Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations
aa r X i v : . [ n li n . S I] N ov Integrable discretisation of hodograph-typesystems, hyperelliptic integrals andWhitham equations
B.G. Konopelchenko and W.K. Schief , Department of Mathematics and Physics “Ennio de Giorgi”, University of Salentoand sezione INFN, Lecce, 73100, Italy School of Mathematics and Statistics, The University of New South Wales, Sydney,NSW 2052, Australia Australian Research Council Centre of Excellence for Mathematics and Statistics ofComplex Systems, School of Mathematics and Statistics, The University of NewSouth Wales, Sydney, NSW 2052, Australia
Abstract
Based on the well-established theory of discrete conjugate nets in dis-crete differential geometry, we propose and examine discrete analogues ofimportant objects and notions in the theory of semi-Hamiltonian systemsof hydrodynamic type. In particular, we present discrete counterparts of(generalised) hodograph equations, hyperelliptic integrals and associatedcycles, characteristic speeds of Whitham type and (implicitly) the corre-sponding Whitham equations. By construction, the intimate relationshipwith integrable system theory is maintained in the discrete setting.
Systems of quasi-linear first-order differential equations of the form u it = λ i ( u ) u ix , i = 1 , . . . , N + 1 , (1.1)where subscripts denote partial derivatives, represent an important subclassof partial differential equations which admit special properties and a varietyof applications [1]. In physics, such systems arise, in particular, as limits ofnonlinear partial differential equations without dissipation or dispersion andas Whitham equations for slow modulations (see, e.g., [2, 3]). The theory of(more general) Hamiltonian quasi-linear systems of hydrodynamic type has beendeveloped by Dubrovin and Novikov [4–6]. It has been established in [7–12] thatthese systems are intimately related to important notions in classical differentialgeometry. In particular, it has been demonstrated by Tsarev [10–12] that asemi-Hamiltonian system of the type (1.1) possesses an infinite set of integralsof motion with densities ψ obeying the system of linear hyperbolic equations ψ u i u k = A ik ψ u i + A ki ψ u k , i = k, (1.2)1herein the coefficients A ik are defined by the system λ iu k = A ik ( λ k − λ i ) , (1.3)and the fluxes ψ ⋆ in the corresponding conservation laws are (uniquely) deter-mined via integration of the compatible system ψ ⋆u i = λ i ψ u i . (1.4)These semi-Hamiltonian systems admit an infinite number of symmetries [10–12] u it α = λ iα ( u ) u ix , α = 2 , , , . . . , (1.5)where each set of characteristic speeds { λ iα } constitutes a solution of (1.3) re-garded as a linear system. The compatibility of the latter (in the sense of anatural Cauchy problem) is equivalent to the existence of the semi-Hamiltonianstructure. In the differential-geometric context, the constituent equations of(1.2) are known as conjugate net equations since these constitute the governinglinear equations in the classical theory of conjugate nets (see, e.g., [13]). More-over, the connection between the density ψ and the flux ψ ⋆ is of Combescuretype (see, e.g., [14]). In modern integrable system terminology, the densities ψ constitute eigenfunctions while the characteristic speeds λ iα represent associatedadjoint eigenfunctions.Remarkably, Tsarev has proven [10, 11] that, locally, all solutions of a semi-Hamiltonian system of the form (1.1) are given implicitly by the algebraic system x + λ i ( u ) t − ω i ( u ) = 0 , i = 1 , . . . , N + 1 (1.6)with { ω i } denoting the general set of adjoint eigenfunctions obeying the linearsystem (1.3). This linearisation technique has come to be known as the gen-eralised hodograph method since, in the case N = 1, the quantities x and t regarded as the unknowns of the system (1.6) obey the classical hodographequations [15] x u + λ ( u ) t u = 0 , x u + λ ( u ) t u = 0 . (1.7)In fact, in this paper, it is shown that such (generalised) hodograph equationsexist for arbitrary N . In summary, the properties of semi-Hamiltonian systemsof the form (1.1) and their solutions are completely encoded in the classicalsurface theory of conjugate nets. This highlights the privileged nature of semi-Hamiltonian systems of hydrodynamic type.The particular class of conjugate nets governed by the compatible hyperbolicequations φ x i x k = 1 x i − x k ( ǫ k φ x i − ǫ i φ x k ) , (1.8)wherein the ǫ i constitute constants, plays a distinguished role in the theory ofsemi-Hamiltonian hydrodynamic-type systems (with the identification u = x ).However, it is important to note that, in general, these special conjugate net2quations are, a priori , unrelated to the conjugate net equations (1.2). In par-ticular, the eigenfunction φ does not necessarily play the role of a density. Thelinear equations (1.8) are known as Euler-Poisson-Darboux equations and havebeen the subject of extensive investigation in classical differential geometry(see, e.g., [16]). Their importance in the one-phase Whitham equations forthe Korteweg-de Vries and nonlinear Schr¨odinger equations has been observedin [17–19] and, in the multi-phase case, in [20]. In fact, the explicit expressionsfor the characteristic speeds in the multi-phase Whitham equations derived inthe pioneering paper [21] and also in [20] contain, as elementary building blocks,particular solutions of the Euler-Poisson-Darboux equations with parameters ǫ i = . Recently it has been demonstrated [22] that the characteristic speedsfor multi-phase Whitham equations may be obtained by means of iterated Dar-boux transformations generated by contour integrals of separable solutions of(extended) Euler-Poisson-Darboux systems. Euler-Poisson-Darboux systems fordifferent values of ǫ i play also a central role in the treatment of various disper-sionless soliton equations and ǫ -systems [23–25].The connection between the Euler-Poisson-Darboux system and the charac-teristic speeds of multi-phase Whitham equations (and therefore the associatedconjugate net system (1.2)) is provided by the observation that the hyperellipticintegrals I b κ ζ k qQ g +1 i =1 ( ζ − x i ) dζ, (1.9)where the contours b κ , κ = 1 , . . . , g are appropriately chosen cycles [20, 21, 26]and k ≤ g −
1, may be regarded as superpositions of separable solutions ofthe Euler-Poisson-Darboux system (1.8) for ǫ i = . The methods recordedin [20–22] are then used in the (algebraic) construction of the characteristicspeeds λ i . For instance, in the case g = 1, the elliptic integral (1.9) may becalculated to be essentially φ O = 2 π √ x − x K r x − x x − x ! , (1.10)where K denotes the complete elliptic integral of the first kind. Each of thethree coordinates x i gives rise to a classical Levy transform [27] λ i = φ − φ O φ O x i φ x i (1.11)of the simplest non-constant solution φ = ( x + x + x ) of the Euler-Poisson-Darboux system (1.8) generated by φ O and these coincide with the characteris-tic speeds for the one-phase Whitham equations [2]. The avatar (1.11) of thesecharacteristic speeds may be found in [19, 28]. It is remarked in passing thatthe action of the Levy transformation on semi-Hamiltonian systems of hydro-dynamic type has been discussed in detail in [29].As indicated above, many systems of hydrodynamic type (1.1) admit dis-persive counterparts which are integrable by means of the Inverse Spectral3ransform (IST) method (see, e.g., [2, 6]). One of the remarkable propertiesof IST integrable equations is that they admit integrable discretisations whichreveal their fundamental properties (see, e.g., [30]). Such discretisations areusually constructed via invariances of the integrable equations under B¨acklund,Darboux or similar discrete transformations (see, e.g., [14]). This method si-multaneously leads to a discretisation of the underlying linear representation(Lax pair [31]).Based on the standard discretisation (see, e.g., [32]) of the conjugate netequations (1.2) and associated adjoint equations (1.5), we here propose a canon-ical integrability-preserving way of discretising the theory outlined in the pre-ceding. In particular, this is shown to lead to integrable discretisations of gener-alised hodograph equations, canonical cycles and associated hyperelliptic inte-grals and characteristic speeds of commuting flows of hydrodynamic type suchas those corresponding to the multi-phase Whitham equations. It is noted thatlarge classes of solutions of the standard discrete conjugate net equations maybe obtained by means of, for instance, the ∂ -bar dressing method [33], Darboux-type transformations [34,35] or the algebro-geometric approach employed in [36].Our approach exploits the existence of a canonical discretisation of the classicalEuler-Poisson-Darboux system (1.8) and associated separable solutions. We are concerned with commuting flows of diagonal systems of hydrodynamictype, that is, compatible systems of first-order equations of the type u it α = λ iα ( u ) u ix , i = 1 , . . . , N + 1 , α = 1 , . . . , N, (2.1)where the subscripts on the functions u i denote derivatives with respect to theindependent variables x and t α . It is emphasised that even though the abovesystems constitute the point of departure, many of the mathematical notionspresented in this paper go beyond these systems and turn out to be of interestin their own right. It is known [11] that diagonal systems of hydrodynamic typecommute if and only if the N sets of characteristic speeds { λ iα } labelled by α obey the same linear system λ iαu k = A ik ( λ kα − λ iα ) , (2.2)where the coefficients A ik may be regarded as being defined by the equations for,say, α = 1. Here, λ iαu k = ∂λ iα /∂u k . Furthermore, it is readily verified that theabove linear equations may also be regarded as the compatibility conditions forthe existence of some functions ψ α defined (up to constants of integration) by ψ αu i = λ iα ψ u i , (2.3)where ψ is a solution of the linear hyperbolic equations ψ u i u k = A ik ψ u i + A ki ψ u k . (2.4)4t is important to note that the coefficients A ik cannot be arbitrary as theseare constrained by the compatibility conditions for the hyperbolic equations(2.4) or, equivalently, the first-order equations (2.2). In fact, the coefficients A ik must be solutions of an integrable system of nonlinear partial differentialequations known as the Darboux system. Indeed, in the context of the geometrictheory of integrable systems (see, e.g., [14] and references therein), the function ψ constitutes an eigenfunction of the conjugate net equations (2.4) and thesets { λ iα } represent adjoint eigenfunctions. The functions ψ α are Combescuretransforms of the eigenfunction ψ and, for reasons of symmetry, it is evidentthat each Combescure transform is a solution of another system of conjugatenet equations with different coefficients. In order to motivate the approach adopted in this paper, we here recall thegeneralised hodograph method developed by Tsarev in [11] for a single systemof hydrodynamic-type equations u it = λ i ( u ) u ix , i = 1 , . . . , N + 1 (2.5)with associated functions A ik defined by (2.2), that is, λ iu k = A ik ( λ k − λ i ) . (2.6)Thus, Tsarev’s theorem states that if { ω i } is another set of adjoint eigenfunc-tions obeying the above linear system then any local solution u ( x, t ) of thenonlinear system ω i ( u ) = λ i ( u ) t + x (2.7)constitutes a solution of the hydrodynamic-type system (2.5). Conversely, anysolution of the hydrodynamic-type system may locally be represented in thismanner. As indicated in the introduction, in the case N = 1, (2.7) may beregarded as a linear system for x and t rather than a nonlinear system for u and u and differentiation of x ( u , u ) and t ( u , u ) leads to the classicalhodograph system [15] x u + λ t u = 0 , x u + λ t u = 0 . (2.8)Here, the coefficients λ i are regarded as known functions of the independentvariables u k . In the original context, this linear system is obtained from thenonlinear two-component system (2.5) N =1 by merely interchanging dependentand independent variables, whereby the Jacobian determinant drops out. Even though the generalised hodograph method encapsulated in the algebraicsystem (2.7) is applicable for all N , an associated system of hodograph-typeequations is not available for N > N − { µ i } constitutes another set of adjoint eigenfunctions then we may locally definea coordinate transformation u = u ( x, t ) (2.9)via the system µ i ( u ) = N X α =1 λ iα ( u ) t α + x (2.10)which coincides with the system (2.7) in the case N = 1.It is now easy to see that Tsarev’s generalised hodograph method is stillvalid in this more general setting so that, locally, the general solution of thehydrodynamic-type system (2.1) is encapsulated in the algebraic system (2.10)regarded as a definition of u . In fact, this observation may be interpreted as acorollary of Tsarev’s theorem since if we select a “time” t α and regard all other t α s as parameters then system (2.10) may be formulated as ω i ( u ) = λ iα ( u ) t α + x, (2.11)where the quantities ω i = µ i − X α = α λ iα t α (2.12)represent linear superpositions of adjoint eigenfunctions, so that, according tothe generalised hodograph method, (2.1) holds for α = α .As in the classical case ( N = 1), the algebraic system (2.10) turns out to beequivalent to a system of first-order differential equations. Indeed, if we regard(2.10) as a definition of some functions µ i then, on substitution into the adjointeigenfunction equations (2.2), it is readily verified that these functions constituteadjoint eigenfunctions if and only if the generalised hodograph equations x u k + N X α =1 λ iα t αu k = 0 , i = k (2.13)are satisfied. By construction, this system of hodograph type is equivalent tothe original hydrodynamic-type system (2.1). It turns out that, just like the characteristic speeds λ iα and the quantities µ i ,the remaining ingredients x and t α of the algebraic system (2.10) have distinctsoliton-theoretic meaning. Thus, we first consider two sets { µ i } and { λ i } ofadjoint eigenfunctions obeying µ iu k = A ik ( µ k − µ i ) , λ iu k = A ik ( λ k − λ i ) (2.14)6or some solution { A ik } of the underlying Darboux system. This system isknown to be invariant under adjoint Darboux transformations [27, 37]. Specifi-cally, for fixed i , the adjoint Darboux transformation D i generated by λ i trans-forms the adjoint eigenfunctions µ l according to D i ( µ i ) = µ i − λ i λ iu i µ iu i D i ( µ k ) = λ i µ k − λ k µ i λ i − λ k , k = i. (2.15)By construction, the above Darboux transforms obey a linear system of thetype (2.14) with coefficients depending on A ik and the adjoint eigenfunctions λ i only. The latter property guarantees that adjoint Darboux transformations maybe iterated in the following purely algebraic manner. Given any N sets of eigen-functions { λ iα } , we begin with the adjoint Darboux transformation D generatedby λ . The quantities D ( µ l ) then constitute new adjoint eigenfunctions. Inparticular, if we focus on the new adjoint eigenfunctions D ( λ l ) then we may usethe adjoint eigenfunction D ( λ ) to define an adjoint Darboux transformationacting on the new adjoint eigenfunctions which we denote by D . This proceduremay be repeated to construct N adjoint Darboux transformations D , . . . , D NN generated by the adjoint eigenfunctions λ , D ( λ ) , D ( D ( λ )) , . . . . On use ofJacobi’s identity for determinants [38], it is then straightforward to verify byinduction that the N th Darboux transform of the adjoint eigenfunction µ N +1 is given by ( D NN ◦ · · · ◦ D )( µ N +1 ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ . . . λ N µ ... ... ... λ N +11 . . . λ N +1 N µ N +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ . . . λ N λ N +11 . . . λ N +1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (2.16)However, the right-hand side of the above expression is completely symmet-ric in both the upper and lower indices. Hence, the N th Darboux transformdepends neither on the order of application of the adjoint Darboux transforma-tions nor on the components of the sets of adjoint eigenfunctions { λ iα } which arechosen to generate the corresponding adjoint Darboux transformations. Moreprecisely, for any permutations ( α , . . . , α N ) and ( i , . . . , i N +1 ) of (1 , . . . , N )and (1 , . . . , N + 1) respectively, the iterated Darboux transform( D i N α N ◦ · · · ◦ D i α )( µ i N +1 ) = µ ( N ) (2.17)is the same. In fact, application of Cramer’s rule shows that x = µ ( N ) (2.18)7orresponds to the unique solution of the algebraic system (2.10) regarded as alinear system for x and t α . Thus, remarkably, by virtue of the commutativity ofthe flows (2.1), the “spatial” independent variable x may be interpreted as theunique N -fold Darboux transform constructed from the characteristic speeds λ iα .The interpretation of the “times” t α is now based on the observation thatthe system (2.10) is implicitly symmetric in x and t α . Indeed, for any fixed α ,the system (2.10) may be reformulated as µ i λ iα = X β = α λ iβ λ iα t β + 1 λ iα x + t α (2.19)so that the roles of x and t α have been interchanged. In fact, the a priori formal symmetry obtained in this manner may indeed be exploited by rewritingthe linear system (2.3) as ψ u i = 1 λ iα ψ αu i , ψ βu i = λ iβ λ iα ψ αu i , β = α. (2.20)The latter implies that the quantities ˜ λ iα = 1 /λ iα , ˜ λ iβ = λ iβ /λ iα and, in fact,˜ µ i = µ i /λ iα constitute adjoint eigenfunctions of the Darboux system associatedwith the eigenfunction ψ α . Hence, for reasons of symmetry, the time t α obtainedby means of Cramer’s rule from (2.19) or, equivalently, the original system (2.10)coincides with the iterated Darboux transform t α = µ ( N,α ) , µ ( N,α ) = ( ˜ D i N α N ◦ · · · ◦ ˜ D i α )(˜ µ i N +1 ) , (2.21)where the Darboux transformations ˜ D i k α k are now generated by the adjoint eigen-functions ˜ λ i k α k . It is also observed that the generalised hodograph system (2.13)may be solved for the derivatives of t α to deduce that t αu k = Λ αk ( λ iβ ) x u k (2.22)for some functions Λ αk and, hence, the N + 1 variables x and t α may also beregarded as Combescure transforms of each other. The formulation of the classical hodograph equations and their generalisation inthe language of (adjoint) eigenfunctions may instantly be utilised to derive theircanonical integrable discrete counterparts. Indeed, the standard integrable dis-cretisation of the conjugate net equations (2.4) turns out to be the fundamentalstructure on which this discretisation technique is based. Thus, if ψ : Z N +1 → R ( n , . . . , n N +1 ) ψ ( n , . . . , n N +1 ) (3.1)8s an eigenfunction obeying the discrete conjugate net equations (see, e.g., [32])∆ ik ψ = A ik ∆ i ψ + A ki ∆ k ψ, (3.2)where the forward difference operators ∆ i are defined by ∆ i ψ = ψ [ i ] − ψ and∆ ik = ∆ i ∆ k = ∆ k ∆ i , then a discrete Combescure transform ψ ⋆ of ψ defined by∆ i ψ ⋆ = λ i ∆ i ψ (3.3)exists if the associated discrete adjoint eigenfunctions λ i constitute solutions ofthe linear system [33, 39] ∆ k λ i = A ik ( λ k [ i ] − λ i [ k ] ) . (3.4)Here, a subscript [ i ] denotes the relative unit increment n i → n i + 1 so that themixed difference operator ∆ ik acts according to ∆ ik ψ = ψ [ ik ] − ψ [ i ] − ψ [ k ] + ψ .It is noted that (3.2) and (3.3) represent the discrete analogues of the linearequations (1.2) and (1.4) defining the densities ψ and fluxes ψ ⋆ associated withthe conservation laws for semi-Hamiltonian systems of hydrodynamic type. Inconnection with an appropriate Cauchy problem, it is convenient to reformulatethe adjoint linear system (3.4) as∆ k λ i = A ik A ik + A ki + 1 ( λ k − λ i ) . (3.5)As in the continuous case, the compatibility conditions for the (adjoint) eigen-function equations (3.2) and (3.4) (or (3.5)) give rise to the same nonlinearsystem of discrete equations for the coefficients A ik which constitutes the stan-dard integrable discretisation of the aforementioned Darboux system [33, 40].The discrete analogues of the classical adjoint Darboux transformations maybe obtained by formally replacing derivatives by differences in the transforma-tion laws (2.15). Indeed, the Darboux transforms of another set of adjointeigenfunctions { µ l } are given by D i ( µ i ) = µ i − λ i ∆ i λ i ∆ i µ i = λ i [ i ] µ i − λ i µ i [ i ] λ i [ i ] − λ i D i ( µ k ) = λ i µ k − λ k µ i λ i − λ k , k = i (3.6)for any fixed i corresponding to the adjoint eigenfunction λ i which generatesthe adjoint Darboux transformation D i . Since iteration of the discrete adjointDarboux transformations only involves the algebraic transformation law (3.6) which coincides with the transformation law (2.15) , the expressions (2.16),(2.17) and (2.21) for the iterated Darboux transforms are also valid in thediscrete case. Moreover, the quantities x and t α defined by x = µ ( N ) , t α = µ ( N,α ) (3.7)9till constitute the unique solution of the linear system (2.10), that is, µ i = N X α =1 λ iα t α + x, (3.8)wherein λ iα and µ i now refer to discrete adjoint eigenfunctions. In analogy withthe continuous case, insertion into the adjoint eigenfunction equations (3.5) for { µ i } leads to the linear system∆ k x + N X α =1 λ iα [ k ] ∆ k t α = 0 , i = k. (3.9)Conversely, any solution of this integrable discretisation of the generalised hodo-graph equations (2.13) provides via (3.8) a set of adjoint eigenfunctions { µ i } .Finally, the discrete generalised hodograph equations adopt the form∆ k t α = Λ αk ( λ iβ [ k ] )∆ k x, (3.10)which demonstrates that, as in the continuous case, the N + 1 variables x and t α may be interpreted as discrete Combescure transforms of each other.As pointed out in the previous section, there exists complete equivalencebetween the hydrodynamic-type system (2.1) and the generalised hodographequations (2.13). In fact, this is verified directly by employing a formulation interms of differential forms (cf. [41]). Indeed, it is seen that the system du i ∧ dx + N X α =1 λ iα du i ∧ dt α = 0 (3.11)reduces to the hydrodynamic-type system (2.1) if x and t α are chosen as theindependent variables. Alternatively, one may select the u i s as the independentvariables so that the generalised hodograph equations (2.13) result. Accordingly,the algebraic system (2.10) encodes the N + 1-dimensional integral manifolds M of the differential system (3.11). Thus, if we interpret a solution ( x, t )( u ) of thegeneralised hodograph equations (2.13) as an N +1-dimensional submanifold M of the space of (in)dependent variables R N +2 then this submanifold M admitsthe parametrisation u ( x ( u ) , t ( u ) , u ) . (3.12)However, locally, we may also utilise the parametrisation( x, t ) ( x, t , u ( x, t )) , (3.13)where u ( x, t ) represents a corresponding solution of the hydrodynamic-type sys-tem (2.1). In the discrete case, the algebraic system (3.8) encapsulates “discreteintegral manifolds” M ∆ in the following sense. Any solution( x, t ) : Z N +1 → R N +1 , n ( x, t )( n ) (3.14)10f the discrete generalised hodograph equations (3.9) may be used to parametrisea “discrete submanifold” M ∆ of R N +1 × δ Z × · · · × δ N +1 Z according to n ( x ( n ) , t ( n ) , u ∆ ) , (3.15)where u ∆ = ( δ n , . . . , δ N +1 n N +1 ) for prescribed lattice parameters δ i . Hence,the variable u ∆ may be regarded as a discretisation of either the independentvariables of the generalised hodograph equations (2.13) or the dependent vari-ables of the system of hydrodynamic type (2.1). The latter corresponds to an“implicit discretisation” of the hydrodynamic-type system with variable spacingbetween the lattice points on the N + 1-dimensional submanifold R N +1 of theindependent variables x and t . It is well known [17–20] that the characteristic speeds λ i associated with themulti-phase averaged Korteweg-de Vries (KdV) equations are related to linearhyperbolic equations of Euler-Poisson-Darboux type. In fact, recently, it hasbeen demonstrated [22] that these characteristics speeds may be generated bymeans of iterated Darboux transformations applied to separable solutions of(extended) Euler-Poisson-Darboux-type systems. It turns out that one mayconstruct canonical discretisations of the multi-phase characteristic speeds ifone carefully defines analogues of the hyperelliptic integrals associated with theunderlying Riemann surfaces of genus g ≥
1. In this section, we demonstratehow one may derive particular classes of discrete characteristic speeds from thediscrete Euler-Poisson-Darboux-type system[ δ i ( n i + ν i ) − δ k ( n k + ν k )]∆ ik φ = δ k ǫ k ∆ i φ − δ i ǫ i ∆ k φ, (4.1)where i = k ∈ { , . . . , g + 1 } , which include those of “averaged KdV” type.Here, the constants δ i are lattice parameters and the constants ǫ i determinethe nature of the contour integrals to be defined in §
5. For ǫ i = , this leadsto analogues of the above-mentioned hyperelliptic integrals. The parameters ν i reflect the fact that it is crucial to maintain the freedom of placing thediscretisation points not necessarily on the vertices of the Z g +1 lattice but,possibly, on the edges, faces etc. Thus, we regard the hyperbolic system (4.1)as a discretisation of the classical Euler-Poisson-Darboux system( x i − x k ) φ x i x k = ǫ k φ x i − ǫ i φ x k (4.2)obtained in the limit x i = δ i ( n i + ν i ), δ i →
0. In the following, the key ideais to introduce an auxiliary continuous variable y and supplement the discreteEuler-Poisson-Darboux system by the differential-difference equations[ y − δ i ( n i + ν i )]∆ i φ y = δ i ǫ i φ y + ( g − i φ. (4.3)The function φ = φ ( n , y ) is well-defined since the semi-discrete Euler-Poisson-Darboux system (4.1), (4.3) remains compatible.11 .1 Separable solutions As in the continuous case [22], we now focus on separable solutions of the semi-discrete Euler-Poisson-Darboux system (4.1), (4.3). Thus, it is readily verifiedthat the ansatz φ sep = ρ ( ζ ) g +1 Y i =1 ρ i ( ζ ) , (4.4)where we have suppressed the dependence of ρ and ρ i on y and n i respectively,leads to the first-order differential/difference equations∆ i ρ i = δ i ǫ i ζ − δ i ( n i + ν i ) ρ i , ρ y = (1 − g ) ζ − y ρ (4.5)with ζ being a (complex) constant of separation. The latter may be solved toobtain ρ = ( ζ − y ) g − (4.6)without loss of generality and, in the continuum limit δ i →
0, the differenceequations (4.5) reduce to ρ ix i = ǫ i ζ − x i ρ i . (4.7)Hence, up to a multiplicative constant, ρ i represents a canonical discretisationof ( ζ − x i ) − ǫ i so that φ sep = ρ ( ζ ) g +1 Y i =1 ρ i ( ζ ) → ( ζ − y ) g − g +1 Y i =1 ( ζ − x i ) − ǫ i (4.8)in the continuum limit. The separable solutions derived in the preceding may be superimposed to obtainlarge classes of solutions of the semi-discrete Euler-Poisson-Darboux system(4.1), (4.3). Here, we consider the contour integrals φ κ = I b κ ( ζ − y ) g − g +1 Y i =1 ρ i ( ζ ) dζ, κ = 1 , . . . , g, (4.9)where the contours b κ on the complex ζ -plane are assumed to be independentof y and “locally” independent of n , that is, we demand that∆ i I b κ f ( n , y ; ζ ) dζ = I b κ ∆ i f ( n , y ; ζ ) dζ (4.10)12or any relevant functions f . Accordingly, the semi-discrete Euler-Poisson-Darboux system (4.1), (4.3) admits vector-valued solutions of the form φ = φ ... φ g , ˆ φ = φ ... φ g . (4.11)The components of these solutions may be used to generate iteratively solu-tions of semi-discrete conjugate net equations with increasingly complex coef-ficients. Thus, the ( g − φ ofthe semi-discrete Euler-Poisson-Darboux system (4.1), (4.3) with respect to theindependent variable y is given by the compact expression φ g − = (cid:12)(cid:12)(cid:12)(cid:12) φ φ y · · · φ ( g − y ˆ φ ˆ φ y · · · ˆ φ ( g − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ φ y · · · ˆ φ ( g − y (cid:12)(cid:12)(cid:12) . (4.12)In the context of classical differential geometry, this Darboux transform is knownas the ( g − y . The Levy transformsof the particular solutions φ = g +1 X i =1 ǫ i δ i n i − ( g − y, φ (4.13)of the semi-discrete Euler-Poisson-Darboux system therefore read φ g − = P g +1 i =1 ǫ i δ i n i | ˆ φ y · · · ˆ φ ( g − y | + ( g − | ˆ φ − y ˆ φ y ˆ φ yy · · · ˆ φ ( g − y || ˆ φ y · · · ˆ φ ( g − y | φ g − = | φ · · · φ ( g − y || ˆ φ y · · · ˆ φ ( g − y | (4.14)so that it is readily verified that φ g − = P g +1 i =1 ǫ i δ i n i | ˆ I g − · · · ˆ I | − | ˆ I g − ˆ I g − · · · ˆ I || ˆ I g − · · · ˆ I | φ g − = | I g − · · · I || ˆ I g − · · · ˆ I | (4.15)with the contour integrals I κk = I b κ ζ k g +1 Y i =1 ρ i ( ζ ) dζ. (4.16)It is observed that, remarkably, the Levy transforms φ g − and φ g − are inde-pendent of y and that, by definition, φ = φ and φ = φ in the case g = 1.13he action of another Levy transformation with respect to the variable n i now produces the g -fold Levy transform λ i = φ ig [ φ ] = φ g − − φ g − ∆ i φ g − ∆ i φ g − (4.17)of φ . Here, the symbol λ i has been chosen to indicate that the set { λ i } willindeed be shown to constitute a set of adjoint eigenfunctions. Since the Levytransform λ i may be formulated as λ i = ∆ i ( φ g − /φ g − )∆ i (1 /φ g − ) , (4.18)we may set H = | ˆ I · · · ˆ I g − || I · · · I g − | , H = − | ˆ I · · · ˆ I g − ˆ I g − || I · · · I g − | ¯Γ = 1 , ¯Γ = g +1 X i =1 ǫ i δ i n i (4.19)to obtain the final expression λ i = ∆ i ( H ¯Γ + H ¯Γ )∆ i H . (4.20)This constitutes a natural discretisation of a particular case of the characteris-tic speeds obtained in an entirely different manner by Tian in the continuouscontext (cf. [20, p. 218] for α = 1, α k = 0 otherwise and H / ∼ K (1)1 / in Tian’snotation). Once again, in the case g = 1, the interpretation H = 1 /I and H = 0 is to be adopted. The connection with discrete characteristic speeds and the associated discretegeneralised hodograph equations (3.9) is now made as follows. By construc-tion, φ g − and φ g − are solutions of the same system of discrete conjugate netequations ∆ ik φ g − = B ik ∆ i φ g − + B ki ∆ k φ g − . (4.21)On the other hand, the classical Levy transforms of an eigenfunction correspond-ing to different “directions” x i may also be regarded as adjoint eigenfunctionsof another system of conjugate net equations [37]. The analogous statement istrue in the discrete case and, accordingly, the quantities λ i constitute adjointeigenfunctions associated with the discrete conjugate net equations∆ ik ψ = A ik ∆ i ψ + A ki ∆ k ψ, (4.22)14herein the coefficients A ik are related to the coefficients B ik by C ik = A ik A ik + A ki + 1 = B ki φ g − k ] ∆ k φ g − φ g − ∆ i φ g − k ] − ∆ k φ g − φ g − . (4.23)The above observation allows us to identify a particular set of discrete character-istic speeds which may be used in the discrete generalised hodograph equations(3.9). However, the definition of the latter requires 2 g + 1 sets of adjoint eigen-functions { λ iα } , each of which represents a solution of the adjoint eigenfunctionequations ∆ k λ i = C ik ( λ k − λ i ) (4.24)satisfied by the Levy transforms λ i .In order to construct canonical sets of adjoint eigenfunctions satisfying (4.24),it is required to introduce an explicit parametrisation of the functions ρ i inthe base separable solution (4.4) of the associated semi-discrete Euler-Poisson-Darboux system. Thus, in terms of Gamma functions [43], the general solutionof the difference equation (4.5) formulated as ρ i [ i ] = ζ − δ i ( n i + ν i − ǫ i ) ζ − δ i ( n i + ν i ) ρ i (4.25)is given by ρ i = ( δ i ) − ǫ i Γ( ξ i − n i − ν i + 1)Γ( ξ i − n i − ν i + ǫ i + 1) , ξ i = ζδ i (4.26)up to a constant of “integration” which may depend on ζ . In fact, the multi-plicative factor has been chosen in such a manner that ρ i → ( ζ − x i ) − ǫ i in thecontinuum limit δ i →
0. This is a consequence of the well-known asymptoticbehaviour lim | z |→∞ z b − a Γ( z + a )Γ( z + b ) = 1 , | arg( z ) | < π (4.27)of ratios of Gamma functions. In fact, the first two terms of the associatedclassical asymptotic expansion [44] readΓ( z + a )Γ( z + b ) = z a − b (cid:20) a − b )( a + b − z + O ( | z | − ) (cid:21) . (4.28)It is noted that (4.26) regarded as a discretisation of a “power function” essen-tially coincides with that considered in [45].It has been pointed out that the Levy transforms λ i = φ ig [ φ ] are independentof the auxiliary variable y . This is due to the fact that the seed solution φ ofthe semi-discrete Euler-Poisson-Darboux system is a polynomial in y of degreeat most g −
1. A canonical way of generating an infinite number of seed solutionswhich admit this property is to expand the separable solution φ = ζ σ φ sep = ζ σ ( ζ − y ) g − g +1 Y i =1 ρ i ( ζ ) , σ = g +1 X i =1 ǫ i − ( g −
1) (4.29)15bout ζ = ∞ to obtain φ = (1 − yζ − ) g − ∞ X m =0 Γ m ( n ) ζ − m . (4.30)The existence of this formal power series in ζ − is readily established by apply-ing the asymptotic expansion (4.28) to the function ρ i as given by (4.26) andreformulating it as an asymptotic series in ζ − , namely ρ i ( ζ ) = ζ − ǫ i (cid:20) ǫ i δ i (cid:18) n i + ν i − ǫ i + 12 (cid:19) ζ − + O ( | ζ | − ) (cid:21) . (4.31)Thus, for instance, the first two coefficients Γ and Γ are seen to beΓ = 1 , Γ = g +1 X i =1 ǫ i δ i (cid:18) n i + ν i − ǫ i + 12 (cid:19) . (4.32)It is evident that the expansion (4.30) is of the form φ = ∞ X α =0 Ξ α ( n , y ) ζ − α , (4.33)where the coefficients Ξ α ( n , y ) are polynomials in y of degree α if α ≤ g − g − α > g −
1. In fact,Ξ α ( n , y ) = g − X k =0 ( − y ) k (cid:18) g − k (cid:19) Γ α,k ( n ) , (4.34)where Γ α,k = Γ α − k if 0 ≤ k ≤ min( α, g −
1) (4.35)and Γ α,k = 0 otherwise. By construction, each coefficient Ξ α constitutes asolution of the semi-discrete Euler-Poisson-Darboux system (4.1), (4.3). Forinstance, Ξ = Γ = 1 represents the trivial constant solution, whileΞ = Γ − ( g − y Γ = φ + c (4.36)turns out to be a linear superposition of the trivial solution and the seed solution φ which has been used to construct the discrete characteristic speeds λ i givenby (4.20). The constant c may be read off (4.32).The general expression (4.12) for the iterated Darboux transform φ g − maybe used to generate the ( g − φ g − [Ξ α ] of any seed solutionΞ α . Since the degree of Ξ α in y is less than g , the Levy transform φ g − [Ξ α ] isindependent of y . Hence, the procedure outlined in § y = 0. As a result, one is immediately ledto the compact expression φ g − [Ξ α ] = (cid:12)(cid:12)(cid:12)(cid:12) Γ α, Γ α, · · · Γ α,g − ˆ I g − ˆ I g − · · · ˆ I (cid:12)(cid:12)(cid:12)(cid:12) | ˆ I g − · · · ˆ I | . (4.37)16n particular, by virtue of (4.36), it may be concluded that φ g − [Ξ ] = φ g − + c (4.38)so that, essentially, the ( g − λ i is retrieved. We may now employ the eigenfunctions φ g − [Ξ α ] and φ g − to generate additional discrete characteristic speeds in themanner described in § φ g − by φ g − [Ξ α ] in (4.17) and (4.18).In terms of the coefficients Γ α,k and the ratios of determinants H k = ( − k +1 | ˆ I · · · ˆ I g − k − ˆ I g − k +1 · · · ˆ I g − || I · · · I g − | , (4.39)these turn out to be λ iα = φ ig [Ξ α ] = ∆ i ( H Γ α, + · · · + H g Γ α,g − )∆ i H (4.40)and encode the discrete characteristic speeds λ i via λ i = λ i + c . (4.41)Accordingly, any choice of 2 g + 1 sets of adjoint eigenfunctions { λ iα } such as α = 1 , . . . , g + 1 gives rise to a discrete system of generalised hodograph equa-tions (3.9) with associated “implicitly defined” discrete commuting flows of hy-drodynamic type. Once again, it is observed that (4.40) represents a naturaldiscretisation of the compact formulation of the corresponding characteristicspeeds recorded in [20]. It has been demonstrated that the characteristic speeds λ iα are independent ofthe auxiliary variable y and, accordingly, the contour integrals I κk = I b κ ζ k g +1 Y i =1 ρ i ( ζ ) dζ (5.1)constitute the main ingredients in their construction. Here, we are concernedwith contours b κ which mimic canonical cycles associated with classical hyper-elliptic integrals [26]. To this end, we make the choice ǫ i = 12 , δ i = δ (5.2)so that the underlying discrete Euler-Poisson-Darboux system (4.1) reduces to2( n i + ν i − n k − ν k )∆ ik φ = ∆ i φ − ∆ k φ (5.3)17nd the continuum limit is represented by δ → δn i in x i = δ ( n i + ν i ) heldconstant as before. Up to the factor ζ k , the integrand of the contour integrals(5.1) may be formulated as ϕ = g +1 Y i =1 p ( ξ, n i , ν i ) , ξ = ζδ , (5.4)where the function p representing all functions ρ i is defined by p ( ξ, n, ν ) = Γ( ξ − n − ν + 1) √ δ Γ( ξ − n − ν + ) (5.5)in agreement with the choice (4.26). We now introduce the ordering n < n < · · · < n g < n g +1 (5.6)and choose ν k = 12 , ν k +1 = 0 (5.7)corresponding to the discretisation points x k = δ ( n k + ) and x k +1 = δn k +1 .Hence, the separable solution (5.4) of the discrete Euler-Poisson-Darboux sys-tem (5.3) becomes ϕ = 1 δ g + g Y k =0 Γ( ξ − n k +1 + 1)Γ( ξ − n k +1 + ) g Y k =1 Γ( ξ − n k + )Γ( ξ − n k + 1) . (5.8)For instance, in the case g = 1, we obtain ϕ = 1 δ Γ( ξ − n + 1)Γ( ξ − n + ) Γ( ξ − n + )Γ( ξ − n + 1) Γ( ξ − n + 1)Γ( ξ − n + ) . (5.9)Since the Gamma function is non-zero but has simple poles at non-positiveintegers, the distribution of zeros and poles of the function p ( ξ, n,
0) is given by p ( ξ, n,
0) = 0 , ξ = . . . , n − , n − p ( ξ, n,
0) = ±∞ , ξ = . . . , n − , n − , (5.10)whereas p ( ξ, n, ) = 0 , ξ = . . . , n − , n − p ( ξ, n, ) = ±∞ , ξ = . . . , n − , n − . (5.11)18igure 1: The distribution of zeros (circles) and poles (crosses) of the function ϕ on the ξ -plane and associated b -cycles for g = 2.Accordingly, the zeros and poles of the functions p which make up ϕ partiallycancel each other in such a manner that, as a function of ξ , ϕ has no zeros orpoles in the region g +1 [ k =1 ( n k − − , n k ) , n g +2 = ∞ . (5.12)It is therefore natural to define the g contours b κ (on the ξ -plane) as closedpaths of counterclockwise orientation which pass through the pairs of intervals( n κ − − , n κ ) and ( n g +1 − , ∞ ) for κ = 1 , . . . , g as indicated in Figure 1.The contour integrals I b κ ζ k ϕ ( ξ ) dζ, κ = 1 , . . . , g, k = 0 , . . . , g − I b κ ζ k qQ g +1 i =1 ( ζ − x i ) dζ (5.14)with the contours b κ essentially becoming the b -cycles employed in [20, 21] inthe limit δ →
0. The intervals ( n κ − − , n κ ) and ( n g +1 − , ∞ ), κ = 1 , . . . , g correspond to the cuts ( x κ − , x κ ) and ( x g +1 , ∞ ) along which the upper andlower sheets of the underlying Riemann surface of genus g are joined. In fact,the ζ -plane represents the union of the half of the upper sheet and the half of thelower sheet which contain the b -cycles. This union is discontinuous between thecuts and, in the discrete case, this is reflected by the presence of poles and zerosbetween the intervals ( n κ − − , n κ ) and ( n g +1 − , ∞ ). As in the continuouscase, the “discrete” b -cycles are “locally” independent of n in the sense of (4.10)so that the “discrete” hyperelliptic integrals (5.13) regarded as functions of n are indeed solutions of the discrete Euler-Poisson-Darboux system (4.1).The discrete hyperelliptic integrals (5.13) may be evaluated explicitly interms of the residues of the meromorphic integrand ϕ since one only requiresthe known relationshipsΓ( ± l ) = (cid:20) (2 l )!( ± l l ! (cid:21) ± √ π, res(Γ( z ) , z = − l ) = ( − l l ! , l ∈ N . (5.15)19pecifically, in the case g = 1, the contour integral φ ∆ = 12 πi I b ϕ ( ξ ) dζ = δ πi I b ϕ ( ξ ) dξ = δ n − X k = n res( ϕ ( ξ ) , ξ = k ) (5.16)is given by φ ∆ = 1 √ δ n − X k = n Γ( k − n + 1)Γ( k − n + ) Γ( k − n + )Γ( k − n + 1) res(Γ( η ) , η = k − n + 1)Γ( k − n + ) . (5.17)This is to be compared with the corresponding elliptic integral (5.14) (dividedby 2 πi ) evaluated at the points x = δn , x = δ ( n + ) , x = δn . (5.18)In terms of the complete elliptic integral K of the first kind [43], this ellipticintegral may be expressed as φ O = 2 π √ x − x K r x − x x − x ! . (5.19)By construction, the latter constitutes an eigenfunction of the continuous Euler-Poisson-Darboux system (4.2) and may be used to generate three adjoint eigen-functions λ , λ , λ by means of the continuous analogue of the Levy transfor-mation (4.18) for g = 1. These turn out to be the characteristic speeds in theone-phase averaged KdV equations derived by Whitham [2, 11]. Thus, the dis-crete elliptic integral φ ∆ gives rise to discrete characteristic speeds of Whithamtype.It is observed that the elliptic integral φ O only depends on the differencesof the x i . In fact, the same applies, mutatis mutandis , to the discrete ellipticintegral (5.16) since the function p ( ξ, n, ν ) is invariant under a shift of ξ and n by the same amount. Accordingly, it is natural to regard the (discrete) ellipticintegrals φ ∆ and φ O as functions of the differences n − n and n − n . Theirgraphs are displayed in Figure 2 and it its seen that there exists virtually nodifference between the discrete and continuous elliptic integrals represented bypoints and a mesh respectively. It is noted that this statement is independent ofthe lattice parameter δ in the sense that, as a function of n , the ratio φ ∆ /φ O doesnot depend on δ . Thus, remarkably, there exists a unique relationship betweenthe discrete and continuous elliptic integrals. We conclude with the remarkthat the summation involved in the determination of the discrete hyperellipticintegrals may be reformulated so that it becomes transparent that the discretehyperelliptic integrals may be expressed in terms of generalised hypergeometricfunctions [43]. It is natural to inquire as to the existence of contour integrals of the type (5.13)which may be regarded as the analogues of hyperelliptic integrals associated20
Figure 2: The (discrete) elliptic integrals φ ∆ (points) and φ O (mesh) plotted asfunctions of the differences n − n and n − n .with an even number 2 g + 2 of branch points. These hyperelliptic integralsarise in connection with the multi-phase averaged nonlinear Schr¨odinger (NLS)equations [46]. In principle, the analogue of the solution (5.8) of the discreteEuler-Poisson-Darboux system, that is,˜ ϕ = g Y k =0 p ( ξ, n k +1 , g +1 Y k =1 p ( ξ, n k , ) , (5.20)is still valid but it is seen that this ansatz does not lead to the distributionof poles and zeros in the case of odd “genus” by formally letting n g +2 → ∞ .However, this situation may be rectified by annihilating the poles of ˜ ϕ andintroducing new poles in the “non-singular” regions by multiplication of ˜ ϕ byan appropriate function of ξ which has zeros and poles at half-integers andintegers respectively. By virtue of the symmetries of the Gamma function, itturns out natural to introduce the “complementary” function ϕ = − ˜ ϕ cot πξ. (5.21)Indeed, in terms of the “complementary” solution q ( ξ, n, ν ) = Γ( n + ν − − ξ ) √ δ Γ( n + ν − ξ ) (5.22)of the difference equation (4.25) which is related to p ( ξ, n, ν ) by q ( ξ, n, ν ) = p ( ξ, n, ν ) tan π ( ξ − ν ) , (5.23)21t is readily verified that ϕ = q ( ξ, n g +2 , ) g Y k =0 p ( ξ, n k +1 , g Y k =1 p ( ξ, n k , ) . (5.24)Hence, the poles and zeros are distributed as required, that is, there are no zerosor poles in the intervals ( n κ − − , n κ ). It is noted that, for convenience, thescaling of q has been chosen in such a manner that it approximates the function( x − ζ ) − / rather than ( ζ − x ) − / in the sense of (4.27). Furthermore, up toa sign, ϕ is symmetric in p and q due to the identity p ( ξ, n, p ( ξ, m, ) = − q ( ξ, n, q ( ξ, m, ) (5.25)for any integers m and n . Once again, in the simplest case g = 1, the contourintegral (5.16), where the contour b passes counterclockwise through the inter-vals ( n − , n ) and ( n − , n ), turns out to be a very good approximation ofthe corresponding elliptic integral φ O = 2 π p ( x − x )( x − x ) K s ( x − x )( x − x )( x − x )( x − x ) ! x = δn , x = δ ( n + ) , x = δn , x = δ ( n + ) (5.26)valid in the classical continuous case. In general, discrete b -cycles are definedas closed paths of counterclockwise orientation passing through the pairs ofintervals ( n κ − − , n κ ) and ( n g +1 − , n g +2 ) for κ = 1 , . . . , g . We conclude with a selection of open problems which naturally arise in connec-tion with the theory presented in this paper. For instance, it has been pointedout in [2, 22, 24, 47] that the theory of semi-Hamiltonian systems of hydrody-namic type is closely related to the analysis of the critical points of appropriatelychosen functions. In the current context, if ψ is an eigenfunction satisfying thediscrete conjugate net equations (4.22) and { µ i } , { λ iα } are associated sets ofadjoint eigenfunctions then one may introduce the corresponding Combescuretransforms ψ α and ˜Θ according to∆ i ψ α = λ iα ∆ i ψ, ∆ i ˜Θ = µ i ∆ i ψ. (6.1)The key function Θ is now defined byΘ = xψ + N X α =1 t α ψ α − ˜Θ , (6.2)where, a priori , x and t α are merely parameters. In analogy with the continuouscase, critical points n c of the function Θ are defined as points n where the22discrete derivatives” of Θ vanish, that is, ∆ i Θ | n = n c = 0. Accordingly, weobtain x ∆ i ψ ( n c ) + N X α =1 t α ∆ i ψ α ( n c ) − ∆ i ˜Θ( n c ) = 0 (6.3)so that the definitions (6.1) imply that x + N X α =1 t α λ iα ( n c ) − µ i ( n c ) = 0 . (6.4)The latter relate x and t α to n c in the same manner (with the index on n c being dropped) as the algebraic system (3.8) which gives rise to the discretegeneralised hodograph equations (3.9). The implications of this observation arecurrently being investigated.In the preceding, we have regarded “complete” hyperelliptic integrals asfunctions of their branch points x i and, in this context, put forward a canonicaldefinition of their discrete analogues. It is natural to inquire as to the existenceof similar analogues of “incomplete” hyperelliptic integrals and their associateddifferential equations. For instance, in the classical case, elliptic integrals arerelated by inversion to the differential equation dζds = p ( ζ − x )( ζ − x )( ζ − x ) (6.5)which essentially defines the elliptic Weierstrass ℘ function [43]. It is evidentthat the approach pursued in this paper suggests that one should examine indetail the properties of the differential equation dζds = δ Γ( ξ − n + )Γ( ξ − n + 1) Γ( ξ − n + 1)Γ( ξ − n + ) Γ( ξ − n + )Γ( ξ − n + 1) , ξ = ζ ( s ) δ (6.6)which may be regarded as a one-parameter deformation of the classical differ-ential equation (6.5). The latter is retrieved in the usual limit δ → §
5, we have confined ourselves to a detailed discussion of the relevance ofthe discrete Euler-Poisson-Darboux-type system (4.1) for ǫ i = . It is easy tosee that, in the classical case, separable solutions of the Euler-Poisson-Darbouxsystem (4.2) for ǫ i = M , where M is a positive integer, are obtained in termsof the superelliptic ( M, N )-curves y M = N Y i =1 ( ζ − x i ) . (6.7)As in the hyperelliptic case M = 2, the corresponding superelliptic integrals arerelevant in the theory of Whitham-type equations. For instance, trigonal curves(3 , N ) appear in connection with the Benney equations and the dispersionlessBoussinesq hierarchy (see, e.g., [48, 49] and references therein). It is thereforedesirable to investigate whether it is possible to extend the theory developed inthis paper to define canonical discrete analogues of superelliptic integrals andassociated discrete characteristic speeds of Whitham type.23 cknowledgment B.G.K. acknowledges support by the PRIN 2010/2011 grant 2010JJ4KBA 003.W.K.S. expresses his gratitude to the DFG Collaborative Research Centre SFB/TRR 109
Discretization in Geometry and Dynamics for its support and hospi-tality.
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