Singular sectors of the 1-layer Benney and dToda systems and their interrelations

Singular sectors of the 1-la y er Benney and dT o da systems and their in terrelations. B. Konop elc henk o 1 , L. Mart ´ ınez Alonso 2 and E. Medina 3 1 Dip artimento di Fisic a, Universit´ a del Salen to an d S ezione INFN 7 3100 Lecce, Italy 2 Dep artamento de F ´ ısic a T e´ oric a I I, Universidad C omplutense E28040 Madrid, S p ain 3 Dep a rtamento de Matem´ atic as, Universidad d e C´ adiz E11510 Puerto R e al, C´ adiz, Sp ain No v ember 2 0 , 2021 Abstract Complete d escription of the singular sectors of the 1-la yer Benn ey system ( classical lo n g wa ve equation) and dT o da system i s presen ted. Associated Euler-Po isson-Darb oux equations E(1/2,1/2 ) and E(-1/2,-1/2) are the mai n to ol in the analysis. A complete li st of solutions o f th e 1-la yer Benney system dep endin g on t wo p arameters and belonging to the singular sector is given. Relation b etw een Euler-Pois son-D arb oux equations E ( ε, ε ) with opposite sign of ε is discussed. 1 In tro d u ction The 1-layer Benney system (clas s ical lo ng w ave equation)      u t + u u x + v x = 0 , v t + ( u v ) x = 0 (1) and dT o da eq uation v xx = ( log v ) tt or eq uiv alently the system      u t + v x = 0 , v t + v u x = 0 (2) are the tw o distinguished in teg rable systems of hydro dynamical type (see e.g. [1, 2 ]). 1-lay er Benney system des crib es lo ng wav es in shallow w a ter with free 1 surface in gravitational field. It is the dis per sionless lim it of t he nonlinear Schr¨ odinger equa tion [3]. Recently , the 1 -lay er B enney ( B (1)) s ystem beca me a crucial ingredient in the analys is of the universality of critica l b ehavior for nonlinear equations [4]. The dT oda equation is the 1+ 1-dimensiona l version of the Boy er-Finley equation from the genera l rela tivit y [5]. It shows up in v a rious problems of fluid mechanics ( se e e.g. [6 ]). It is known als o (see e.g.[7]) that the ho dogra ph eq uations of the dT o da hierar ch y determine the lar ge N -limit of the Hermitian model in random matr ix theory . In gener al, these tw o systems ar e an excellent lab ora to ry for studying prop erties of integrable hydro dynamical type systems. In the present w o rk we analy ze the structure s of the s et o f ho dograph eq ua- tions of the B (1) hiera rch y a nd dT o da hierarch y in terms of its Riemann inv ari- ants. These ho do graph so lutions descr ibe the c ritical p oints ∂ W ∂ β i = 0 , i = 1 , 2 , (3) of a function W = W ( t , β 1 , β 2 ) whic h dep end linearly on the co o rdinates t , where t denotes the flow parameters of the Benney hiera rch y and dT oda hier- arch y res pec tively and ob ey an Euler-Poisson-Dar b oux equations E ( ε, ε ) [8] ( β 1 − β 2 ) ∂ 2 W ∂ β 1 ∂ β 2 = ε  ∂ W ∂ β 1 − ∂ W ∂ β 2  . (4) where for the Benney system one has ε = 1 / 2 and for the dT o da system ε = − 1 / 2. The equation (4) and its m ultidimensiona l version a re well known for a long time in class ical geometry [8]. Its relev ance to the theory of Whitham equations has be e n demonstrated recently in the pap ers [9]-[11]. Here w e will use clas sical notation E ( ε, µ ) for the Euler-Poisson-Dar b oux equation prop osed in [8] where suc h equations w ith different ε a nd µ hav e b een studied to o. If we deno te by M the s et of s o lutions ( t , β ) ( β 1 6 = β 2 ) of the ho dograph equations (3) , we may distinguis h a regular a nd a singular s ector in M M = M reg ∪ M sing , such that given ( t , β ) ∈ M ( t , β ) ∈ M reg if det  ∂ 2 W ( t , β ) ∂ β i ∂ β j  6 = 0 , ( t , β ) ∈ M sing if det  ∂ 2 W ( t , β ) ∂ β i ∂ β j  = 0 . The elements of M reg , corr esp ond to the case when the system (3) is uniquely solv able a nd hence, it defines a unique so lutio n β ( t ). The singular class M sing represents degenerate cr itica l p oints of the function W and are the p oints on which the implicit s olutions β ( t ) of the ho dograph eq ua tions exhibit “gra die nt catastrophe” b ehaviour. As we will see in this pap er , the Euler-Poisson-Dar b oux equation is o f grea t help to a nalyze the str ucture of M sing . As the illustr ation of the gener a l result a complete list o f so lutions of the 1- lay er Benney hierarch y 2 from M sing depe nding on tw o parameter s is pre s ented. W e also discuss the relation b etw een Euler-Poisso n- Darb oux e quations with opp osite a and Euler- Poisson-Darb oux equations for symmetries and densities of in tegr als o f motio n for integrable hydrody na mical type systems. 2 1-la y er Benney hierarc h y a nd its singular sec- tor The B (1) sy stem (1) is a member of a disper sionless integrable hierar ch y of deformations of the curve (see e.g. [14, 15]). p 2 = ( λ − β 1 ) ( λ − β 2 ) . (5) where u = − ( β 1 + β 2 ) , v = 1 4 ( β 1 − β 2 ) 2 . The flows β ( t ) ar e characterized by the following condition: There exists a family of functions S ( λ, t , β ) satisfying ∂ t n S ( λ, t , β ( t )) = Ω n ( λ, β ( t )) , n ≥ 1 . (6) where Ω n ( λ, β ) =  λ n p ( λ − β 1 ) ( λ − β 2 )  ⊕ p ( λ − β 1 ) ( λ − β 2 ) . (7) where ⊕ denotes the s ta ndard pro jection o n the po sitive pow ers of λ .F unctions S which s a tisfy (6) are referred to as action functions in the theory of dispe rsionless int eg rable systems (see e.g. [16]). Notice tha t for n = 1 (6) r eads p = ∂ S ∂ x , x := t 1 , so that the sytem (6) is equiv alent to ∂ t n p = ∂ x Ω n , (8) and, in terms of Riemann inv ariants β , it can b e rewritten in the hydrodyna mica l form ∂ t n β i =  λ n p ( λ − β 1 ) ( λ − β 2 )  ⊕    λ = β i ∂ x β i , i = 1 , 2 . (9) The t 2 -flow of this hie r arch y is the B (1) system (1) ( t := t 2 )          ∂ t β 1 = 1 2 (3 β 1 + β 2 ) β 1 x , ∂ t β 2 = 1 2 (3 β 2 + β 1 ) β 2 x . (10) F or v > 0 the B (1) system is hyperb olic while for v < 0 it is elliptic. 3 It was prov ed in [12] that the s y stem (3) fo r the cr itical po ints of the function W ( t , β ) := I γ d λ 2 i π V ( λ, t ) p ( λ − β 1 ) ( λ − β 2 ) , (11) where V ( λ, t ) = P n ≥ 1 t n λ n , is a s ystem of ho dogra ph equations for the Be nney hierarch y . Moreov er , the a ction function for the cor r esp onding solutions is g iven by S ( λ, t , β ) = X n ≥ 1 t n Ω n ( λ, β ) = h ( λ, t , β ) p ( λ − β 1 )( λ − β 2 ) . (12) where h ( λ, t , β ) :=  V ( λ, t ) p ( λ − β 1 )( λ − β 2 )  ⊕ . Obviously , the function W s a tisfies the Euler - Poisson-Darb oux equa tion E (1 / 2 , 1 / 2 ). W ritten e x plicitly , W represents its e lf the series W = x 2 ( β 1 + β 2 ) + t 2 8 (3 β 2 1 + 2 β 1 β 2 + 3 β 2 2 ) + t 3 16  5 β 3 1 + 3 β 2 1 β 2 + 3 β 1 β 2 2 + 5 β 3 2  + t 4 128 (35 β 4 1 + 2 0 β 3 1 β 2 + 1 8 β 2 1 β 2 2 + 2 0 β 1 β 3 2 + 3 5 β 4 2 ) + · · · . (13) The ho dog r aph equations (3) with t n = 0 for n ≥ 5 take the form          8 x + 4 t 2 (3 β 1 + β 2 ) + 3 t 3  5 β 2 1 + 2 β 1 β 2 + β 2 2  + t 4 8 (140 β 3 1 + 6 0 β 2 1 β 2 + 1 8 β 1 β 2 2 + 2 0 β 3 2 ) = 0 , 8 x + 4 t 2 ( β 1 + 3 β 2 ) + 3 t 3  β 2 1 + 2 β 1 β 2 + 5 β 2 2  + t 4 8 (140 β 3 2 + 6 0 β 2 2 β 1 + 1 8 β 2 β 2 1 + 2 0 β 3 1 ) = 0 . (14) Detailed a nalysis of eq uations (14) will b e perfo rmed in sec tio n 3 . Here, we would like to ma ke tw o observ ations. First, one is tha t the formulae (14 ) p oint out on the po ssible alternative interpretation o f the times t 2 , t 3 , t 4 ,... of the B (1 ) hier arch y . Namely , taking t 2 = 0 in the for mulae (14), we s ee that t 3 and t 4 are par ameters appea ring in the initial data β 1 ( x, t 2 = 0) and β 2 ( x, t 2 = 0). Thu s , one can view hodog raph equatio ns (3) (in particular, equations (14)) as equations describing the time evolution of the family of initial data for the B (1) system , parametr ized by the v a riables t 3 , t 4 , t 5 ,... Second obs erv ation concerns with the elliptic version o f the B (1) system. In this ca se β 2 = β 1 and the system (10) reduce s to the single equa tio n ∂ t β = 1 2 (3 β + β ) β x , t := t 2 , β := β 1 . (15) This equatio n is equiv a lent to the nonlinear Beltra mi equation β ¯ z = 2 i − 3 β − β 2 i + 3 β + β β z , (16) 4 where z = x + i t . This fact indicates that the theory o f q uasi-confor mal mappings (see e.g. [1 7]) ca n b e relev ant for the analys is of pro per ties of the elliptic B (1 ) system (1) ( v < 0). Hence, since the elliptic B (1) system is the quasiclass ic al limit [3] of the fo cusing nonlinear Schr¨ odinge r (NLS) equation i ǫ ψ t + ǫ 2 2 ψ xx + | ψ | 2 ψ = 0 , with ψ = A exp  i ǫ S  , u = ǫ 2 i  ψ x ψ − ψ x ψ  , v = − | ψ | 2 , ǫ → 0 , the quasicon- formal ma pping can b e useful a lso in the study o f the s ma ll dis pe r sion limit of the fo cusing NLS equation (compa r e with [4]). In order to analyse s ing ular sector of the 1 -lay er Benney hierarch y w e first observe that due to the Euler-Poisson- Darb oux equatio n for given ( t , β ) ∈ M , as a consequence o f (4) one has ∂ 2 W ∂ β 1 ∂ β 2 = 0 . (17) Consequently det  ∂ 2 W ∂ β i ∂ β j  = ∂ 2 W ∂ β 2 1 · ∂ 2 W ∂ β 2 2 . (18) Thu s , we hav e Prop ositi on 1. Given ( t , β ) ∈ M then 1. ( t , β ) ∈ M r e g if and only if ∂ 2 W ∂ β 2 1 6 = 0 and ∂ 2 W ∂ β 2 2 6 = 0 . 2. ( t , β ) ∈ M sing if and only at le ast one of the derivatives ∂ 2 W ∂ β 2 1 , ∂ 2 W ∂ β 2 2 , vanishes. F urthermore, using (4) it follows easily that at any p oint ( t , β ) ∈ M all mixed deriv atives ∂ i β 1 ∂ j β 2 W can b e expressed in ter ms of linear combination o f deriv a tives ∂ n β 1 W and ∂ m β 2 W . Hence if we define M sing n 1 ,n 2 as the set of p o ints ( t , β ) ∈ M such that ∂ n i +2 W ∂ β n i +2 i 6 = 0 , ∂ k W ∂ β k i = 0 , ∀ 1 ≤ k ≤ n i + 1 , ( i = 1 , 2) , (19) it follows that M sing = [ n 1 + n 2 ≥ 1 M sing n 1 ,n 2 , where M sing n 1 ,n 2 \ M sing n ′ 1 ,n ′ 2 = ∅ , for ( n 1 , n 2 ) 6 = ( n ′ 1 , n ′ 2 ) 5 W e may c ha racterize the c la sses M sing n 1 ,n 2 of the singular sec to r in ter ms of the b ehaviour of S ( λ ) at λ = β i ( i = 1 , 2) . Indeed the deriv ative ∂ k +1 β 1 W w ith k ≥ 1 is pro p o rtional to the integral I γ d λ 2 i π V ( λ, t ) ( λ − β 1 ) k +1 p ( λ − β 1 ) ( λ − β 2 ) = I γ d λ 2 i π h ( λ ) ( λ − β 1 ) k +1 = 1 k !  ∂ k λ h ( λ )     λ = β 1 , and a similar result follows for the deriv atives ∂ k +1 β 2 W with k ≥ 2. As a cons e- quence we hav e Prop ositi on 2. A p oint ( t , β ) ∈ M b elongs to the singularity class M sing n 1 ,n 2 if and only if S ( λ, t , β ) ∼ ( λ − β i ) 2 n i +3 2 as λ → β i , ( i = 1 , 2) (20) 3 Explicit determination of singular sectors It is ea sy to see that the singular classes M sing n 1 ,n 2 can b e de ter mined by means of a system of n 1 + n 2 constraints for the co ordina tes t . Indeed, the p oints ( t , β ) of M sing n 1 ,n 2 are character ized by the equations ∂ k W ∂ β k i = 0 , ∀ 1 ≤ k ≤ n i + 1 , i = 1 , 2 , (21) and ∂ n i +2 W ∂ β n i +2 i 6 = 0 , i = 1 , 2 . (22) Now the observ ation is that the jacobia n ma tr ix of the the system of tw o equa - tions ∂ n i +1 W ∂ β n i +1 i = 0 , i = 1 , 2 (23) is not singular a s ∆ :=           ∂ n 1 +2 W ∂ β n 1 +2 1 ∂ n 2 +2 W ∂ β 1 ∂ β n 2 +1 2 ∂ n 1 +2 W ∂ β n 1 +1 1 ∂ β 2 ∂ n 2 +2 W ∂ β n 2 +2 2           6 = 0 . (24) Indeed, we notice that as a co nsequence of (4) the deriv a tives outside the diag- onal of ∆ are line a r combinations o f the deriv atives { ∂ k β i W , 1 ≤ k ≤ n i + 1 , i = 1 , 2 } , so that fr o m (21)-(2 2) we hav e ∆ = ∂ n 1 +2 W ∂ β n 1 +2 1 · ∂ n 2 +2 W ∂ β n 2 +2 2 6 = 0 . 6 Therefore, one can solve (23 ) and g et a solution β ( t ). Substituting this solution in the remaining e q uations (2 1 ) gives n 1 + n 2 constraints o f the form f k ( t ) = 0 , k = 1 , . . . , n 1 + n 2 . It is not difficult to determine the solutions of (21)-(22) in tw o simple cases: with one parameter t 3 ( t 4 = t 5 = · · · = 0 ), and with t wo parameters t 3 , t 4 ( t 5 = t 6 = · · · = 0 ). W e hav e that in this c ase M sing = M sing 10 ∪ M sing 01 with M sing 10 defined by 1. x = − 45 t 4 t 3 3 + 1 80 t 2 t 2 4 t 3 + √ 15(8 t 2 t 4 − 3 t 2 3 ) p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 360 t 3 4 , β 1 = − 5 t 3 t 4 + √ 15 p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 20 t 2 4 , β 2 = − 3 t 3 t 4 + √ 15 p t 2 3 (3 t 2 3 − 8 t 2 t 4 ) 12 t 2 4 , 2. x = − 45 t 4 t 3 3 + 1 80 t 2 t 2 3 t 3 − √ 15(8 t 2 t 4 − 3 t 2 3 ) p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 360 t 3 4 , β 1 = − 5 t 3 t 4 + √ 15 p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 20 t 2 4 , β 2 = − 3 t 3 t 4 + √ 15 p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 12 t 2 4 , and M sing 01 by 3. x = − 45 t 4 t 3 3 + 1 80 t 2 t 2 4 t 3 − √ 15(8 t 2 t 4 − 3 t 2 3 ) p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 360 t 3 4 , β 1 = − 3 t 3 t 4 + √ 15 p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 12 t 2 4 , β 2 = − 5 t 3 t 4 + √ 15 p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 20 t 2 4 , 4. x = − 45 t 4 t 3 3 + 1 80 t 2 t 2 4 t 3 + √ 15(8 t 2 t 4 − 3 t 2 3 ) p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 360 t 3 4 , β 1 = − 3 t 3 t 4 + √ 15 p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 12 t 2 4 , β 2 = − 5 t 3 t 4 + √ 15 p t 2 4 (3 t 2 3 − 8 t 2 t 4 ) 20 t 2 4 . 4 Singular sector of the elliptic B (1) system Now, we will consider the elliptic B (1) s ystem (1). Sing ular sector M sing has in this case a structure which is quite different from tha t of the hyperb olic system. 7 Indeed, since β 1 = β 2 , the function W for real x, t 2 , t 3 , . . . is the rea l v alued function W ( t , β , β ) = W ( t , β , β ) , and the ho dogra ph equa tion (3) has the fo rm of the Cauch y-Riemann condition ( β = β 1 ) ∂ W ∂ β = 0 . (25) Regular and singular sector s M reg and M sing are defined as the sets ( t , β , β ) of solutions of equation (25 ) such that the hermitian form d 2 W = ∂ 2 W ∂ β 2 d β 2 + 2 ∂ 2 W ∂ β ∂ β d β d β + ∂ 2 W ∂ β 2 d β 2 , is nondeg enerate or de g enerate, resp ectively . F or unreduced so lutio ns ( β 6 = β ), the co rresp onding Euler - Poisson-Darb oux equa tion implies that ∂ 2 W ∂ β ∂ β = 0 , and, hence           ∂ 2 W ∂ β 2 ∂ 2 W ∂ β ∂ β ∂ 2 W ∂ β ∂ β ∂ 2 W ∂ β 2           =      ∂ 2 W ∂ β 2      2 . (26) Thu s , one has Prop ositi on 3. F or unr e duc e d s olut ions of the el liptic B (1) system, the r e gular se ctor M r e g is define d by the c ondition ∂ W ∂ β = 0 , ∂ 2 W ∂ β 2 6 = 0 . (27) A simila r ana lysis to that of the h yp er b o lic case readily leads to Prop ositi on 4. Singular se ctor M sing of the el liptic B (1 ) system (1) is t he union of t he subsp ac es M sing n , ( n = 1 , 2 , 3 , . . . ) define d as M sing n = ( ( t , β , β ) ∈ M sing : ∂ k W ∂ β k = 0 , k = 1 , . . . , n + 1; ∂ ( n +2) W ∂ β ( n +2) 6 = 0 ) (28) Solutions b elonging to M sing n ar e define d on a subsp ac e of c o dimension 2 n in the sp ac e of p ar ameters x , t 2 , t 3 , ... . 8 So, in the elliptic case, gr adient catastro phe happens in the p oint ( x, t ) at fixed parameters t 2 , t 3 ,... Similar to the hyperb olic case, the subspace M sing n is not empty if at lea st n parameter s t 2 , t 3 ,..., t n +1 are different from zer o in the formula (13). It is instructive to rewrite the formula (13) for the function W in terms of the r eal and imaginary pa rt of β 1 . i.e. β 1 = U + i V : W = x U + t 2 ( U 2 − 1 2 V 2 ) + t 3 ( U 3 − 3 2 U V 2 ) + t 4 ( U 4 − 3 U 2 V 2 + 3 8 V 4 ) + t 5 ( U 5 − 5 U 3 V 2 + 15 8 U V 4 ) + · · · . (29) This formula explicitly shows the character of elliptic singular ities exhibited for the function W for v ar ious v a lues of para meters t 2 , t 3 , .... Basic equations (25), (13) and also conditio ns (28) defining subspa ces M sing n can b e easily rewritten in terms of the origina l v ar iables u and v . Since ∂ W ∂ β = − ∂ W ∂ u + i √ − v ∂ W ∂ v , the ho do graph equa tio n (25) beco mes (for v 6 = 0) ∂ W ∂ u = 0 , ∂ W ∂ v = 0 , ( 3 0) while the Euler-Poisson-Dar b oux equation and equatio n (30) take the form ∂ 2 W ∂ u 2 − v ∂ 2 W ∂ v 2 = 0 . (31) F or the subspace M sing 1 conditions (2 8 ) are ∂ W ∂ β = 0 , ∂ 2 W ∂ β 2 = 0 , ∂ 3 W ∂ β 3 6 = 0 . (32) Since ∂ 2 W ∂ β 2 = ∂ 2 W ∂ u 2 − 2 i √ − v ∂ 2 W ∂ u ∂ v + v ∂ 2 W ∂ v 2 + 1 2 ∂ W ∂ v , one concludes taking in to account equatio n (30) and (3 1) that the second con- dition (3 2) is satisfied if and only if ∂ 2 W ∂ u 2 = 0 , ∂ 2 W ∂ v 2 = 0 , ∂ 2 W ∂ u ∂ v = 0 . (33) Thu s , the subspace M sing 1 is characterized by the conditions (30), (33) and b y requirement of nonv anishing third order deriv atives of W . 9 In or der to compar e these conditions with those of pa p e r [4], we firs t obs erve that the B (1) system (1) is conv er ted into the sy stem (1.8) b y the s ubs titution u → v , v → − u . Then, with the choice W = f ( u, v ) + x v − u v t, the ho dograph equations (30 ) b eco me equations (2.4 ) of [4] and equa tion (31 ) is reduced to their equa tion (2.5 ). Fina lly , with such a ch o ice, the conditions (33) ar e converted to the condition (2.12) fro m the pap er [4]. Finally , w e note tha t acco r ding to the prop o sition 4 for the subspace M sing 1 , the co dimension of the co rresp onding s ubspace of ( x, t 2 , t 3 , . . . ) is equal to t wo and the function W with t n = 0, n ≥ 4 i.e. W = x U + t 2 ( U 2 − 1 8 V 2 ) + t 3 ( U 3 − 3 2 U V 2 ) , exhibits the elliptic um bilic singular ity accor ding to Thom’s clas sification [20] (see a lso [17]-[19]). These results r epro duce those orig ina lly obtained in the pap er [4] (formula (4.2)) 5 5. dT o da hierarc hy . Now let us cons ider the function W T ( x, β 1 , β 2 ) = Z dλ 2 π i V T ( x, λ ) r (1 − β 1 λ ) (1 − β 2 λ ) (34) where V T ( x, λ ) = P n ≥ 0 λ n x n . C r itical p oints for this function ar e defined by the eq ua tions ∂ W T ∂ β 1 = 0 , ∂ W T ∂ β 2 = 0 . (35) It is a s imple c heck to s ee that W T ob eys the Euler-Poisson-Da r b oux eq ua tion of the t y p e E ( − 1 / 2 , − 1 / 2) 2( β 1 − β 2 ) ∂ 2 W T ∂ β 1 ∂ β 2 = − ( ∂ W T ∂ β 1 − ∂ W T ∂ β 2 ) . (36) W ritten explicitly the function W T is the ser ie s W T = − 1 2 x 0 ( β 1 + β 2 ) − 1 8 x 1 ( β 1 − β 2 ) 2 − 1 16 x 2 ( β 1 + β 2 )( β 1 − β 2 ) 2 − 1 128 x 3 (5 β 2 1 +6 β 1 β 2 +5 β 2 1 )( β 1 − β 2 ) 2 + ... (37) while the ho dogra ph equations take the form x 0 + 1 2 x 1 ( β 1 − β 2 ) + 1 8 x 2 (3 β 2 1 − 2 β 1 β 2 − β 2 2 ) + . . . = 0 , x 0 − 1 2 x 1 ( β 1 − β 2 ) + 1 8 x 2 (3 β 2 2 − 2 β 1 β 2 − β 2 1 ) + . . . = 0 . 10 These ho dog raph equatio ns provide us with the so lutions o f the sy s tem ∂ β 1 ∂ x 1 = 1 2 ( β 1 − β 2 ) ∂ β 1 ∂ x 0 , ∂ β 2 ∂ x 1 = − 1 2 ( β 1 − β 2 ) ∂ β 2 ∂ x 0 . (38) In terms of the v ar ia bles u = − ( β 1 + β 2 ) , v = 1 4 ( β 1 − β 2 ) 2 one has the dT o da system (2). Conside r ing the higher times x 2 , x 3 , ... .one g ets the whole dT o da hierarch y . Similar to the Benney case the function W T is the ge ne r ating function for classical singula rities for functions of tw o v ariables . Indeed, in the v ariables X = 1 2 ( β 1 + β 2 ) , Y = 1 2 ( β 1 − β 2 ) it is o f the form W T = − x 0 X − 1 2 x 1 Y 2 − 1 2 x 2 X Y 2 − 1 8 x 3 (4 X 2 + Y 2 ) Y 2 + ... (39) The third term here r epresents the par ab olic umbilic s ing ularity both for hy- per b olic a nd elliptic cases. The formulas for the dT o da hier arch y pre sented here coincide with those given in the pap er [2 1] after the identification V T ( x, λ ) = − 2 T λ + λV ′ H ( t, λ ) . (40) i.e. x 0 = − 2 T , x n = nt n , n = 1 , 2 , 3 , ... . It is ob v io us that the descriptions of the regular and singula r secto rs o f the dT o da hierar chy completely co incide with those of 1-layer Benney hierarch y . 6 6. In terrelations b et w een the Euler-P oisson- Darb oux equations with diffe ren t indices and those for function W and d ensities of in tegrals of motion. 1-lay er Benney hierarch y and dT o da hierarchy are tw o exa mples of h ydr o dy- namical t yp e s ystems for which functions W ob ey the Euler-Poisson-Dar b oux equations L ε W ε := h ∂ 2 ∂ x∂ y − ε x − y  ∂ ∂ x − ∂ ∂ y i W ε = 0 , (41) with different indexes ε . Such linear equations are well studied ( see e.g. [8]). The op erator s L ε hav e a num be r o f r e mark able prop erties . One of them ( probably missed b efore) is g iven by the iden tity L ε +1 L µ = L µ +1 L ε (42) for arbitrar y indices ε and µ . This identit y implies, for instance , that for any so lution W ε the function L µ W ε with a rbitrary µ o bey s the Euler-Poisson- Darb oux eq uation with index ε + 1 , more precisely L µ W ε = ε ( ε − µ ) W ε +1 . 11 In particular, a t ε = − 1 2 and µ = 0 o ne has L 1 2 L 0 = L 1 L − 1 2 . In terms o f the op erator s e L ε defined a s e L ε = ( x − y ) L ε the last relation takes the form ∂ x ∂ y e L − 1 2 = e L 1 2 ∂ x ∂ y . (43) This identit y clea rly demonstra tes the duality b etw een the Eule r -Poisson-Darb oux equations with indices 1 2 and - 1 2 and consequently b etw een 1- lay er Benney and dT o da hierar chies. Dualit y b etw een the functions W and densities of in teg rals of motions is the another t yp e of duality typical for the so -called ε integrable h ydr o dynamical t yp e systems. Indeed, due to the Tsarev’s re s ult [22] , a symmetry w i of a semi-Hamiltonian h ydr o dynamical system ∂ β i ∂ t = λ i ( β ) ∂ β i ∂ x , i = 1 , ..., n, (44) i.e. a s o lution of the system ∂ β i ∂ τ = w i ( β ) ∂ β i ∂ x , i = 1 , ..., n (45) which commutes with the s ystem (44), are de fined by the s y stem ∂ w k ∂ β i w i − w k = ∂ λ k ∂ β i λ i − λ k , i 6 = k . (46) Such w i provide us with the s o lutions of the systems (4 4) via the ho dogra ph equations Ω i := − x + λ i ( β ) t + w i = 0 , i = 1 , ..., n. (47) F or such s ystem (44) densities P of integrals o f mo tion ob ey the equa tions [22] ∂ 2 P ∂ β i ∂ β k = ∂ λ i ∂ β k λ i − λ k ∂ P ∂ β i − ∂ λ k ∂ β i λ i − λ k ∂ P ∂ β k , i 6 = k . (48) Let us define ε -systems as those ( for pa rticular class of such sy stems see e.g. [23]) for which ∂ λ i ∂ β k λ i − λ k = ∂ λ k ∂ β i λ i − λ k = ε β i − β k (49) F or such s ystems densities of in tegr als ob ey Euler- Poisson-Darb oux equations ∂ 2 P ∂ β i ∂ β k = ε β i − β k ∂ P ∂ β i − ε β i − β k ∂ P ∂ β k , i 6 = k . (50) A t the same time the e q uations for w i bec ome ∂ w k ∂ β i = − ε w i − w k β i − β k , i 6 = k (51) 12 Symmetry of these equatio ns with res p ect to the transp ositio n of indices i and k implies that ∂ w k ∂ β i = ∂ w i ∂ β k . Hence w i = ∂ f W ∂ β i , i = 1 , ..., n, for a certain function f W . Thus, equations (51) are the Euler- Poisson-Darb oux equations of the t yp e E ( − ε, − ε ) for the function f W ∂ 2 f W ∂ β i ∂ β k = −  ε β i − β k ∂ f W ∂ β i − ε β i − β k ∂ f W ∂ β k  , i 6 = k . (52) The fact that the generating function for symmetries of the Whitham equa tions and so me o ther integrable hydro dynamical systems ob ey the Euler -Poisson- Darb oux equations has b een obse rved ea rlier in the pap ers [9, 13, 23]. Also the duality b etw een the E ule r -Poisson-Darb oux equatio ns for the densities o f int eg rals of motions and ge ne r ating functions of symmetries has b een noted b e- fore to o. Howev er the demonstratio n pre s ented ab ov e seems to be differen t from those discuss ed ear lier. In addition o ne can note that equations (49 ) imply that for ε -systems a lso λ i = ∂ g ∂ β i with so me function g . As the result the ho dog raph equations (47) for the ε - systems take the form Ω i = − x + t ∂ g ∂ β i + ∂ f W ∂ β i = ∂ W ∂ β i = 0 , i, ..., n (53) where W = − x ( β 1 + β 2 ) + g t + f W . Th us, ho dogr aph equatio ns for the integrable hydrodynamica l type systems ar e nothing but the equations defining the critical po ints of the function W . It seems that this fact has b een missing in the previous publications. Moreover, due to the equations (49) the function g also ob eys the E ( − ε, − ε ) Euler-Poisson- Darb oux equation a nd , hence , the function W do es the sa me. Note that pa rticular class of ε -systems fo r whic h λ i are linea r functions o f β i has been discusse d in [23]. So, for integrable h ydr o dynamical systems o f the ε type, the densities of int eg rals and the functions W ( a s well as the functions f W generating sym- metries) play a dual role o b eying the Euler-Poisson- Darb oux equations with opp osite sign of the index ε . 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