The multicomponent 2D Toda hierarchy: dispersionless limit

The m ulticomp onen t 2D T o da hierarc h y: disp ersionless limit Man uel Ma ˜ nas and Luis M art ´ ınez Alonso Departamen to de F ´ ısica T e´ orica I I, U nive rsidad Complutense 28040-Madrid, Spain emails: man uel.manas@fis.ucm.es, luism@fis.ucm.es No v em b er 21, 2018 Abstract The factorization problem of the mult i-c o mpo nen t 2D T o da hier a rch y is used to analy z e the disper sionless limit of this hierarch y . A disp ersive version o f the Whitham hierarch y defined in t er ms of scalar Lax a nd Orlov–Sch ulman op erators is in tro duced and the correspo nding additiona l symmetries and string equations are discussed. Then, it is shown how KP a nd T oda pictures o f the disp ersionles s Whitham hierarch y emerge in the dispersio nless limit. Moreov er, the additional sy mmetries a nd string equations for the disper sive Whitham hierarch y are studied in this limit. 1 In tro duction In [1] the theor y of the mult i-co mpo nen t T o da hierarchy [2] was a nalyzed from the p oint of view of a factorization problem g = W − 1 ¯ W (1) in a n infinite-dimensio nal gr oup and a natura l formulation of t he additional symmetries and the string equations of the hierarch y was given. In the present work we use this formulation to study the disper sionless limit of the solutions of (1). As it is known in the theory of rando m matrix models [3]-[5], the s tudy the large N limit can b e per formed in terms of the disp ers io nless limit o f the str ing equations satisfied by the solution of the underlying int eg r able sys tem. Notice that in recent years the formalis m of string eq uations [6] for dis per sionless integrable systems [7] has b een muc h dev elop ed [8]. O ur present work is motiv ated b y the applications o f m ulti-comp onent int eg r able hierarchies [2, 9] to the study of t he lar ge N limit of t he tw o-ma trix model [10]-[13], as well as mo dels of r andom matrices with exter nal sour ce and no n-intersecting B rownian mo tions [14]-[19]. A co mmon feature of these models is that they have an asso ciated family of multiple orthogo nal p oly no mials which is in turn characterized by a matrix Riemann-Hilbert (MRH) problem whic h is a bas ic ingr edient to analyz e the large N limit [19]-[22]. On the o ther hand, MRH problems a lso provide solutions of r eductions of multi-compo ne nt int eg r able hierarchies of KP or T o da t yp e. These reductions corre spo nd to so lutions o f factorization problems (1) constra ined by cer tain type s o f string equations . In our a na lysis we intro duce matrix wav e functions and scalar Lax and Or lov–Sc hulman ope r ators [23] asso ciated to the solutions o f (1). W e prov e that the ro ws of the matrix wa ve functions sa tisfy auxiliary linear systems inv olving the sc alar Lax oper ators, which constitute the disp ers ive v ersio ns of the genus zero disp e rsionless Whitham hie r archies [24]. In order to study the disp ersionle s s limit, w e assume the T ak asaki– T akebe qua s i-classical ansatz [25, 2 6] for the rows of the matrix wav e functions. Th us, w e prov e that in the disp e rsionless limit the auxilia ry linear systems reduce to systems of Hamilton–Ja cobi equations that are shown to be equiv alen t to the disper sionless Whitham hierarchies. In pa r ticular, tw o natural pictures (KP and T o da types) of the disp ersionles s Whitham hierarchies emerg e in o ur analysis. An impor tant adv antage of our a pproach is that it yields a natur al metho d for characterizing str ing equations and additio na l symmetries in the disp ersionless limit. In pa rticular, we character ize the disp ersive analo gues o f the so luble string equa tio ns discussed in [2 7]. The layout o f the pap er is a s follows. In § 1 .1 we pres en t a summary of the relev ant parts o f [1 ] neede d in the subsequent analysis. Then, in § 2 we discuss the disp ersive Whitham hierarchies. W e introduce a set of scalar 1 Lax and Orlov–Sch ulman op er a tors, a nd v ector wa ve functions to deduce the corr e s po nding a uxiliary linear systems, as well as additio na l symmetries and string equations of disp ersive type. Finally , in § 3 we dis cuss the aforementioned disp ersio nless limits. W e find the Hamilton–Ja cobi type equatio ns, a nd then derive the K P and T o da pictures of the disp ersionless Whitham hierar c hy . W e conclude the pap er by consider ing the disp ersio nless counterparts of disp ers ive string eq ua tions. 1.1 Reminder As in our previo us work [1] we only cons ider forma l ser ies expans ions in the Lie g r oup theor e tic set up without any assumption on their conv er g ency . Let us remind s o me notations and res ults from [1]. Giv en Lie alg ebras g 1 ⊂ g 2 , a nd X , Y ∈ g 2 then X = Y + g 1 means X − Y ∈ g 1 . F or any Lie groups G 1 ⊂ G 2 and a , b ∈ G 2 then a = G 2 · b sta nds for a · b − 1 ∈ G 2 . Let M N ( C ) denote the asso cia tiv e algebra of complex N × N co mplex matrices we will co nsider the linear space o f sequences f : Z → M N ( C ). The s hift op era to r Λ acts o n these s e q uences as (Λ f )( n ) := f ( n + 1). A sequence X : Z → M N ( C ) acts b y left multiplication in this space of sequences, and therefore we may consider op erators of the type X Λ j , ( X Λ j )( f )( n ) := X ( n ) · f ( n + j ). Moreov er , defining the pro duct ( X ( n )Λ i ) · ( Y ( n )Λ j ) := X ( n ) Y ( n + i )Λ i + j and extending it linea rly we have that the set g of La urent series in Λ is an asso ciative a lgebra, which under the standa rd commutator is a Lie algebra. This Lie algebr a has the following imp ortant splitting g = g + ∔ g − , (2) where g + = n X j ≥ 0 X j ( n )Λ j , X j ( n ) ∈ M N ( C ) o , g − = n X j < 0 X j ( n )Λ j , X j ( n ) ∈ M N ( C ) o , are Lie subalgebr as of g with tr ivial intersection. The g roup o f linear in vertible elements in g will b e denoted b y G and has g as its Lie a lgebra, then the splitting (2 ) leads us to consider the fo llowing facto r ization o f g ∈ G g = g − 1 − · g + , g ± ∈ G ± (3) where G ± hav e g ± as their Lie a lg ebras. E xplicitly , G + is the se t of inv ertible linear oper ators of the form P j ≥ 0 g j ( n )Λ j ; while G − is the set of inv ertible linear o per ators of the form 1 + P j < 0 g j ( n )Λ j . Now we intro duce tw o sets of indexes, S = { 1 , . . . , N } a nd ¯ S = { ¯ 1 , . . . , ¯ N } , of the same ca rdinality N . In what follows we will use letters k , l and ¯ k, ¯ l to denote elements in S and ¯ S , r esp ectively . F ur thermore, we will use letters a, b, c to denote elements in S := S ∪ ¯ S . W e define the following op erato rs W 0 , ¯ W 0 ∈ G W 0 := N X k =1 E kk Λ s k e P ∞ j =0 t jk Λ j , (4) ¯ W 0 := N X k =1 E kk Λ − s ¯ k e P ∞ j =1 t j ¯ k Λ − j (5) where s a ∈ Z , t j a ∈ C are deformatio n parameters, tha t in the sequel will play the role of discrete and co nt inuous times, r esp e c tiv ely . Given a n element g ∈ G and a se t of defo rmation par ameters s = ( s a ) a ∈S , t = ( t j a ) a ∈S , j ∈ N we will consider the factorization problem S ( s , t ) · W 0 · g = ¯ S ( s , t ) · ¯ W 0 , S ∈ G − and ¯ S ∈ G + , (6) and will confine ourse lves to the zer o char ge se ctor | s | := P a ∈S s a = 0. W e define the dr essing o r Sato op erators W , ¯ W as follows W := S · W 0 , ¯ W := ¯ S · ¯ W 0 , (7) 2 so that the facto rization pr oblem in G reads W · g = ¯ W (8) Observe that S, ¯ S have expansio ns of the form S = I n + ϕ 1 ( n )Λ − 1 + ϕ 2 ( n )Λ − 2 + · · · ∈ G − , ¯ S = ¯ ϕ 0 ( n ) + ¯ ϕ 1 ( n )Λ + ¯ ϕ 2 ( n )Λ 2 + · · · ∈ G + . (9) The La x op erators L, ¯ L, C kk , ¯ C kk ∈ g are defined by L := W · Λ · W − 1 , ¯ L := ¯ W · Λ · ¯ W − 1 , (10) C kk := W · E kk · W − 1 , ¯ C kk := ¯ W · E kk · ¯ W − 1 (11) and hav e the following expa nsions L = Λ + u 1 ( n ) + u 2 ( n )Λ − 1 + · · · , ¯ L − 1 = ¯ u 0 ( n )Λ − 1 + ¯ u 1 ( n ) + ¯ u 2 ( n )Λ + · · · , C kk = E kk + C kk, 1 ( n )Λ − 1 + C kk, 2 ( n )Λ − 2 + · · · , ¯ C kk = ¯ C kk, 0 ( n ) + ¯ C kk, 1 ( n )Λ + ¯ C kk, 2 ( n )Λ 2 + · · · . (12) Now we int r o duce s o me further notation 1. ∂ j a := ∂ ∂ t j a , for a = S and j = 1 , 2 , . . . 2. Given K = ( a, b ) the ba sic char ge pr eserving shift op era to rs T K are defined as follows ( T K f )( s a , s b ) := f ( s a + 1 , s b − 1) . W e define the Orlov–Sch ulman op erators [23] for the m ulti-comp onent 2D T o da hier arch y by M := W n W − 1 , ¯ M := ¯ W n ¯ W − 1 . (13) One prov es at once that • The Orlov–Sc hulman o per ators satisfy the following commutation relatio ns [ L, M ] = L, [ L, C kk ] = 0 , [ ¯ L, ¯ M ] = ¯ L, [ ¯ L, ¯ C kk ] = 0 , (14) • The following expa nsions hold M = M + N X k =1 C kk ( s k + ∞ X j =1 j t j k L j ) , M = n + g − ¯ M = ¯ M − N X k =1 ¯ C kk ( s ¯ k + ∞ X j =1 j t j ¯ k ¯ L − j ) , ¯ M = n + g + Λ . (15) 1.1.1 Additional symmetries Suppo se that the op er a tor g in (8) depends o n an additional para meter b ∈ C . Then, the basic ob jects of the m ulti-co mpone nt T o da hier arch y inher it a dep endence on b . F or co nvenience a nd for the time b eing we use the following equiv alent factoriza tion problem W · h = ¯ W · ¯ h, with g = h · ¯ h − 1 . (16) 3 Observe that ∂ b W · W − 1 + W ( ∂ b h · h − 1 ) W − 1 = ∂ b S · S − 1 + W ( ∂ b h · h − 1 ) W − 1 = ∂ b ¯ S · ¯ S − 1 + ¯ W ( ∂ b ¯ h · ¯ h − 1 ) ¯ W − 1 = ∂ b ¯ W · ¯ W − 1 + ¯ W ( ∂ b ¯ h · ¯ h − 1 ) ¯ W − 1 . (17) Now, let us supp ose that h and ¯ h s atisfy ∂ b h · h − 1 = F (0) = N X l =1 F l ( n, Λ) E ll , ∂ b ¯ h · ¯ h − 1 = ¯ F (0) = N X l =1 ¯ F l ( n, Λ) E ll , (18) then from (17)we get ∂ b W · W − 1 = ∂ b S · S − 1 = − H − , ∂ b ¯ W · ¯ W − 1 = ∂ b ¯ S · ¯ S − 1 = H + , H ± ∈ g ± . where H := F − ¯ F , F := N X l =1 F l ( M , L ) C ll , ¯ F := N X l =1 ¯ F l ( ¯ M , ¯ L ) ¯ C ll . (19) Hence it follows that Prop ositio n 1 . Given a dep endenc e on an additional p ar ameter b ac c or ding to (16) , (1 8) and (19) then 1. The dr essing op er ators W and ¯ W satisfy ∂ b W = − H − · W, ∂ b ¯ W = H + · ¯ W , 2. The L ax and Orlov–Schulman op er ators satisfy ∂ b L = [ − H − , L ] , ∂ b M = [ − H − , M ] , ∂ b C kk = − [ H − , C kk ] , ∂ b ¯ L = [ H + , ¯ L ] , ∂ b ¯ M = [ H + , ¯ M ] , ∂ b ¯ C kk = [ H + , ¯ C kk ] . (20) A key obser v a tio n is Prop ositio n 2 . Given op er ators R, ¯ R ∈ g satisfying R · g = ¯ R and s u ch that RW − 1 0 ∈ g − , ¯ R ¯ W − 1 0 ∈ g + . (21) Then R = ¯ R = 0 1.2 W a v e functions The wa ve functions of the multi-compo nen t 2 D T o da hiera rch y ar e defined b y ψ = W · χ, ¯ ψ = ¯ W · χ. (22) where χ ( z ) := { z n I N } n ∈ Z , Note that Λ χ = z χ . The following as ymptotic ex pansions are a co nsequence of (9) ψ = z n ( I N + ϕ 1 ( n ) z − 1 + · · · ) ψ 0 ( z ) , ψ 0 := N X k =1 E kk z s k e P ∞ j =1 t jk z j , z → ∞ , ¯ ψ = z n ( ¯ ϕ 0 ( n ) + ¯ ϕ 1 ( n ) z + · · · ) ¯ ψ 0 ( z ) , ¯ ψ 0 := N X k =1 E kk z − s ¯ k e P ∞ j =1 t j ¯ k z − j , z → 0 . (23) 4 Prop ositio n 3 . 1. Given op er ators of the form F := N X k =1 F k C kk , F k := X i ≥ 0 ,j ∈ Z F kij M i L j , ¯ F := N X k =1 F ¯ k ¯ C kk , F ¯ k := X i ≥ 0 ,j ∈ Z F ¯ kij ¯ M i ¯ L j , with c omplex-value d sc alar c o efficients, we have F ( ψ ) = ( ψ ) N X k =1 ← − − − − − − − − F k  z d d z , z  E kk , ¯ F ( ¯ ψ ) = ( ¯ ψ ) N X k =1 ← − − − − − − − − F ¯ k  z d d z , z  E kk . (24) wher e ( ψ ) ← − − − − − − − − F k  z d d z , z  := X i ≥ 0 ,j ∈ Z F kij z j  z d d z  i ( ψ ) , ( ¯ ψ ) ← − − − − − − − − F ¯ k  z d d z , z  := X i ≥ 0 ,j ∈ Z ¯ F kij z j  z d d z  i ( ¯ ψ ) 2. Given op er ators P := N X k =1 P k C kk , P k := X i ≥ 0 ,j ∈ Z P kij M i L j , Q := N X k =1 Q k C kk , Q k := X i ≥ 0 ,j ∈ Z Q kij M i L j , ¯ P := N X k =1 P ¯ k ¯ C kk , P ¯ k := X i ≥ 0 ,j ∈ Z P ¯ kij ¯ M i ¯ L j , ¯ Q := N X k =1 Q ¯ k ¯ C kk , Q ¯ k := X i ≥ 0 ,j ∈ Z ¯ Q ¯ kij ¯ M i ¯ L j , with c omplex-value d sc alar c o efficients, we have P Q ( ψ ) = N X k =1  ( ψ ) ← − − − − − − − − P k  z d d z , z   ← − − − − − − − − Q k  z d d z , z  , ¯ P ¯ Q ( ¯ ψ ) = N X k =1  ( ¯ ψ ) ← − − − − − − − − P ¯ k  z d d z , z   ← − − − − − − − − Q ¯ k  z d d z , z  . Pr o of. 1. It is easy to find fro m (10) a nd (15) that M i L j C kk ( ψ ) = W n i Λ j E kk ( χ ) = W  { n i z n + j I N } n ∈ Z  E kk , ¯ M i ¯ L j ¯ C kk ( ¯ ψ ) = ¯ W n i Λ j E kk ( χ ) = ¯ W  { n i z n + j I N } n ∈ Z  E kk . Now observe that the a ction o f X = P j ′ ∈ Z X j ′ Λ j ′ on { n i z n + j } n ∈ Z is X  { n i z n + j } n ∈ Z  = n X j ′ ∈ Z X j ′ ( n )( n + j ′ ) i z n + j + j ′ o n ∈ Z or equiv alently z j  z d d z  i ( X · χ ) =  z j  z d d z  i  X j ′ ∈ Z X j ′ ( n ) z n + j ′   n ∈ Z . Thu s, the formulae M i L j C kk ( ψ ) = z j  z d d z  i ( ψ ) E kk , ¯ M i ¯ L j ¯ C kk ( ¯ ψ ) = z j  z d d z  i ( ¯ ψ ) E kk , (25) hold. 2. It is a consequence of the identities M i 1 L j 1 M i 2 L j 2 = M i 1 ( M + j 1 ) i 2 L j 1 + j 2 , ¯ M i 1 ¯ L j 1 ¯ M i 2 ¯ L j 2 = ¯ M i 1 ( ¯ M + j 1 ) i 2 ¯ L j 1 + j 2 , z j 2  z d d z  i 2 z j 1  z d d z  i 1 = z j 1 + j 2  z d d z + j 1  i 2  z d d z  i 1 , for any i 1 , i 2 ≥ 0 and j 1 , j 2 ∈ Z . Therefore,  ( ψ ) ← − − − − − − − −  z d d z  i 1 z j 1  ← − − − − − − − −  z d d z  i 2 z j 2 = ( ψ ) ← − − − − − − − − − − − − − − − − − − − − −  z d d z  i 1  z d d z + j 1  i 2 z j 1 + j 2 = M i 1 L j 1 M i 2 L j 2 ( ψ ) . 5 2 The disp ersiv e Whitham hierarc h ies As w e will see certain families of equations of the m ulti-comp onent 2 D T o da hiera rch y , asso ciated with an y given row of the dressing op erators, beco me the Whitham hierarchies under appropriate disper sionless limits. Consequently , these families will be referred to as the dis per sive Whitham hierar ch ies. F o r simplicity and without los s of generality , we will work with the first r ow of the dressing op erator s. It will b e useful to intro duce the following shift op era tors T a := ( T (1 ,a 0 ) , a = 1 , T ( a, 1) , a 6 = 1 ¯ T a := ( T ( ¯ 1 ,a 0 ) , a = ¯ 1 , T ( a, ¯ 1) , a 6 = ¯ 1 , (26) where for the cases a = 1 and a = ¯ 1, the index a 0 stands for any fixed elements in S − { 1 } a nd ¯ S − { ¯ 1 } , resp ectively . These tw o types of shift op era tors, tha t we refer as bar ed and unb a r ed, lead to tw o algebr as of shift op erators , and also to tw o different families of Hamilton– Jacobi equa tio ns, s ee (47) and (48). W e also define the scalar dressing o per ators K a :=      1 + ϕ 1 , 11 T − 1 1 + ϕ 2 , 11 T − 2 l + · · · , a = 1 ϕ 1 , 1 k + ϕ 2 , 1 k T − 1 k + · · · , a = k 6 = 1 , ¯ ϕ 0 , 1 k + ¯ ϕ 1 , 1 k T − 1 ¯ k + · · · , a = ¯ k, (27) ¯ K a :=      1 + ϕ 1 , 11 ¯ T − 1 1 + ϕ 2 ,ll ¯ T − 2 1 + · · · , a = 1 , ϕ 1 ,lk + ϕ 2 ,lk ¯ T − 1 k + · · · , a = k 6 = 1 ¯ ϕ 0 , 1 k + ¯ ϕ 1 , 1 k ¯ T − 1 ¯ k + · · · , a = ¯ k , (28) where ϕ i , ¯ ϕ i are the matrix co efficients of (9). Thu s, we may now int r o duce the asso ciated scala r Lax op erator s L a := K a ◦ T a ◦ K − 1 a = W a ◦ T a ◦ W − 1 a = ( T 1 + L 1 , 0 + L 1 , − 1 T − 1 1 + · · · , a = 1 , L a, 1 T a + L a, 0 + L a, − 1 T − 1 a + · · · , a 6 = 1 ¯ L a := ¯ K a ◦ ¯ T a ◦ ¯ K − 1 a = ¯ W a ◦ ¯ T a ◦ ¯ W − 1 a = ( ¯ T 1 + ¯ L 1 , 0 + ¯ L 1 , − 1 ¯ T − 1 1 + · · · , a = 1 , ¯ L a, 1 ¯ T a + ¯ L a, 0 + ¯ L a, − 1 ¯ T − 1 a + · · · , a 6 = 1 , (29) where W a := K a ◦ W 0 ,a , W 0 ,a := exp( T a ) , T a := ∞ X j =1 t j a T j a , (30) ¯ W a := ¯ K a ◦ ¯ W 0 ,a , ¯ W 0 ,a := exp( ¯ T a ) , ¯ T a := ∞ X j =1 t j a ¯ T j a . (31) Similarly , we define the co rresp onding scalar Orlov–Sch ulman o per ators by M a := n − ν a + sg( a ) W a ◦ s a ◦ W − 1 a , ¯ M a := n − ν a + sg( a ) ¯ W a ◦ s a ◦ ¯ W − 1 a , (32) where sg( a ) := ( 1 , a ∈ S , − 1 , a ∈ ¯ S , ν a := ( 1 , a ∈ S − { 1 } , 0 , a 6∈ S − { 1 } . F r om the identities [ T a , sg( a ) s a ] = sg( a ) T a , [ ¯ T a , sg( a ) s a ] = sg( a ) ¯ T a , it follows that [ L a , M a ] = sg( a ) L a , [ ¯ L a , ¯ M a ] = sg( a ) ¯ L a , 6 Prop ositio n 4 . The Orlov–Schulman op er ators satisfy M a = n − ν a + sg ( a )  s a + ∞ X j =1 j t j a L j a + ∞ X i =1 m ai T − i a  , (33) ¯ M a := n − ν a + sg ( a )  s a + ∞ X j =1 j t j a ¯ L j a + ∞ X i =1 ¯ m ai ¯ T − i a  . (34) Pr o of. These formu la e follow fro m W 0 ,a s a W − 1 0 ,a = s a + [ T a , s a ] = s a + ∞ X j =1 j T j a , ¯ W 0 ,a s a ¯ W − 1 0 ,a = s a + [ ¯ T a , s a ] = s a + ∞ X j =1 j ¯ T j a , and the fact that ther e are expansions o f the for m K a s a K − 1 a = s a + ∞ X i =1 m ai T − i a , ¯ K a s a ¯ K − 1 a = s a + ∞ X i =1 ¯ m ai ¯ T − i a . W e further int ro duce the v ector w ave functions Ψ a := ( ψ 1 k , a = k , ¯ ψ 1 k , a = ¯ k, (35) Prop ositio n 5 . We have the identities [ F a ( M a , L a )](Ψ a ) = [ F a ( ¯ M a , ¯ L a )](Ψ a ) = (Ψ a ) ← − − − − − − − − − − F a  z d d z , z sg a  = ( E 11 F k ( M , L ) C kk ( ψ ) , a = k , E 11 F ¯ k ( ¯ M , ¯ L − 1 ) ¯ C kk ( ¯ ψ ) , a = ¯ k . (36) Pr o of. F rom the definitions (4),(5) and (7) W 1 k n i Λ j = S 1 k W 0 ,kk n i Λ j = S 1 k ( W 0 ,kk nW − 1 0 ,kk ) i Λ j W 0 ,kk = S 1 k ( n + s k + ∞ X j ′ =1 j ′ t j ′ k Λ j ′ ) i Λ j W 0 ,kk , ¯ W 1 k n i Λ j = ¯ S 1 k ¯ W 0 ,kk n i Λ j = ¯ S 1 k ( ¯ W 0 ,kk n ¯ W − 1 0 ,kk ) i Λ j ¯ W 0 ,kk = ¯ S 1 k ( n − s ¯ k − ∞ X j ′ =1 j ′ t j ′ ¯ k Λ − j ′ ) i Λ j ¯ W 0 ,kk , Now, observe that Λ − 1 ( n + s k ) W 0 ,kk = T − 1 k (( n + s k ) W 0 ,kk ) = ¯ T − 1 k (( n + s k ) W 0 ,kk ) , Λ( n − s ¯ k ) ¯ W 0 ,kk = T − 1 ¯ k (( n − s ¯ k ) ¯ W 0 ,kk ) = ¯ T − 1 ¯ k (( n − s ¯ k ) ¯ W 0 ,kk ) , together with Pr op osition 3 imply the result. 2.1 Auxiliary linear systems Our next ana lysis uses the following complex algebr as t a := n X j ∈ Z c j T j a o , ¯ t a := n X j ∈ Z c j ¯ T j a o , (37) 7 and their subalg e bras          t a, + = t a,> := n X j > 0 c j T j a o t a, ≤ := n X j ≤ 0 c j T j a o , a 6 = 1 t 1 , + = t 1 , ≥ := n X j ≥ 0 c j T j 1 o t 1 ,< := n X j < 0 c j T j 1 o (38)                              ¯ t a, + = ¯ t a,> := n X j > 0 c j ( ¯ T j a − 1) o , ¯ t a,< := n X j < 0 c j ( ¯ T j a − 1) o , a 6 = 1 , ¯ 1 ¯ t 1 , + = ¯ t 1 , ≥ := n X j ≥ 0 c j ¯ T j 1 o , ¯ t 1 ,< := n X j < 0 c j ¯ T j 1 o , ¯ t ¯ 1 , + = ¯ t ¯ 1 ,> := n X j > 0 c j ( ¯ T j ¯ 1 − 1) o ¯ t ¯ 1 ,< := n X j < 0 c j ( ¯ T j ¯ 1 − 1) o , a ′ 6 = 1 , ¯ t ¯ 1 , + = ¯ t ¯ 1 ,> := n X j > 0 c j ¯ T j ¯ 1 o ¯ t ¯ 1 , ≤ := n X j ≤ 0 c j ¯ T j ¯ 1 o , a ′ = 1 . (39) W e will denote b y ( T a, + , T a,< , T a,> , T a, ≤ , T a, ≥ ) the pr o jections o f an op era tor T a induced by the corre spo nding splittings. The following imp or tant result links the op erato rs ( M k , L k ) with the o per ators ( M , L ) . Here the s plittings for each shift alg ebra t a or ¯ t a are tho se indicated by (38) and (39), Prop ositio n 6 . The fol lowing r elations hold ( F ( M k , L k ) + ( E 11 W ) = F ( ¯ M k , ¯ L k ) + ( E 11 W ) = E 11 ( F ( M , L ) C kk ) + W , F ( M k , L k ) + ( E 11 ¯ W ) = F ( ¯ M k , ¯ L k ) + ( E 11 ¯ W ) = E 11 ( F ( M , L ) C kk ) + ¯ W , (40) ( F ( M ¯ k , L ¯ k ) + ( E 11 W ) = F ( ¯ M ¯ k , ¯ L ¯ k ) + ( E 11 W ) = E 11 ( F ( ¯ M , ¯ L − 1 ) ¯ C kk ) − W , F ( M ¯ k , L ¯ k ) + ( E 11 ¯ W ) = F ( ¯ M ¯ k , ¯ L ¯ k ) + ( E 11 ¯ W ) = E 11 ( F ( ¯ M , ¯ L − 1 ) ¯ C kk ) − ¯ W . (41) Pr o of. See App endix B. If we set F ( x , y ) = y j in Pro p os ition 6 and r ecall that ∂ j a W = B j a W , ∂ j a ¯ W = B j a ¯ W , with B j k = ( C kk L j ) + , B j ¯ k = ( ¯ C kk ¯ L − j ) − [1] we deduce Theorem 1. The fol lowing sc alar line ar systems hold ∂ j a ( E 11 W ) = ( L j a ) + ( E 11 W ) = ( ¯ L j a ) + ( E 11 W ) , ∂ j a ( E 11 ¯ W ) = ( L j a ) + ( E 11 ¯ W ) = ( ¯ L j a ) + ( E 11 ¯ W ) (42) The linea r s ystem (42 ) determines a set of commuting flows for ( W, ¯ W ) which, as we will show in the next Section, leads to the Whitham hierar ch y in the disp ersio nless limit. F or that r eason this system will b e refer red to as the dis p er s ive Whitham hierarch y of flows. 2.2 Additional symm etr ies and string equations Using Pr opo sition 1 we deduce the following results on the additional symmetries Prop ositio n 7 . Given an additional symmetry ∂ b E 11 W = − E 11  N X k =1  F k ( M , L ) C kk − F ¯ k ( ¯ M , ¯ L − 1 ) ¯ C kk   − · W, ∂ b E 11 ¯ W = E 11  N X k =1  F k ( M , L ) C kk − F ¯ k ( ¯ M , ¯ L − 1 ) ¯ C kk   + · ¯ W , (43) 8 then we have ∂ b (Ψ a ) = − F a ( M a , L a )(Ψ a ) +  X a ′ ∈S F a ′ ( M a ′ , L a ′ ) +  (Ψ a ) = − F a ( ¯ M a , ¯ L a )(Ψ a ) +  X a ′ ∈S F a ′ ( ¯ M a ′ , ¯ L a ′ ) +  (Ψ a ) . Pr o of. F rom (43) w e get ∂ b E 11 W = − N X k =1 E 11 F k ( M , L ) C kk · W + E 11 h  N X k =1 F k ( M , L ) C kk  + +  N X k =1 ¯ F k ( ¯ M , ¯ L − 1 ) ¯ C kk  − i · W, ∂ b E 11 ¯ W = − N X k =1 E 11 ¯ F ¯ k ( ¯ M , ¯ L − 1 ) ¯ C kk · ¯ W + E 11 h  N X k =1 F k ( M , L ) C kk  + +  N X k =1 F ¯ k ( ¯ M , ¯ L − 1 ) ¯ C kk  − i · ¯ W . Now, from Pr op ositions 5 and 6 w e conclude that ∂ b ( E 11 W ) = − N X k =1 F k ( M k , L k )( W 1 k ) E 1 k +  N X k =1 ( F k ( M k , L k ) + + F ¯ k ( M ¯ k , L ¯ k ) +  ( E 11 W ) = − N X k =1 F k ( ¯ M k , ¯ L k )( W 1 k ) E 1 k +  N X k =1 ( F k ( ¯ M k , ¯ L k ) + + F ¯ k ( ¯ M ¯ k , ¯ L ¯ k ) +  ( E ll W ) , ∂ b ( E 11 ¯ W ) = − N X k =1 ¯ F k ( M k , L k )( ¯ W 1 k ) E lk +  N X k =1 ( F k ( M k , L k ) + + F ¯ k ( M ¯ k , L ¯ k ) +  ( E 11 ¯ W ) = − N X k =1 ¯ F k ( ¯ M k , ¯ L k )( ¯ W 1 k ) E 1 k +  N X k =1 ( F k ( ¯ M k , ¯ L k ) + + F ¯ k ( ¯ M ¯ k , ¯ L ¯ k ) +  ( E 11 ¯ W ) . and the re sult follows. As a consequence we hav e Prop ositio n 8 . If the string e quation E 11 N X k =1 F k ( M , L ) C kk = E 11 N X k =1 F ¯ k ( ¯ M , ¯ L − 1 ) ¯ C kk (44) is satisfie d, then F a ( M a , L a )(Ψ a ) =  X a ′ ∈S F a ′ ( M a ′ , L a ′ ) +  (Ψ a ) , F a ( ¯ M a , ¯ L a )(Ψ a ) =  X a ′ ∈S F a ′ ( ¯ M a ′ , ¯ L a ′ ) +  (Ψ a ) , for al l a ∈ S . Pr o of. The str ing equations (44) imply the in v ariance conditions ∂ b E 11 W = ∂ b E 11 ¯ W = 0 . (45) Now, recalling Pr op osition 7 we g et the desired result. 3 The disp ersionless limit W e co nsider here the disp e r sionless limit of the multi-component 2D T o da hierar ch y . F o r that a im w e use the vector w av e functions (3 5) at a g iven fixe d v alue n 0 of the discrete v ar iable n . Th us, from Theorem 1 the following auxilia r y linear s y stem follows ∂ j a (Ψ b ) = ( L j a ) + (Ψ b ) = ( ¯ L j a ) + (Ψ b ) a ∈ S , j = 1 , 2 , . . . . (46) 9 Let us now introduce slow v ariables by t sl ,j a = ǫt j a , s sl ,a = ǫs a , where ǫ is a sma ll r eal parameter and s sl ,a are ass umed to b e contin uous v aria bles. F or the sake of simplicit y , we w ill hence fo rth denote by ( t j a , s a ) these slow v ariables. Moreover, we a s sume that the wa ve functions hav e the quasi-cla ssical form Ψ a = exp  S a ǫ  , S a = S a, 0 + ǫ S a, 1 + · · · . with S a = T a +      ǫϕ 1 , 11 z − 1 + O ( z − 2 ) a = 1 , ǫ log ϕ 1 ,k 1 + O ( z − 1 ) a = k 6 = 1 ǫ log ¯ ϕ 0 ,k 1 + O ( z ) a = ¯ k . T a :=      ( ǫn 0 + s 1 ) log z + P ∞ j =1 t j l z j , a = 1 , ( ǫn 0 + s k − ǫ ) log z + P ∞ j =1 t j k z j , a = k 6 = 1 , ( ǫn 0 − s ¯ k ) log z + P ∞ j =1 t j ¯ k z − j , a = ¯ k . F r om these expres s ions we deduce that as ǫ → 0 ϕ 1 , 11 = O ( ǫ − 1 ) , log ϕ 1 , 1 k = O ( ǫ − 1 ) , k 6 = 1 , log ¯ ϕ 0 , 1 ¯ k = O ( ǫ − 1 ) . As a consequence the co efficients in the op erator s L a , ¯ L a are T aylor series in ǫ while those of the Orlov–Sc hulman op erators M a , ¯ M a hav e at most a simple po le in ǫ = 0. W e in tro duce so me new v ar iables σ a := s a , a 6 = 1 , σ 1 := X a ∈S s a , ¯ σ a := s a , a 6 = ¯ 1 , ¯ σ 1 := X a ∈S s a , Observe that ∂ ∂ σ a = ∂ ∂ s a − ∂ ∂ s 1 , a 6 = 1 , ∂ ∂ ¯ σ a = ∂ ∂ s a − ∂ ∂ s ¯ 1 , a 6 = ¯ 1 . The zero charge condition implies that σ 1 = σ ¯ 1 = 0. Then, we define ∂ a :=        ∂ ∂ σ a , a 6 = 1 , − ∂ ∂ σ a 0 , a = 1 , , ¯ ∂ a :=        ∂ ∂ ¯ σ a , a 6 = ¯ 1 , − ∂ ∂ ¯ σ a 0 , a = ¯ 1 . Notice that Prop ositio n 9 . In t he limit ǫ → 0 we have that T j a (exp( S b /ǫ )) = ex p( T j a ( S b ) /ǫ ) = ex p( j ∂ a ( S b, 0 ) + O ( ǫ )) exp( S b /ǫ ) , ¯ T j a (exp( S b /ǫ )) = ex p( ¯ T j a ( S b ) /ǫ ) = ex p( j ¯ ∂ a ( S b, 0 ) + O ( ǫ )) exp( S b /ǫ ) , ∂ j a (exp( S b /ǫ )) = ( ∂ j a ( S b, 0 ) + O ( ǫ )) exp( S b /ǫ ) . 3.1 Hamilton–Jacobi equations and disp ersionless Whitham hierarchies As ǫ → 0 it follows that ( L a ) j + (Ψ b ) =  P j a  e ∂ a S b, 0  + O ( ǫ )  Ψ b , ( ¯ L a ) j + (Ψ b ) =  ¯ P j a  e ∂ a S b, 0  + O ( ǫ )  Ψ b , 10 where P j a and ¯ P j a are p olynomials P j 1 ( Z ) = Z j + P j 1 ,j − 1 Z j − 1 + · · · + P j 1 , 0 , P j a ( Z ) = P j a,j Z j + · · · + P j a, 1 Z, a 6 = 1 ¯ P j ¯ 1 ( Z ) = ¯ P j ¯ 1 , j Z j + ¯ P j ¯ 1 , j − 1 Z j − 1 + · · · + ¯ P j ¯ 1 , 1 Z − (1 − δ 1 a 0 ) j X i =1 ¯ P j ¯ 1 ,i , ¯ P j a ( Z ) = ¯ P j a,j Z j + · · · + ¯ P j a, 1 Z − (1 − δ a 1 ) j X i =1 ¯ P j a,i , a 6 = ¯ 1 . Hence, as ǫ → 0 we ge t from (46) the following Ha milton–Jacobi type equations Prop ositio n 1 0. The fol lowing e quations holds ∂ j a ( S b, 0 ) = P j a  e ∂ a S b, 0  , (47) ∂ j a ( S b, 0 ) = ¯ P j a  e ¯ ∂ a S b, 0  . (48) Next we show how these equations lea d tot he tw o pictures of the Whitham hiera rch y describ ed in the Appendix A. 3.1.1 KP and T o da disp ersionless limits from the Ha milt on–Jacobi equations F r om the basic equation ∂ Ψ b ∂ t 11 = ( L 1 ) + (Ψ b ) , we get the imp orta n t formula ∂ t 11 ( S b, 0 ) = e ( ∂ s 1 − ∂ s a )( S b, 0 ) + q a , a 6 = 1 . (49) Where q a is an appropriate function defined in terms of deriv atives o f the leading co efficient of ϕ 1 , 11 . Obse r ve that a family of equations as (49) o nly o ccurs for the time t 11 and not for the times t 1 a with a 6 = 1 . T his is a consequence of the fact that we have chosen the fir st row in the matrix wa ve functions, and we ar e dea ling with the shifts of type T a . The KP-picture disp ersionless limit Definition 1. We intr o duc e the disp ersionless L ax functions in the KP pictur e, z a = z a ( s , t ) by the implicit r elations p = ∂ x S a, 0 ( z a ) , x := t 11 , and the c orr esp onding disp ersionless Orlov–Schulman functions by m a := ∂ S a, 0 ∂ z     z = z a . This definition implies e ∂ 1 S 1 , 0    z = z 1 = p − q a 0 , e ∂ a S a, 0   z = z a = 1 p − q a , a 6 = 1 . The nex t Prop ositio n exhibits the a symptotic form of these functions Prop ositio n 1 1. The disp ersionless L ax and Orlov–Schulman functions s atisfy z sg a a =    p + ℓ 1 , 0 + O ( p − 1 ) , p → ∞ , a = 1 , ℓ a, 1 p − q a + ℓ a, 0 + O ( p − q a ) , p → q a a 6 = 1 , m a = ( n 0 + sg( a ) s a ) z − 1 a + ∞ X j =1 j t j a z j − 1 a + z − 1 a ( P ∞ j =1 µ aj ( p − q a ′ ) − j , a = 1 , P ∞ j =1 µ aj ( p − q a ) j , a 6 = 1 . 11 Pr o of. Particular cases of (36) ar e M a ( ψ a ) = z d ψ a d z , L a ( ψ a ) = z sg a ψ a , which together with (29) a nd (33 ) imply z sg a = ( e ∂ l S 1 , 0 + ℓ 1 , 0 + ℓ 1 , − 1 e − ∂ l S 1 , 0 + ℓ 1 , − 2 e − 2 ∂ l S 1 , 0 + · · · , a = 1 ℓ a, 1 e ∂ a S a, 0 + ℓ a, 0 + ℓ a, − 1 e − ∂ a S a, 0 + ℓ a, − 2 e − 2 ∂ a S a, 0 + · · · , a 6 = 1 , ∂ S a, 0 ∂ z = ( n 0 + sg( a ) s a ) z − 1 + ∞ X j =1 j t j a z j − 1 + z − 1 ∞ X j =1 µ aj e − j ∂ a S a, 0 and the ev aluation at z = z a gives the desir ed r esult. Therefore for a 6 = 1 we have ( ∂ a ( S b, 0 ))    z = z b = − log( p − q a ) , a 6 = 1 ( ∂ j a ( S b, 0 ))    z = z b = P j a  1 p − q a  =: Ω j a , a 6 = 1 ( ∂ j 1 ( S b, 0 ))    z = z b = P j 1 ( p − q a ′ ) =: Ω j 1 , j > 1 Then we hav e that d S b, 0 = m b d z b + p d x − X a 6 =1 log( p − q a )d s a + X j,a ′ Ω j a d t j a where the Σ ′ indicates the sum over the s et of indexes ( j, a ) where j = 1 , 2 , · · · a nd a ∈ S excluding the case j = 1 a nd a = l . Th us the functions d S b, 0 determine a solution of the zero- genus Whitham hierarch y with 2 N punctures in the KP picture (see App endix A). The T o da -picture disp e rsionless limit W e aga in consider equa tion (4 9) ∂ t 11 ( S b, 0 ) = e ( ∂ s 1 − ∂ s a )( S b, 0 ) + q a , which implies e − ∂ a 0 S b, 0 = e − ∂ a S b, 0 + Q a , a, a 0 6 = 1 , Q a := q a − q a 0 Definition 2. In the T o da r epr esentation the disp ersionless L ax function z a = z a ( s , t ) is given by the implicit r elation p = e − ∂ x S b, 0    z = z b , x := − σ a 0 , and the disp ersionless O r lov–Schulman fun ction by m a := z ∂ S a, 0 ∂ z     z = z a . Observing that e ∂ a S b, 0 = 1 e − ∂ a 0 S b, 0 − Q a we conclude e ∂ x S 1 , 0    z = z 1 = p, e ∂ a 0 S a 0 , 0    z = z a ′ = p − 1 , e ∂ a S a, 0    z = z a = 1 p − Q a , a 0 6 = 1 a 6 = 1 , a 0 . (50) Hence, we deduce 12 Prop ositio n 12. The disp ersionless L ax and Orlov–Schulman fun ct ions in the T o da-pictur e disp ersionless limit satisfy z sg a a =          p + ℓ 1 , 0 + O ( p − 1 ) , p → ∞ , a = 1 , ℓ 2 , 1 p − 1 + O (1) , p → 0 , a = a 0 , ℓ a, 1 p − Q a + O (1) , p → Q a , a 6 = 1 , a 0 . m a = ( n 0 + sg( a ) s a ) + ∞ X j =1 j t j a z j a +      P ∞ j =1 µ 1 j p − j , a = 1 , P ∞ j =1 µ aj p j , a = a 0 , P ∞ j =1 µ aj ( p − Q a ) j , a 6 = 1 , a 0 . Pr o of. Pro c eed a s in the pr o of o f Prop ositio n 11 and use (50). As in the KP case we g e t now ( ∂ a ( S b, 0 ))    z = z b = − log( p − Q a ) , a 6 = 1 , a 0 , ( ∂ j a ( S b, 0 ))    z = z b = P j a  1 p − Q a  =: Ω j a , a 6 = 1 , a 0 , ( ∂ j 1 ( S b, 0 ))    z = z b = P j 1 ( p ) =: Ω j 1 , ( ∂ j a 0 ( S b, 0 ))    z = z b = P j a 0  p − 1  =: Ω j a 0 . Hence we hav e that d S b, 0 = m b d log z b + log p d x − X a 6 =1 ,a ′ log( p − Q a )d s a + X j ≥ 1 ,a ∈S Ω j a d t j a , and therefor e the functions S b, 0 determine a solutio n of the zero-g en us Whitham hier arch y with 2 N punctures in the T o da picture (se e App endix A). An alt ernativ e T o da-picture dispersionless limit F rom the ba sic equation ∂ Ψ b ∂ t 1 ¯ 1 = ( ¯ L ¯ 1 ) + (Ψ b ) , we deduce ∂ 1 ¯ 1 S b, 0 = r 1 e ( ∂ s ¯ 1 − ∂ s 1 )( S b, 0 ) = r a  e ( ∂ s ¯ 1 − ∂ s a )( S b, 0 ) − 1  , a 6 = 1 , ¯ 1 for some functions r a . Hence, e ¯ ∂ a S b, 0 = e ¯ ∂ 1 S b, 0 e ¯ ∂ 1 S b, 0 + ρ a , ρ a := r a r 1 . Now, we take a 0 = 1 and define Definition 3. The disp ersionless L ax functions z a ar e define d by the implicit r elation ∂ x S a, 0 | z = z a = log p, x = − ¯ σ 1 while the disp ersionless O rlov–Schulman functions ar e define d by m a := z a ∂ S a, 0 ∂ z     z = z a . Observe that e ∂ 1 S 1 , 0    z = z 1 = p, e ∂ ¯ 1 S ¯ 1 , 0    z = z ¯ 1 = p − 1 , e ∂ a S a, 0    z = z a = p p + ρ a , a 6 = 1 , ¯ 1 . 13 Prop ositio n 1 3. The disp ersionless L ax and Orlov–Schulman functions ar e of the fol lowing form z sg a a =          p + ¯ ℓ 1 , 0 + O ( p − 1 ) , p → ∞ , a = 1 ¯ ℓ ¯ 1 , 1 p − 1 + ¯ ℓ ¯ 1 , 0 + O ( p ) , p → 0 , a = ¯ 1 ¯ ℓ a, 1 p p + ρ a + ¯ ℓ a, 0 + O  p p + ρ a  − 1  , p → − ρ a , a 6 = 1 , m a = ( n 0 + sg ( a ) s a ) + ∞ X j =1 j t j a ¯ z j − 1 a +        P ∞ j =1 µ aj p − j , a = 1 , P ∞ j =1 µ aj p j , a = ¯ 1 , P ∞ j =1 µ aj  p p + ρ a  j , a 6 = 1 , ¯ 1 . Pr o of. Particular cases of (36) ar e ¯ M a (Ψ a ) = z dΨ a d z , ¯ L a (Ψ a ) = z sg a Ψ a , which together with (29) a nd (33 ) imply z sg a = ( e ¯ ∂ 1 S 1 , 0 + ¯ ℓ 1 , 0 + ¯ ℓ 1 , − 1 e − ¯ ∂ 1 S 1 , 0 + ¯ ℓ 1 , − 2 e − 2 ¯ ∂ 1 S 1 , 0 + · · · , a = 1 , ¯ ℓ a, 1 e ¯ ∂ a S a, 0 + ¯ ℓ a, 0 + ¯ ℓ a, − 1 e − ¯ ∂ a S a, 0 + ¯ ℓ a, − 2 e − 2 ¯ ∂ a S a, 0 + · · · , a 6 = 1 , z ∂ S a, 0 ∂ z = ( n 0 + sg( a ) s a ) + ∞ X j =1 j t j a z j + ∞ X j =1 ¯ µ aj e − j ¯ ∂ a S a, 0 . and the ev aluation of these expressions at z = z a gives the re sult. Therefore, ∂ x ( S b, 0 )    z = z b = log p , ∂ a ( S b, 0 )    z = z b = log  p p + ρ a  , a 6 = 1 , ¯ 1 , ( ∂ j a ( S b, 0 ))    z = z b = ¯ P j a  p p + ρ a  =: Ω j a , a 6 = 1 , ¯ 1 , ∂ j ¯ 1 ( S b, 0 )    z = z b = ¯ P j ¯ 1 ( p − 1 ) = : Ω j ¯ 1 , ( ∂ j 1 ( S b, 0 ))    z = z b = ¯ P j 1 ( p ) =: Ω j 1 , j > 1 . In this wa y we hav e d S b, 0 = m b d log z b + log ( p )d x + X a 6 = ¯ 1 , 1 log  p p + ρ a  d s a + X j ≥ 1 ,a ∈S Ω j a d t j a . As we will show at the e nd of this section the functions S b, 0 determine a solution of the zero -genus Whitham hierarch y with 2 N punctures in the T o da picture . 3.2 The disp ersionless limits of the string equations Let us consider o p er a tors o f the fo r m F a ( M a , L a ) = X i ≥ 0 ,j ∈ Z F aij M i a L j a , F a ( ¯ M a , ¯ L a ) = X i ≥ 0 ,j ∈ Z F aij ¯ M i a ¯ L j a . In o r der to formulate their disp er s ionless limits it is conv enient to ass ume that the co efficients s atisfy as ǫ → 0 that F aij = F aij, 0 ǫ i + O ( ǫ i +1 ) . 14 Recalling (36 ) and observing that  z d d z  i = z i d i d z i + i X i ′ =2  i ′ 2  z i ′ − 1 d i ′ − 1 d z i ′ − 1 , ∂ i Ψ a ∂ z i =  ǫ − i  ∂ S a, 0 ∂ z  i + O ( ǫ − i +1 )  Ψ a we get (Ψ a ) ← − − − − − − − − − − F a  z d d z , z sg a  = " X i ≥ 0 ,j ∈ Z F aij, 0 z sg( a ) j  ∂ S a, 0 ∂ z  i + O ( ǫ ) # Ψ a . Hence, F a, 0 ( z a , m a ) :=  lim ǫ → 0 (Ψ a ) ← − − − − − − − − − − F a  z d d z , z sg a  Ψ − 1 a  z = z a =          X i ≥ 0 ,j ∈ Z F aij, 0 z i +sg( a ) j a m i a , KP , X i ≥ 0 ,j ∈ Z F aij, 0 z sg( a ) j a m i a , T o da. W e define F a, 0+ :=        h lim ǫ → 0 F a ( M a , L a ) + (Ψ a ) Ψ a i z = z a , unbared cases h lim ǫ → 0 F a ( ¯ M a , ¯ L a ) + (Ψ a ) Ψ a i z = z a , bared cases Prop ositio n 1 4. Given F a ( M a , L a ) = X j ∈ Z f aj T j a , or F a ( ¯ M a , ¯ L a ) = X j ∈ Z ¯ f aj ¯ T j a , with f ai = f ai | 0 + O ( ǫ ) as ǫ → 0 , their disp ersionless limits ar e F a, 0 =                                             P j ∈ Z f aj | 0 ( p − q a ) j , a 6 = 1 , P j ∈ Z f 1 j | 0 ( p − q a ′ ) j a = 1 , KP,        P j ∈ Z f aj | 0 ( p − Q a ) j , a 6 = 1 , a ′ , P j ∈ Z f 1 j | 0 p j , a = 1 , P j ∈ Z f a ′ j | 0 p − j , a = a ′ , T o da ,        P j ∈ Z f aj | 0  p p + ρ a  j , a 6 = 1 , ¯ 1 , P j ∈ Z f 1 j | 0 p j , a = 1 , P j ∈ Z f ¯ 1 j | 0 p − j , a = ¯ 1 , Alternative T o da. Mor e over, F a, 0+ =                                             P j > 0 f aj | 0 ( p − q a ) j , a 6 = 1 , P j ≥ 0 f 1 j | 0 ( p − q a ′ ) j a = 1 , KP,        P j > 0 f aj | 0 ( p − Q a ) j , a 6 = 1 , a ′ , P j ≥ 0 f 1 j | 0 p j , a = 1 , P j > 0 f a ′ j | 0 p − j , a = a ′ , T o da,        P j > 0 f aj | 0  p p + ρ a  j − 1  , a 6 = 1 , ¯ 1 , P j ≥ 0 f 1 j | 0 p j , a = 1 , P j > 0 f ¯ 1 j | 0 p − j , a = ¯ 1 , Alternative T o da. 15 In p articular Ω j a = ( ( z sg( a ) j a ) + , KP, T o da, ( ¯ z sg( a ) j a ) + , Alternative T o da. Pr o of. The for m ulae follow from the identit y (Ψ a ) ← − − − − − − − − − − F a  z d d z , z sg a  = F a ( M a , L a )(Ψ a ) = X j ∈ Z f aj e j ∂ a S a, 0 + O ( ǫ ) Ψ a = F a ( ¯ M a , ¯ L a )(Ψ a ) = X j ∈ Z f aj e j ¯ ∂ a S a, 0 + O ( ǫ ) Ψ a . As a consequence Prop ositio n 1 5. If the string e quations (44) hold, their c orr esp onding disp ersionless limits F a, 0 ( z a , m a ) = X b ∈S F b, 0+ , ∀ a ∈ S (51) ar e satisfie d. Pr o of. It follows from Prop os itio n 8. These disp ersionless string equations ar e o f the t yp e co ns idered in [8 ] for the disp ersio nle s s Whitham hier- arch y . Moreov er, given a decomp osition S = I ∪ J into tw o disjoint subsets, w e may take P a, 0 =    z ℓ a a , a ∈ I , − m a ℓ a z ℓ a − 1 , a ∈ J , Q a, 0 =    m a ℓ a z ℓ a − 1 , a ∈ I , z ℓ a a , a ∈ J . The corr esp onding disp ersive string eq uations are E 11 ( X k ∈ ( I ∩ S ) L ℓ k C kk − X k ∈ ( J ∩ S ) ℓ − 1 k M L − ℓ k +1 C kk ) = E 11 ( X k ∈ ( I ∩ ¯ S ) ¯ L − ℓ k ¯ C kk − X k ∈ ( J ∩ ¯ S ) ℓ − 1 k ¯ M ¯ L ℓ k − 1 ¯ C kk ) , E 11 ( X k ∈ ( I ∩ S ) ℓ − 1 k M L − ℓ k +1 C kk + X k ∈ ( J ∩ S ) L ℓ k C kk ) = E 11 ( X k ∈ ( I ∩ ¯ S ) ℓ − 1 k ¯ M ¯ L ℓ k − 1 ¯ C kk + X k ∈ ( J ∩ ¯ S ) ¯ L − ℓ k ¯ C kk ) . If J = ∅ w e get E 11 N X k =1 L ℓ k C kk = E 11 N X k =1 ¯ L − ℓ k ¯ C kk , E 11 N X k =1 ℓ − 1 k M L − ℓ k +1 C kk = E 11 N X k =1 ℓ − 1 k ¯ M ¯ L ℓ k − 1 ¯ C kk . F o r p ositive integers ℓ a the disp ers ionless limits o f these disp ersive string equations describ e the algebr aic orbits of the genus 0 Whitham hier a rch y [24]. The first of these disper sive string equa tio ns g ives the multigraded reduction as discussed in [1]. W e r eturn to the eq uiv alence o f the alter native T o da and T o da pictures. First we notice that within alter - native T o da picture we have L a urent expa ns ions in Z a Z a := p p + ρ a = 1 − ρ a ζ a , ζ a := 1 p + ρ a . The functions Z j a are singular at p = − ρ a and lim p →∞ Z j a = 1, th us the linea r combinations of factors Z j a − 1 which app ear in the co nstruction of the Ω j a , lead to singular functions in p = − ρ a normalized to 0 a t infinity . Hence, if we expr ess z a as Laurent series in ζ a , the function Ω j a is just the s ingular part corresp onding to the 16 pro jection to pow er series in ζ a with non constant term. Thus, we re c over the T o da picture of the gen us 0 Whitham hierarchy (see Appendix A). Given op erators P a and Q a as in Pr o po sition 3 we get for the comm utato r (Ψ a ) ← − − − − − [ P a , Q a ]  z d d z , z sg a  = (Ψ a )  ← − − − − − − − − − − P a  z d d z , z sg a  , ← − − − − − − − − − − − Q a  z d d z , z sg a   =  X i 1 ,i 2 ≥ 0 j 1 ,j 2 ∈ Z P ai 1 j 1 , 0 Q ai 2 j 2 , 0 sg( a )( i 2 j 1 − i 1 j 2 ) z sg( a )( j 1 + j 2 )+ i 1 + i 2 − 1  ∂ S a, 0 ∂  i 1 + i 2 − 1 ǫ + O ( ǫ 2 )  Ψ a so that for the K P picture we find  lim ǫ → 0 (Ψ a ) ← − − − − − − − ǫ − 1 [ P a , Q a ]  z d d z , z  Ψ − 1 a  z = z a = X i 1 ,i 2 ≥ 0 j 1 ,j 2 ∈ Z P ai 1 j 1 , 0 Q ai 2 j 2 , 0 sg( a )( i 2 j 1 − i 1 j 2 ) z sg( a )( j 1 + j 2 )+ i 1 + i 2 − 1 a m i 1 + i 2 − 1 a = { P 0 ,a , Q 0 ,a } 0 , while for the T o da picture w e have  lim ǫ → 0 (Ψ a ) ← − − − − − − − ǫ − 1 [ P a , Q a ]  z d d z , z  Ψ − 1 a  z = z a = X i 1 ,i 2 ≥ 0 j 1 ,j 2 ∈ Z P ai 1 j 1 , 0 Q ai 1 j 1 , 0 sg( a )( i 2 j 1 − i 1 j 2 ) z sg( a )( j 1 + j 2 ) a m i 1 + i 2 − 1 a = { P 0 ,a , Q 0 ,a } 1 . Thu s, [ P a , Q a ] = ǫ = ⇒ { P a, 0 , Q a, 0 } 0 = 1 or { P a, 0 , Q a, 0 } 1 = 1 . App endix A: Wh itham hierarc hies in the zero gen us case The zero -genus Whitham hierarchies [24] are systems of flows on a phase sp ac e c M 0 of data asso c ia ted to algebraic Riemann surfaces of ge nus 0. The p o in ts of c M 0 are given (Γ , Q a , z − 1 a ), wher e Γ is an algebra ic Riema nn sur face of g en us 0, Q a are N p oints (punctures) o f Γ and z − 1 a denote lo cal co or dinates aro und each Q a such that z − 1 a ( Q a ) = 0. In order to for m ulate Whitham flows o n c M 0 it is conv enient to introduce a meromorphic function p = p ( Q ) on Γ such that the lo c a l co o rdinates have asymptotic expans io ns of the form z a =                p + ℓ 1 , 0 + ∞ X n =1 ℓ 1 , − n p n , a = 1 , ℓ a, 1 p − q a + ∞ X n =0 ℓ a, − n ( p − q a ) n , a = 2 , . . . , N , (52) where p ( Q a ) = q a with q 1 = ∞ . In genera l the p oints of the phase space c M 0 are characterized by an infinite nu mber o f par ameters w := ( w i ) of the set ( q a , ℓ a,n ). Howev er, under appropriate reduction conditions on the form of Γ only a finite num b er of these par ameters ar e indep endent and cons titute a co ordinate system for c M 0 . Example: Algebraic or bits If we restrict to zero-genus Riema nn surfaces Γ of the form λ = p n 1 + n 1 − 1 X n =0 u 1 n p n + N X i =2 n q X n =1 u in ( p − q i ) n , (53) 17 we may take Q a = ( λ a , p a ) = ( ∞ , q a ) ( a = 1 , . . . , q ) , with corresp onding lo cal co ordina tes given by z a = λ 1 /n a . The function p ( λ, p ) = p is mer omorphic on Γ a nd the lo ca l co ordinates have a symptotic expansions of the form (5 2). In this case the p oints of the phas e space c M 0 are characterized by the parameter s w = ( { q a } N a =2 , { u 1 n } n 1 n =0 , . . . , { u N n } n N N =1 ) The Whitham flows w ( t ) = ( w i ( t )) are introduced through sets Ω := { Ω A ( w , p ) } of functions, with mero- morphic differentials ∂ p Ω A ( w , p ) d p , which sa tisfy the conditions: 1. One o f the functions Ω A 0 is indep endent of the data w . 2. Ther e ex is t lo cal functions S a ( t , z a ) around the puncture s s a tisfying ∂ A S a ( t , z a ) = Ω A ( w ( t ) , p ( t , z a )) . (54) Here ∂ A := ∂ / ∂ t A and t denotes the s et o f flow pa rameters t A . The first condition only demands to include a function of the form Ω( p ) in Ω . On the o ther ha nd, it is o bvious that the seco nd conditio n is satisfied if and o nly if the following Z akharov–Shabat equatio ns are sa tisfied ∂ B Ω A − ∂ A Ω B + { Ω A , Ω B } = 0 , (55) where ∂ t := P i ∂ t w i ∂ w i for t = t A , t B , and { , } denotes the Poisson brack et { F , G } := ω ( p )  ∂ p F ∂ x G − ∂ x F ∂ p G  , ω ( p ) := ( ∂ p Ω A 0 ( p )) − 1 , x := t A 0 . W e may write (54) a s d S a = m a d Ω A 0 ( z a ) + X A Ω A d t A , m a := ω ( z a ) − 1 ∂ S a ∂ z a , (56) which implies d Ω A 0 ( z a ) ∧ d m a = X A d Ω A ∧ d t A , (57) and by equating the coe fficie n ts of d p ∧ d x in both mem b ers of (57) yields { z a , m a } = ω ( z a ) . (58) Moreov er , if we ide ntify the co efficients o f d p ∧ d t A and d x ∧ d t A in (57) we get ( ∂ p z a ∂ A m a − ∂ A z a ∂ p m a = ω ( z a ) ∂ p Ω A , ∂ x z a ∂ A m a − ∂ A z a ∂ x m a = ω ( z a ) ∂ x Ω A , so that taking (58) int o account we deduce the system of Lax e q uations ∂ A z a = { Ω A , z a } , ∂ A m a = { Ω A , m a } . (59) As it was s hown in [ ? ]-[27] imp ortant cla s ses of solutions of the zer o -genus Whitham hiera rch y can be obtained from systems o f cano nical pair s o f constrains (string equa tio ns) o f the form ( P 1 ( z 1 , m 1 ) = P 2 ( z 2 , m 2 ) = · · · = P N ( z N , m N ) , Q 1 ( z 1 , m 1 ) = Q 2 ( z 2 , m 2 ) = · · · = Q N ( z N , m N ) , (60) where ( P a , Q a ) are N pair s of ca nonically co njugate functions { P a ( p, x ) , Q a ( p, q ) } = ω ( p ) . (61) In particula r this t yp e of methods a pplies for finding solutions for algebra ic o r bits. Indeed these solutions a re asso ciated to string eq uations generated by P a ( p, x ) = p n a , Q a ( p, x ) = x n a p n a − 1 + f a ( p ) . (62) 18 The KP picture The KP pictur e o f the zero -genus Whitham hier arch y with N punctures [24] is for m ulated by as suming ℓ 1 , 0 ≡ 0 in the asy mptotic expa nsions (52 ) and by taking the following functions Ω A Ω na :=      ( z n a ) ( a, +) , n ≥ 1 , − log( p − q a ) , n = 0 , a = 2 , . . . , N . (63) Here ( · ) ( a, +) stand for the pro jectors on the subspaces ge ne r ated by { p n } ∞ n =0 (case a = 1) and { ( p − q a ) − n } ∞ n =1 (cases a ≥ 2). In this case A 0 = (1 , 1) , x = t 1 , 1 , Ω A 0 = p, and the Poisson bra ck et is g iven by { F , G } := ∂ p F ∂ x G − ∂ x F ∂ p G. The functions Ω A satisfy the c o mpatibilit y conditio ns (5 5) s o that there exist functions S a such that d S a = m a d z a + p d x − X a 6 =1 log( p − q a ) d t 0 a + N X a =1 X n ≥ 1 Ω na d t na . (64) The T o da picture A simple redefinition of the meromorphic function p ( Q ) used to define the K P flows o f the Whitham hier arch y with N punctures supplies a different picture (the T o da pictur e) of the hierar c hy . Indeed if we set p T o da = p KP − q a 0 , for a given index a 0 , then now u 1 , 1 = q a 0 and we may take A 0 := (0 , a 0 ) , x := − t 0 ,a 0 , Ω A 0 := − log p. Thu s the Poisson brack et is given by { F , G } := p  ∂ p F ∂ x G − ∂ x F ∂ p G  , and the functions S a satisfy d S a = m a d log z a + log p d x − X a 6 =1 ,a 0 log( p − q a ) d t 0 a + N X a =1 X n ≥ 1 Ω na d t na . ( 65 ) App endix B: Pro of of Prop osition (6) The pro of of Pro po sition (6) r equires the following Lemma. Lemma 1. 1. Given T = P j ∈ Z c j T j a ∈ t a , a 6 = 1 , then ( T ( E 11 W ) = T > ( W 1 k ) E 1 k + T ≤ ( W 11 ) E 1 + g − W 0 , T ( E 11 ¯ W ) = g + ¯ W 0 , a = k 6 = 1 , (66) ( T ( E 11 W ) = T ≤ ( W 11 ) E 11 + g − W 0 , T ( E 11 ¯ W ) = T > ( ¯ W 1 k ) E 1 k + g + ¯ W 0 , a = ¯ k , (67) 2. Given T = P j ∈ Z c j T j l ∈ t 1 , then ( T ( E 11 W ) = T ≥ ( W 11 ) E 11 + T < ( W 1 l 0 ) E 1 l 0 + g − W 0 , T ( E 11 ¯ W ) = g + ¯ W 0 , a 0 = l 0 6 = 1 , (68) ( T ( E 11 W ) = T ≥ ( W 11 ) E 11 + g − W 0 , T ( E 11 ¯ W ) = T < ( ¯ W 1 l 0 ) E 1 l 0 + g + ¯ W 0 , a 0 = ¯ l 0 , (69) 19 3. Given T = P j ∈ Z c j ( ¯ T j a − 1) + c 0 ∈ ¯ t a , a 6 = ¯ 1 , 1 , t hen ( T ( E 11 W ) = T > ( W 1 k ) E 1 k + c 0 W 11 E 11 + g − W 0 , T ( E 11 ¯ W ) = T < ( ¯ W 11 ) E 11 + g + ¯ W 0 , a = k 6 = 1 , (70) ( T ( E 11 W ) = c 0 W 11 E 11 + g − W 0 , T ( E 11 ¯ W ) = T > ( ¯ W lk ) E lk + T < ( ¯ W 11 ) E 11 + g + ¯ W 0 , a = ¯ k 6 = ¯ 1 , (71) 4. Given T = P j ∈ Z c j ¯ T j 1 ∈ ¯ t 1 , then ( T ( E 11 W ) = T ≥ ( W 11 ) E 11 + g − W 0 , T ( E 11 ¯ W ) = T < ( ¯ W 11 ) E 11 + g + ¯ W 0 , (72) 5. Given T = P j ∈ Z c j ( ¯ T j ¯ 1 − 1) + c 0 ∈ ¯ t ¯ 1 , with a 0 6 = 1 , then ( T ( E 11 W ) = T < ( W 1 l 0 ) E 1 l 0 + c 0 W ll E 11 + g − W 0 , T ( E 11 ¯ W ) = T > ( ¯ W 11 ) + g + ¯ W 0 , a 0 = l 0 6 = 1 , (7 3) ( T ( E 11 W ) = c 0 W 11 E 11 + g − W 0 , T ( E 11 ¯ W ) = T > ( ¯ W 11 ) E 11 + T < ( ¯ W 1 l 0 ) E 1 l 0 + g + ¯ W 0 , a 0 = ¯ l 0 , (74) 6. Given T = P j ∈ Z c j ¯ T j ¯ 1 ∈ ¯ t ¯ 1 with a 0 = 1 , then ( T ( E 11 W ) = T ≤ ( W 11 ) E 11 + g − W 0 , T ( E 11 ¯ W ) = T > ( ¯ W 11 ) E 11 + g + ¯ W 0 . (75) Pr o of. W e only prove 1 ) since the others r elations a r e pr ov en similar ly . F ro m (66 ) observe that T j k ( E 11 W ) = T j k ( E 11 S ) T j k ( W 0 ) = T j k ( E 11 S )( E kk Λ j + I N − E kk − E 11 + E 11 Λ − j ) W 0 = ( T j k ( S 1 k ) E 1 k Λ j + E 11 T j k ( S )( I N − E kk − E 11 ) + T j k ( S 11 ) E 11 Λ − j ) W 0 , T j k ( E 11 ¯ W ) = T j k ( E 11 ¯ S ) ¯ W 0 and therefor e T j k ( E 11 W ) E k ′ k ′ ∈ g − W 0 , k ′ 6 = k , 1 , T j k ( E 11 W ) E kk = T j k ( S 1 k ) E 1 k Λ j W 0 ∈ g − W 0 if j ≤ 0 , T j k ( E 11 W ) E 11 = T j k ( S 11 ) E 11 Λ − j W 0 ∈ g − W 0 if j > 0 , T j k ( E 11 ¯ W ) ∈ g + ¯ W 0 . Now, we chec k (67). Notice that T j ¯ k ( E 11 W ) = T j ¯ k ( E 11 S ) T j k ( W 0 ) = T j ¯ k ( E 11 S )( I N − E 11 + E 11 Λ − j ) W 0 = ( E 11 T j ¯ k ( S )( I N − E 11 ) + T j ¯ k ( S 11 ) E 11 Λ − j ) W 0 , T j ¯ k ( E 11 ¯ W ) = T j ¯ k ( E 11 ¯ S ) T j ¯ k ( ¯ W 0 ) = T j ¯ k ( E 11 ¯ S )( E kk Λ − j + I N − E kk ) ¯ W 0 = ( E 11 T j ¯ k ( ¯ S )( I N − E kk ) + T j ¯ k ( ¯ S 1 k ) E 1 k Λ − j ) ¯ W 0 , and therefor e T j ¯ k ( E 11 W ) E k ′ k ′ ∈ g − W 0 , k ′ 6 = 1 , T j ¯ k ( E 11 W ) E ll = T j ¯ k ( S ll ) E 11 Λ − j W 0 ∈ g − W 0 , if j > 0 , T j ¯ k ( E 1 ¯ W ) E k ′ k ′ ∈ g + ¯ W 0 , k ′ 6 = k , T j ¯ k ( E 11 ¯ W ) E kk = T j ¯ k ( ¯ S 1 k )Λ − j ¯ W 0 ∈ g + ¯ W 0 , if j ≤ 0 , 20 Pr o of of Pr op osition 6. The pro of of these r esults relies on the previo us Lemma 1 and Prop os itions 2, 5. Let us go into details. W e first co nsider (40). F rom (66) we find for k 6 = 1 F ( M k , L k ) > ( E 11 W ) = F ( M k , L k )( W 1 k ) E 1 k + g − W 0 , so that, as we prove in Prop osition 5, we deduce F ( M k , L k ) > ( E 11 W ) = E 11 ( F ( M , L ) C kk ) W + g − W 0 = E 11 ( F ( M , L ) C kk ) + W + g − W 0 . Therefore, R := F ( M k , L k ) > ( E 11 W ) − E 11 ( F ( M , L ) C kk ) + W ∈ g − W 0 and from (8) a nd (66) we get Rg = F ( M k , L k ) > ( E 11 ¯ W ) − E 11 ( F ( M , L ) C kk ) + ¯ W ∈ g + ¯ W 0 . so that Pr opo sition 2 implies the first formula en (4 0). Now, fr om (70) we get for k 6 = 1 F ( ¯ M k , ¯ L k ) > ( E 11 W ) = F ( ¯ M k , ¯ L k )( W 1 k ) E 1 k + g − W 0 . Hence Pro po sition 5 ensure s tha t R := F ( ¯ M k , ¯ L k ) > ( E 11 W ) − E 11 ( F ( M , L ) C kk ) + W ∈ g − W 0 and from (70) we deduce Rg = F ( M k , L k ) > ( E 11 ¯ W ) − E 11 ( F ( M , L ) C kk ) + ¯ W ∈ g + ¯ W 0 . In this wa y , Prop ositio n 2 leads to the last formula in (40). The pro of o f (41) follows simila rly . Ac kno wledgemen ts The authors wish to tha nk the Spanish Ministerio de Ciencia e Inno v aci´ on, resea rch pro ject FIS200 8-002 0 0, and acknowledge the suppo r t received from the E urop ean Science F oundatio n (ESF) and the activity entitled Metho ds of Inte gr able Systems, Ge ometry, Applie d Mathematics (MISGAM). MM wish to thank P rof. v an Mo erb eke and Pr of. Dubrovin for their warm hospitality , acknowledge economical supp ort fro m MISGAM and SISSA and r e c kons different conversations with P . v an Mo er beke, T. Gr av a, G. Carlet and M. Caffasso . MM also ackno wledges to P r of. L iu fo r his invitation to visit the China Mining and T echnology Universit y at Beijing. References [1] M. Ma˜ nas, L. Mart ´ ınez Alo nso, and C. ´ Alv a r ez-F ern´ andez, The mu lti-c omp onent 2D T o da Hier ar chy: dis- cr ete flows and string e quations a rXiv:0 809.27 20 . T o b e published in Inv erse Problems. [2] K . Ueno and K . T ak asa ki, Adv. Stud. Pure Ma th. 4 (198 4) 1. [3] P . Di F ra ncesco, P . Ginsparg and Z. Zinn-Justin, Phys. Rept. 254 (199 5) 1. [4] B. Eynard, An Intr o duction to Ra ndom Matric es , lectures g iven at Saclay , October 2000, ht tp:// w w w- spht .cea .fr/articles/ t01/01 4 / . [5] L. Bono ra a nd C. S. Xiong, Phys. Lett. B34 7 (1 9 95) 4 1 . [6] K . T ak asa k i, Commun. Math. Phys. 181 (1 996) 1 31. [7] K . T ak asa k i and T. T akebe, Rev. Math. Phys. 7 (19 9 5) 74 3 . [8] P . B. Wiegmann a nd A. Z abro din, Comm. Math. P h ys . 213 (200 0) 523 . M. Mineev-W einstein, P .B. Wiegma nn, and A. Zabro din, P hys. Rev. Lett. 8 4 (2 000) 5 106 I. M. Krichev er, M. Mineev-W einstein, P . B. Wiegma nn, and A. Zabro din, Physica D19 8 (2 0 04) 1 . I. M. Krichev er, A. Marsha ko v and A. Zabro din, Comm. Math. Phys. 259 (2005 ) 1. M. Ma ˜ na s , L. Mar t ´ ınez Alonso, and E. Medina , J. Phys. A: Math. Gen. 35 (200 2) 401 . 21 F. Guil, M. Ma ˜ nas, a nd L. Mart ´ ınez Alonso, J. Phys. A: Ma th. Gen. 3 6 (2 0 03) 4047 . F. Guil, M. Ma ˜ nas, a nd L. Mart ´ ınez Alonso, J. Phys. A: Ma th. Gen. 3 6 (2 0 03) 6457 . L. Ma rt ´ ınez Alonso and M. Ma ˜ nas, J. Math. Phys. 44 (20 03) 3294 . M. Ma ˜ na s , J. P h ys. A: Math. Gen. 37 (2 004) 9 195. M. Ma ˜ na s , J. P h ys. A: Math. Gen. 37 (2 004) 1 1191. [9] E . Date, M. Jimbo, M. Ka shiwara, and T. Miw a, J. Phys. So c. Japan 40 (1981 ) 3 8 06. M. J. Bergvelt and A. P . E . ten K ro o de, Pacific J. Math. 171 2 3-88 (19 9 5). M. Ma ˜ na s , L. Mar t ´ ınez Alonso, and E. Medina , J. Phys.A: Math. Gen. 33 (200 0) 2 871. M. Ma ˜ na s , L. Mar t ´ ınez Alonso, and E. Medina , J. Phys.A: Math. Gen. 33 (200 0) 7 181. V. G. Kac and J . W. v an de Leur , J. Math. Phys. 44 (2 0 03) 3 2 45. [10] M. Bertola, B. Eynard and J. Harnad, Co mm. Math. Phys. 229 (200 2) 7 3. [11] M. Bertola, B. Eynard and J. Harnad, Co mm. Math. Phys. 243 (200 3) 1 93. [12] A. B. J . Kuijlaars and K. T-R McLa ughlin , J. Co mput. Appl. Math. 178 (20 05) 31 3. [13] M. Adler and P . V an Mo e rb e ke, Co mm. Pure and Appl. Math. J . 50 (1997) 24 1 ; Ann. Math. 1 49 (1 999) 921 . [14] M. Adler, P . v a n Mo er beke, a nd P . V anha eck e, Commun. Math. P h ys . 28 6 (20 09) 1 . M.Adler. J. Del´ epine and P . v an Mo erb eke, Comm. Pur e and Appl. Ma th. 62 (2009 ) 334. [15] P . M. Bleher a nd A. B. J. Kuijlaar s, Int. Math. Research Notices 200 4 (2004 ) 109 ; Comm. Math. Phys. 252 (2004 ) 43; Comm. Math. P h ys. 27 0 (2 007) 4 81. [16] L . Mart ´ ınez Alonso, E. Medina, Multiple ortho gonal p olynomia ls, string e quations and the lar ge- n limit arXiv: 0812.3 817 . T o b e published in J . Phys. A. [17] W. V an Assche, J. S. Geronimo , a nd A. B. J. K uijlaars, Riema nn -Hilb ert pr oblems for multiple ortho gonal p olynomia ls , In: Sp e cial F unctions 2000: Curr ent Persp e ctives and F ut u r e Dir e ctions (J. Bustoz et al., eds.), Kluw er, Dordrech t, (2 001) 23. [18] E . Dae ms a nd A. B. J. Kuijlaa rs, J. of Approx. Theory 146 (20 07) 91. [19] E . Daems , Asymptotics for non-int erse cting Br ownian motions using multiple ortho gonal p olynomial s , Ph.D. thesis, K.U.Leuven, 2 0 06, URL http://hdl.handle.net/1979 / 324 . [20] A.S. F ok as, A.R. Its and A.V. Kitaev, Commun. Math. P h ys. 147 (199 2) 3 95. [21] P . Deift, Ortho gonal Polynomials and R andom Matric es: A R iemann-Hilb ert appr o ach , Cour a nt Lecture Notes in Mathematics V ol. 3 , Amer. Math. So c., Providence R.I. 1 9 99. [22] P . Deift and X. Z hou, Ann. Math. 2 (199 3) 137 . [23] A. Y u O rlov and E. I. Sch ulman, Lett. Math. Phys. 12 (19 86) 171. [24] I . M. Krichev er, Comm. Pure. Appl. Math. 4 7 (1 994) 437 . [25] K . T a k asa ki, “ Disper sionless in tegra ble hierar chies revis ited”, talk deliv ere d at SISSA at septem b er 2005 (MISGAM progra m). [26] K . T ak asa ki and T. T akeb e , Physica D2 3 5 (2 0 07) 109. [27] L . Mar t ´ ınez Alonso, E. Medina a nd M. Ma ˜ nas, J. Math. P h ys . 47 (2 006) 0 83512 . M. Ma ˜ na s , E. Medina a nd L. Mart ´ ınez Alonso , J. Phys.A: Math. Gen. 39 (200 6) 2 349. 22