Hamiltonian structure of reductions of the Benney system

Hamiltonian structure of reductions of the Benne y system John Gibbons ∗ , P ao lo Lorenzoni ∗∗ , Andrea Raimondo ∗ * Departmen t of Mathematics, Imperial C olle ge 180 Queen’ s G ate, London SW7 2AZ, UK j.gibbo ns@imperia l.ac.uk, a.raimond o@imperial . ac.uk ** Dipartimento di Matematica e Applicazi oni Uni versit ` a di Milano-Bicocca V ia Roberto Cozzi 53, I-20125 Milano, Italy paolo.lo renzoni@unimib .it Abstract W e sho w ho w t o constru ct the Hamiltonian structures of any reduction of the Benney chain (dKP) start ing from the famil y of conformal maps associated to it. Introduction The Benney moment chain [4], gi ven by the equations A k t = A k +1 x + k A k − 1 A 0 x , k = 0 , 1 , . . . , with A k = A k ( x, t ) , is t he most famous example o f a chain of hydrodynami c type, which gen- eralizes the classical systems of hydrodynamic type in the case when t he dependent variables (and the equations they ha ve to satisfy) are infinitely many . A n − component reducti on of t he Benney chain is a restriction of the infinite dim ensional system to a suitable n − dimensional submanifold , t hat is A k = A k ( u 1 , . . . , u n ) , k = 0 , 1 , . . . The reduced systems are syst ems of hydrodynamic type in the variables ( u 1 , ..., u n ) that parametrize the submanifold: u i t = v i j ( u ) u j x , i = 1 , . . . , n. 1 Benney reductio ns were introduced in [11], and there it was proved that s uch system s are integrable via the generalized hodograph transformation [20]. In particular , this method requires the system to be diagonalizable, that is, there exists a set of coordin ates λ 1 , . . . , λ n , called Riemann in variants , such that the reduction takes diagonal form: λ i t = v i ( λ ) λ i x . The functions v i are called characteristic velocit ies . A more compact descr iption of the B enney chain can be gi ven by introd ucing the f ormal series λ = p + + ∞ X k =0 A k p k +1 . In this picture, as follows f rom [16, 17], the Benney chain can be written as the single equa- tion λ t = pλ x − A 0 x λ p , which is the equation of th e second flow of the dispersionless KP hierarchy . This equation related with the Benne y chain also appea rs in [18]. Clearly , in the case of a reduction, the coefficients of this series depend on a finite numb er of v ariables ( u 1 , . . . , u n ) . In this case, the series can be thought a s the asym ptotic expansion for p 7→ ∞ of a suitable function λ ( p, u 1 , . . . , u n ) depending piecewise analytically on the parameter p . It turns o ut [11, 12] that such a function satisfies a system of chordal Loe wner equations, describing families of conform al maps (with respect to p ) in the com plex upper half plane. The analytic properties of λ characterize t he reduction. More precisely , i n the case of an n − reduction the ass ociated fun ction λ po ssess n distinct c ritical points o n the real axis, these are the characteristic velocities v i of the reduced syst em, and the corresponding critical v alu es can be chosen as Riemann in variants. Some examples of such reductions, discus sed bel ow , hav e known Hami ltonian s tructures, but the most general result is far weaker , all such reductions are semi-Hami ltonian [20, 11]. The aim of this paper is to in vestigat e the relations between the analytic properti es of the function λ ( p, u 1 , . . . , u n ) and the Hamiltoni an structures of the associated reduction. As is well known, such structures are associated t o pseudo-riemannian metrics, and i n particular , local Hamiltonian structures are associated to flat metrics. Our approach is general, in the sens e that it applies to all B enney reductions . Conse- quently , it reveals a unified structure for the Hamiltonian structure of such reduced systems. The main resul t of the p aper provides the Hamiltonian structures of a Benney reduction di- rectly in terms of th e funct ion λ ( p, u 1 , . . . , u n ) and its in verse with respect t o p , denot ed by p ( λ, u 1 , . . . , u n ) . The Hamiltonian operator then takes the form Π ij = ϕ i λ ′′ ( v i ) δ ij d dx + Γ ij k λ k x + 1 2 π i n X k =1 Z C k ∂ p ∂ λ λ i x ( p ( λ ) − v i ) 2  d dx  − 1 ∂ p ∂ λ λ j x ( p ( λ ) − v j ) 2 ϕ k ( λ ) dλ, 2 where Γ ij k λ k x = ϕ j λ i x − ϕ i λ j x ( v i − v j ) 2 i 6 = j , Γ ii k λ k x = ϕ i  1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) − 1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 2  λ i x + 1 2 ϕ ′ i λ i x − X k 6 = i λ ′′ ( v i ) λ ′′ ( v k ) ϕ i λ k x ( v i − v k ) 2 . Here ϕ 1 , . . . , ϕ n are arbitrary functions of a si ngle variable, C k are suitable closed contours on a complex domain, and λ ′′ ( p ) = ∂ 2 λ ∂ p 2 ( p ) , λ ′′′ ( p ) = ∂ 3 λ ∂ p 3 ( p ) , . . . The paper is organized as follows. In Section 1 we re vi e w the concepts o f integrability for diagonali zable systems of hydrodynami c t ype and the Ham iltonian formalism for these systems, both in the local and non local case. In Section 2 w e i ntroduce the Benney chain, its reductions, and we discus s the prop erties of these sy stems. Section 3 is dedicated to the representation of Benney reductions in the λ picture and to the relations with th e Lo e w ner e volution. The study of the Hamiltonian properties of reductions of Benney is add ressed in Sections 4 and 5: in the former we use a direct approach, st arting from the reduction itself, in the latter we describe these results from t he point of view of the function λ associat ed with the reduction. In th e last secion we di scuss two examples where calculations can be expressed in details. 1 Systems of hydr odynamic type 1.1 Semi-Hamiltonian systems In (1 + 1) dimensi ons, s ystems of hydr odynami c type are quasil inear first order PDE of the form u i t = v i j ( u ) u j x , i = 1 , . . . , n. (1.1) Here and below sums over repeated indices are assumed if not oth erwise stated. W e s ay that th e s ystem (1. 1) is diagonalizable i f t here exist a set of coordinates λ 1 , . . . , λ n , called Riemann in variants , such that the matrix v i j ( λ ) takes diagonal form: λ i t = v i ( λ ) λ i x . (1.2) The function s v i are called characteristic velocities . W e recall that the Riemann i n variants λ i are not defined uniquely , but up to a change of coordinates ˜ λ i = ˜ λ i ( λ i ) . (1.3) 3 A diagonal system of PDEs of hydrodyn amic type (1.2) is called semi-Hamilt onian [20] if the coef ficients v i ( u ) satisfy the system of equations ∂ j  ∂ k v i v i − v k  = ∂ k  ∂ j v i v i − v j  ∀ i 6 = j 6 = k 6 = i, (1.4) where ∂ i = ∂ ∂ λ i . The equation s (1.4) are the integrability con ditions both for the system ∂ j w i w i − w j = ∂ j v i v i − v j , (1.5) which provides the c h aracteristic velocities of the symmetries u i τ = w i ( u ) u i x i = 1 , ..., n of (1.1), and for the system ( v i − v j ) ∂ i ∂ j H = ∂ i v j ∂ j H − ∂ j v i ∂ i H , which provides the densities H of conserv ati on laws of (1.1). The properties of being diag- onalizable and semi-Hamilton ian i mply the integrability of the system: Theor em 1 [20](Generalized hodograph transforma tion) Let λ i t = v i λ i x (1.6) be a diagonal semi-Hamiltoni an system of hydr odyna mic type, and let ( w 1 , . . . , w N ) be the characteristic velocities of one of its symmetr ies. Then, t he function s ( λ 1 ( x, t ) , . . . , λ N ( x, t )) determined by the system of equations w i = v i x + t, i = 1 , . . . , N , (1.7) satisfy (1.6). Mor eover , every smo oth sol ution of t his system is local ly obtaina ble in t his way . 1.2 Hamiltonian f ormalism A class of Hamilto nian formali sms for system s of hydrodynami c type (1.1) w as introd uced by Dubrovin and Novikov in [6, 7]. They cons idered local Ham iltonian operators of th e form P ij = g ij ( u ) d dx − g is Γ j sk ( u ) u k x (1.8) and the associated Poisson brackets { F , G } := Z δ F δ u i P ij δ G δ u j dx (1.9) where F = R g ( u ) dx and G = R g ( u ) dx are function als not dependi ng on t he d eri vati ves u x , u xx , ... 4 Theor em 2 [6] If det g ij 6 = 0 , then the formula (1.9) with (1.8) defines a P oisson brac ket if and only i f the tens or g ij defines a flat pseud o-riemannian metric and the coef ficients Γ j sk ar e the Christoffel symbols of the associated Levi-C i vita connection. Non-local extensions of the bracket (1. 9), related to metrics of constant curvature, were considered by Ferapontov and Mokhov in [19]. Further generalizations were considered by Ferapontov in [9], where he introdu ced the nonlocal dif ferential operator P ij = g ij d dx − g is Γ j sk u k x + X α ǫ α ( w α ) i k u k x  d dx  − 1 ( w α ) j h u h x , ǫ α = ± 1 . (1.10) The index α can take values on a finite or infinite – e ven continuous – set. Theor em 3 If det g ij 6 = 0 , then t he formula (1 .9) with (1.8) defines a P oisson brac ket if and only if the tensor g ij defines a pseud o-riemannian metric, the coefficients Γ j sk ar e the Christoffel symbols of the associated Levi-Civita connection ∇ , and the affinors w α satisfy the conditions  w α , w β  = 0 , g ik ( w α ) k j = g j k ( w α ) k i , ∇ k ( w α ) i j = ∇ j ( w α ) i k , R ij k h = X α n ( w α ) i k ( w α ) j h − ( w α ) j k ( w α ) i h o , wher e R ij k h = g is R j sk h ar e the components of the Riemann curvatur e tensor of the metric g . In the case of zero curvature, operator (1.10) reduces to (1.9). Let us focus ou r atten- tion on semi -Hamiltonian sys tems. In [9] Ferapontov conjectured that any diagonalizable semi-Hamilton ian sys tem is always Hami ltonian with respect to suitable, possi bly non lo- cal, Hamiltonian operators. Moreover he propos ed the follo wing construction to define such Hamiltonian operators: 1. Consider a diagonal system (1.2). Find the general solution of the system ∂ j ln √ g ii = ∂ j v i v j − v i , (1.11) which is compatibl e for a semi-Hamiltonian sys tem, and comp ute the curvature tens or of the metric g . 5 2. If the non vanishing compon ents of the curvature tensor can be writt en in terms o f solutions w i α of the linear system (1.5): R ij ij = X α ǫ α w i α w j α , ǫ α = ± 1 , (1.12) then it turns out that the system (1.1) is Hamiltonian with respect to the Hamiltonian operator P ij = g ii δ ij d dx − g ii Γ j ik ( u ) u k x + X α ǫ α w i α u i x  d dx  − 1 w j α u j x , (1.13) which is the form of (1.10) in case of diagonal matrices. 2 Benney r edu ctions A natural generalizatio n of n − component systems of hyd rodynamic t ype (1.1) can be ob- tained by allowing the number of equations and variables to be infinite. These sy stems are known as hydrodynamic chains , a n d the best known e xampl e is the Benney chain [4]: A k t = A k +1 x + k A k − 1 A 0 x , k = 0 , 1 , . . . . (2.1) In this setting, the v ariables A n are usually called m oments. In [4] Benne y p rove d that this system admi ts an infinite series of conserved quantities, wh ose densities are polynom ial in the moments. The first fe w of them are H 0 = A 0 , H 1 = A 1 , H 2 = 1 2 A 2 + 1 2  A 0  2 . . . A n − component r educti on of th e Benney chain (2.1) is a restriction of t he infini te di - mensional system to a suit able n − dimension al su bmanifold in the space of the mom ents, that is: A k = A k  u 1 , . . . , u n  , k = 0 , 1 , . . . (2.2) where u i = u i ( x, t ) are the ne w dependent variables. These are re garded as coordinates on the subm anifold specified b y (2.2), and all the equations of t he chain ha ve to be satisfied o n this subm anifold. In addition, we require the x − deriv atives u i x to be linearly independent 1 , in the sense that n X i =1 α i ( u 1 , . . . , u n ) u i x = 0 ⇒ α i ( u 1 , . . . , u n ) = 0 , ∀ i. (2.3) Thus, the infinite dimens ional system reduces to a system with finitely many dependent var i ables (1.1). It wa s shown in [11] that all Benne y reductions ar e diagonali zable and pos- sess the semi-Hamiltonian property , hence they are integrable via the generalized hodograph 1 If this constraint is relaxed, solutions such as described in [15] may be obtained. 6 method. On t he other hand, we m ay consider wheth er a diagonal sys tem of hydrodynam ic type λ i t = v i ( λ ) λ i x , i = 1 , . . . , n. (2.4) is a reduction of Benney (note that we do not impose the semi-Hamiltoni an condition). A direct substitution in the chain (2.1) leads, after collecting the λ i x and making use of (2.3), to the system v i ∂ i A k = ∂ i A k +1 + k A k − 1 ∂ i A 0 , i = 1 , . . . , n, (2.5) where ∂ i A 0 = ∂ A 0 ∂ λ i . The consistency conditions ∂ j ∂ i A k +1 = ∂ i ∂ j A k +1 , i 6 = j, k = 0 , 1 , . . . reduce to the 3 2 n ( n − 1) equati ons ∂ i v j = ∂ i A 0 v i − v j (2.6a) i 6 = j , ∂ 2 ij A 0 = 2 ∂ i A 0 ∂ j A 0 ( v i − v j ) 2 (2.6b) which are called the Gibbons-Tsare v system . It has been shown that this system is in in vo- lution, hence it characterizes a n -component reduction of Benney . Moreover , if a sol ution of (2.6 ) is kno wn, all the higher moment s can be found, making recursive use of conditions (2.5). Theor em 4 [11] A diagonal system of hydr odynamic type ( 2 .4) is a r eduction of t he Benne y moment chain (2.1) if and only if ther e exist a fu nction A 0 ( λ 1 , . . . , λ n ) such that A 0 and the v 1 , . . . , v n of the system satisfy the Gibbons -Tsar ev syst em (2.6). In this case, system (2.4) is automaticall y semi-Hamiltonian. It w as noticed in [11] that a generic solution of the Gibbons-Tsarev system depends o n n arbitrary functions of one v ariable. Essenti ally , this is due to the f act that in the system (2.6) the deri vativ es ∂ i v i , ∂ 2 ii A 0 (2.7) are not specified. This leads to a free dom of 2 n functions of a single v ariable, which r edu ces to n all owing for the freedom of reparametrization (1.3) in the definiti on of Riemann in- var i ants. Thus, for any fixed integer n , the Benney moment chain possesses infinit ely many integrable n -component reductions, parametrized by n arbitrary functions of one variable. In the n ext sections we will see how the knowledge of the ‘ di agonal’ terms (2.7 ) pl ays an important role in determi ning the Hamiltoni an structure of a Benney re duction. If these terms are specified, the Gibbon s-Tsare v sys tem becomes a system of pfaf fian type, and a generic solution depends on n arbitrary constants. 7 Example 2.1 The 2 − component Zakhar ov r eductio n [22], is obtained by imposing on the moments the constraints A k = u 1  u 2  k , k = 0 , 1 , . . . , wher e ( u 1 , u 2 ) ar e the new dependent variables. The r esulting classi cal shallow water wave system, first solved by Riemann, is known to be the dispersionless limit of the 2 − compon ent vector NLS equation. U nder the change of dependent coordinates λ 1 = u 2 + 2 √ u 1 λ 2 = u 2 − 2 √ u 1 , the system takes the diagonal f orm (1.2), with velocities v 1 = 3 4 λ 1 + 1 4 λ 2 v 2 = 1 4 λ 1 + 3 4 λ 2 . It is easy to chec k that these velocities satisfy the Gibbons-Tsar ev system with A 0 = ( λ 1 − λ 2 ) 2 16 . 3 The λ picture an d chordal Loewne r e quations 3.1 Reductions in the λ pictur e A more compact description of the Benney chain can be given by introdu cing [16] a formal series λ ( p, x, t ) = p + ∞ X k =0 A k ( x, t ) p k +1 . (3.1) It is well known t hat the moments satisfy the Benney chain (2.1) if and only if λ satisfies λ t = pλ x − A 0 x λ p =  λ , 1 2  λ 2  ≥ 0  , (3.2) where ( ) ≥ 0 denotes the polynomial part of the ar gum ent, and {· , ·} is the canonical Poisson bracket on the ( x, p ) sp ace. Equation 3.2 corresponds to the Lax equation of t he second flow of the dispersionless KP hierarchy . Remark 1 If we intr odu ce the in verse of the series λ wit h r espect to p , and denote it as p ( λ ) = λ + ∞ X k =0 H k λ k +1 , then it is easy to chec k that the following equation holds p t = ∂ x  1 2 p 2 + A 0  . 8 Equivalently , its coef ficients satis fy H k t = ∂ x H k +1 x − 1 2 k − 1 X i =0 H i H k − 1 − i ! , which is the Benne y chain written in cons ervation law form using the coor dinat e set H n . It is easy to show that every H k is polynomial in the moments A 0 , . . . , A k . The use of the formal series (3.1) is to be u nderstood as an al gebraic model for describin g the underlying integrable system in a mo re com pact way . Howe ver , to describe the s ystem in more detail we mus t impo se more st ructure on λ . Following [12, 21], rather than considering a form al series in the parameter p , we instead consider a piece wis e analytic functi on for the var i able p . In particul ar , we let λ + be an analytic function defined on I m ( p ) > 0 , and λ − an analytic function on I m ( p ) < 0 . W e also require the normalization λ ± = p + O  1 p  , p 7→ ∞ . (3.3) Let us define, on the real axis, the jump function f ( p, x, t ) = 1 2 π i ( λ − ( p, x, t ) − λ + ( p, x, t )) , and suppose f is a function of real p which is Holder conti nuous and satisfying th e conditions Z + ∞ −∞ p n f dp < ∞ , n = 0 , 1 , . . . . Then, using Plemelj’ s formula for boundary values of a nalytic functions, we may take λ ± ( p ) = p − π Z + ∞ −∞ f ( p ′ ) p − p ′ dp ′ ∓ iπ f ( p ) . What we obtain ed is th at, with hypotheses above, the functions λ + and λ − are Borel sum s of the series (3.1) i n the upper and lo wer half plane re s pectiv ely . On t he other hand, λ ± will hav e, at p 7→ ∞ , the formal asympto tic series (3.1) , where A n ( x, t ) = Z + ∞ −∞ p n f ( p, x, t ) dp. Thus, to any solution of Benne y’ s equations we can associate a pair of functions λ ± ( p ; x, t ) . In particular , a real v alu ed f leads to real v al ued moments. In t his case, using the Schwa rz reflection principle, we can restrict o ur attention to the functio n λ + ; thi s is t he case s tudied in [10, 21, 1, 2, 3]. On the other hand, it will be us eful b elow to consider the analytic continuation of λ + into th e lower half plane, in the neighborhood o f specified points in the real axis. Such a continuation may or may not coi ncide with λ − , the Schwarz reflection of λ + . In particular important examples such a continuation may be de veloped consistently , giving the structure of a Riemann surface . 9 Remark 2 Other nor malizatio ns, mo r e general than (3.3) ar e allowed, based on the fact that for any differ entiable functi on ϕ of a single vari able, the composed function ϕ ( λ + ) r emains a solution of (3.2) , t he associa ted r educt ion bein g the same. In concr ete e xamples, it is sometimes mor e con venient to make use a differ ent normal isation. Let us consider now the relations between solutions of (3. 2) and Benney reduction. In this case, we ha ve that λ + is associated with a n component reduction if and only if it depends on the var iables x, t via n independent functions. As any reduction is diagonalizable, it is not restrictiv e to take as these v ariables the Riemann in variants. Thus, we hav e λ + ( p, x, t ) = λ + ( p, λ 1 ( x, t ) , . . . , λ n ( x, t )) , (3.4) with λ i t = v i λ i x . (3.5) Remarkably , the characteristic v elocit ies of the reduction turn out to be the critical points of the function λ + associated with it. More precisely , we hav e Theor em 5 Let λ + , solution of (3.2), satisfy conditions (3.4) with (3.5). Let us denote ϕ i ( λ 1 , . . . , λ n ) = λ + ( v i , λ 1 , . . . , λ n ) , i = 1 . . . n, and suppose that the ρ i ar e not constant function s. Then, the velocities v i satisfy ∂ λ + ∂ p ( v i ) = 0 , i = 1 . . . n, and the corr espond ing criti cal values ϕ i can be chosen as Riemann in varia nts for the system (3.5). Pr oof Considering equation (3.2) at p = v i , we obtain the system of n equations ϕ i t = v i ϕ i x − A 0 x ∂ λ + ∂ p ( v i ) . As λ 1 , . . . , λ n can be chosen as coordinates, by the chain rule we get n X j =0 ∂ ϕ i ∂ λ j λ j t = v i n X j =0 ∂ ϕ i ∂ λ j λ j x − ∂ λ + ∂ p ( v i ) n X j =0 ∂ A 0 ∂ λ j λ j x , and this, after substitut ing ( 3 .5) into it, is equiv alent to ∂ ϕ i ∂ λ j  v j − v i  + ∂ λ + ∂ p ( v i ) ∂ A 0 ∂ λ j = 0 i, j = 1 . . . , n, (3.6) due to the independence of the λ j x . Particularly , for i = j the system above reduces to ∂ λ + ∂ p ( v i ) ∂ A 0 ∂ λ i = 0 . (3.7) 10 Further , if A 0 does not depend on λ i , the function λ + is also independent of the same λ i . In this c ase, the associated system (3.5 ) reduces to a n − 1 reduction. On the other hand, if the system is a proper n − component reductio n then ∂ i A 0 6 = 0 and the characteristic velocities are critical points for λ + . Substitut ing back (3.7) into (3.6), w e obtain ϕ i = ϕ i ( λ i ) . Thus, if the critical v alues ϕ i are not constant functions, it is p ossible to choose them as Riemann in variants. The con verse of the Theorem abo ve is also true: if λ + is a solution of (3.2) satisfying λ + ( p, x, t ) = λ + ( p, λ 1 ( x, t ) , . . . , λ n ( x, t )) , (3.8) and with n dist inct critical points v 1 , . . . , v m , then by ev aluatin g equation (3.2) at p = v i we obtain the diagonal system ϕ i t = v i ϕ i x , where ϕ i = λ + ( v i ) . Thus, critical po ints are characteristic velocities. Moreov er , the exis- tence of a function λ associated with a reductio n selects a natural set of Riemann in variants, the critical v alues of λ . Unless otherwise stated, th ese are the coordinates we wil l consider below . Remark 3 It might hap pen that t he functi on λ + possesses m crit ical po ints, with m > n . This is the case, for instance, in Remark 2, wher e criti cal points of the function ϕ have to be added. Then, substi tuting the critical poin ts int o (3.2) we obtain an m component diagonal system. However , in this c a se w e have that m − n of the critical values have trivial dynamics for they are in dependent of x, t . Consequently , the m component syst em r educes to an n component one. Example 3.2 Consi der u i = u i ( x, t ) , i = 1 , 2 . The functi on λ + = p + u 1 p − u 2 , (3.9) rational in p, satisfies equation (3.2) if and only i f u 1 , u 2 satisfy the 2 component Zakhar ov r eduction of Example 2.1. 3.2 Reductions and Loewner equations It was shown in [13, 21] that the sol ution of the i nitial value prob lem of an n reductio n is given b y a In verse Scattering T ransform procedure (which leads to a p articular form of Tsare v ’ s generalized hodograph formula (1.7)), provided that ∂ λ + ∂ p ( p ) 6 = 0 , I m ( p ) > 0 . 11 It is thus necessary that λ + ( p ) be an u nivalent conformal map from t he upper half p lane to som e image region. In [11, 12] i t was prove d that these conformal maps have to be solutions of a system of so called chordal Lo e wn er equations . In fact, if a sol ution of equation (3.2) i s associated with a n − component reducti on of Benney , then conditions (3.4) holds. Substitutin g into equation (3.2), if v i are the characteristic velocities associat ed with t he reduction, we obtain N X i =1  ( v i − p ) ∂ λ + ∂ λ i + ∂ A 0 ∂ λ i ∂ λ + ∂ p  λ i x = 0 . (3.10) As the λ i x are independent, then it follows th at ∂ λ + ∂ λ i = ∂ i A 0 p − v i ∂ λ + ∂ p , i = 1 , . . . , n. (3.11) This is a system of n chordal Loe wner equations (see for example [8]). When the function λ + is chosen w ith the normali zation (3.3), t his system describes the e volution of families of univ alent conformal maps from t he upper complex half plane t o the upper half plane with n slits, when the end points o f the slits are allo wed to move along prescribed mutually non intersecting Jordan arcs. Using the implicit function theorem i t is possible to sho w that the in verse function p satisfies an analogous system ∂ p ∂ λ i = − ∂ i A 0 p − v i , i = 1 , . . . , n. (3.12) v 1 × v 2 × v n × p × λ + ( v 1 ) × λ + ( v 2 ) × λ + ( v n ) λ + Figure 1: n − slit Loewner ev o lution on the upper half plane. 12 For n > 1 , the consis tency conditions of (3.11) (or (3.12) equiv alently) turn out to be the Gibbons-Tsarev system. On the other hand, and more generally , we can consider a set of n Loewner equati ons, ∂ λ + ∂ λ i = a i p − b i ∂ λ + ∂ p , i = 1 , . . . , n, ( 3 .13) for arbitrary functions a i , b i . The consistency conditions ∂ 2 λ + ∂ λ i ∂ λ j = ∂ 2 λ + ∂ λ j ∂ λ i are then equiv alent to the set of equations ∂ i a j = ∂ j a i (3.14) ∂ i a j = 2 a i a j ( b i − b j ) 2 (3.15) ∂ i b j = a i b i − b j , (3.16) where i 6 = j . The first of these equations im plies locally th e existence of a function A 0 ( λ 1 , . . . , λ n ) such that a i = ∂ i A 0 . Consequently , equations (3.15), (3.16) become the Gibbons-Tsare v system (2.6), with b i = v i . So, t o any so lution of a system o f n chordal Loewner equations th ere corresponds a n -component reduction of the Benney c h ain. Example 3.3 The d ispersionless Bouss inesq re duction, which i s a 2 − component Gelfand- Dikii r eduction, is given by A 0 t = A 1 x A 1 t = − A 0 A 0 x , can be described in the λ pictur e using the polynomi al function λ + = p 3 + 3 A 0 p + 3 A 1 . The characteristic velocities are v 1 = − √ − A 0 , v 2 = √ − A 0 , and the Riemann in variants ar e given by λ 1 = λ + ( v 1 ) = 3 A 1 + 2 ( − A 0 ) 3 2 , λ 2 = λ + ( v 2 ) = 3 A 1 − 2( − A 0 ) 3 2 . After the r enormaliz ation ˜ λ ( λ + ) = 3 p λ + = 3 p p 3 + 3 A 0 p + 3 A 1 we obt ain a famil y of Schwarz-Christoffel maps as in F igure 3.3. It is easy to verify that the critical points of the function ˜ λ ( λ + )( p ) are the same as λ + ( p ) , while the corr esponding new Riemann in variants ar e ˜ λ i = 3 √ λ i . 13 v 1 × v 2 × p × ˜ λ 1 × ˜ λ 2 λ + Figure 2: The di spersionless Boussinesq reduction. As a consequence of the Loewner system (3.11) satisfied by a Benney re duction, it fol- lows immediately that th e critical points of λ + ( p ) are simp le. Indeed, taking t he limit of the i − th equation of the system, for p → v i giv es 1 = ∂ 2 λ + ∂ p 2 ( v i ) ∂ i A 0 , (3.17) where we used the identity ∂ λ ∂ λ i | p = v i = dλ i dλ i − ∂ λ ∂ p | p = v i ∂ v i ∂ λ i = 1 . Thus, ∂ 2 λ + ∂ p 2 ( v i ) 6 = 0 , hence the v i are simple. Suppose n ow that λ + admits an analytic conti nuation in s ome n eighborhood of v i . Hence- forth, to si mplify t he notations, the subscript + will be dropped from λ + , and we will denote both the analytic function λ + and its analytic cont inuation simply by λ . Moreove r , we will write λ ′ ( p ) = ∂ λ ∂ p ( p ) , λ ′′ ( p ) = ∂ 2 λ ∂ p 2 ( p ) , . . . Then, the function p ( λ ) has the series dev elop ment near λ = λ i , p ( λ ) = v i + √ 2 p λ ′′ ( v i ) p λ − λ i + O  λ − λ i  , (3.18) 14 which becomes a T aylor expansion i n the com plex local parameter t = √ λ − λ i . Further- more, we ha ve 1 λ ′ ( p ) = 1 λ ′′ ( v k ) 1 p − v k − 1 2 λ ′′′ ( v k ) λ ′′ ( v k ) 2 (3.19) +  1 4 λ ′′′ ( v k ) 2 λ ′′ ( v k ) 3 − 1 6 λ ′′′′ ( v k ) λ ′′ ( v k ) 2  ( p − v k ) + O  ( p − v k ) 2  . This e x pansion will be useful in Section 5. Finally , we i ntroduce two sets of contours, in the p and λ pl ane respectiv ely , that we wi ll need later for describi ng the Hamilt onian structure of th e reduction s. W e d efine Γ i as a clos ed and sufficiently small contour in the p − plane around v i , and C i as the image of Γ i according to the analytical continuation of λ . Thus, Γ i and C i are well defined; in particular , it follo ws from expansion (3.18) that λ i – the tip of the slit – is a square root branch point for p ( λ ) , hence C i encircles it twice. 3.3 Symmetries of the Benney r eductions A well-known method [13] of obtaining a countable set of symmetries of the B enney reduc- tion is based on the Lax representation of the dKP hierarchy , λ t n = { λ , h n } = ( h n ) p λ x − ( h n ) x λ p , n = 1 , 2 , . . . where h n = 1 n ( λ n ) ≥ 0 . W e assume, unless ot herwise stated, that λ is normalized as in (3 .3). If the solution λ possesses n criti cal values ( v 1 , . . . , v n ) , the hierarchy can be reduced to λ i t n = w i n λ i x , i = 1 , . . . , n, n = 1 , 2 , . . . , with w i n =  ∂ h n ∂ p  | p = v i . (3.20) These are, by construction, components of the symmetries of the reduct ions, the fir st few of them being w i 1 = 1 , w i 2 = v i , w i 3 = ( v i ) 2 + A 0 . . . i = 1 , . . . , n. Further , the abov e characteristic velocities can be obtained as the coef ficients of the ex- pansion at λ = ∞ of the g enerating functions W i ( λ ) = 1 p ( λ ) − v i . (3.21) Pr oposition 3.1 The functions W i ( λ ) ar e solut ions of the linear system ∂ j W i ( λ ) W j ( λ ) − W i ( λ ) = ∂ j v i v j − v i . (3.22) 15 Mor eover , ex p anding W i ( λ ) at λ = ∞ we get W i ( λ ) = ∞ X n =1 w i n λ n , (3.23) wher e the coefficients of the series ar e th e symmetries w i n given in (3.20) . Remark 4 The fi rst cond ition (3.22) h olds in a ny normali zation, but the expansion (3.23) assumes the normalizatio n (3.3) . Pr oof In order to prov e (3.22), knowing that ∂ p ∂ λ i = ∂ i A 0 v i − p , we can write ∂ j W i ( λ ) = ∂ j  1 p − v i  = − 1 ( p − v i ) 2  ∂ p ∂ λ j − ∂ v i ∂ λ j  = − ∂ j A 0 ( p − v i ) 2  1 v j − p − 1 v j − v i  = ∂ j A 0 ( p − v i )( p − v j )( v j − v i ) . On the other hand W j ( λ ) − W i ( λ ) = ( v j − v i ) ( p − v i )( p − v j ) . and so ∂ j W i ( λ ) W j ( λ ) − W i ( λ ) = ∂ j A 0 ( v j − v i ) 2 = ∂ j v i v j − v i . In order to prove (3.23), chos en the normalization (3.3), we ha ve w i n = 1 n lim p → v i d dp  ( p − v i ) 2 ( λ n ) + ( p − v i ) 2  = 1 n res p = v i  ( λ n ) + ( p − v i ) 2 dp  . The function ( λ n ) + ( p − v i ) 2 has poles only at p = v i and p = ∞ and t herefore res p = v i  ( λ n ) + ( p − v i ) 2 dp  = − res p = ∞  ( λ n ) + ( p − v i ) 2 dp  = − res p = ∞  ( λ n ) ( p − v i ) 2 dp  16 where the last identity is due to res p = ∞  ( λ n ) − ( p − v i ) 2 dp  = 0 . Changing v ariable we obtain − res p = ∞  ( λ n ) ( p − v i ) 2 dp  = − 1 2 π i Z Γ ∞ λ n dp dλ ( p ( λ ) − v i ) 2 dλ = n 2 π i Z Γ ∞ λ n − 1 p ( λ ) − v i dλ, where Γ ∞ is a suf ficientl y small contour around p = ∞ . Thus, w i n = Z Γ ∞ W i ( λ ) λ n − 1 dλ, var y ing n w e obtain the coef ficients of the expansion (3.23).  Remark 5 It will be useful below to consider , as a generating function of the symmetries, w i ( λ ) = ∂ p ∂ λ ( p ( λ ) − v i ) 2 = − ∞ X n =1 n w i n λ n +1 , (3.24) which is nothing b ut the λ − derivative of (3.21 ) . W e finally observe th at, due to linearity of (3.22), the functions z i = n X k =1 Z C k w i ( λ ) ϕ k ( λ ) dλ still satisfy the s ystem for the symmetries and t herefore, applying the generalized hodograph method, we can write the general solution of the Benney reduction in the impl icit form z i = v i x + t, i = 1 , . . . , n. The in verse scattering solutions found in [10, 21] are of this form. 4 Hamilton ian structure o f the r e ductions As it was shown in [11], any re duction of Benney is a diago nalizable and semi-Hamil tonian system of hydrodynamic type. Howe ver , very lit tle is known about the Ham iltonian structu re of these systems , whether local or nonl ocal. A few examples are known explicitly . These include t he Gel fand-Dikii and Zakharov reductions, which arise as dispersionl ess limits of known dis persiv e Hamiltonian systems. In thi s section, we us e Ferapontov’ s procedure sketched in Section 1.2 for s emi-Hamiltoni an diagonal systems in order to determine the metric associated with a generic reduction. 17 Theor em 6 The general solution of the system (1.11) in the case of Benne y r eductions is g ii = ∂ i A 0 ϕ i ( λ i ) (4.1) wher e the functions ϕ i ( λ i ) ar e n arbitrary f unctions of one vari able, the functions λ 1 , . . . , λ n being the Riemann in variants of the system. Pr oof . From the system (1.11), and making use of both the Gibbons-Tsarev equations (2.6), which hold for any Benne y reduction, we obt ain ∂ j ln √ g ii = ∂ j v i v j − v i = ∂ j A 0 ( v j − v i ) 2 = ∂ j ln p ∂ i A 0 (4.2) from which we obtain the general solution (4.1).  In the case ϕ i = 1 t he rotation coef ficients β ij := ∂ i √ g j j √ g ii (4.3) are symmetric: β ij = 1 2 ∂ i ∂ j A 0 p ∂ i A 0 ∂ j A 0 = p ∂ i A 0 ∂ j A 0 ( v i − v j ) 2 . (4.4) W e no w focus our attention on this case, that is we consider the Egorov (potential) metric g ii = ∂ i A 0 . (4.5) Remark 6 W e noti ce that the choice of potential metric is not re strictive, as any oth er metric (4.1) can be written in potential form under a change of coo r dinates λ i 7− → ϕ i ( λ i ) , (4.6) which is e xactly the fr eedom we h ave in the definit ion of the Riemann in vari ants. On the other hand, the choice of the Riemann in var iants determines a uni que metric which is potential in those coor dinates. In order to find the Christoffel sym bols and the curv atu re tensor o f the m etric (4.5), one could compute t hese objects – as usual – starting from their definitions. Howe ver , for a Benney reduction, this procedure can be shortened. Indeed, using the Gibbons-Tsarev system (2.6), the connection and the curvature can be written as si mple algebraic combinatio ns of the quantities v i , ∂ i A 0 , δ ( v i ) , δ (lo g p ∂ i A 0 ) , i = 1 , . . . , n, 18 where we introduced the shift operator δ = n X k =1 ∂ ∂ λ k . W e ha ve Pr oposition 4.2 The symbols Γ ij k = − g is Γ j sk = − 1 2 g is g j l  ∂ s g lk + ∂ k g ls − ∂ l g sk  , wher e Γ k ij ar e the Chris toffel symbols as sociated to the diagonal metri c g ii = ∂ i A 0 , for a Benne y r educti on ar e given by Γ ij k = 0 , i 6 = j 6 = k (4.7a) Γ ij i = − Γ j i i = 1 ( v i − v j ) 2 , i 6 = j (4.7b) Γ ii k = − ∂ k A 0 ∂ i A 0 1 ( v k − v i ) 2 , i 6 = k (4.7c) Γ ii i = X k 6 = i ∂ k A 0 ∂ i A 0 1 ( v i − v k ) 2 − δ (ln √ ∂ i A 0 ) ∂ i A 0 . (4.7d) Pr oof F or t he metric g ij = δ ij ∂ i A 0 , we get Γ ij k = − 1 2 1 ∂ i A 0 ∂ i A 0  δ j k ∂ ik A 0 + δ ij ∂ k j A 0 − δ ik ∂ j k A 0  , and equati ons (4.7) are obtained from these b y su bstituti ng, whenev er is allowed, the Gi bbons- Tsare v equation s (2.6b).  In par ticular , t he curvature can be expressed solely via the shift operator δ , acting on v i and ln √ ∂ i A 0 . Pr oposition 4.3 The non vanishi ng components of the curvatur e t ensor of the metric (4.5) for a Benney r eduction can be writt en in terms of the quanti ties δ ( v i ) , δ (ln √ ∂ i A 0 ) . Mor e pr ecisely , we have the following identity R ij ij = δ (ln √ ∂ i A 0 ) + δ (ln p ∂ j A 0 ) ( v i − v j ) 2 − 2 δ ( v i ) − δ ( v j ) ( v i − v j ) 3 . (4.8) 19 Pr oof Since the rotation coeffi ci ents of the metric (4.5) are symm etric, it is easy to check that R ij ij = δ ( β ij ) . Using the Gibbons-Tsarev system (2.6) we get (4.8). Moreov er , i t is well kn own that for a semi-Hamilton ian s ystem the other components of the Riemann tensor are identically zero.  Formula (4.8) presents a compact way of describing t he curv ature tensor of the Poisson structure (1.13) associated with a Benney reduction. Howe ver , we should notice here that the knowledge of the curvature is not sufficient to write the Poiss on bracket. Indeed, what we ne ed is a decompositio n (1.12) of t he c urva ture in terms of th e symmetries of the system. From formula (4.8) this decompositio n looks non-tri vial; we wil l address this problem i n t he next section using a dif ferent approach. 5 Hamilton ian structure in the λ pic tur e The purpose of this Sec t ion is to derive an explicit formulat ion for the Ham iltonian structure of a reduction of Benney in terms of t he function λ ( p ) , which defi nes the reduction itself. In particular , we will show ho w the dif ferential geometric objects associated wi th Fer apontov’ s Poisson operator of type (1.13) can be expressed, i n the case o f a Benney reduction with associated λ , in terms of the set of data v i , λ ′′ ( v i ) , λ ′′′ ( v i ) , λ ′′′′ ( v i ) , i = 1 , . . . , n, where v i , the characteristic velocities of the re duction, are the critical points of λ . Moreover , we will describe the quadratic expansion of the curv ature associated with the metric. Let us cons ider a Benney reduction with associated function λ ( p ) . In this case, as already mentioned, a set of Riemann in variants is n aturally selected, the critical values of λ ( p ) . From (3.17) and (4.1), it is immediate to check t hat the component s of the metri c which is po tential in those coordinates can be expressed in terms of λ as g ii = ∂ i A 0 = 1 λ ′′ ( v i ) = res p = v i  dp λ ′ ( p )  . (5.1) This r esu lt was already known in the ca se of dispersion less Gelfand-Dikii [5] reductions. Howe ver , it holds for all Benney reductions. 5.1 Completing the Loewner system W e m ove now our attent ion from the metric to the Chris toffe l symbol s and t he curvature tensor , l ooking for a way to describ ing t hese objects in terms of λ and i ts critical points. 20 Howe ver , t his step is not i mmediate. Indeed, from equations (4.7) and (4.8) we need to find an expression in the λ picture for the quantities δ ( v i ) , δ (lo g p ∂ i A 0 ) , we will see that the right object to look at is F ( p ) = ∂ p ∂ λ + n X i =1 ∂ p ∂ λ i , (5.2) obtained from the in verse function of λ with respect to p , that is p = p ( λ, λ 1 , . . . , λ n ) . (5.3) The function F is thus d etermined once the function λ ( p, λ 1 , . . . , λ n ) is known. Before discussing the C hristoffel symbols and the curvature, we will consider the properties of this function in detail. First of all, using the Loe wner equations (3.12) a n d the expression (3.17), we can write F ( p ) in the following form: F ( p ) = 1 λ ′ ( p ) − n X i =1 ∂ i A 0 p − v i = 1 λ ′ ( p ) − n X i =1 res p = v i  1 λ ′ ( p )  1 p − v i . (5.4) From its expansion (3.19), the function 1 λ ′ ( p ) in v i has a simple pole, th erefore, F ( p ) i s analytic at p = v i . Using this fact, we can pro ve the following Theor em 7 Let λ ( p, λ 1 , . . . , λ n ) be a s olution of equat ion (3.2), and let v 1 , . . . , v n be its critical points. Defin ing the function F ( p ) as above (5.2), we have F ( v i ) = δ ( v i ) (5.5) ∂ F ∂ p ( v i ) = δ (ln p ∂ i A 0 ) . (5.6) Pr oof W e have already sh own that F ( p ) is analyti c at p = v i . In order to prove (5.5) and (5.6), we consider the system of n + 1 differential equations ∂ p ∂ λ i = ∂ i A 0 v i − p i = 1 , . . . , n (5.7) ∂ p ∂ λ = n X k =0 ∂ k A 0 p − v k + F ( p ) . 21 The conditions ∂ 2 p ∂ λ i ∂ λ j − ∂ 2 p ∂ λ j ∂ λ i = 0 i 6 = j (5.8) giv e no thing but the Gibb ons-Tsarev system, h ence are s atisfied for any reduction. So, we concentrate our attention on the remaining n cons istency conditions ∂ 2 p ∂ λ∂ λ i − ∂ 2 p ∂ λ i ∂ λ = 0 , (5.9) which – by construction – are satisfied, to obtain s ome information about F ( p ) . E xpanding both sides we obtain ∂ 2 p ∂ λ∂ λ i = ∂ i A 0 ( p − v i ) 2 ∂ p ∂ λ = ∂ i A 0 ( p − v i ) 2 " F ( p ) + n X k =1 ∂ k A 0 p − v k # , ∂ 2 p ∂ λ i ∂ λ = ∂ F ∂ p ∂ p ∂ λ i + ∂ F ∂ λ i + n X k =1  ∂ k ∂ i A 0 p − v k + ∂ k A 0 ( p − v k ) 2  ∂ i v k − ∂ p ∂ λ i  = ∂ F ∂ p ∂ i A 0 v i − p + ∂ F ∂ λ i + n X k =1  ∂ k ∂ i A 0 p − v k + ∂ k A 0 ( p − v k ) 2  ∂ i v k + ∂ i A 0 p − v i  , substitut ing the Gibbons-Tsarev equations ( 2.6) i n the above formulae and rearranging (5.9), we find that F ( p ) − δ ( v i ) ( p − v i ) 2 + ∂ F ∂ p ( p ) − 2 δ (lo g √ ∂ i A 0 ) p − v i − 1 ∂ i A 0 ∂ F ( p ) ∂ λ i = 0 . (5.10) Multiply ing by ( p − v i ) 2 and taking the limit for p → v i , we get F ( v i ) = δ ( v i ) . Then, taking the residue of the right hand side of (5.10) at p = v i giv es ∂ F ∂ p ( v i ) = δ (log p ∂ i A 0 ) .  Thus, specifying the function F turns out to be the analogue, in the λ picture, of com- pleting t he sys tem (2.6), from which we were able to express t he Christo f fel symbol s and th e curva t ure tensor of the metric. It fol lows from its defi nition, that F ( p ) i s obtained from 1 λ ′ ( p ) by subtractin g off its si ngu- larities at p = v i . The impo rtance of this function is that it d escribes the in variant properties 22 of the reduction. For instance, if a reduction admits a function λ associated wit h it such that F ( p ) = 1 , then t he reducti on is Galilean in variant. The ca s e F ( p ) = p corresponds in stead to the scaling in variance of t he sy stem. As s een in example below , the function F , h ence these in var iances, are strongly related with the curv ature of th e Poisson brack et. Example 5.4 The 2 − component Zakhar ov r eductio n is k n own to possess both Galilean and scaling in varia nce. Using the tec hni que above, the f ormer can be e xpl ained by sa ying th at, for the function λ ( p ) given in Example 3.2, it follows F ( p ) = 1 . Thus, we get δ ( v i ) = n X k =1 ∂ v i ∂ λ k = 1 , wher e the Ri emann in varia nts ar e the criti cal values of λ ( p ) . F or t he scaling in vari ance, one can pr oceed as follows: define the f unction ϕ ( p ) = ln λ ( p ) , wher e λ ( p ) is the sam e as befor e. This new f unction has the same critical point s v i as λ , plus t he p oles of λ ( p ) , which play no r ol e in th e deeri vation of the r eduction (see Remark 3). Hence, it is associat ed with the Zakhar ov r educti on, and in this case F ( p ) = p . Thus δ ( v i ) = n X k =1 ∂ v i ∂ ϕ k = v i , (5.11) wher e the ϕ i = ln λ i ar e the natural Riemann inv a riants associa ted with ϕ ( p ) , i.e . its critical values. It is elementary to show that (5.11) corresponds to t he scali ng in vari ance with r espect to the λ i . Remark 7 System (5.7) was consider ed for the first time in connection with a Benne y r educ- tion by K okotov and K o r otkin [ 14], in the pa rticular case of t he N − component Zakhar ov r eduction, wher e F ( p ) = 1 . In the next sectio n, we wi ll us e t he function F to describe the Christoffel symbols and the curv ature in terms of λ . Nev ertheless, the latter can be e xpressed directly in terms of F using the following residu e formula Pr oposition 5.4 In terms of the function F , the non vanishi ng components of the Riemann tensor satisfy the following identity: R ij ij = X k = i,j res p = v k  F ( p ) ( p − v i ) 2 ( p − v j ) 2 dp  = 1 2 π i Z Γ i ∪ Γ j F ( p ) ( p − v i ) 2 ( p − v j ) 2 dp (5.12) wher e Γ i and Γ j ar e two sufficiently small contours ar ound p = v i and p = v j . Pr oof . From (4.8) and Theorem 7 i t follows imm ediately that R ij ij = 1 ( v i − v j ) 2  ∂ F ∂ p ( v i ) + ∂ F ∂ p ( v j )  − 2 F ( v i ) − F ( v j ) v i − v j  (5.13) 23 It is easy to check the chain of identities: 1 ( v i − v j ) 2  ∂ F ∂ p ( v i ) + ∂ F ∂ p ( v j )  − 2 F ( v i ) − F ( v j ) v i − v j  = lim p → v i d dp  F ( p ) ( p − v j ) 2  + lim p → v j d dp  F ( p ) ( p − v i ) 2  = lim p → v i d dp  ( p − v i ) 2 F ( p ) ( p − v j ) 2 ( p − v i ) 2  + lim p → v j d dp  ( p − v j ) 2 F ( p ) ( p − v i ) 2 ( p − v j ) 2  = res p = v i  F ( p ) ( p − v i ) 2 ( p − v j ) 2 dp  + res p = v j  F ( p ) ( p − v i ) 2 ( p − v j ) 2 dp  . In the last identity we used the fac t that F ( p ) is re gular at p = v i , for all i .  5.2 Christoffel symbols and Cur vature tensor 5.2.1 Pote ntial metric W e are no w a ble t o complete the des cription of the Poisson bra cket a ssociated with a Benney reduction, in the case where the metric is potential in the coordinate used. Pr oposition 5.5 The Christoffel symbols (4.7) of the potentia l metric (5.1) can be written, in terms of the function λ , as Γ ij k = 0 , i 6 = j 6 = k (5.14a) Γ ij i = − Γ j i i = 1 ( v i − v j ) 2 , i 6 = j (5.14b) Γ ii k = − λ ′′ ( v i ) λ ′′ ( v k ) 1 ( v k − v i ) 2 , i 6 = k (5.14c) Γ ii i = 1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) − 1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 2 , (5.14d) The curvatur e tens or (4 .8) of the potential metric (5. 1) can be written, in terms of the function λ ( p ) as R ij ij = 3 ( v i − v j ) 4  1 λ ′′ ( v i ) + 1 λ ′′ ( v j )  + 1 ( v i − v j ) 3  λ ′′′ ( v i ) λ ′′ ( v i ) 2 − λ ′′′ ( v j ) λ ′′ ( v j ) 2  + 1 ( v i − v j ) 2  1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 3 − 1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) 2 + 1 4 λ ′′′ ( v j ) 2 λ ′′ ( v j ) 3 − 1 6 λ ′′′′ ( v j ) λ ′′ ( v j ) 2  + X k 6 = i,j 1 λ ′′ ( v k ) 1 ( v i − v k ) 2 ( v j − v k ) 2 . (5.15) 24 Pr oof Starting from the definition (5.2) of F , and us ing (3.19), we can write F ( v i ) = − 1 2 λ ′′′ ( v i ) λ ′′ ( v i ) 2 − X k 6 = i 1 λ ′′ ( v k ) 1 v i − v k , (5.16) ∂ F ∂ p ( v i ) = 1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 3 − 1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) 2 + X k 6 = i 1 λ ′′ ( v k ) 1 ( v i − v k ) 2 . (5.17) Then, by Theorem 7 , and subst ituting the above expressions for F into (4.7) and (4.8), we obtain (5.14) and (5.15) respectiv ely .  Remark 8 W e note t hat (5.14d) i s a consta nt multi ple o f the Schwarzian d erivative of λ ′ ( p ) , evaluated at p = v i . Recalling the expression (5.1) for the the metric, form the Propositi on above we find that the wh ole Poisson opera t or asso ciated with a Benney reduction of symbol λ depend s only on the critical points of λ and on the value of its second, th ird and fourth deriv atives eva l uated at these points. W e have now all we need to write the non local t ail of the Hamiltonian s tructure asso ciated to the metric g ii = ∂ i A 0 . Pr oposition 5.6 The non-vanishing compon ents of the R iemann tensor of the metric (5.1) admit the following quadratic e xpansion R ij ij = 1 2 π i Z C w i ( λ ) w j ( λ ) dλ. (5.18) wher e C = C 1 ∪ · · · ∪ C n with C i described as above , an d the functions w i ( λ ) = ∂ p ∂ λ ( p ( λ ) − v i ) 2 , ar e the generating functions of the symmetries (3.24) . Consequently the non local tail of the Hamiltonia n st ructur e associated to the metric g ii = ∂ i A 0 is given by 1 2 π i Z C w i ( λ ) λ i x  d dx  − 1 w j ( λ ) λ j x dλ. Pr oof W e prove the Proposition showing that the integra l in (5.18) is t he same as the right hand side of (5.15). First, writing t he integral 1 2 π i Z C w i ( λ ) w j ( λ ) dλ 25 in terms of the va ri able p we obt ain R ij ij = 1 2 π i Z Γ 1 λ ′ ( p ) ( p − v i ) 2 ( p − v j ) 2 dp = n X k =1 res p = v k 1 λ ′ ( p ) ( p − v i ) 2 ( p − v j ) 2 dp ! . (5.19) Using (3.19), the integrand can be e xpand ed, for k = 1 . . . . , n , as 1 ( p − v i ) 2 ( p − v j ) 2  1 λ ′′ ( v k ) 1 p − v k − 1 2 λ ′′′ ( v k ) λ ′′ ( v k ) 2 +  1 4 λ ′′′ ( v k ) 2 λ ′′ ( v k ) 3 − 1 6 λ ′′′′ ( v k ) λ ′′ ( v k ) 2  ( p − v k ) + . . .  . Thus, for k 6 = i, j we get res p = v k 1 λ ′ ( p ) ( p − v i ) 2 ( p − v j ) 2 dp ! = 1 λ ′′ ( v k ) 1 ( v k − v i ) 2 ( v k − v j ) 2 , while res p = v i 1 λ ′ ( p ) ( p − v i ) 2 ( p − v j ) 2 dp ! = 3 ( v i − v j ) 4 1 λ ′′ ( v i ) + 1 ( v i − v j ) 3 λ ′′′ ( v i ) λ ′′ ( v i ) 2 + 1 ( v i − v j ) 2  1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 3 − 1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) 2  , res p = v j 1 λ ′ ( p ) ( p − v i ) 2 ( p − v j ) 2 dp ! = 3 ( v j − v i ) 4 1 λ ′′ ( v j ) + 1 ( v j − v i ) 3 λ ′′′ ( v j ) λ ′′ ( v j ) 2 + 1 ( v j − v i ) 2  1 4 λ ′′′ ( v j ) 2 λ ′′ ( v j ) 3 − 1 6 λ ′′′′ ( v j ) λ ′′ ( v j ) 2  , From these, formula (5.15) for the curv ature tensor follows. The last statement of the Propo- sition is a consequence of the general theory of Ferapontov .  Remark 9 Alternatively , one can pr ove the above r esult usi ng starting fr om the functio n F , namely deforming the inte gral 1 2 π i Z Γ i ∪ Γ j F ( p ) ( p − v i ) 2 ( p − v j ) 2 dp which is shown in Pr opositi on 5.4 to be equal to the curvatur e, in to (5.18) . In or der to do so, it is sufficient to verify the following identit y − 1 2 π i Z Γ i ∪ Γ j P n k =1 ∂ k A 0 p − v k ( p − v i ) 2 ( p − v j ) 2 dp = 1 2 π i Z Γ − (Γ i ∪ Γ j ) 1 ∂ λ ∂ p ( p − v i ) 2 ( p − v j ) 2 dp, 26 that can b e pro ved by straightfo rwar d computation. Indeed, since the left hand side is t he sum of r esid ues at p = v i and p = v j of a functio n having a pol e of or d er 3 at p = v i , v j we obtain l.h.s = −  lim p → v i + lim p → v j  1 2 d 2 dp 2 n X k =1  ( p − v i ) ∂ k A 0 ( p − v j ) 2 ( p − v k )  = X k 6 = i,j ∂ k A 0 ( v i − v j ) 2  2 ( v i − v j )( v i − v k ) + 1 ( v i − v k ) 2 − 2 ( v i − v j )( v j − v k ) + 1 ( v j − v k ) 2  = X k 6 = i,j ∂ k A 0 ( v i − v k ) 2 ( v j − v k ) 2 . On the other h and the right ha nd side is the sum of re s idues at p = v k , for k 6 = i, j of a function having simple poles at p = v k : r .h.s. = X k 6 = i,j lim p → v k ( p − v k ) 1 ∂ λ ∂ p ( p − v i ) 2 ( p − v j ) 2 = X k 6 = i,j ∂ k A 0 ( v i − v k ) 2 ( v j − v k ) 2 = l.h.s. Recalling the results abov e, we ha ve the following Theor em 8 The r eduction of Benne y associated with the functio n λ ( p, λ 1 , . . . , λ n ) i s Ham il- tonian with the Hamiltoni an s tructur e Π ij = λ ′′ ( v i ) δ ij d dx + Γ ij k λ k x + 1 2 π i Z C ∂ p ∂ λ λ i x ( p ( λ ) − v i ) 2  d dx  − 1 ∂ p ∂ λ λ j x ( p ( λ ) − v j ) 2 dλ, (5.20) wher e Γ ij k λ k x = λ i x − λ j x ( v i − v j ) 2 i 6 = j , Γ ii k λ k x =  1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) − 1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 2  λ i x − X k 6 = i λ ′′ ( v i ) λ ′′ ( v k ) λ k x ( v i − v k ) 2 and C = C 1 ∪ · · · ∪ C n . Her e, t he v i ar e the critical points o f λ , and the λ i the critical values. In this coor dinates, the metric g ij = δ ij λ ′′ ( v i ) is potential . 5.2.2 The general case As we poi nted o ut previously , any metric (4.1) associated with a reduction can be put in potential form, after a suitable change of the Riemann in variants. Howe ver , it is often con- venient to write the expression of the Poisson operators generated by these metrics in terms 27 of the Riemann in variants sel ected by λ . Thu s, we consider the metrics g ii = ∂ i A 0 ϕ i ( λ i ) = 1 ϕ i ( λ i ) λ ′′ ( v i ) , (5.21) where ϕ i = ϕ i ( λ i ) are arbitrary functions, and we proceed as before. Pr oposition 5.7 The Christoffel symbols appearing in the Hamil tonian structur e ar e given by Γ ij k = 0 , i 6 = j 6 = k Γ ij i = ϕ j ( v i − v j ) 2 , i 6 = j Γ ij j = − ϕ i ( v i − v j ) 2 , i 6 = j Γ ii k = − λ ′′ ( v i ) λ ′′ ( v k ) ϕ i ( v k − v i ) 2 , i 6 = k Γ ii i = ϕ i  1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) − 1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 2  + 1 2 ϕ ′ i . Her e we denote ϕ ′ i = dϕ dλ i ( λ i ) . The nonlocal tail appearing in the Hamiltonian structur e is then given by 1 2 π i n X k =1 Z C k ∂ p ∂ λ λ i x ( p ( λ ) − v i ) 2  d dx  − 1 ∂ p ∂ λ λ j x ( p ( λ ) − v j ) 2 ϕ k ( λ ) dλ. (5.22) Pr oof The proof of the formul a for the Γ ij k is a straig htforward computation . Let us prove the second st atement. Since nothing new is in volved i n such com putations we wi ll sk ip the details. For the metric (5.21), the non vanishing components of the curva ture tensor are R ij ij = 3 ( v i − v j ) 4  ϕ i ∂ i A 0 + ϕ j ∂ j A 0  − 2 ( v i − v j ) 3  ϕ i ∂ i v i − ϕ j ∂ j v j  + 1 ( v i − v j ) 2  ϕ i ∂ i ln p ∂ i A 0 + ϕ j ∂ j ln p ∂ j A 0  + X k 6 = i,j ϕ k ∂ k A 0 ( v i − v k ) 2 ( v j − v k ) 2 + 1 2 ϕ ′ i + ϕ ′ j ( v i − v j ) 2 . (5.23) 28 Expression (5.23) can be written, in terms of λ , as R ij ij = 3 ( v i − v j ) 4  ϕ i λ ′′ ( v i ) + ϕ j λ ′′ ( v j )  + 1 ( v i − v j ) 3  ϕ i λ ′′′ ( v i ) λ ′′ ( v i ) 2 − ϕ i λ ′′′ ( v j ) λ ′′ ( v j ) 2  + 1 ( v i − v j ) 2  ϕ i  1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 3 − 1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) 2  + ϕ j  1 4 λ ′′′ ( v j ) 2 λ ′′ ( v j ) 3 − 1 6 λ ′′′′ ( v j ) λ ′′ ( v j ) 2  + X k 6 = i,j 1 λ ′′ ( v k ) ϕ k ( v i − v k ) 2 ( v j − v k ) 2 + 1 2 ϕ ′ i + ϕ ′ j ( v i − v j ) 2 , ( 5.24) The equivalence bet ween (5.22) and the rig ht h and si de of (5.24) can be o btained by rewr iting the integrals abo ve in the p − plane, 1 2 π i n X k =1 Z Γ k ϕ k ( λ ( p )) 1 λ ′ ( p ) ( p − v i ) 2 ( p − v j ) 2 dp. (5.25) and usin g the sam e ar gu ments o f the main theorem, except that for ev ery k , in the integral around Γ k we have to consider also the contrib uti on o f the function ϕ k ( λ ( p )) which e xpands, at p = v k , as ϕ k ( λ ( p )) = ϕ k + ϕ ′ k 2 λ ′′ ( v k ) ( p − v k ) 2 + . . .  It follows from the above that we ha ve Theor em 9 The r eduction of Benne y associated with the functio n λ ( p, λ 1 , . . . , λ n ) i s Ham il- tonian with the family of Hamiltonian structur es Π ij = ϕ i λ ′′ ( v i ) δ ij d dx + Γ ij k λ k x + 1 2 π i n X k =1 Z C k ∂ p ∂ λ λ i x ( p ( λ ) − v i ) 2  d dx  − 1 ∂ p ∂ λ λ j x ( p ( λ ) − v j ) 2 ϕ k ( λ ) dλ, (5.26) with Γ ij k λ k x = ϕ j λ i x − ϕ i λ j x ( v i − v j ) 2 i 6 = j , Γ ii k λ k x = ϕ i  1 6 λ ′′′′ ( v i ) λ ′′ ( v i ) − 1 4 λ ′′′ ( v i ) 2 λ ′′ ( v i ) 2  λ i x + 1 2 ϕ ′ i λ i x − X k 6 = i λ ′′ ( v i ) λ ′′ ( v k ) ϕ i λ k x ( v i − v k ) 2 , wher e ϕ 1 , . . . , ϕ n ar e arbitrary functio ns of a s ingle var iable. Her e, the v i ar e the crit ical points o f λ , the coor di nates λ i the cor r esponding c r itical val ues, a nd t he C k ar e the contours defined above . 29 6 Finite nonloca l tail: som e examples In th e expression (5.26) for the Poisson operator , the components of the curvature are ex- pressed as integrals o f fun ctions around suit able contours in a com plex domain. A natural question to ask is wheth er this int egral can be reduced to a finite s um, and we wil l s how now som e examples where this is pos sible. For simplicit y , we will consider th e case when ϕ 1 ( λ ) = · · · = ϕ n ( λ ) = λ k , for k ∈ Z . In this case the curvature can be expressed as R ij ij = 1 2 π i Z C  ∂ p ∂ λ  2 λ k ( p ( λ ) − v i ) 2 ( p ( λ ) − v j ) 2 dλ, k ∈ Z . Essentially , th e fi ni te expansion app ears whene ver is possible to substitut e the the contour C with a contour a rou nd λ = ∞ and a finite number of other mark ed points . W e il lustrate this special situation in two simple e x amples. 6.1 2-component Zakhar ov red uction In th is case (see examples 2.1, 3.2), since λ is a single-valued rational function of p, it is con venient to work in the p -pl ane. In order to calculate the curvature, t he non vanishing components of the Riemann tensor are giv en by R 12 12 = 2 X i =1 res p = v i λ ( p ) k 1 λ ′ ( p ) ( p − v 1 ) 2 ( p − v 2 ) 2 dp ! , k ∈ Z . The abelian dif ferential λ ( p ) k 1 λ ′ ( p ) ( p − v 1 ) 2 ( p − v 2 ) 2 dp has poles at the points p = v 1 , p = v 2 , as well as: if k > 2 p = ∞ , p = A 1 A 0 ( poles of λ ) if k < 0 p = s 1 = 1 2 A 1 + ( A 2 1 − 4 A 3 0 ) 1 / 2 A 0 , p = s 2 = 1 2 A 1 − ( A 2 1 − 4 A 3 0 ) 1 / 2 A 0 ( zeros of λ ) , while for k = 0 , 1 , 2 there are no other poles. Since t he sum of the residues of an abelian diffe rential on a com pact R iemann surface is zero, we can substitute t he sum of residues at p = v 1 , v 2 with, respectiv ely - zero if k = 0 , 1 , 2 , 30 - minus the sum of residues at p = ∞ , p = A 1 A 0 if k > 2 - minus the sum of residues at p = s 1 , p = s 2 , if k < 0 . Summarizing, we hav e R ij ij = 0 , k = 0 , 1 , 2 , R ij ij = − res p = ∞ + res p = A 1 A 0 ! λ ( p ) k 1 λ ′ ( p ) ( p − v i ) 2 ( p − v j ) 2 dp, k > 2 , R ij ij = −  res p = s 1 + res p = s 2  λ ( p ) k 1 λ ′ ( p ) ( p − v i ) 2 ( p − v j ) 2 dp, k < 0 . Moreover , as a counterpart in the λ -plane of the abov e formulae we have R ij ij = 0 , k = 0 , 1 , 2 , R ij ij = − 2 res λ = ∞  w 1 ( λ ) w 2 ( λ ) λ k dλ  , k > 2 , R ij ij = − 2 res λ =0  w 1 ( λ ) w 2 ( λ ) λ k dλ  , k < 0 , and this shows that the residues ha ve to be comput ed around marked points, which not de- pend on the dynamics of the reduction. Ex panding w 1 ( λ ) and w 2 ( λ ) near λ = ∞ , we get w 1 ( λ ) = − ∞ X k =1 k w 1 k λ k +1 = − 1 λ 2 − 2 v 1 λ 3 − 3(2 A 3 0 + A 2 1 + 2 A 1 A 3 2 0 ) A 2 0 1 λ 4 − 4( A 3 1 + 6 A 1 A 3 0 + 3 A 9 2 0 + 3 A 2 1 A 3 2 0 ) A 3 0 1 λ 5 + . . . w 2 ( λ ) = − ∞ X k =1 k w 2 k λ k +1 = − 1 λ 2 − 2 v 2 λ 3 − 3 (2 A 3 0 + A 2 1 − 2 A 1 A 3 2 0 ) A 2 0 1 λ 4 − 4 ( A 3 1 + 6 A 1 A 3 0 − 3 A 9 2 0 − 3 A 2 1 A 3 2 0 ) A 3 0 1 λ 5 + . . . 31 and near λ = 0 w 1 ( λ ) = ∞ X h =0 z 1 − h λ h = − 2 A 2 0 ( − p A 2 1 − 4 A 3 0 + A 1 )  A 1 − p A 2 1 − 4 A 3 0 + 2 A 3 2 0  2 p A 2 1 − 4 A 3 0 + − 8 A 3 0  p A 2 1 − 4 A 3 0 ( A 3 0 − A 2 1 ) + A 3 1 − 3 A 1 A 3 0 + 2 A 9 2 0  ( A 2 1 − 4 A 3 0 ) 3 2  A 1 − p A 2 1 − 4 A 3 0 + 2 A 3 2 0  3 λ + . . . w 2 ( λ ) = ∞ X h =0 z 2 − h λ h = − 2 A 2 0 ( − p A 2 1 − 4 A 3 0 + A 1 )  A 1 − p A 2 1 − 4 A 3 0 − 2 A 3 2 0  2 p A 2 1 − 4 A 3 0 + − 8 A 3 0  p A 2 1 − 4 A 3 0 ( A 3 0 − A 2 1 ) + A 3 1 − 3 A 1 A 3 0 − 2 A 9 2 0  ( A 2 1 − 4 A 3 0 ) 3 2  A 1 − p A 2 1 − 4 A 3 0 − 2 A 3 2 0  3 λ + . . . and taking into ac count that the coef ficients of the expansion are charac teristic ve locities of symmetries, we easily obtain the quadratic expansion of the Riemann tensor . For k > 2 we hav e R 12 12 = X i + j = k − 1  w 1 i w 2 j + w 1 j w 2 i  , while for k < 0 , we obtai n R 12 12 = X i + j = k + 1  z 1 i z 2 j + z 1 j z 2 i  , which can be put in the canonical form (1.12) after a li near cheng e of b asis o f the sym metries. The expressions of these expansions in the Rieman inv ariants can be found by using formulae giv en in Example 2.1. 6.2 Dispersionless Boussinesq r eduction The case of the dispersionl ess Boussinesq reduction can be treated in a si milar way . From Example 3.3, we will consider a function λ which is polynomial in p , λ = p 3 + 3 A 0 p + 3 A 1 , thus meromorph ic on the Riemann sphere. The choice of a differe nt norm alisation reflects in the expansions below , wh ere we hav e to consid er an expansion in t he local parameter t = λ − 1 3 . For simplicit y l et us consider only the case k ≥ 0 . W e obs erve that, apart from the poles at p = v 1 and p = v 2 , we ha ve o nly an additional pole at infinity (starting from k = 2 ). Follo wi ng the same procedure used in the Zakharov case we obtain R ij ij = 0 , k = 0 , 1 , R ij ij = 3 res t =0  w 1 ( t ) w 2 ( t ) t − (3 k +4) dt  , k > 2 . 32 The expansions of w 1 ( t ) and w 2 ( t ) near t = 0 are give n by w 1 ( t ) = ∞ X k =0 k w 1 k t k +4 = t 4 + 2 ( − A 0 ) 1 2 t 5 + 4  A 1 − ( − A 0 ) 3 2  t 7 + 5  2 A 1 ( − A 0 ) 1 2 − A 2 0  t 8 + . . . w 2 ( t ) = ∞ X k =0 k w 2 k t k +4 = t 4 − 2 ( − A 0 ) 1 2 t 5 + 4  A 1 + ( − A 0 ) 3 2  t 7 + 5  − 2 A 1 ( − A 0 ) 1 2 − A 2 0  t 8 + . . . From these formulas we immediately get the quadratic expansion of the R i emann tensor: k = 0 : R 12 12 = 0 , k = 1 : R 12 12 = 0 , k = 2 : R 12 12 = 3( v 1 + v 2 ) = 0 , More generally , we hav e R 12 12 = 3 2 X i + j = k − 1 ( w 1 3 i w 2 3 j + w 1 3 j w 2 3 i ) k > 2 . 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