The Integration Algorithm of Lax equation for both Generic Lax matrices and Generic Initial Conditions

The In tegration Algorithm of Lax equation for b oth Generic Lax matric es and Generi c Initi al Cond itions Wissam Chemissany a , Pietr o F r ´ e b and Alexa nder S. Sorin c a University of L et hbridg e, Physics Dept., L ethbridge A lb erta, Canada T1K 3M4 wissam.chem issany@uleth.ca b Italian Emb assy in the R ussian F e der ation, Denezhny Per eulok, 5, 12 1 0 02 Mosc ow, Russia pietro.fre@ esteri.it and Dip a rtime nto di Fisic a T e oric a, Universit´ a di T orino, & INFN - Sezion e di T orino via P. Giuria 1, I-10125 T orino, Italy fre@to.infn .it c Bo go liub ov L a b or atory of The or e tic al Physics, Joint Institute for Nucle ar R ese ar ch, 141980 Dubna, Mosc ow R e gion, Russia sorin@theor .jinr.ru Abstract Sev eral physic al applications of Lax equation require its general solution for generic La x matric es and generic not necessarily diago nalizable initial conditions. In the presen t pap er w e complete the analysis started in [arXiv:0903. 3771 ] o n th e in tegration of Lax equations with b oth generic Lax op er ators and generic initial conditions. W e presen t a complete general in tegration formula holding tru e for an y (d iagonaliz able or non diagonaliza ble) initial Lax matrix and giv e an original rigorous mathematical pro of of its v alidity relying on no previously published results. † This w or k is supported in part by the Italia n Ministry o f Univ ersity (MIUR) under contracts PRIN 2007- 0240 4 5. F urthermore the work of A.S. was partially s uppo rted b y the RFBR Grants No. 0 9-02- 12417 -ofi m , 09-02 -0072 5-a, 09-02 -9134 9-NNIO a; DF G grant No 43 6 R US/113/ 669, and the Heisen b erg- Landau Progr am. W.C. is suppo rted in part b y the Natural Sciences and E ngineering Research Council (NSER C) of Cana da. 1 In tro duction Lax equation app ears in a v ariet y o f phy sical-mathematical problems whic h turn out to constitute integrable dynamical systems. An in tegration algorithm for generic matrix La x equation w as o r iginally deriv ed in the mathematical literature in [1], [2] and it was applied in the con text of sup ergr a vit y to cosmic billiards in [3], [4], [5] a nd to blac k-holes in [6], [7], [8] on the basis of their 3 D description pioneered in [9]. The la tter application to the case of blac k-holes rev ealed that the in tegration algorithm of [1], [2] did not co ve r the case of non-diagonalizable initial conditions. Suc h a situation o ccurs in particular when the initial Lax op erator at time t = 0 is nilp otent and this case is extremely r elev an t for Ph ysics since it corresp onds to extremal blac k-ho les [9]. An extension of the in tegration algorithm to the case of nilp oten t op erators was presen ted in [8]. In the app endix of that pap er it w as also conjectured a general fo rm ula, whic h w as v erified for some nontrivial cases, that prov ides the in tegrat io n of Lax equation for completely g eneric initial data. In the presen t pap er w e recall the general formula o f [8] a nd we provide t he rig orous mathematical pro o f that indeed it solve s Lax equation with a r bitrary initial conditions (diagonalizable or non diagonalizable). Our pro of is completely original and independen t from the algorithm discussed in [1],[2]. 2 The integration algorit hm for g eneric Lax matrices and generic in itial co ndition s Let us consider Lax equation 1 d dt L ( t ) + [ L > ( t ) − L < ( t ) , L ( t ) ] = 0 (2.1) where L ( t ) denotes a time-dep enden t generic N × N matrix and remo ve an y h yp othesis on the nature of the initial conditions. W e a r e in terested in writing an integration algo rithm for eq.(2.1 ) whic h should hold true for generic initia l matrices L (0) ≡ L 0 indep enden tly from the fact that L 0 b e diagonalizable o r non diagonalizable, nilp otent or not, symmetric or non symmetric with resp ect to an y definite or indefinite metric η . It turns out that suc h an in tegration a lgorithm exists, is v ery simple and equally simple and elegant is the formal pro of of its v alidit y . F ollow ing [8] w e first presen t the in tegration form ula and then provide the mathemat- ical pro of that it satisfies Lax equation. 1 Hereafter, we denote b y L > ( L < ) the upp er (low er) triangular par t including the diagona l of the matrix L , and L T is the transp osed matrix. 1 2.1 In tegration form ula The matrix elemen ts L pq ( t ) of the time-ev olving Lax op erator which coincides with the giv en L 0 at time t = 0 a re constructed as follow s [8 ]: L pq ( t ) = 1 p D p ( t ) D p − 1 ( t ) D q ( t ) D q − 1 ( t ) p X k =1 q X ℓ =1 M pk ( t ) ( C ( t ) L 0 ) k ℓ f M q ℓ ( t ) . (2.2) The building blo c ks a pp earing in eq.(2.2) are defined as follows . W e hav e M ik ( t ) := ( − 1) i + k Det          C 1 , 1 ( t ) . . . C 1 ,i − 1 ( t ) . . . . . . . . . d C k , 1 ( t ) . . . [ C k ,i − 1 ( t ) . . . . . . . . . C i, 1 ( t ) . . . C i,i − 1 ( t )          , 1 ≤ k ≤ i ; 2 ≤ i ≤ N , M 11 ( t ) := 1 (2.3) where the hats on the en tries corresp onding to the k - th row mean that suc h a ro w has b een suppressed giving rise to a squared ( i − 1) × ( i − 1 ) matrix of which one can calculate the determinan t. Similarly: f M ik ( t ) := ( − 1) i + k Det     C 1 , 1 ( t ) . . . d C 1 ,k ( t ) . . . C 1 ,i ( t ) . . . . . . . . . . . . . . . C i − 1 , 1 ( t ) . . . [ C i − 1 ,k ( t ) . . . C i − 1 ,i ( t )     , 1 ≤ k ≤ i ; 2 ≤ i ≤ N ; f M 11 ( t ) := 1 , (2.4) where the hatted k - th column is deleted just as in eq.(2.3) it was deleted the k -th row . In the ab o v e formulae w e hav e used the follo wing definitions: C ( t ) := e − 2 t L 0 (2.5) and D i ( t ) := D et     C 1 , 1 ( t ) . . . C 1 ,i ( t ) . . . . . . . . . C i, 1 ( t ) . . . C i,i ( t )     , D 0 ( t ) := 1 . (2.6) 2.2 The theorem and its pro of In o r der to prov e the integration formula (2.2) w e first recast it into another equiv alent form whic h is not only the most con v enien t for the formal pr o of but also the simplest and b est suited for computer implemen tatio n. 2 Theorem 2.1 The solution of L ax e quation (2.1) w ith generic initial c ondition L (0) = L 0 is given by L ( t ) = Q ( C ) L 0 ( Q ( C )) − 1 (2.7) wher e the N × N matrix Q( C ( t )) is Q ij ( C ) := 1 p D i ( t ) D i − 1 ( t ) i X k =1 M ik ( t ) ( C 1 2 ( t )) k ,j ≡ 1 p D i ( t ) D i − 1 ( t ) Det     C 1 , 1 ( t ) . . . C 1 ,i − 1 ( t ) ( C 1 2 ( t )) 1 ,j . . . . . . . . . . . . C i, 1 ( t ) . . . C i,i − 1 ( t ) ( C 1 2 ( t )) i,j     (2.8) and C 1 2 ( t ) ≡ e − t L 0 (se e, e q. (2.5)). Pro of 2.1 The matr ix Q ( C ) satisfies three prop erties 2 X > ( t ) := Q ( C ) C 1 2 ( t ) ≡ upper tr iang ul ar , (2.9) ( X < ( t )) − 1 := Q ( C ) ( C 1 2 ( t )) − 1 ≡ low er triang ul ar , (2.10) ( X > ( t )) ii = ( X < ( t )) ii (2.11) whic h are crucial in what follows. Therefore, the matrix e − t L 0 ≡ C 1 2 ( t ) admits the follo wing t w o different represen tations: e − t L 0 = ( Q ( C )) − 1 X > ( t ) , (2.12) e − t L 0 = X < ( t ) Q ( C ) (2.13) resulting from eqs. (2 .9) and (2.10). D ifferen tiating with resp ect to time t , one a f ter the other equations (2.12), (2.1 3) and (2.7), then using t hem one can straightforw ar dly deriv e the following relations: Q ( C ) d dt ( Q ( C )) − 1 = − L ( t ) −  d dt X > ( t )  ( X > ( t )) − 1 , (2.14) Q ( C ) d dt ( Q ( C )) − 1 = + L ( t ) + ( X < ( t )) − 1  d dt X < ( t )  , (2.15) d dt L ( t ) +  Q ( C ) d dt ( Q ( C )) − 1 , L ( t )  = 0 . (2.16) 2 The matrix ( X > ( t )) ij (2.9) is upp er triang ular, since at j < i ther e is a co incidence of tw o columns in the determinant originating from (2 .8) in eq. (2.9). The matrix ( X < ( t )) − 1 ij (2.10) is low er triangular, since at j > i the a ll e nt ries of the last column in the determinan t originating from (2.8) in eq. (2.10) bec ome equal to zero. 3 A simple insp ection of eqs.(2.14 – 2 .15) with the use of relation ( 2 .11) leads to the conclusion that the quantit y Q ( C ) d dt ( Q ( C )) − 1 is a tra celess matrix whic h is expressed in terms of the Lax o p erator L ( t ) as follow s Q ( C ) d dt ( Q ( C )) − 1 = L > ( t ) − L < ( t ) , (2.17) then with this matrix Q ( C ) d dt ( Q ( C )) − 1 equation (2.16) repro duces L a x equation (2.1) and form ulae (2.7 – 2.8) giv e indeed its g eneral solution for generic initial conditions L 0 . This ends the pro of of our prop osition. ♦ No w, we prese nt the useful expression of the in v erse matrix ( Q ( C )) − 1 : ( Q ( C )) − 1 j i := 1 p D i ( t ) D i − 1 ( t ) i X ℓ =1 ( C 1 2 ( t )) j,ℓ f M iℓ ( t ) ≡ 1 p D i ( t ) D i − 1 ( t ) Det        C 1 , 1 ( t ) . . . C 1 ,i ( t ) . . . . . . . . . C i − 1 , 1 ( t ) . . . C i − 1 ,i ( t ) ( C 1 2 ( t )) j, 1 . . . ( C 1 2 ( t )) j,i        (2.18) whic h fo llo ws from eq. (2.2). It is a very simple exercise t o v erify that indeed  Q ( C ) ( Q ( C )) − 1  pq = 1 p D p ( t ) D p − 1 ( t ) D q ( t ) D q − 1 ( t ) p X k =1 q X ℓ =1 M pk ( t ) C k ℓ ( t ) f M q ℓ ( t ) = δ p,q (2.19) since it is equal to zero if p 6 = q , due to a coincidence o f t w o ro ws or columns in the determinan ts originating from (2 .8 – 2.18) in eq. (2.19). It is in teresting to note one more prop erty of the ma t r ix Q ( C ) ( 2 .8): Q T ( C T ) = ( Q ( C )) − 1 (2.20) whic h is obvious if one compares equations ( 2 .8) and (2.18). When L 0 is symmetric the same is true of C T and this implies that Q ( t ) is orthogonal, namely Q ( t ) ∈ SO(N). Similarly when L 0 is pseudo-ortho g onal η -metric symmetric, Q ( t ) is pseudo-orthogonal, Q ( t ) ∈ SO(p , N − P). Y et Q ( t ) ex ists in general also for non η -symmetric initial data. In this case Q ( t ) ∈ gl (N). 2.3 The Generalized Linear Sy stem Finally , based on the know ledge of the constructed general solution (2.7 – 2.8) with generic initial data L 0 , w e commen t briefly on a w ay of a generalization ( suggested by this solution 4 with the aim to repro duce it) of Ko dama–Y e algorithm, whic h w as based on the in v erse scattering metho d applied to the case of the diagonalizable L 0 [2]. A link to w ards the inv erse scattering metho d is giv en b y the follo wing observ ation. Relations (2.7) and (2.17) corresponding to solution Q ( C ) (2.8), b eing iden tically rewritten as follows : L ( t ) Q ( C ) = Q ( C ) L 0 , (2.21) d dt Q ( C ) = − ( L > ( t ) − L < ( t )) Q ( C ) , (2.22) represen t a generalized linear system for the matrix Q ( C ), whose consistency is provide d b y Lax equation (2.1). This generalized linear system could b e a start ing p o in t t o apply the in v erse scattering metho d to cons truct the solution for Q ( C ). But, there is a subtlet y: up to no w the inv erse scattering metho d was alw a ys applied o nly when L 0 is a diagonal or a dia gonalizable matrix; it was nev er applied b efore to the case of non-diagonalizable L 0 . Just the classical case of diagonal (diagonalizable) L 0 w as considered in [2] where t he corresp onding solution to the linear system was constructed. The v ery existence of our solution Q ( C ) (2.8) f o r generic L 0 giv es an evidence tha t a g eneralizatio n of the algorithm of [2] t o the case of the generalized linear systems (2.21 – 2.2 2) with generic L 0 should exist as well. It is indee d the case, and the generalization is rather straightforw ard. So it turns out that one can rep eat a ll the steps of the algorithm, starting from the generalize d linear system (2.21 – 2.22), in such a w ay that all the in termediate op eratio ns ar e formulated in purely matrix terms; thus they are insensitiv e to whether L 0 is a diagonalizable or a non diagonalizable matrix: this pro ces s ends with the solution presen ted in (2.8). 3 Conclus ions In the presen t pap er w e completed t he analysis started in [8] on the integration of Lax equations with b oth generic Lax op erators and generic initial conditions b y presen ting a simple, original rig o rous pro of of the v alidity o f the in tegration algo r ithm prop osed in [8]. The presen ted pro of is constructiv e, and moreo v er it is inte resting fo r it s o wn sak e since it op ens a new deep insigh t in to the structure of t he corresp o nding in tegrable T o da-like systems whic h w e plan to discuss else where in connection with relev ant phy sical problems like the classification and construction of Sup ergra vit y Blac k Hole solutions. W e also clarified the relation b et w een our new in tegrat io n algorithm and t he inv erse scattering framew or k adopt ed b y Ko dama et al fo r the in tegration of Lax equation in the diagonalizable case. Aknolwdgemen ts W e would lik e to express our gratitude to o ur colla b orators J. R osseel, T. v an Riet and M. T rigia n te for the many in teresting discussions and exc hanges of useful information. A ve ry sp ecial akno wledgemen t one o f us (W.C.) w ould lik e to express to Y. Ko dama fo r the many rep eated, essen tial and enligh tening discussions. 5 References [1] Y. Ko dama and J. Y e, T o da hier ar c h y with indefinite metric , Ph ysica D91 (19 96) 321, [arXiv:solv-in t/95 05004]. [2] Y. Ko dama and J. Y e, Iso-sp e ctr al de formations of gener al m a trix and their r e ductions on Lie algebr as , Commun. Mat h. Phys . 178 (1996) 765, [arXiv:solv-in t/95060 05]. [3] P . F r ´ e and A.S. Sorin, Inte gr ability of Sup er gr avity Bil liar ds and the gener alize d T o da lattic e e quations , Nucl. Ph ys. B733 (200 6) 334, [arXiv:hep-th/05101 56]. [4] P . F r ´ e a nd A.S. Sorin, The arr ow of time and the Weyl gr oup: al l s up er gr avity bil liar ds ar e inte g r able, [arXiv:071 0.1059], to app ear in Nuclear Ph ysics B . [5] P . F re and J. Rosseel, O n ful l-fle dge d sup er g r avity c osmolo gies and their Weyl gr oup asymptotics, [arXiv:0805.43 39]. [6] P . F r´ e, A.S. Sorin, Sup er gr avity Black Holes and Bi l liar ds and Liouvil le inte gr able structur e of dual Bor el algebr as , [7] W. Chemissan y , J. Rosseel, M. T rigiante, T. V an Riet, The ful l inte gr ation of b lack hole solutions to symmetric sup er gr avity the ories , [arXiv:0903.27 7 7]. [8] P . F re and A. S. Sorin, The Inte gr ation Algorithm for Nilp otent Orbits of G/H ⋆ L ax systems: for Extr emal Black Holes, [arXiv:0903.3 771]. [9] E. Berg sho eff, W. Chemis sany , A. Plo egh, M. T rig ian te and T. V an Riet, Gener- ating Ge o desic Flows and Sup er gr a v ity Solutions, ” Nucl. Ph ys. B812 (200 9 ) 3 43, 6