The double scaling limit method in the Toda hierarchy

The double scaling limit metho d in the T o da hierarc h y ∗ L. Mart ´ ınez Alonso 1 and E. Medina 2 1 Dep artamento de F ´ ısic a T e´ oric a II, Universidad C o mplutense E28040 Madrid, Sp ain 2 Dep artamento de M a tem´ atic as, Universidad de C´ adiz E11510 Puerto R e al, C´ adi z , Sp ain Abstract Critical p oints of semiclassical expans ions of solutions to the disp er s ionful T o da hierarc h y are considered and a double scaling limit metho d of regularization is formulate d . The analogues of the critical p oints characte rized by the strong conditions in the Hermitian matrix mo d el are analyzed and the prop ert y of doubling of equations is pr o v ed. A wide family of sets of critical p oints is in tro d uced and the corresp ond ing double scaling limit expansions are d iscu ssed. Key wor ds: T o da hierarc hy . Double scaling limit. Hermitian m atrix mo del. P ACS numb e r: 02.30. Ik. ∗ Partially suppo r ted by MEC pro ject FIS2005- 00319 and ESF pro gramme MISGAM 1 1 In tro ductio n In a recen t w ork [1] w e considered the large N expans ion of the Hermitian matrix mo del Z N ( t ) = Z H . exp  N g tr( X k ≥ 1 t k H k )  , H = H † , (1) from the p oint of view of the disp ersionful T o da hierarc hy [2] ǫ ∂ L ∂ t j = [( L j ) + , L ] , L = Λ + u + v Λ − 1 , Λ := e xp ( ǫ ∂ x ) , ǫ := g/ N , (2) where ( ) + denotes the p olynomial part in Λ. If t h e equilibr ium mea su re in the large N limit is supp orted on a single int erv al (one-cut case), th en the partition function b ecomes a tau-fu nction τ ( x, t ) of th is hierarc h y at x = g . T h e corresp onding solutio n of (2) is c h aracterized b y t wo string equations whic h can b e solv ed in terms of a semiclassic al expansion u = X k ≥ 0 ǫ k u ( k ) ( x, t ) , v = X k ≥ 0 ǫ 2 k v (2 k ) ( x, t ) . (3) The co efficien ts ( u ( k ) , v (2 k ) ) can b e expressed as rational fun ctions of the leading terms ( u (0) , v (0) ) and their deriv ativ es with r esp ect to x . A metho d to d etermine the expansions (3) wa s pro vided in [1]. It uses t wo functions R := X k ≥ 0 R k ( u, v ) z k , T := X k ≥ 0 T k ( u, v ) z k , R 0 = T 0 = 1 , whic h are relate d to the resolv ent R := ( z − L ) − 1 of the Lax op erator L thr ou gh th e equation  z − 2 R (1 + T ) Λ  R + = R , so that they generate the tr ace densities Res( R Λ − j ) for j ≥ 0, where Res( P c k Λ k ) := c 0 . These functions are determined by the equations          T + T [ − 1] + 2 z ( u − z ) R = 0 , T 2 − 4 z 2 v [1] R R [1] = 1 . (4) where w e are denoting F [ r ] ( x ) := F ( x + r ǫ ) . As it w as s ho wn in [1] th e string equations for ( u, v ) reduce to the system            I γ dz 2 π i z V z ( z , t ) R ( z , u, v ) = 0 , I γ dz 2 π i V z ( z , t ) T [ − 1] ( z , u, v ) = − 2 x, (5) 2 where V ( z , t ) := P ∞ j =1 t j z j , and γ is a large p ositiv ely orien ted closed path. F or ǫ = 0 the system (5) reduces to a pair of h o dograph t yp e equations f v ( t , u (0) , v (0) ) = 0 , f u ( t , u (0) , v (0) ) = 2 x, (6) where f ( t , u, v ) := 1 2 π i I γ dz V z ( z , t )  ( z − u ) 2 − 4 v  1 2 = − 2 v t 1 − 4 u v t 2 + . . . . (7) These equations determine ( u (0) , v (0) ) and represent the planar limit contribution to the partition function of the Hermitian mo d el [6]. Ho wev er, near critical p oints of (6) the functions ( u (0) , v (0) ) are multi-v alued and ha ve singular x -deriv ativ es ( gr adient c atastr ophe ), so that the semiclassical expansion (3) breaks do wn. In terms of the Hermitian matrix mod el this s ituation corresp onds to the critical p oints of the large N expansion of Z N ( t ) and, as it is w ell-kno wn [3]-[12], a double sc aling limit metho d of regularization is av ailable. This metho d leads to an imp ortan t nonp erturb ativ e approac h to t w o-dimensional quan tum gra vity . Recen t researc h [13]-[16] shows that the double scaling limit expansions are also r elev an t in the asymptotic analysis of solutio n s of disp ersionful in tegrable systems. They supp ly goo d appro xi- mations in the trans ition regions fr om the semiclassical to the oscillatory regimes afte r the time of gradien t catastrophe. These applications motiv ate the interest in reconsidering the double scaling limit metho d from the p oin t of view of the theory of string equatio ns for integrable systems of dis- p ersionfu l type. The form ulation of the double scali n g limit within the theory of the d isp ersionf u l T o da hierarc hy wa s a lready addressed in [1 ], but only the case corresp ondin g to u = 0 and e ven p oten tials V ( − z ) = V ( z ) w as analyzed. Th e p u rp ose of the pr esent pap er is to analyze the general case th at exhibits a m u c h r ic her structure of c ritical b ehavio ur s (see [ 10]-[12 ] for its relev ance in the Hermitian matrix mo d el of r andom surfaces and quant u m gra vit y). According to the strategy of the double scaling m etho d we lo ok at the systems (4)-(5) as singular p erturb ation problems in the s mall p arameter ǫ and consider th e semiclassica l expansions near critical p oints as outer expansions of the solutions of (4)-(5). Consequent ly the problem consists in c haracterizing appropriate inner expansions x = x c + ǫ 2 n x, u = u c + X j ≥ 2 ǫ j u ( j ) , v = v c + X j ≥ 2 ǫ j v ( j ) , ǫ = ǫ 2 n +1 (8) with co efficients dep endin g on stretc h ed v ariables ( x, t ). This t yp e of expansions w ere determined in the theory of random m atrix mo dels through sev eral metho d s of analysis of d iscrete string equations for orthogonal p olynomials in the large N limit. T he presen t w ork pro vides an alternativ e sc heme based on the c haracterization of solutions R = X j ≥ 0 ǫ j R ( j ) , T = X j ≥ 0 ǫ j T ( j ) , of the resolv en t trace equ ations ( 4 ) of the d isp ersionfu l T o da hierarc h y . This sc heme establishes the existence of the series (8) to all orders in ǫ an d leads to the c haracterization of the s ublead- ing co efficien ts ( u (2) , v (2) ) in te r ms of ordinary differenti al equations inv olving the Gel’fand-Dikii p olynomials of the resolv ent trace expansion of th e Sc hr¨ odinger op erator. W e analyze tw o differen t classes of critical p oints. F irstly , w e consider the sets C n of criti cal p oints wh ic h un d erlie the Hermitian matrix mo del of quant u m gra vity [10]-[11] and we determine 3 inner expansions (8) which satisfy the doubling pr op erty c ± G n, ± ( u ± ) = x, u ± := v (2) ± √ v c u (2) , with G n, ± b eing the Gel’fand-Dikii p olynomials. Secondly , w e c haracterize a wide family of sets C lmn of critical p oints where l ≥ 1 , m = 1 , . . . , l and n ≥ 2, describ e the branc hing b eha vior of ( u (0) , v (0) ) near the critical p oin ts. Th en w e dev elop the double scaling limit metho d for the case n = 2 with stretc hed v ariables x = x c + ¯ ǫ 4 ¯ x, t j = t c,j + ¯ ǫ 4 ¯ t j , j ≥ 1 , ¯ ǫ = ǫ 1 / 5 . F or C 112 w e fi nd that the subleading co efficien ts satisfy v (2) = ± √ v c u (2) , with v (2) b eing determined b y the P ainlev ´ e I equatio n . F or the other cases C lm 2 , l ≥ 2, we find again the doubling pr op ert y . Indeed, the fun ctions u ± := v (2) ± √ v c u (2) , turn out to b e determined b y a p air of decoupled P ainlev ´ e I equ ations. The double scaling limit exp ansions at critical p oin ts of the types C lmn can b e applied to the asymptotic analysis of solutions of the disp ersionful T o da hierarch y . As an illustrativ e examp le w e end our w ork by applying our results to regularize some critical processes of ideal models of Hele-Sha w flows [18]-[20] based on the T o d a hierarc hy . Our sc heme ca n b e form ulated with other sca ling a ns ¨ atze f or stretc hed v ariables li ke those of Galilea n t yp e used in [13]-[16]. F u r thermore, it can b e applied to the s tudy of the double scaling limit of the normal matrix mod el wh ich is related to another solution of the tw o-dimens ional disp ers ionf ul T o da hierarch y [21]-[23] determined by a pair of string equations. W e also exp ect the methods of t h is p ap er to b e applicable to disp ersionful versions of the m ulti-comp onen t KP hierarc hies w hic h arise in random matrix theory and Dyson Bro wn ian motions [26]. The la yout o f the pap er is as follo ws: in section 2 w e sho w ho w in ner expans ions of the gen- erating functions R and T can b e determined from th e resolv ent trac e equations (4). In sect ion 3 we charact erize the double scaling limit expansions at critical p oin ts satisfying strong cond i- tions. Finally , we dev ote section 4 to in tro d u ce families C lmn of critical p oint s and to dev elop th e corresp ondin g d ouble scaling limit metho d . 2 Inner e xpansions o f the re s olv en t function s Let us consider the ho d ograph system (6) f v = 0 , f u = 2 x. (9) Notice that the fun ction f d efi ned in (7) satisfies f uu − v f vv = 0 . (10) 4 The h o dograph sys tem determin es a solution of the disp ersionless T o d a hierarch y pro vided that the implicit function theorem applies for obtaining ( u ( x, t ) , v ( x, t )) from (9). F or example (9) imp lies u t 1 = v x , v t 1 = v u x , so that v xx − (log v ) t 1 t 1 = 0 , whic h is the h yp erb olic v ersion of the long wa ve limit of th e T o da lattice equation. It is also easy to see that the solutions of (9) verify u t 2 = 2 ( u u t 1 + v t 1 ) , v t 2 = 2 ( v u ) t 1 , whic h is the disp ersionless limit of the fo cusing nonlinear Sc hr ¨ odinger equation [16]. The set of critical p oin ts ( x c , t c , u c , v c ) of (9) is c haracterized b y the system f v = 0 , f u = 2 x f 2 uv − f uu f vv = 0 , (11) Our main aim in this w ork is to c h aracterize inner expansions for solutio ns ( u, v ) of the system (4)-(5) near critical p oin ts of (9). 2.1 Symmetric v ariables Let us start b y analyzing the resolv ent trace system (4). It is form in v arian t und er the change of v ariables x = x c + ǫ 2 n x, ǫ = ǫ 2 n +1 , n ≥ 2 . (12) Indeed since ǫ ∂ x = ǫ ∂ x w e can rewr ite (4) as          T + T [ − 1] + 2 z ( u − z ) R = 0 , T 2 − 4 z 2 v [ 1] R R [1] = 1 , (13) where w e are denoting F [ ¯ r ] ( ¯ x ) := F ( ¯ x + r ¯ ǫ ) . W e will pr o v e th at if ( u, v ) are expansions of the form u = u c + X j ≥ 2 ǫ j u ( j ) , v = v c + X j ≥ 2 ǫ j v ( j ) , (14) then the solution of (13) can b e expressed as T = X j ≥ 0 ǫ j T ( j ) , R = X j ≥ 0 ǫ j R ( j ) , (15) where the co efficien ts ( T ( j ) , R ( j ) ) are differentia l p olynomials in the co efficien ts of the expansions of ( u, v ). 5 An imp ortant prop ert y of the com bined system (4)-(5) is its inv ariance un der the transformation b u ( ǫ, x ) := u ( − ǫ, x + ǫ ) , b v ( ǫ, x ) := v ( − ǫ, x ) , (16) b T ( ǫ, z , x ) := T ( − ǫ, z , x + 2 ǫ ) , b R ( ǫ, z , x ) := R ( − ǫ, z , x + ǫ ) . Th u s it is con v enient to int r o duce the symmetric v ariables w ( ǫ ) := u ( ǫ, x − ǫ 2 ) = u [ − 1 / 2] ( ǫ ) , (17) U ( ǫ ) := T ( ǫ, z , x − ǫ ) = T [ − 1] ( ǫ ) V ( ǫ, z , x ) := R ( ǫ, z , x − ǫ 2 ) = R [ − 1 / 2] ( ǫ ) , (18) whic h transf orm under (16) in the simple form b w ( ǫ ) = w ( − ǫ ) , b U ( ǫ ) := U ( − ǫ ) , b V ( ǫ ) = V ( − ǫ ) . (19) In terms of ( w, v , U , V ) the system (13) r eads          U [ 1] + U + 2 z ( w [ 1 / 2] − z ) V [1 / 2] = 0 , U 2 − 4 z 2 v V [ − 1 / 2] V [1 / 2] = 1 . (20) Moreo v er, from (20) one easily d educes the follo wing linear equation for U ( z − w [ − 1 / 2] ) ( z − w [1 / 2] ) ( z − w [3 / 2] ) ( U [ 1] − U ) = v [1] ( z − w [ − 1 / 2] ) ( U [ 2] + U [1] ) (21) − v ( z − w [ 3 / 2] )) ( U [ − 1] + U ) . It can b e expressed as  X k ≥ 1 ǫ k J k  U = 0 , (22) where J k are linear differen tial op erators dep ending on the co efficients of the expansions of ( w, v ) of the form J k = ( z − u c ) r c J k , 0 + ( z − u c ) J k , 1 + r c J k , 2 + J k , 3 , where r c = r c ( z ) := ( z − u c ) 2 − 4 v c , and J k ,i are z -in d ep endent. F or example, the first few are J 1 = ( z − u c ) r c ∂ x , J 2 = 1 2 ( z − u c ) r c ∂ 2 x , (23) J 3 = z − u c 6 r c ∂ 3 x − 3 r c u (2) ∂ x − ( z − u c ) ( v c ∂ 3 x + 4 v (2) ∂ x + 2 v (2) x ) − 4 v c (2 u (2) ∂ x + u (2) x ) . W e will now use b oth (20) and (21) to determine the exp ansions of U and V . 6 If w e s ubstitute in (20) the expans ions w = u c + X j ≥ 2 ǫ j w ( j ) , v = v c + X j ≥ 2 ǫ j v ( j ) , V = X j ≥ 0 ǫ j V ( j ) , U = X j ≥ 0 ǫ j U ( j ) , then b y identifying the co efficien ts of the p ow ers of ǫ w e fin d V (0) z = 1 r 1 / 2 c , U (0) = z − u c r 1 / 2 c , V (1) z = U (1) = 0 , and the recurrence system ( j ≥ 2) U ( j ) − ( z − u c ) V ( j ) z = − 1 2 X k + i = j, k ≥ 1 1 k ! ∂ k x U ( i ) + ( z − u c ) X k + i = j, k ≥ 1 1 2 k ( k )! ∂ k x V ( i ) z − X i 1 + i 2 + k 1 + k 2 = j i 1 ≥ 2 1 2 k 1 + k 2 k 1 ! k 2 ! ( ∂ k 1 x w ( i 1 ) ) ∂ k 2 x V ( i 2 ) z ! , U (0) U ( j ) − 4 v c V (0) z V ( j ) z = − 1 2 X i + k = j, 1 ≤ i, k ≤ j − 1 U ( i ) U ( k ) + X i 1 + i 2 + k 1 + k 2 = j 1 ≤ i 1 , i 2 ≤ j − 1 , k 1 + k 2 ev en ( − 1) k 1 2 k 1 + k 2 − 1 k 1 ! k 2 ! v c ∂ k 1 x V ( i 1 ) z ! ∂ k 2 x V ( i 2 ) z ! + X i 1 + i 2 + k 1 + k 2 + l = j l ≥ 2 , k 1 + k 2 ev en ( − 1) k 1 2 k 1 + k 2 k 1 ! k 2 ! v ( l ) ∂ k 1 x V ( i 1 ) z ! ∂ k 2 x V ( i 2 ) z ! These equations are linear with resp ect to U ( j ) and V ( j ) z . F urtherm ore the determinan t o f their corresp ondin g co efficients is r 1 / 2 c . 7 Th u s one obtains U (2) = 1 r 3 2 c [4 v c w (2) + 2( z − u c ) v (2) ] , V (2) z = 1 r 3 2 c [2 v (2) + ( z − u c ) w (2) ] , U (3) = 1 r 3 2 c [4 v c w (3) + 2( z − u c ) v (3) ] , V (3) z = 1 r 3 2 c [2 v (3) + ( z − u c ) w (3) ] , U (4) = 1 r 3 2 c h 4 v c w (4) + v c 2 w (2) xx + 4 v (2) w (2) + 2( z − u c ) v (4) i + + 1 r 5 2 c h 4 v 2 c w (2) xx + 24 v c v (2) w (2) + ( z − u c )  2 v c v (2) xx + 6 v (2) 2 + 6 v c w (2) 2 i , V (4) z = 1 r 3 2 c  2 v (4) + 1 4 v (2) xx + w (2) 2 + ( z − u c ) w (4)  + 1 r 5 2 c h 2 v c v (2) xx + 6 v c w (2) 2 + 6 v (2) 2 + ( z − u c )  v c w (2) xx + 6 v (2) w (2) i , and b y using induction one p ro ve s that the coefficient s of the expansions of U and V are of the form U (2 j ) = j X l =1 α (2 j ) l ( z − u c ) + β (2 j ) l r l + 1 2 c , j ≥ 3; U (2 j +1) = j X l =1 α (2 j +1) l ( z − u c ) + β (2 j +1) l r l + 1 2 c , j ≥ 2 , (24) V (2 j ) z = j X l =1 γ 2 j l ( z − u c ) + η 2 j l r l + 1 2 c , j ≥ 3; V (2 j +1) z = j X l =1 γ (2 j +1) l ( z − u c ) + η (2 j +1) l r l + 1 2 c , j ≥ 2 . Moreo v er b y sub stituting (24) in the linear equation (21) and taking into accoun t (23) it can b e pro ve d by in duction that the functions α ( i ) l , β ( i ) l , γ ( i ) l and η ( i ) l are differen tial p olynomials in ( v (2) ,..., v ( i − 2 l +2) , w (2) ,..., w ( i − 2 l + 2) ). 2.2 In v arian t solutions The solution ( U , V ) of the system (20) is uniqu ely determined by ( w , v ). Hence if w e assume that ( w , v ) are ev en fu nctions of ǫ , then as a consequence of the in v ariance of (20) und er the transformation (16) w e deduce that ( U , V ) are eve n functions of ǫ to o. In this w a y w e hav e w = u c + X j ≥ 1 ǫ 2 j w (2 j ) , v = v c + X j ≥ 1 ǫ 2 j v (2 j ) , U = X j ≥ 0 ǫ 2 j U (2 j ) , V = X j ≥ 0 ǫ 2 j V (2 j ) . (25) These solutions will b e hen ceforth called invariant solutions of the resolv ent trace equations. 8 As w e ha ve s een in the ab o v e subs ection the coefficient s U (2 j ) and V (2 j ) can be expanded in p o wers of r c with leading terms U (2 j ) = ( z − u c ) α (2 j ) j + β (2 j ) j r j + 1 2 c + O  1 r j − 1 2 c  , V (2 j ) z = ( z − u c ) γ (2 j ) j + η (2 j ) j r j + 1 2 c + O  1 r j − 1 2 c  . (26) F urthermore, by ident ifyin g the co efficient of ǫ 2 j in t h e first equ ation o f the s y s tem (2 0) and by taking in to account that ( z − u c ) 2 = r c + 4 v c it follo ws that γ (2 j ) j = β (2 j ) j 4 v c , η (2 j ) j = α (2 j ) j . (27) No w we prov e the connection b et w een these co efficien ts and the Gel’fand-Dikii differen tial p olyno- mials of the KdV theory . Theorem 1. The functions G j, ± := β (2 j ) j 2 v c ± α (2 j ) j √ v c , (28) ar e the Gel’fand-Dikii differ ential p olynomials in u ± := v (2) ± √ v c u (2) , r esp e ctively. Pr o of. By ident ifyin g the co efficien t of ǫ 2 j + 1 in (22) we get X k +2 l =2 j +1 J k U (2 l ) = 0 . (29) F rom (23) and by taking into acco unt that ( z − u c ) 2 = r c + 4 v c it is clear that only the terms J 1 U (2 j ) and J 3 U (2 j − 2) con tribute to the co efficient of 1 r j − 1 2 c in (29). Th us we get the recursion relations ∂ x α (2 j ) j = ( v c ∂ 3 x + 4 v (2) ∂ x + 2 v (2) x ) α (2( j − 1)) j − 1 + (2 u (2) ∂ x + u (2) x ) β (2( j − 1)) j − 1 , (30) ∂ x β (2 j ) j = ( v c ∂ 3 x + 4 v (2) ∂ x + 2 v (2) x ) β (2( j − 1)) j − 1 + 4 v c (2 u (2) ∂ x + u (2) x ) α (2( j − 1)) j − 1 , whic h lead at once to the w ell-kno wn th ir d-order differen tial equation for the Gelfand-Dikii differ- en tial p olynomials ∂ x G j, ± = ( v c ∂ 3 x + 4 u ± ∂ x + 2 u ± ,x ) G j − 1 , ± . (31) F rom (28) we ha v e th at G 1 , ± = ± 2 √ v c u ± and by u s ing (31) we find G 2 , ± = ± 2 √ v c  v c u ± , xx + 3 u 2 ±  , G 3 , ± = ± 2 √ v c  v 2 c u ± , xxxx + 10 v c u ± ,xx u ± + 5 v c u 2 ± , x + 10 u 3 ±  , G 4 , ± = ± 2 √ v c  v 3 c u ± , xxxxxx + 14 v 2 c u ± u ± ,xxxx + 28 v 2 c u ± ,xxx u ± ,x + 21 v 2 c u 2 ± , xx +70 v c u 2 ± u ± , xx + 70 v c u ± u 2 ± , x + 35 u 4 ±  . 9 3 Strong con d itions for criti c al p oin ts and the do u- bling prop er ty In what follo ws the notation F ( c ) will represen t the v alue of a function F at a critica l p oin t ( x c , t c , u c , v c ) of (9). W e w ill a lso s u pp ose th at v c 6 = 0. The follo wing sets of critical p oin ts were considered in the applications of the Hermitian matrix mo del to quantum gra vity [12]-[10]. Definition 1. Given n ≥ 2 we denote b y C n the set of critic al p oints of the ho do gr aph system (9) which satisfy t he ( strong ) c onditions ( ∂ k v f ) ( c ) = ( ∂ u ∂ l v f ) ( c ) = 0 , 1 ≤ k ≤ n, 1 ≤ l ≤ n − 1 , (32) and suc h that (( ∂ n +1 v f ) ( c ) , ( ∂ u ∂ n v f ) ( c ) ) 6 = (0 , 0) . Due to (10) it is clear that C n is also determined by the condition that all deriv ativ es ( ∂ k u ∂ l v f ) ( c ) with ( k , l ) 6 = (1 , 0) u p to order n v anish and suc h that at least one n + 1-order deriv ativ e is different from zero. An alternativ e c haracterizatio n of C n can b e formulated in terms of the in tegrals I k ( t c , u c , v c ) := 1 2 π i I γ dz V z ( z , t c ) r k +1 / 2 c , J k ( t c , u c , v c ) := 1 2 π i I γ dz ( z − u c ) V z ( z , t c ) r k + 1 / 2 c . (33) Indeed, I k and J k are prop ortional to ( ∂ k +1 v f ) ( c ) and ( ∂ u ∂ k v f ) ( c ) resp ectiv ely . Hence it follo ws that Lemma 1. ( x c , t c , u c , v c ) ∈ C n if and only i f I 0 = 0 , J 0 = − 2 x c , I k = J k = 0 , for 1 ≤ k ≤ n − 1 , ( I n , J n ) 6 = (0 , 0) . (34) Let us consider the system of string equations (5) at p oin ts ( x, t c , u, v ) near a giv en critical p oint ( x c , t c , u c , v c ) ∈ C n . In terms of sym m etric v ariables it reads            I γ dz 2 π i z V z ( z , t c ) V ( z ) = 0 , I γ dz 2 π i V z ( z , t c ) U ( z ) = − 2 x. (35) Then if we set x = x c + ǫ 2 n x, ǫ = ǫ 2 n +1 , and assume (25) we obtain a r ecur siv e metho d for determining the co efficien ts of w and v . In deed (35) is equiv alent to            I γ dz 2 π i z V z ( z , t c ) V (2 j ) ( z ) = 0 , I γ dz 2 π i V z ( z , t c ) U (2 j ) ( z ) = − 2 x c δ j 0 − 2 δ j n x. (36) 10 F or 0 ≤ j ≤ n − 1 th ese equations are iden tically satisfied b ecause of (34). F or j = n w e get from (26)-(27) that the equations (36 ) red uce to      J n β (2 n ) n + 4 v c I n α (2 n ) n = 0 , I n β (2 n ) n + J n α (2 n ) n = − 2 x, (37) or, equiv alen tly , in terms of th e Gel’fand-Dikii p olynomials (28) w e obtain a p air of decoupled ordinary differentia l equations for u ± := v (2) ± √ v c u (2)  J n ± 2 √ v c I n  G n, ± ( u ± ) = ∓ 2 √ v c x, (38) Th u s if the condition J n ± 2 √ v c I n 6 = 0 , is satisfied, w e h a v e the so-called doubling pr op erty arising in the one-cut case of the Hermitian matrix mo del [10]-[11]. The first few cases of (38) are ( f ( c ) uvv ± √ v c f ( c ) vv v )  v c u ± , xx + 3 u 2 ±  = 12 x, n = 2 ; ( f ( c ) uvv v ± √ v c f ( c ) vv v v )  v 2 c u ± , xxxx + 10 v c u ± , xx u ± + 5 v c u 2 ± , x + 10 u 3 ±  = 120 x, n = 3 ; ( f ( c ) uvv v v ± √ v c f ( c ) vv v v v )  v 3 c u ± , xxxxxx + 14 v 2 c u ± u ± , xxxx + 28 v 2 c u ± , xxx u ± ,x + 21 v 2 c u 2 ± , xx +70 v c u 2 ± u ± , xx + 70 v c u ± u 2 ± , x + 35 u 4 ±  = 1680 x , n = 4 . Finally for j > n the system (36) yields th e pair of equations j X l = n  J l γ (2 j ) l + I l η (2 j ) l  = 0 , j X l = n  J l α (2 j ) l + I l β (2 j ) l  = 0 , (39) whic h d etermin e eac h pair ( w (2( j − n +1) , v (2( j − n +1) ) recursiv ely . 4 F urther criti c al p oin ts and regu larized expansi ons Let us go bac k to the sys tem (11) for critical p oint s of the ho d ograph equations (9) . It is equiv alent to f v = 0 , f u = 2 x, f uv = σ √ v f vv , σ = ± 1 . (40) Let us consider solutions of the ho dograph equations near critical p oints and assume that f v ( t c , u, v c ) 6≡ 0 (similar results are obtained by in terc hanging th e roles of u and v ). Then there exists an in teger l ≥ 1 verifying ( ∂ k u f v ) ( c ) = 0 , (0 ≤ k < l ) , ( ∂ l u f v ) ( c ) 6 = 0 . (41) 11 As a consequence the first ho d ograph equ ation f v ( t c , u, v ) = 0 can b e u sed to eliminate the v ariable u as a function of v n ear ( u c , v c ). Indeed, from W eierstrass’ prep aration theorem it follo ws that near ( u c , v c ) there exists a factorization f v ( t c , u, v ) = ( A 0 ( t c , v ) + . . . + A l − 1 ( t c , v ) u l − 1 + u l ) g ( t c , u, v ) , (42) where A j (0 ≤ j < l ) are analytic fun ctions of v , and g is an analytic function of ( u, v ) wh ic h do es n ot v anish near ( u c , v c ). Hence the s olutions of the ho dograph equation f v ( t c , u, v ) = 0 n ear ( u c , v c ) are giv en b y the ro ots of the p olynomial factor in (42) A 0 ( t c , v ) + . . . + A l − 1 ( t c , v ) u l − 1 + u l = 0 , (43) and consequen tly they are c haracterized by a Puiseux series u ( t c , v ) = u c + X k ≥ 1 a k ( t c ) ( v − v c ) k m , (44) for a certain int eger 1 ≤ m ≤ l . If w e now in tro duce the fun ction H ( t c , w ) := f u ( t c , u ( t c , v c + w m ) , v c + w m ) , ( w := ( v − v c ) 1 m ) (45) then at t = t c the second ho d ograph equation in (9) reads H ( t c , w ) = 2 x . F urthermore it is easy to see that as a consequence of the system (40) w e hav e H ( c ) w = f ( c ) uu u ( c ) w + f ( c ) uv v ( c ) w = σ √ v c f ( c ) vv ( σ √ v c u ( c ) w + v ( c ) w ) . On the other hand by differen tiating the identit y f v ( t c , u ( t c , v c + w m ) , v c + w m ) ≡ 0 we get f ( c ) vu u ( c ) w + f ( c ) vv v ( c ) w = f ( c ) vv ( σ √ v c u ( c ) w + v ( c ) w ) = 0 . Hence, w e d educe that H ( c ) w = 0 . (46) In this wa y if we assume that there exists an integ er n ≥ 2 such that ( ∂ k w H ) ( c ) = 0 , (1 ≤ k < n ) , ( ∂ n w H ) ( c ) 6 = 0 , (47) then the ho dograph equation H ( t c , w ) = 2 x d etermines w as a fun ction of x with a branc h p oin t of order n − 1 at x = x c Definition 2. We wil l denote by C lmn , ( l ≥ 1 , 1 ≤ m ≤ l , n ≥ 2) the set of critic al p oints of (9) char acterize d by (40) - (41) and such that 1. u = u ( t c , v ) has br anching or der m − 1 at v = v c . 2. The c orr esp onding fu nction H ( t c , w ) define d by (45) satisfies (47) . 12 Example If w e s et t n = 0 for n 6 = 1 , 3 then th e system (40) reads t c 1 + 3 t c 3 ( u 2 c + 2 v c ) = 0 , 6 t c 3 u c v c + x c = 0 , u c = σ √ v c , σ = ± 1 . (48) Let us consider the critical p oin ts ( x c , t c 1 , t c 3 , u c , v c ) with t c 3 6 = 0. T hey are giv en b y v c = − t c 1 9 t c 3 , u c = σ √ v c , , where ( x c , t c 1 , t c 3 ) are constrained by the equation 3 x c = 2 σ t c 1  − t c 1 9 t c 3  1 / 2 . As w e are assu m ing that v c 6 = 0, w e hav e that t c 1 6 = 0 and u c 6 = 0 so that (4 1 ) is s atisfied b y l = 1. Moreo v er the ho dograph equation t c 1 + 3 t c 3 ( u 2 + 2 v ) = 0 , (49) leads to u ( t c , w ) =  − 2  w − v c 2  1 / 2 , w := v − v c , v c = − t c 1 9 t c 3 6 = 0 . F urthermore H ( t c , w ) := − 12 t c 3 u ( t c , w ) ( w + v c ) verifies H w w ( t c , 0) 6 = 0. T herefore, it follo ws that ( x c , t c 1 , t c 3 , u c , v c ) ∈ C 112 . The follo win g statemen ts will b e useful for the subsequent discussion. T h ey are easily pro v ed b y differentiat ing (45 ) and the identi ty f v ( t , u ( w ) , v c + w m ) = 0. Lemma 2. Given ( x c , t c , u c , v c ) ∈ C lmn with v c 6 = 0 then • If l = 1 then f ( c ) vv do es not vanish and H ( c ) w w = − σ v − 1 / 2 c (3 f ( c ) vv + 4 v c f ( c ) vv v − 4 σ √ v c f ( c ) uvv ) . (50) • If l ≥ 2 then f ( c ) uu = f ( c ) uv = f ( c ) vv = 0 and H ( c ) w w =      v c f ( c ) uvv ( u ( c ) w ) 2 + 2 v c f ( c ) vv v u ( c ) w + f ( c ) uvv , f or m = 1 , v c f ( c ) uvv ( u ( c ) w ) 2 , for m ≥ 2 , (51) wher e u ( c ) w satisfies      v c f ( c ) vv v ( u ( c ) w ) 2 + 2 f ( c ) uvv u ( c ) w + f ( c ) vv v = 0 , for m = 1 , f ( c ) vv v u ( c ) w = 0 , for m ≥ 2 . (52) As a consequence we deduce the follo w ing cond itions for critical p oint s w ith n = 2 13 Prop osition 1. Given ( x c , t c , u c , v c ) ∈ C lm 2 with v c 6 = 0 , then it fol lows that 1. If ( x c , t c , u c , v c ) ∈ C 112 , then 3 σ f ( c ) vv + 4 σ v c f ( c ) vv v − 4 √ v c f ( c ) uvv 6 = 0 . (53) 2. If ( x c , t c , u c , v c ) ∈ C l 12 with l ≥ 2 then f ( c ) 2 uvv − v c f ( c ) 2 vv v 6 = 0 . (54 ) 3. If ( x c , t c , u c , v c ) ∈ C lm 2 with l , m ≥ 2 then f ( c ) uvv 6 = 0 , and f ( c ) vv v = 0 . (55) Pr o of. F or l = 1 we h a ve that m = 1 so that (53) is a consequence of (50). F or l ≥ 2 and m = 1 , b y su b stituting the solution of (52) into (51) w e obtain the cond ition (54). Finally su pp ose that l ≥ 2 , m ≥ 2. Then ac cordin g to (52) and (51) we ha ve that u ( c ) w f ( c ) vv v = 0, and H ( c ) w w = v c ( u ( c ) w ) 2 f ( c ) uvv . Therefore the conditions for H ( c ) w w 6 = 0 are f ( c ) vv v = 0 and f ( c ) uvv 6 = 0 wh ich pro ves (55). Let us consid er th e system of string equations (5) at p oints ( x, t , u, v ) near a giv en critical p oin t ( x c , t c , u c , v c ) ∈ C lmn . In terms of sym m etric v ariables it reads            I γ dz 2 π i z V z ( z , t ) V ( z ) = 0 , I γ dz 2 π i V z ( z , t ) U ( z ) = − 2 x. (56) One ma y in tro d uce stretc hed v ariables not only for x but also for t . The most symmetrical c hoice is x = x c + ǫ 2 n x ; t j = t c,j + ǫ 2 n t j , j ≥ 1; ǫ = ǫ 2 n +1 . (57) Notice that ǫ ∂ t j = ǫ ∂ t j so that (12), (57) and u = u c + X j ≥ 2 ǫ j u ( j ) , v = v c + X j ≥ 2 ǫ j v ( j ) , (58) are consisten t w ith th e Lax equations (2). W e will n ext concen trate on the case n = 2 and will pro vide a recurs iv e metho d for determining expansions of the form (58) n ear critical p oint s in C lm 2 . Thus w e set x = x c + ¯ ǫ 4 ¯ x, t j = t c,j + ¯ ǫ 4 ¯ t j , j ≥ 1 , ¯ ǫ = ǫ 1 / 5 . F rom (56), equating th e coefficients of order ǫ j one find s • F or 0 ≤ j ≤ 3: I γ dz 2 π i V z ( z , t c ) V ( j ) z = 0 , I γ dz 2 π i V z ( z , t c ) U ( j ) = − 2 x c δ j 0 , (59) 14 • F or j ≥ 4:              I γ dz 2 π i V z ( z , t c ) V ( j ) z + I γ dz 2 π i V z ( z , t ) V ( j − 4) z = 0 , I γ dz 2 π i V z ( z , t c ) U ( j ) + I γ dz 2 π i V z ( z , t ) U ( j − 4) = − 2 x δ j 4 . (60) Let us first analyze the system (59). F or j = 0 it redu ces to the ho dograph equations f ( c ) v = 0 , f ( c ) u = 2 x c , and for j = 1 is trivially verified s in ce U (1) ≡ V (1) ≡ 0. F urth er m ore, it is straightforw ard to see that for j = 2 b oth equations in (59) redu ce to ( σ √ v c u (2) + v (2) ) f ( c ) vv = 0 . (61) Hence to pro ceed furth er it is required to distin gu ish th e t wo types of critical points corresp ond ing to C 112 and C lm 2 with l ≥ 2. Case C 112 In this case f ( c ) vv 6 = 0 and (61) implies u (2) = w (2) = − σ √ v c v (2) . F or j = 3 b oth equations in (59 ) lead to w (3) = − σ √ v c v (3) . By setting j = 4 in (60) we fin d that w (4) = − σ √ v c  v (4) + 1 8 v (2) xx + 1 2 v c v (2) 2  − σ f v ( t , u c , v c ) √ v c f ( c ) vv + 1 6 f ( c ) vv h f ( c ) uvv − σ √ v c f ( c ) vv v i  v (2) xx + 6 v c v (2) 2  . Moreo v er v (2) = − σ √ v c u (2) m ust satisfy the P ainlev ´ e I equation Γ ( c ) ( v (2) xx + 6 v c v (2) 2 ) = 6  2 x − f u ( t , u c , v c ) + σ √ v c f v ( t , u c , v c )  . (62) where Γ ( c ) := − σ √ v c 2 (3 f ( c ) vv + 4 v c f ( c ) vv v − 4 σ √ v c f ( c ) uvv ) . Notice that since ( x c , t c , u c , v c ) ∈ C 112 then Prop osition 1 implies Γ ( c ) 6 = 0. 15 T o pro ceed fu rther one uses in duction in the recurrence system for U ( j ) , V ( j ) to pro v e that th e only co efficien ts in the expansions of U ( j ) , V ( j ) dep end ing on v ( j ) , v ( j − 1) , v ( j − 2) , w ( j ) , w ( j − 1) and w ( j − 2) are those corresp onding to l = 1 or 2. Moreo v er one finds α ( j ) 1 = 2 v ( j ) + A ( j ) 1 , α ( j ) 2 = 2 v c v ( j − 2) xx + 12 v (2) v ( j − 2) + 12 v c w (2) w ( j − 2) + A ( j ) 2 , β ( j ) 1 = 4 v c w ( j ) + v c 2 w ( j − 2) xx + 4 v (2) w ( j − 2) + 4 v ( j − 2) w (2) + B ( j ) 1 , β ( j ) 2 = 4 v 2 c w ( j − 2) xx + 24 v c ( v (2) w ( j − 2) + v ( j − 2) w (2) ) + B ( j ) 2 , γ ( j ) 1 = w ( j ) + C ( j ) 1 , γ ( j ) 2 = v c w ( j − 2) xx + 6 v (2) w ( j − 2) + 6 v ( j − 2) w (2) + C ( j ) 2 , η ( j ) 1 = 2 v ( j ) + 1 4 v ( j − 2) xx + 2 w (2) w ( j − 2) + D ( j ) 1 , η ( j ) 2 = 2 v c v ( j − 2) xx + 12 v c w (2) w ( j − 2) + 12 v (2) v ( j − 2) + D ( j ) 2 . where A ( j ) i , B ( j ) i , C ( j ) i and D ( j ) i , j ≥ 5, i = 1 , 2 are differen tial p olynomials in v (2) , . . . , v ( j − 3) , w (2) , . . . , w ( j − 3) . Then, for j ≥ 5 s u bstituting (24) in to the first equation in (60 ) yields expressions w ( j ) = − σ √ v c v ( j ) + K j ( t , v (2) , . . . , v ( j − 2) ) , where K j are different ial p olynomials in v (2) , . . . , v ( j − 2) . Moreo ver, b oth equations in (60) imply that v ( j − 2) m ust verify a second order linear differentia l equation of the form Γ ( c ) ( v ( j − 2) xx + 12 v c v (2) v ( j − 2) ) = H j − 2 ( t , v (2) , . . . , v ( j − 3) ) , (63) where H j − 2 are different ial p olynomials in v (2) ,..., v ( j − 3) . In this wa y , w e h a v e a scheme for deter- mining the co efficien ts u ( j ) , w ( j ) in (14). W e notice that if { v ( l ) , l ≥ 2 } is a solution of (62) -(63) ( j ≥ 5), then as a consequen ce of the sym metry transformation (16)-(1 9 ) w e ha ve that { ( − 1) l v ( l ) , l ≥ 2 } is also a solution of these equations. Cons equ en tly , th e differen tial p olynomials H 2 l +1 , l ≥ 1, in th e righ t hand side of (63) are od d p olynomials in v (2 j +1) , 1 ≤ j ≤ l − 1 and their deriv ative s (for example H 3 ( t , v (2) ) = 0). Hence we ma y set v (2 j +1) ≡ 0 for all j ≥ 1. Analogously , sin ce the equations (62) -(63) can b e written in terms of { w ( l ) , l ≥ 2 } , w e m a y set w (2 j +1) ≡ 0 for all j ≥ 1. 16 Case C l m 2 , l ≥ 2 In this case the system (59) is trivially satisfied. Moreo v er, for j = 4 we get from (60) that        ∆ ( c ) ( v c v (2) xx + 3 v (2) 2 + 3 v c w (2) 2 ) = 6  f ( c ) uvv ( f u ( t , u c , v c ) − 2 ¯ x ) − v c f ( c ) vv v f v ( t , u c , v c )  , ∆ ( c ) ( v c w (2) xx + 6 w (2) v (2) ) = 6  f ( c ) uvv f v ( t , u c , v c ) − f ( c ) vv v ( f u ( t , u c , v c ) − 2 ¯ x )  , (64) where w e are denoting ∆ ( c ) := v c ( f ( c ) vv v ) 2 − ( f ( c ) uvv ) 2 . F rom Prop osition 1 w e h av e that ∆ ( c ) 6 = 0 for critical p oin ts in C lm 2 with l ≥ 2. In terms of the v ariables u ( ± ) = v (2) ± √ v c u (2) , the sys tem (6 4 ) decouples int o the t wo Pa inlev´ e I equations ( f ( c ) uvv ± √ v c f ( c ) vv v )( v c u ± , xx + 3 ( u ± ) 2 ) = 6 (2 x − f u ( t , u c , v c ) ∓ √ v c f v ( t , u c , v c )) . Th u s the doubling prop erty is satisfied in this case. In general, for j ≥ 5 (60) leads to a second ord er lin ear system for w ( j − 2) , v ( j − 2) of the form: 2 v c ( v ( j − 2) xx + 6 w (2) w ( j − 2) ) + 12 v (2) v ( j − 2) = M j − 2 ( t, w (2) , . . . , w ( j − 3) , v (2) , . . . , v ( j − 3) ) , (65) v c w ( j − 2) xx + 6 w (2) v ( j − 2) + 6 v (2) w ( j − 2) = N j − 2 ( t, w (2) , . . . , w ( j − 3) , v (2) , . . . , v ( j − 3) ) . where M j − 2 and N j − 2 are different ial p olynomials in w (2) , . . . , w ( j − 3) and v (2) , . . . , v ( j − 3) . Thus, w e h a v e a recursiv e pr o cedure to construct the co efficien ts w ( j ) , v ( j ) , j ≥ 2. Again, due to the symmetry (16)-(19), it is clear that if ( w ( j ) , v ( j ) ) , ( j ≥ 2) is solution of (65) then (( − 1) j w ( j ) , ( − 1) j v ( j ) ) , ( j ≥ 2) is also a solution of (65). Hence for o dd j the d ifferential p olynomials M j − 2 , N j − 2 are o dd p olynomials in w (2 l +1) , v (2 l +1) , l ≥ 1 and th eir x -deriv ativ es. Therefore, w e m a y set w (2 l +1) = 0, v (2 l +1) = 0 for all l ≥ 1. 4.1 Critical pro c esses in ideal Hele -Sh a w flo ws In v iew of the prop erties of asymp totic solutions of the Kd V equation [13]-[15] and the NLS equation [16] it should b e exp ected that the inn er expansions provi d ed by the double scaling metho d will b e relev ant when the solutions of the disp ersionless or the disp ersionful T o da hierarchies reac h a p oint of gradient catastrophe. W e n ext consider an application to an ideal mo del of Hele-Sha w flo ws sup plied b y the T o da hierarc hy . A Hele -Sh a w cell is a narro w gap b et w een t wo plates filled with t wo flu ids: sa y oil sur roundin g one or sev eral bub bles of air. In the set-up considered in [18 ] (see also [19]-[20]) air is injected in tw o fixed p oin ts of a simply-connected air b ubble making the bubble b reak into tw o emergent bubbles. Before the b reak-off the interface oil-air remains free of cusp-lik e singularities and dev elops a smo oth nec k. T he r ev ersed ev olution describ es the merging of t w o bub bles. T h e analysis of [18] concludes that b efore the merging, the local structure of a small p art of the in terface con taining the tips of the bubbles is describ ed b y a cur ve Y = Y ( X ) whic h falls into u niv ersal classes c haracterized by t wo ev en inte gers (4 n , 2) , n ≥ 1 , and a fin ite n umb er 2 n of deformation p arameters t k . Assuming 17 symmetry of the curve with resp ect to the X -axis, the general solution for the curv e in th e (4 n, 2) class is Y ( z ) :=  P 2 n k =1 ( k + 1) t k +1 z k p ( z − a )( z − b )  ⊕ p ( z − a )( z − b ) , X = z . (66) where a and b are the p ositions of the b ubbles tips. Due to the ph ysical assumptions of the problem, the expansion Y ( z ) = 2 n X k =1 ( k + 1) t k +1 z k + ∞ X k =0 Y n z n , z → ∞ , (67) m ust satisfy the conditions Y 0 = t (physic al time) and Y 1 = 0 (bu bble merging condition) whic h determine the p ositions a , b of the tips. How ever if Y 1 6 = 0 the ev olution pro cess leads to a critical p oint in wh ic h cusp-like singularities app ear. As it w as s ho wn in [18] the p ositions of the bu bbles tips are determined by the pair of ho dograph equations ∞ X k =1 k t k r k − 1 ( u, v ) = 0 , ∞ X k =1 k t k r k ( u, v ) + 2 x = 0 , (68) where t k = 0 for k > 2 n , Y 1 = 2 x and r := z p ( z − u ) 2 − 4 v = X k ≥ 0 r k ( u, v ) z k , a := u − 2 √ v , b := u + 2 √ v . (69) These are pr ecisely the ho d ograph equations (6) . Thus the d ou b le scaling limit metho d can b e used to regularize th e solutions of (66) -(68) at critical p oints in terms of inn er exp ansions of s olutions of (5) . As an example let us analyze the critical pro cess of a merging of tw o bubbles studied in section VI I of [18 ] . W e set t := t 1 , t 2 = 0, t n = 0, n > 3 and fix t 3 to a give n constan t v alue c . Th u s th e Hele-Sha w interface is lo cally c h aracterized by the curv e Y ( X ) = 3 c ( X + u ) p ( X − u ) 2 − 4 v , (70) and (68) reduces to t + 3 c ( u 2 + 2 v ) = 0 , 6 c v u + x = 0 . (71 ) This is the ho dograph sys tem (48) and its solution is giv en b y v = t 2 2 2 / 3 9 c 3 p 9 √ 81 c 2 x 4 + 4 c x 2 t 3 − 81 c x 2 − 2 t 3 − t 18 c (72) + 3 p 9 √ 81 c 2 x 4 + 4 c t 3 x 2 − 81 c x 2 − 2 t 3 18 3 √ 2 c , u = − x 6 c v . As we hav e seen ab o v e the critical p oin ts ( x c , t c , t c 3 , u c , v c ) with t c 6 = 0 are in C 112 so that our sc heme can b e applied to provi d e a disp ers ive r egularizatio n of (72) n ear critical p oints. Setting x = x c in (72) one fin d s that the outer ap p ro ximations for u and v near t = t c are giv en b y u ∼ u out ( t ) := u c − 1 3 r 1 c ( t c − t ) , v ∼ v out ( t ) := v c + u c 3 r 1 c ( t c − t ) , as t → t − c , (73) 18 where u c = σ √ v c . S ince we are considering a critical p oint in C 112 w e introd u ce the str etc hed v ariables x = x c + ǫ 4 x, t = t c + ǫ 4 t and consider the inner expansions for u and v u ∼ u in := u c + ǫ 2 u (2) ( x − u c t ) , v ∼ v in ( t ) := v c + ǫ 2 v (2) ( x − u c t ) , as t → 0 , (74) where u (2) = − v (2) /u c and v (2) v erifies th e P ainlev ´ e I equation v (2) xx + 6 u 2 c ( v (2) ) 2 = 2 3 c u c ( x − u c t ) . (75) The inn er appro ximation at x = 0 must matc h the outer appr o ximation in an o verla p interv al whic h has b oth t − t c small and t large. W riting the outer approximat ions (73) in terms of th e inner v ariable t w e h a v e u out ( t ) = u c − ǫ 2 1 3 r − t c , v out ( t ) = v c + ǫ 2 u c 3 r − t c . Hence, it is clear that matc hin g requires a solution v (2) ( x − u c t ) of (75) satisfying v (2) ( − u c t ) ∼ u c 3 r − t c , as t → − ∞ . 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