Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation

Ho dograph solutions of the disp ersionless coupled KdV hierarc hies, critic al p oin ts and the Euler-P oisson-Darb oux equation B. Konop elchenk o 1 , L. Mart ´ ınez Alonso 2 and E. Medina 3 1 Dip artimento di Fisic a, Un iversit´ a di L e c c e and Sezione INFN 7 3100 Lecce, Ita l y 2 Dep artamento de F ´ ısic a T e´ oric a II, Universidad Co mplutense E28040 Madrid, Sp ain 3 Dep artamento de M a tem´ atic as, Universidad de C´ ad i z E11510 Puerto R e al, C´ adiz, Sp ain Marc h 1, 202 2 Abstract It is sho wn that the ho dograph solutio ns of the d isp ersionless coupled KdV (dcKdV) hierarc hies describ e critical and degenerate critical p oin ts of a scalar fu n cti on whic h ob eys the E u ler-P oisson- Darb oux equation. Singular sectors of eac h dcKdV hierarch y are found to b e describ ed by solutions of higher gen us dcKdV hierarc hies. Concrete solutions exhibiting sh oc k t yp e singularities are present ed. Key wo r ds: In tegrable systems. Ho dograph equations. Euler-Poisson- Darb oux equation. P A CS numb er: 02.30.Ik. 1 1 In tro ducti on In the p resen t pap er we study hierarchies of h ydr odyn amical systems describing quasiclassical deformations of hyp erelliptic cur ves [1 , 2] p 2 = u ( λ ) , u ( λ ) := λ m − m − 1 X i =0 λ i u i , m ≥ 1 . (1) These hierarchies are of in terest for sev eral reasons. First, there are hierarchies of imp ortant h ydro dynamical t yp e systems among them. F or m = 1 one has the Burgers-Hopf hierarch y [3, 4] asso cia ted with the disp ersionless KdV equ a tion u t = 3 2 u u x . F or m = 2 it is the hierarc h y of higher equations for the 1-la yer Benney system (classsical long w a ve equ at ion)      u t + u u x + v x = 0 v t + ( u v ) x = 0 . (2) The system (2) and the corresp ond in g hierarch y are qu asiclassical limits of the nonlinear S c hr¨ od in ge r (NLS) equation and the NLS hierarc hy [5]. F or m ≥ 3 these hierarc hies turn to d esc rib e the singular sectors of the ab o v e m = 1 , 2 hierarc h ies [1]. Second, all these hierarchies are the disp ersionless limits of inte grable coupled Kd V (cKdV) hierarc hies [6]-[8] associated to Sc hr¨ odinger sp ectral p roblems ∂ xx ψ = v ( λ, x ) ψ , (3) with p ote ntia ls wh ic h are p olynomials in the sp ect ral parameter λ v ( λ, x ) := λ m − m − 1 X i =0 λ i v i ( x ) m ≥ 1 , The cKdV h ierarc hies hav e b een studied in [6]-[ 8], they ha ve bi-Hamilto nian structures and , as a consequence of this prop ert y , the disp ersionless expansions of their solutions p ossess in teresting features su c h as the q u asi-t rivialit y p roper ty [9]-[1 0 ]. Mo reo v er, the cKdV hierarc h ies arise also in the study of the singular secto rs of th e K d V and AKNS hierarc hies [11, 12]. Henceforth w e will refer to the hierarchies of h yd rod y n amica l systems asso ciated with the curv es (1) for a fixed m as the m -th disp ersionless coupled KdV (dcKdV m ) hierarc hies. The Hamilt onian structures of the dcKdV m hierarc hies ha ve been studied in [13]. At last, it sh ould b e noticed that the dcKdV m hierarc hies are closely co nn ected with the h igher gen us Whitham hierarchies introd uced in [14]. In our analysis of the h odograph equations for the dcKdV m hierarc hies w e use Riemann in- v arian ts β i (ro ots of the p ol ynomial u ( λ ) in (1)) which pro vide a sp eci ally con ve nient s ystem of co ordinates. W e sho w that the dcKdV m ho dograph equations ha ve th e form ∂ W m ( t , β ) ∂ β i = 0 , i = 1 , . . . , m, (4) where t = ( t 1 , t 2 , . . . ) are times of the hierarc hy a nd W m ( t , β ) := I γ d λ 2 i π P n ≥ 0 t n λ n p Q m i =1 (1 − β i /λ ) . (5) 2 Here γ denotes a large p ositiv ely orient ed circle | λ | = r . Th us, the ho dograph solutio ns of the dcKdV m hierarc hies describ e critic al p oints of the fu nctio ns W m ( t , β ). These functions turn to b e v ery sp ecia l as they satisfy a w ell-kno w n system of equations in differentia l ge ometry: the Euler-P oisson-Darb oux (EPD) equati ons [15] 2 ( β i − β j ) ∂ 2 W m ∂ β i ∂ β j = ∂ W m ∂ β i − ∂ W m ∂ β j . (6) The system (6) h as also app eared in the theory of the Whitham equations arising in the small disp ersion limit of the KdV equations [17]-[19], and in the theory of h ydro dynamic c hains [20]. W e also study the singular sect ors M sing m of th e spaces of hodograph solutions for the dcKdV m hierarc hies. They are giv en by the p oin ts ( t , β ) suc h that rank  ∂ 2 W m ( t , β ) ∂ β i ∂ β j  < m. (7) The v arieties M sing m pro vide us with sp ecial classes of degenerate critica l p oin ts of th e function W m within the general theory of critical p oin ts deve lop ed by V. I. Arnold and others about f ou r t y y ears ago [23, 24]. The u se of equations (4)-(6) simplify drastically the analysis of the structure of these singular sec tors. In particular, we pro ve th at there is a n ested sequence of sub v arieties M sing m ⊃ M sing m, 1 ⊃ M sing m, 2 ⊃ · · · M sing m,q ⊃ · · · , (8) whic h r epresen ts subsets of the singular sector M sing m of the dcKdV m hierarc h y with increasing singular degree q , s uc h that eac h M sing m,q is d et ermined b y a cla ss of hod og raph solutions of the dcKdV m +2 q hierarc h y . The pap er is organized as f ol lo ws. The dcKdV m hierarc hies are describ ed in Sectio n 2. Equa- tions (4 )-(6) are der ived in Section 3. Section 4 deals with the analysis of the sin gula r sectors of the dcKdV m hierarc hies in te rms of th ei r asso ciat ed ho dograph equatio ns. The relation b et wee n singular p oin ts of the dcKdV m ho dograph equations and solutions of higher dcKdV m +2 q ho do- graph equatio ns is stated in S ec tion 4. Some concrete examples inv olving sho c k singularities of the Burgers-Hopf equatio n and the 1-la yer Benney system are presente d in Section 5. 2 The dcKdV m hierarc hies Giv en a p ositiv e in teger m ≥ 1 we consider th e set M m of algebraic cu r v es (1). F or m = 2 g + 1 (o dd case) and m = 2 g + 2 (ev en case) these cur v es are, generica lly , hyperelliptic Riema nn surfaces of gen us g . W e will denote b y q = ( q 1 , . . . , q m ) an y of th e t w o sets o f p aramet ers u := ( u 0 , . . . , u m − 1 ) or β := ( β 1 , . . . , β m ) whic h determine the curves (1) u ( λ ) = λ m − m − 1 X i =0 λ i u i = m Y i =1 ( λ − β i ) . (9) Ob viously , for an y fixed β all the p erm utations σ ( β ) := ( β σ (1) , . . . , β σ ( m ) ) represent the same elemen t of M m . Note also that u i = ( − 1) m − i − 1 s m − i ( β ) , (10) 3 where s k are th e element ary sym metric p olynomials s k = X 1 ≤ i 1 <... 0 the 1-la yer Benney system is hyper b olic while for v < 0 it is elliptic. Finally , w e consider the BH 1 hierarc h y . Its associated cur ve is giv en by p 2 − u ( λ ) = 0 , u ( λ ) = λ 3 − λ 2 u 2 − λ u 1 − u 0 = ( λ − β 1 ) ( λ − β 2 ) ( λ − β 3 ) , u 1 = β 1 + β 2 + β 3 , u 2 = − ( β 1 β 2 + β 1 β 3 + β 2 β 3 ) , u 3 = β 1 β 2 β 3 . The first flow ta ke s the forms                    ∂ t 1 u 0 = 1 2 u 2 u 0 x + u 0 u 2 x , ∂ t 1 u 1 = u 0 x + 1 2 u 2 u 1 x + u 1 u 2 x , ∂ t 1 u 2 = u 1 x + 3 2 u 2 u 2 x . ⇐ ⇒                    ∂ t 1 β 1 = 1 2 (3 β 1 + β 2 + β 3 ) β 1 x , ∂ t 1 β 2 = 1 2 ( β 1 + 3 β 2 + β 3 ) β 2 x , ∂ t 1 β 3 = 1 2 ( β 1 + β 2 + 3 β 3 ) β 3 x . (25) 3 Ho dograph equati ons for dcKdV m hierarc hies and the Euler-P oiss on-Darb oux equation Let us introdu ce the function W m ( t , q ) := I γ d λ 2 i π U ( λ, t ) R ( λ, q ) = X n ≥ 0 t n R n +1 ( q ) , (26) where γ denotes a large p ositiv ely orien ted circle | λ | = r , U ( λ, t ) := P n ≥ 0 t n λ n and R ( λ, q ) is the function defined in (16). Theorem 1. If the functions q ( t ) = ( q 1 ( t , . . . , q m ( t )) satisfy the system of ho do gr aph e quations ∂ W m ( t , q ) ∂ q i = 0 , i = 1 , . . . , m, (27) then q ( t ) is a solution of the dcKdV m hier ar chy. Pr o of. W e are going to p ro ve that the function S ( z , t , q ( t )) = X n ≥ 0 t n Ω n ( z , q ( t )) =  U ( λ ( z ) , t ) R ( λ ( z ) , q ( t ))  ⊕ p ( z , q ( t )) , (28) is an action fun ct ion for th e dcKdV m hierarc h y . By differen tiating (28) with resp ect to t n w e h a v e that ∂ n S = Ω n + ( U ∂ n R ) ⊕ p + ( U R ) ⊕ ∂ n p, (29) W e no w u se the coordin at es β = ( β 1 , . . . , β m ) so that w e ma y tak e adv ant age of the id en tities ∂ β i p = − 1 2 p λ − β i , ∂ β i R = 1 2 R λ − β i . (30) 7 Th us w e deduce that ( U ∂ n R ) ⊕ p + ( U R ) ⊕ ∂ n p = 1 2 m X i =1 h U R λ − β i  ⊕ − ( U R ) ⊕ λ − β i i p ∂ n β i . (31) On the other hand ∂ W m ( t , β ) ∂ β i = 1 2 I γ d λ 2 i π U ( λ, t ) R ( λ, β ) λ − β i = 1 2 I γ d λ 2 i π ( U ( λ, t ) R ( λ, β )) ⊕ λ − β i . (32) Hence the hodograph equations (27) can b e w r itte n as ( U ( λ, t ) R ( λ, β ( t ))) ⊕ | λ = β i = 0 , i = 1 , . . . , m. (33) Th us w e hav e th at ( U ( λ, t ) R ( λ, β ( t )) ⊕ is a p olynomial in λ whic h v anish at λ = β i ( t ) for all i . As a consequence ( U R ) ⊕ λ − β i =  ( U R ) ⊕ λ − β i  ⊕ =  U R λ − β i  ⊕ . Then from (29) and (31) w e deduce that ∂ n S = Ω n and therefore the statemen t follo ws. Using (26) w e obtain that the ho dograph equations (27) can b e expressed as X n ≥ 0 t n ∂ R n +1 ( q ) ∂ q i = 0 , i = 1 , . . . , m. (34) F urthermore, from (21), (22 ) and (33) the hodograph equations (27) can b e also wr itt en as [1] X n ≥ 0 t n ω n,i ( β ) = 0 , i = 1 , . . . , m, (35) whic h represen t the hod og raph transform for th e dcKd V m hierarc h y of flo ws in h ydro dynamic form. Notice also that we ma y sh ift the time p arameters t n → t n − c n in (34) to get solutions dep endin g on an arbitrary n umb er of constan ts. It is easy to see that the generating function R ( λ, β ) := s λ m u ( λ ) = s λ m Q m i =1 ( λ − β i ) , is a symm et ric solution of th e EPD equation 2 ( β i − β j ) ∂ 2 R ∂ β i ∂ β j = ∂ R ∂ β i − ∂ R ∂ β j . (36) Consequent ly , the same prop ert y is satisfied by W ( t , β ) for all t . Th us, we h a ve pro ved 8 Theorem 2. The solutions ( t , β ) of th e ho do gr aph e quations ∂ W m ( t , β ) ∂ β i = 0 , i = 1 , . . . , m, (37) ar e the critic al p oints of the solution W m ( t , β ) := I γ d λ 2 i π U ( λ, t ) p Q m i =1 (1 − β i /λ ) of the EPD e quation 2 ( β i − β j ) ∂ 2 W m ∂ β i ∂ β j = ∂ W m ∂ β i − ∂ W m ∂ β j . (38) Let us denote b y M m the variety of p oin ts ( t , β ) ∈ C ∞ × C m whic h satisfy the ho dograph equations (37). F rom (32) it is clear that for an y p ermutati on σ of { 1 , . . . , m } the functions F i ( t , β ) := ∂ W m ( t , β ) ∂ β i , (39) satisfy F i ( t , σ ( β )) = F σ ( i ) ( t , β ) . (40) Then, it is clear that M m is in v ariant un der the action of the group of per mutations ( t , β ) ∈ M m = ⇒ ( t , σ ( β )) ∈ M m . If ( t , β ) is a solution of (37) suc h that β i 6 = β j for all i 6 = j then it will b e called an unr e duc e d solution of (37). In th is case the EPD equ at ion (38) implies that ∂ 2 W m ( t , β ) ∂ β i ∂ β j = 0 , ∀ i 6 = j. (41) Giv en 2 ≤ r ≤ m , a solutio n ( t , β ) of (37) such that exactly r of its comp onent s are equal will b e called a r - r e duc e d solution of (37). The formulat ion (2 7) of the ho dograph equations for the d cKdV m hierarc hies allo ws us to apply the theory of critical p oin ts of fun cti ons to analyze the solutions of these hierarc hies, while (38) indicates that th e fu nctio ns W m are of a v ery sp ecial cla ss. The EPD equation (38) arose in the stu d y of cyc lids [15], where solutions W of the ab o ve form ha v e b een found to o. Muc h lat er it app eared in the theory of Whitham equations describing the small disp ersion limit of the Kd V equation [17, 19]. W e n ot e that h odograph equations of a form close to (2 7) h a ve b een presen ted in [20] and [2 2 ]. F urthermore, linear equations of the EPD type and their connection with h ydro dynamic c hains ha v e b een stu died in [21] too. Finally , w e emp h asize that the functions W m dep end on the parameters t 1 , t 2 , . . . (times of the hierarc h y). Since ”degenerate critica l p oints app ear naturally in cases when the functions dep end on p arame ters ” [23, 24], one s h ould exp ect the exist ence of families of degenerate critic al p oin ts for the functions W m . Their connection with the singular secto rs in the sp ac es of solutions for dcKdV m will b e considered in the next section. 9 T o illustrate the statement s giv en ab o ve w e next pr esen t some simp le examples. F or the dcKdV 2 hierarc h y we hav e W 2 ( t , β ) = x 2 ( β 1 + β 2 ) + t 1 8 (3 β 2 1 + 2 β 1 β 2 + 3 β 2 2 ) + t 2 16  5 β 3 1 + 3 β 2 1 β 2 + 3 β 1 β 2 2 + 5 β 3 2  + t 3 128 (35 β 4 1 + 20 β 3 1 β 2 + 18 β 2 1 β 2 2 + 20 β 1 β 3 2 + 35 β 4 2 ) + · · · The ho dograph equations with t n = 0 for n ≥ 4, take the form          8 x + 4 t 1 (3 β 1 + β 2 ) + 3 t 2  5 β 2 1 + 2 β 1 β 2 + β 2 2  + t 3 8 (140 β 3 1 + 60 β 2 1 β 2 + 36 β 1 β 2 2 + 20 β 3 2 ) = 0 , 8 x + 4 t 1 ( β 1 + 3 β 2 ) + 3 t 2  β 2 1 + 2 β 1 β 2 + 5 β 2 2  + t 3 8 (140 β 3 2 + 60 β 2 2 β 1 + 36 β 2 β 2 1 + 20 β 3 1 ) = 0 . (42) F or the dcKdV 3 hierarc h y we hav e W 3 ( t , β ) = x 2 ( β 1 + β 2 + β 3 ) + t 1 8  3 β 2 1 + 3 β 2 2 + 3 β 2 3 + 2 β 1 β 2 + 2 β 1 β 3 + 2 β 2 β 3  + t 2 16  5 β 3 1 + +5 β 3 2 + 5 β 3 3 + 3 β 2 1 β 2 + 3 β 2 1 β 3 + 3 β 1 β 2 2 + 3 β 2 2 β 3 + 3 β 1 β 2 3 + 3 β 2 β 2 3 + 2 β 1 β 2 β 2  + · · · The ho dograph equations with t n = 0 for n ≥ 3 are                8 x + 4 t 1 (3 β 1 + β 2 + β 3 ) + t 2 (15 β 2 1 + 3 β 2 2 + 3 β 2 3 + 6 β 1 β 2 + 6 β 1 β 3 + 2 β 2 β 3 ) = 0 , 8 x + 4 t 1 ( β 1 + 3 β 2 + β 3 ) + t 2 (3 β 2 1 + 15 β 2 2 + 3 β 2 3 + 6 β 1 β 2 + 2 β 1 β 3 + 6 β 2 β 3 ) = 0 , 8 x + 4 t 1 ( β 1 + β 2 + 3 β 3 ) + t 2 (3 β 2 1 + 3 β 2 2 + 15 β 2 3 + 2 β 1 β 2 + 6 β 1 β 3 + 6 β 2 β 3 ) = 0 . (43) 4 Singular sectors of dcKdV m hierarc hies W e sa y that ( t , β ) ∈ M m is a regular p oint if it is a nond egenerate critica l p oin t of the fu nction W m . T hat it is to s a y , if it satisfies [23 , 24] det  ∂ 2 W m ( t , β ) ∂ β i ∂ β j  6 = 0 . (44) The set of regular p oin ts of M m will b e denoted b y M reg m and the p oin ts of its complemen tary set M sing m := M m − M reg m , where the second d ifferen tial of W m is a degenerate quadratic form, will b e called singular p oints. W e will also refer to M reg m and M sing m as the r eg ular and singular sectors of the dcKdV m hierarc h y . So M sing m describ es familie s of degenerate critical p oint s of the 10 function W m . Near a r eg ular p oin t the v ariet y M reg m can b e uniquely describ ed as ( t , β ( t )) where β ( t ) is a solution of the dcKdV m hierarc h y . The aim of this section is to an alyze the structure of M sing m b y taking adv an tage of the s p ecial prop erties o f the set of co ordinates β . In ge neral, the singular s ec tors of dcKdV m hierarc hies with m ≥ 2 con tain b oth reduced and unredu ce d p oin ts. F or example, t he ho dograph equations (42) of the dcKdV 2 hierarc h y ha v e reduced singular p oints giv en by ( x, t 1 , t 2 , t 3 , β 1 = β 2 ) where 72 xt 2 3 = − 9 t 2 2 + 36 t 1 t 2 t 3 + (8 t 1 t 3 − 3 t 2 2 ) q 9 t 2 2 − 24 t 1 t 3 , and β 1 = β 2 = − 3 t 2 + p 9 t 2 2 − 24 t 1 t 3 12 t 3 . F urthermore, there are also unred u ced s in gular p oin ts ( x, t 1 , t 2 , t 3 , β 1 , β 2 ) determined b y the con- strain t 360 xt 3 3 = − 45 t 3 t 3 2 + 180 t 1 t 2 3 t 2 + √ 15 (8 t 1 t 3 − 3 t 2 2 ) q t 2 3  3 t 2 2 − 8 t 1 t 3  , and β 1 = − 3 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  12 t 2 3 , β 2 = − 5 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  20 t 2 3 F rom (41) it f ollo ws at once that Theorem 3. L et ( t , β ) b e an unr e duc e d solution of the ho do gr aph e quations (3 7 ) , then ( t , β ) is a singular p oint if and o nly if at le ast one of the derivatives ∂ 2 W m ( t , β ) ∂ β 2 i , i = 1 , . . . , m, vanishes. Notice that s ince the function W m satisfies the E PD equation (38) , its p artia l deriv ative s at unredu ce d p oints ( t , β ) ∂ q W m ( t , β ) ∂ β q 1 1 · · · ∂ β q m m , q := q 1 + · · · + q m , can alw a ys b e expressed as a linear combinatio n of diagonal deriv ativ es ∂ k i β i W m with k i ≤ q i . Th us, for eac h ve ctor q = ( q 1 , . . . , q m ) ∈ N m with at least one q i ≥ 1 it is n at ur al to int ro duce an associated su b v ariet y M sing m, q of M sing m defined as the s et of unr educed solutions ( t , β ) of th e ho dograph equations (37) suc h that ∂ k i W m ( t , β ) ∂ β k i i = 0 , ∀ k i ≤ q i + 1 . (45) In particular, for q = (0 , . . . , 0 , q ) with q ≥ 1 w e denote by M sing m,q the sub v ariet y associated to q = (0 , . . . , 0 , q ). That is to sa y , M sing m,q is the set of solutions ( t , β ) of the ho dograph equations (37) suc h that ∂ 2 W m ( t , β ) ∂ β 2 m = ∂ 3 W m ( t , β ) ∂ β 3 m = . . . = ∂ q +1 W m ( t , β ) ∂ β q +1 m = 0 . (46) 11 These sub v arieties define a nested sequence M sing m ⊃ M sing m, 1 ⊃ M sing m, 2 ⊃ · · · M sing m,q ⊃ · · · , (47) and repr ese nt sets of p oin ts wh ose sin gular degree increases with q . Moreo ver, d ue to the co v ariance of the functions F i = ∂ β i W m under p erm utations there is no need of in tro ducing alternativ e se- quences of the f orm (46) b ase d on systems of equations corresp onding to the r emaining co ordinates β j for j 6 = m . The next result states that the v arieties M sing m,q of the dcKd V m hierarc h y are closely related to the (2 q + 1)-reduced solutions of th e dcKd V m +2 q hierarc h y . Notice that giv en 2 ≤ r ≤ m , the ho d og raph equations for r -red uced solutions β m − r + 1 = β m − r + 2 = . . . = β m , of the d cKdV m hierarc h y reduce to the s y s te m F i ( t , β ) = 0 , i = 1 , . . . , m − r + 1 , of m − r + 1 equations for th e m − r + 1 unknowns ( β 1 , . . . , β m − r + 1 ). Now we pro ve Theorem 4. If ( t , β ) ∈ M sing m,q wher e t = ( t 0 , t 1 , . . . ) and β = ( β 1 , . . . , β m ) , then if we define t ( m +2 q ) := ( t q , t q +1 , . . . ) , β ( m +2 q ) := ( β 1 , . . . , β m , 2 q z }| { β m , . . . , β m ) , it fol lows tha t ( t ( m +2 q ) , β ( m +2 q ) ) is a (2 q + 1) -r e duc e d solution of the ho do gr aph e quations for the dcKdV m +2 q hier ar chy. Pr o of. T o pro of this statemen t w e will u s e sup erscripts ( m ) and ( m + 2 q ) to distinguish ob jects corresp onding to different hierarchies. By assumption w e ha ve th at ( t ( m ) , β ( m ) ) ∈ M sing m,q . Th us, taking (30) in to ac count, w e hav e that (46) can b e rewritten as                F ( m ) i ( t ( m ) , β ( m ) ) := I γ d λ 2 i π U ( m ) ( λ, t ( m ) ) R ( m ) ( λ, β ( m ) ) λ − β ( m ) i = 0 , i = 1 , . . . , m F ( m ) m,j ( t ( m ) , β ( m ) ) := I γ d λ 2 i π U ( m ) ( λ, t ( m ) ) R ( m ) ( λ, β ( m ) ) ( λ − β ( m ) m ) j = 0 , j = 2 , . . . , q + 1 . (48) No w a (2 q + 1)-reduced solution of the ho dograph equations for the dcKdV m +2 q is c h aract erized b y F ( m +2 q ) i ( t ( m +2 q ) , β ( m +2 q ) ) := I γ d λ 2 i π U ( m +2 q ) ( λ, t ( m +2 q ) ) R ( m +2 q ) ( λ, β ( m +2 q ) ) λ − β ( m +2 q ) i = 0 , (49) 12 where i = 1 , . . . , m . But it is clear that R ( m +2 q ) ( λ, β ( m +2 q ) ) = λ q ( λ − β ( m ) m ) q R ( m ) ( λ, β ( m ) ) (50) Hence if we set t ( m +2 q ) i := t ( m ) i + q , i ≥ 0 , w e ha v e U ( m ) ( λ, t ( m ) ) = x ( m ) + λ t ( m ) 1 + · · · + λ q − 1 t ( m ) q − 1 + λ q U ( m +2 q ) ( λ, t ( m +2 q ) ) . (51) Then it follo ws th at F ( m +2 q ) i ( t ( m +2 q ) , β ( m +2 q ) ) = I γ d λ 2 i π U ( m ) ( λ, t ( m ) ) R ( m ) ( λ, β ( m ) ) ( λ − β ( m ) i )( λ − β ( m ) m ) q , i = 1 , . . . , m. (52) F urthermore, for an y giv en i = 1 , . . . , m w e ha ve F ( m ) i ( t ( m ) , β ( m ) ) = I γ d λ 2 i π U ( m ) ( λ, t ( m ) ) R ( m ) ( λ, β ( m ) ) λ − β ( m ) i = I γ d λ 2 i π ( λ − β ( m ) m ) q U ( m ) ( λ, t ( m ) ) R ( m ) ( λ, β ( m ) ) ( λ − β ( m ) i ) ( λ − β ( m ) m ) q = q X k =0 c 1 ,k ( β ( m ) ) I i,k ( t ( m ) , β ( m ) ) , and F ( m ) m,j ( t ( m ) , β ( m ) ) = I γ d λ 2 i π U ( m ) ( λ, t ( m ) ) R ( m ) ( λ, β ( m ) ) ( λ − β ( m ) m ) j = I γ d λ 2 i π ( λ − β ( m ) i ) ( λ − β ( m ) m ) q − j U ( m ) ( λ, t ( m ) ) R ( m ) ( λ, β ( m ) ) ( λ − β ( m ) i ) ( λ − β ( m ) m ) q = q − j +1 X k =0 c j,k ( β ( m ) ) I i,k ( t ( m ) , β ( m ) ) , j = 2 , . . . , q + 1 . where the fu nctio ns c j,k ( β ( m ) ) are the co efficien ts of the p olynomials      ( λ − β ( m ) m ) q = P q k =0 c 1 k ( β ( m ) ) λ k ; ( λ − β ( m ) i ) ( λ − β ( m ) m ) q − j = P q − j +1 k =0 c j k ( β ( m ) ) λ k , j = 2 , . . . , q + 1 . (53) 13 and I i,k ( t ( m ) , β ( m ) ) := I γ d λ 2 i π λ k U ( m ) ( λ, t ( m ) ) R ( m ) ( λ, β ( m ) ) ( λ − β ( m ) i ) ( λ − β ( m ) m ) q (54) No w, for any giv en i = 1 , . . . , m the sys tem (46) implies      F ( m ) i ( t ( m ) , β ( m ) ) = 0 , F ( m ) m,j ( t ( m ) , β ( m ) ) = 0 , j = 2 , . . . , q + 1 , and, as a consequence, w e dedu ce the follo wing system of q homogeneous linear equations for the q f unctions I i,k ( t ( m ) , β ( m ) ) q − j +1 X k =0 c j,k ( β ( m ) ) I i,k ( t ( m ) , β ( m ) ) = 0 , j = 1 , . . . , q + 1 . Because of the linea r indep endence of the p olynomials (53) these equations are li nearly indep en- den t and, therefore, all the fu nctio ns I i,k ( t ( m ) , β ( m ) ) v anish . Finally , from (52) w e conclud e that I i, 0 ( t ( m ) , β ( m ) ) = 0 is equiv alent to F ( m +2 q ) i ( t ( m +2 q ) , β ( m +2 q ) ) = 0 and the state ment follo ws. 5 Examples dcKdV 1 hierarc h y The ho dograph equation for the d cKd V 1 hierarc h y with t n = 0 for all n ≥ 3 reduce to 8 x + 12 t 1 β 1 + 15 t 2 β 2 1 = 0 . (55) The singular v ariet y M sing 1 , 1 for (55) is determined by adding to (5 5) the equation 2 t 1 + 5 t 2 β 1 = 0 , (56) so that f or t 2 6 = 0 w e ha ve β 1 = − 2 t 1 5 t 2 . Su b stituting this result in (55) w e fi n d a constrain t for the flo w parameters x = 3 10 t 2 1 t 2 , whic h is the sh oc k region for the solution of (55) given by β 1 = 2 15 t 2  − 3 t 1 + q 3 (3 t 2 1 − 10 t 2 x )  . (57) There are t wo sec tors M sing 1 , 1 ,k ( k = 1 , 2) in M sing 1 , 1 M sing 1 , 1 , 1 : x = t 1 = t 2 = 0 , β 1 arbitrary; (58) M sing 1 , 1 , 2 : ( x, t 1 , t 2 , β 1 ) suc h that t 2 6 = 0, x = 3 10 t 2 1 t 2 and β 1 = − 2 5 t 1 t 2 14 T o see the r ela tionship with the dcKdV 3 hierarc h y we notice that x (3) = t 1 , t (3) 1 = t 2 , and β (3 = ( β 1 , β 1 , β 1 ) = − 2 5 x (3) t (3) 1 (1 , 1 , 1) , whic h is a 3-reduced solution of the fi rst flo w (25) of the dcKdV 3 hierarc h y . The dcKdV 1 ho dograph equation with t n = 0 for all n ≥ 6 is 693 t 5 β 5 1 + 630 t 4 β 4 1 + 560 t 3 β 3 1 + 480 t 2 β 2 1 + 384 t 1 β 1 + 256 x = 0 . Let us first consider the singular v ariet y M sing 1 , 1 with t n = 0 f or all n ≥ 4. It is is determined by the equati ons 560 t 3 β 3 1 + 480 t 2 β 2 1 + 384 t 1 β 1 + 256 x = 0 , 1680 t 3 β 2 1 + 960 t 2 β 1 + 384 t 1 = 0 . Th us an op en subset of M sing 1 , 1 can b e parametrized by the equatio ns x = − 25 t 3 2 + 105 t 1 t 2 t 3 + √ 5 p 125 t 6 2 − 1050 t 1 t 3 t 4 2 + 2940 t 2 1 t 2 2 t 2 3 − 2744 t 3 1 t 3 3 245 t 2 3 , β 1 = − 2  − 25 t 3 2 + 70 t 1 t 2 t 3 + √ 5 q  5 t 2 2 − 14 t 1 t 3  3  35 t 3  14 t 1 t 3 − 5 t 2 2  . It determines the follo wing 3-reduced solution of th e tw o first flo ws of the d cKdV 3 hierac h y ( x (3) = t 1 , t (3) 1 = t 2 , t (3) 2 = t 3 ) β (3) 1 = β (3) 2 = β (3) 3 = − 2 − 25 ( t (3) 1 ) 3 + 70 x (3) t (3) 1 t (3) 2 + √ 5 r  5 ( t (3) 1 ) 2 − 14 x (3) t (3) 2  3 ! 35 t (3) 2  14 x (3) t (3) 2 − 5 ( t (3) 1 ) 2  . Next, for the sector M sing 1 , 2 if w e set t n = 0 for all n ≥ 5, we obtain the equatio ns 630 t 4 β 4 1 + 560 t 3 β 3 1 + 480 t 2 β 2 1 + 384 t 1 β 1 + 256 x = 0 , 2520 t 4 β 3 1 + 1680 t 3 β 2 1 + 960 t 2 β 1 + 384 t 1 = 0 , 7560 t 4 β 2 1 + 3360 t 3 β 1 + 960 t 2 = 0 . 15 F rom these equations w e find t 1 = 5  − 49 t 3 3 + 189 t 2 t 3 t 4 + √ 7 p 343 t 6 3 − 2646 t 2 t 4 t 4 3 + 6804 t 2 2 t 2 3 t 2 4 − 5832 t 3 2 t 3 4  1701 t 2 4 , x = 5  − 98 t 4 3 + 378 t 2 t 4 t 2 3 + 2 √ 7 q  7 t 2 3 − 18 t 2 t 4  3 t 3 − 243 t 2 2 t 2 4  10206 t 3 4 , β 1 = − 2  − 49 t 3 3 + 126 t 2 t 3 t 4 + √ 7 q  7 t 2 3 − 18 t 2 t 4  3  63 t 4  18 t 2 t 4 − 7 t 2 3  . Then the associated 5-redu ced solution o f th e tw o first fl o ws of the dcKdV 5 hierarc h y ( x (5) = t 2 , t (5) 1 = t 3 , t (5) 2 = t 4 ) is give n b y β i = − 2 − 49 ( t (5) 1 ) 3 + 126 x (5) t (5) 1 t (5) 2 + √ 7 r  7 ( t (5) 1 ) 2 − 18 x (5) t (5) 2  3 ! 63 t (5) 2  18 x (5) t (5) 2 − 7 ( t (5) 1 ) 2  , i = 1 , . . . , 5 . dcKdV 2 hierarc h y Let us consider the ho dograph equations for the dcKdV 2 hierarc h y with t n = 0 for all n ≥ 3. F rom (42) w e ha ve th at they tak e the form      8 x + 4 t 1 (3 β 1 + β 2 ) + 3 t 2  5 β 2 1 + 2 β 1 β 2 + β 2 2  = 0 , 8 x + 4 t 1 ( β 1 + 3 β 2 ) + 3 t 2  β 2 1 + 2 β 1 β 2 + 5 β 2 2  = 0 . (59) The singular v ariet y M sing 2 is determined b y (59) to gether with the additional co nd it ion (det( ∂ β i β j W m ( t , β )) = 0) − (2 t 1 + 3 t 2 ( β 1 + β 2 )) 2 + 9(2 t 1 + t 2 (5 β 1 + β 2 ))(2 t 1 + t 2 ( β 1 + 5 β 2 )) = 0 . (60) There elemen ts of M sing 2 are x = t 1 = t 2 = 0 , ( β 0 , β 1 ) arb itrary; (61) ( x, t 1 , t 2 , β 1 , β 2 ) s uc h that t 2 6 = 0, x = t 2 1 3 t 2 and β 1 = β 2 = − t 1 3 t 2 The sub v arieties M sing 2 ,q are all equal and giv en b y x = t 1 = t 2 = 0 , ( β 0 , β 1 ) arbitrary with β 0 6 = β 1 . 16 Notice that the constraint x = t 2 1 3 t 2 determines the sho c k region for the follo wing solution of (59) β 1 = − t 1 + √ 2 p t 2 1 − 3 t 2 x 3 t 2 , β 2 = − t 1 − √ 2 p t 2 1 − 3 t 2 x 3 t 2 . (62) Let us no w consider the s y s te m of ho dograph equ ations (42) f or the dcKdV 2 hierarc h y with t n = 0 for all n ≥ 4. The singular v ariet y M sing 2 is n o w d ete rmined b y (42) and the condition (det( ∂ β i β j W m ( t , β )) = 0) 32 t 2 1 + 96 t 2 ( β 1 + β 2 ) t 1 + 702 t 2 3 β 2 1 β 2 2 + 72  3 t 2 2 + t 1 t 3  β 1 β 2 + 12  3 t 2 2 + 13 t 1 t 3   β 2 1 + β 2 2  + 486 t 2 t 3  β 2 β 2 1 + β 2 2 β 1  + 90 t 2 t 3  β 3 1 + β 3 2  + 180 t 2 3  β 2 β 3 1 + β 3 2 β 1  + 45 t 2 3  β 4 1 + β 4 2  = 0 . One finds the follo wing six sectors in M sing 2 1. x = − 9 t 3 2 + 36 t 1 t 3 t 2 + (8 t 1 t 3 − 3 t 2 2 ) p 9 t 2 2 − 24 t 1 t 3 72 t 2 3 , β 1 = β 2 = − 3 t 2 + p 9 t 2 2 − 24 t 1 t 3 12 t 3 , 2. x = − 9 t 3 2 + 36 t 1 t 3 t 2 − (8 t 1 t 3 − 3 t 2 2 ) p 9 t 2 2 − 24 t 1 t 3 72 t 2 3 , β 1 = β 2 = − 3 t 2 + p 9 t 2 2 − 24 t 1 t 3 12 t 3 , 3. x = − 45 t 3 t 3 2 + 180 t 1 t 2 3 t 2 + √ 15(8 t 1 t 3 − 3 t 2 2 ) q t 2 3  3 t 2 2 − 8 t 1 t 3  360 t 3 3 , β 1 = − 5 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  20 t 2 3 , β 2 = − 3 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  12 t 2 3 , 4. x = − 45 t 3 t 3 2 + 180 t 1 t 2 3 t 2 − √ 15(8 t 1 t 3 − 3 t 2 2 ) q t 2 3  3 t 2 2 − 8 t 1 t 3  360 t 3 3 , β 1 = − 3 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  12 t 2 3 , β 2 = − 5 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  20 t 2 3 , 17 5. x = − 45 t 3 t 3 2 + 180 t 1 t 2 3 t 2 − √ 15(8 t 1 t 3 − 3 t 2 2 ) q t 2 3  3 t 2 2 − 8 t 1 t 3  360 t 3 3 , β 1 = − 5 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  20 t 2 3 , β 2 = − 3 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  12 t 2 3 6. x = − 45 t 3 t 3 2 + 180 t 1 t 2 3 t 2 + √ 15(8 t 1 t 3 − 3 t 2 2 ) q t 2 3  3 t 2 2 − 8 t 1 t 3  360 t 3 3 , β 1 = − 3 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  12 t 2 3 , β 2 = − 5 t 2 t 3 + √ 15 q t 2 3  3 t 2 2 − 8 t 1 t 3  20 t 2 3 . 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