to two diff erent Q ’ s. One can theref ore use the theorem w ith a simple form of H (our choice of H ′ ) and t hen app ly a trans formatio n to gene rate a sol ution associa ted with a more co mplicate d choice of H . C hoosin g H ′ =  R 0 0 L  or H ′ =  L I N 0 L  , and Γ =  I N − K 0 I M  (3.47) ( I N is the N × N unit matrix, M = N in th e second case), yields the next resu lt. Cor ollary 3.1 Let X ′ and Y ′ solve ¯ d X ′ = (d X ′ ) P , ¯ d Y ′ = (d Y ′ ) P , L Y ′ = Y ′ P (3.48) and R X ′ = X ′ P r especti vely L X ′ + Y ′ = X ′ P . (3.49) Then Φ = Y ′ ( X ′ − K Y ′ ) − 1 with a constan t matrix K solves (3.41) with Q = R K − K L r espec tively Q = I + [ L, K ] . (3.50) 7 Remark. Solution s of the modified pseudod ual equatio n (3.20) are obta ined as follo ws. Let G be an m × N matrix solution of ¯ d( GX ) = d( GX ) P and ( GX ) P − ∆ ( G X ) = C X , (3.51) where C and ∆ satisfy d C = 0 and ¯ d∆ = (d∆) ∆ . Requiring (3.37), (3.38) and (3.40), one finds that G solv es ¯ d G + G Q dΦ = d (∆ G ) . (3.52) W ith Q as specified abo ve, it follo ws that g = ( GV ) − 1 , pro vided the in verse exis ts, solves (3.22) with φ ′ = U T Φ V . As a conse quence , g solve s (3.20). Remark. A rela tion with the bDT is est ablish ed as f ollo ws. If th e alg ebraic Riccat i equ ation (3 .44) hold s, assuming d -constant L and R we hav e in addition to (3.36) also ¯ dΦ = ( L − Φ Q ) d Φ . Now let Q = V U T . Setting Θ = U T Φ and ˜ Θ = Φ V , we obtain (3.27) with ∆ = R and ˜ ∆ = L . W ith the identification Ω = Φ , we find that (3.28) holds if S = 0 . Furthermore , (3.30 ) yields φ ′ = 0 . 4 Examples In some example s presented belo w , the graded algebra will be take n of the form Ω ( A ) = A ⊗ C V ( C n ) where V ( C n ) is the ext erior algebra of C n . It is then suf ficient to define the m aps d and ¯ d on A . They ext end to Ω( A ) in an obvio us way , treating elements of V ( C n ) as constan ts. ξ 1 , . . . , ξ n denote s a basis of V 1 ( C n ) . 4.1 Self-dual Y ang-Mills (sdYM) equation Let A be the alg ebra of smooth complex fun ctions of comple x varia bles y , z and their comple x conjuga tes ¯ y , ¯ z . L et d f = ∓ f y ξ 1 + f z ξ 2 , ¯ d f = f ¯ z ξ 1 + f ¯ y ξ 2 (4.1) for f ∈ A . This deter mines a BDGA. Then (2.6), fo r an m × m matrix φ with e ntries in A , is equ i v alent to φ ¯ y y ± φ ¯ z z + [ φ y , φ z ] = 0 , (4.2) which is a potential form of the (Euclidean or split signatur e) sdYM equati on (see e.g. [27]). Writing J instead of g , the Miur a tran sformatio n (2.3) becomes J ¯ y J − 1 = φ z and J ¯ z J − 1 = ∓ φ y , and the pseud odual of (4.2) tak es the form ( J ¯ y J − 1 ) y ± ( J ¯ z J − 1 ) z = 0 , ( 4.3) which is anoth er well-kno wn potential form of the sdYM equatio n. T urn ing to DTs, we choose ∆ , ˜ ∆ constant. T hen (3.8) reads ψ ¯ z = ∓ ( φ y ψ + ψ y ∆) = J ¯ z J − 1 ψ ∓ ψ y ∆ , (4.4) ψ ¯ y = φ z ψ + ψ z ∆ = J ¯ y J − 1 ψ + ψ z ∆ . (4.5) For an in vert ible solution θ of this sys tem, a new solution is gi ve n by (3.13), r espect i vel y J ′ = ( θ ∆ θ − 1 − C ′ ) J K − 1 via (3.7). (3.23) becomes ˜ ψ ¯ z = ± ( ˜ ψ φ y − ˜ ∆ ˜ ψ y ) = − ˜ ψ J ¯ z J − 1 ∓ ˜ ∆ ˜ ψ y , (4.6) ˜ ψ ¯ y = − ˜ ψ φ z + ˜ ∆ ˜ ψ z = − ˜ ψ J ¯ y J − 1 + ˜ ∆ ˜ ψ z . (4.7) 8 (3.25) reads ˜ ∆ Ω − Ω ∆ = ˜ ψ ψ and (3.26) takes the form Ω ¯ z = ∓ (Ω y ∆ + ˜ ψ y ψ ) , Ω ¯ y = Ω z ∆ + ˜ ψ z ψ . (4 .8) In this way one recov ers correspondi ng formulae in [28]. Corollar y 3.1 pro vides a more easily applied constr uction of exact solutions (see also [22]). 4.2 Pseudodual chiral model hierar chy Let M be a space with coordina tes x 1 , x 2 , . . . . On smooth function s on M we define d f = X n ≥ 1 f x n d x n , ¯ d f = X n ≥ 1 f x n +1 d x n . (4.9) Hence d is t he or dinary ext erior deri vati v e. Let A be th e alge bra of m × m matric es of smoot h functi ons and Ω( A ) = A ⊗ C ∞ ( M ) V ( M ) , where V ( M ) is the algeb ra of differ ential forms on M . T his det ermines a BDGA (Ω( A ) , d , ¯ d) and (2.6) reproduc es the hierarc hy of the g l ( m, C ) ‘pseudod ual chiral model’ in 2 + 1 dimens ions, φ x n +1 ,x m − φ x m +1 ,x n = [ φ x n , φ x m ] , m, n = 1 , 2 , . . . . (4.10) The fi rst (non-tri via l) equatio n is a well-kno wn reducti on of the sdYM equation. φ can be restricted to any Lie subalg ebra of g l ( m, C ) , but corres pondin g condition s then ha v e to be imposed on the solutio n genera ting methods. The method of section 3.5 has been applied in [22]. In the su ( m ) case, a varian t of coroll ary 3.1 has been used in partic ular to constru ct multiple lump solutio ns. 4.3 The potential KP (pKP) equation On smooth funct ions of x, y , t we define d f = [ ∂ x , f ] ξ 1 + 1 2 [ ∂ y + ∂ 2 x , f ] ξ 2 , ¯ d f = 1 2 [ ∂ y − ∂ 2 x , f ] ξ 1 + 1 3 [ ∂ t − ∂ 3 x , f ] ξ 2 . (4.11) Besides smooth fun ctions of x, y , t with v alues in some asso ciati v e algebra , A must also conta in powers of the partia l deri v ati ve operato r ∂ x . T hen (2.6) beco mes the (noncommutati v e) pKP equ ation, and (3.20) with ∆ = − ∂ x the (noncommutat i ve) mKP equation. Concerning DTs, we set ∆ = ∆ ′ = ˜ ∆ = C ′ = − ∂ x , an d M = N = ˜ N = I . Then (3.13) and (3.14) take the form φ ′ = φ + θ x θ − 1 and ψ ′ = ψ x − θ x θ − 1 ψ , respecti v ely . Equat ion (3.8) becomes ψ y = ψ xx + 2 φ x ψ , ψ t = ψ xxx + 3 φ x ψ x + 3 2 ( φ y + φ xx ) ψ , (4.12) a familiar Lax pair for the pKP equa tion. The same equations, with ψ replaced by g , are obtained from (3.19), which means that the DT ψ 7→ ψ ′ acts on mKP solut ions. T urnin g to bDTs, (3.23) reads ˜ ψ y = − ˜ ψ xx − 2 ˜ ψ φ x , ˜ ψ t = ˜ ψ xxx + 3 ˜ ψ x φ x − 3 2 ˜ ψ ( φ y − φ xx ) . (4.13) (3.25) become s Ω x = − ˜ ψ ψ , and (3.26) yields Ω y = ˜ ψ x ψ − ˜ ψ ψ x , Ω t = − ˜ ψ xx ψ + ˜ ψ x ψ x − ˜ ψ ψ xx − 3 ˜ ψ φ x ψ . (4.14) These are well-kn o wn formulae, see e.g. [26, 29]. The ξ 1 -part of (3.45) is Z y − Z xx = 2 Z x ( P + ∂ x ) . Choosing P = − I N ∂ x , this is the heat equ ation Z y = Z xx . The ξ 2 -part of (3.45) then becomes the second heat hierarchy equation , Z t = Z xxx . Setting R = ˜ R − I N ∂ x in (3.38), turns it into X x = ˜ R X + Q Y . (4.15) No w theor em 3.1 e xpress es a res ult for the pKP equation [3 0, 31] that e xtends to the whole pKP hierarchy , see the ne xt subsect ion. 9 4.4 Kadomtsev-P etviashvili hierar ch y On smooth funct ions of v ariable s x and t = ( t 1 , t 2 , . . . ) we define d f = [ E λ , f ] ξ 1 + [ E µ , f ] ξ 2 , ¯ d f = [( λ − 1 − ∂ x ) E λ , f ] ξ 1 + [( µ − 1 − ∂ x ) E µ , f ] ξ 2 (4.16) where E λ is the Miwa shift operat or with an indete rminate λ , i.e. E λ f = f [ λ ] E λ where f ± [ λ ] ( x, t ) = f ( x, t ± [ λ ]) with [ λ ] = ( λ, λ 2 / 2 , λ 3 / 3 , . . . ) . F urthermor e, ∂ x is the partial deri v ati ve operat or with respec t to x . L et A contain the algebra of m × m m atrices of smooth functions . The abov e expressi ons for d f , ¯ d f requ ire that A als o contains the Miwa shift operat ors and powers of ∂ x . (2.6) is equiv alent to the follo wing functio nal representatio n [32, 33] of the (matrix) poten tial KP hierarc hy , ( φ − [ λ ] − φ − [ µ ] ) x = ( µ − 1 − φ + φ − [ µ ] ) − [ λ ] ( λ − 1 − φ + φ − [ λ ] ) − ( λ − 1 − φ + φ − [ λ ] ) − [ µ ] ( µ − 1 − φ + φ − [ µ ] ) . (4.17) In parti cular , φ t 1 = φ x . The linear system (3.40), which is (3.45), takes the form ( Z − Z − [ λ ] )( P + ∂ x − λ − 1 ) + Z x = 0 . (4.18) Choosin g P = − I N ∂ x and applyi ng a Miwa shift , this reduces to λ − 1 ( Z − Z − [ λ ] ) = Z x , (4 .19) which is the linear heat hierarchy Z t n = ∂ n x ( Z ) , n = 2 , 3 , . . . . C hoosin g moreov er R = ˜ R − I N ∂ x , (3.38) takes the form (4.15). Now theorem 3.1 reproduces theor em 4.1 in [30]. S ee also [31, 34, 35] for exa ct solution s obtained in this way . 4.5 2-dimensional T oda lattice (2dTL ) equation On smooth funct ions of x, y and an addit ional discrete variab le, w e set d f = [Λ , f ] ξ 1 + [ ∂ y , f ] ξ 2 , ¯ d f = [ ∂ x , f ] ξ 1 − [Λ − 1 , f ] ξ 2 , (4.20) where Λ is the shift operator in the discrete v ariable . A must also contain powers of Λ . Now (2.6) leads to the nonco mmutati v e 2dTL equation ˜ φ xy = ( ˜ φ + − ˜ φ )( I + ˜ φ y ) − ( I + ˜ φ y )( ˜ φ − ˜ φ − ) where ˜ φ := φ Λ (4.21) and ˜ φ + = Λ( ˜ φ ) , ˜ φ − = Λ − 1 ( ˜ φ ) . For a commutat i ve algebra A , this tak es the form (log(1 + v )) xy = v + − 2 v + v − (4.22) in terms of v = ˜ φ y . Choosin g ∆ = ν Λ − 1 with a cons tant ν , (3.20) tak es the form ( g x g − 1 ) y = g + g − 1 ( I − ν g y g − 1 ) − ( I − ν g y g − 1 ) g ( g − ) − 1 . (4.23) If g = e q with a scala r function q , this is the modified T oda equation [36] q xy = (1 − ν q y )( e q + − q − e q − q − ) . (4.24) Inspec tion of (3.8) suggest s ∆ = ν Λ − 1 , w hich turns it into ψ x = ( ˜ φ + − ˜ φ ) ψ + ν ( ψ + − ψ ) and ν ψ y = − ˜ φ y ψ − + ( ψ − ψ − ) . Setting M = M ′ = ∆ ′ = Λ − 1 and C ′ = 0 in secti on 3.2, (3.13) and 10 (3.14) yield the familiar DT ˜ φ ′ = ˜ φ + θ ( θ − ) − 1 , ψ ′ = ν ψ − θ ( θ − ) − 1 ψ − . For the itera ted t ransfor mation (with ∆ k = M k = Λ − 1 ), we obtai n in quasideter minant notation (see also [37, 38]) ˜ φ [ N +1] = ˜ φ + N X k =1 θ [ k ] ( θ − [ k ] ) − 1 where θ [ k +1] =           θ 1 · · · θ k +1 θ − 1 · · · θ − k +1 . . . . . . . . . θ ( k − ) 1 · · · θ ( k − ) k +1           , ψ [ N +1] =            θ 1 · · · θ N ν N ψ θ − 1 · · · θ − N ν N − 1 ψ − . . . . . . . . . . . . θ ( N − ) 1 · · · θ ( N − ) N ψ ( N − )            . (4.25) If ψ [ N +1] is in vertible , then it solv es the no ncommutati ve (or non-Abelian ) modified 2dTL eq uation (4.23), and for ν = 0 it solv es the ordinar y non commutati ve 2dTL eq uation . In a similar way , one reco ver s the bDT in [38]. In the approach of section 3.5, setting X = Λ ˜ X , R = ˜ R Λ − 1 , P = Λ − 1 , we ha ve ˜ Φ = Φ Λ = Y ˜ X − 1 , ˜ Z x = ˜ Z + − ˜ Z , ˜ Z y = ˜ Z − − ˜ Z , ˜ Z =  ˜ X Y  , (4.26) and ˜ R ˜ X + Q Y = ˜ X + . Extending the last relation to H ˜ Z = ˜ Z + , we obtain ˜ Z n = e x ( H − I N + M )+ y ( H − 1 − I N + M ) H n ˜ Z 0 , (4.27) assuming H in vertible . Using (3 .46) with (3.47), a nd impo sing rank( Q ) = m on Q then giv en by (3.50), one can no w compute expl icit solutions of (4.21) in the algebra of m × m matrices . 4.6 Lotka-V olterra (L V ) lattice equation Let d f = [Λ 2 , f ] ξ 1 + [Λ , f ] ξ 2 , ¯ d f = ˙ f ξ 1 + [Λ − 1 , f ] ξ 2 , (4.28) with the shift operato r Λ and ˙ f = f t . Intro ducing a = ϕ + − ϕ − I where ϕ = φ Λ 2 , (2.6) becomes the (nonco mmutati v e) L V lattic e equation ˙ a = a + a − a a − . (4.2 9) In terms of b = g + g − 1 and c = ˙ g g − 1 , the modified Miura transf ormation (3.19 ) with ∆ = λ Λ − 2 reads a = − b − − λ ( b b − − b − ) , c = 2 − λ + ( λ − 1)( b + b − ) − λ bb − . (4.30) Since ˙ b = c + b − b c by definition of b an d c , we obtain ˙ b = ( λ − 1)( b + b − b b − ) − λ ( b + b 2 − b 2 b − ) , (4.31) a nonco mmutati v e version of the modified L V lattice [39]. For a DT , appro priate choice s are ∆ = λ Λ − 2 and M = − ∆ − 1 . In the approac h of section 3.5, we set X = Λ 2 ˜ X , R = ˜ R Λ − 2 , P = − ˜ P − 1 Λ − 2 , (4.32) 11 with const ant ˜ P . Then (3.38) tak es the form ˜ R ˜ X + Q Y = − ˜ X ++ ˜ P − 1 , (4.33) and (3.40) leads to ˙ ˜ X = − ( ˜ X ++ − ˜ X ) ˜ P − 1 , ( ˜ X + − ˜ X ) = ( ˜ X + − ˜ X ) − ˜ P , (4.34) and the same equat ions for Y . T hese equat ions are solved by ˜ X n = A + B ˜ P n e t ( ˜ P − 1 − ˜ P ) , Y n = C + D ˜ P n e t ( ˜ P − 1 − ˜ P ) , (4 .35) with const ant matrices A, B , C, D . Inserting this in (4.33), we find the constrai nts QC + ˜ RA = − A ˜ P − 1 , QD + ˜ RB = − B ˜ P . (4.36) If Q = V U t , then via ϕ n = U t Y n ˜ X − 1 n V and a n = ϕ n +1 − ϕ n − 1 we obtai n solutions of the scalar L V lattice equati on. An exten sion of (4.33) in the sense of (3.43) turns out to be too restricti ve. 5 Final remarks Any BDGA formula tion of an inte grable system p rov ides us with a Lax pair which is linear in the spectr al parameter , a situation well kno wn from the (anti-) self-dua l Y ang-Mil ls system. The restriction to a special form of Lax pairs is what enabled us to work out calculatio ns, that appear ed in the literature for specific integr able systems, in a univ ersal way . 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