Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy
aa r X i v : . [ m a t h - ph ] D ec HIGHER GENERA CATALAN NUMBERS AND HIROTAEQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGERHIERARCHY
G. CARLET, J. VAN DE LEUR, H. POSTHUMA, AND S. SHADRIN
Dedicated to the memory of Boris Dubrovin
Abstract.
We consider the Dubrovin–Frobenius manifold of rank 2 whose genusexpansion at a special point controls the enumeration of a higher genera general-ization of the Catalan numbers, or, equivalently, the enumeration of maps on sur-faces, ribbon graphs, Grothendieck’s dessins d’enfants, strictly monotone Hurwitznumbers, or lattice points in the moduli spaces of curves. Liu, Zhang, and Zhouconjectured that the full partition function of this Dubrovin–Frobenius manifoldis a tau-function of the extended nonlinear Schr¨odinger hierarchy, an extensionof a particular rational reduction of the Kadomtsev–Petviashvili hierarchy. Weprove a version of their conjecture specializing the Givental–Milanov method thatallows to construct the Hirota quadratic equations for the partition function, andthen deriving from them the Lax representation.
Contents
Introduction 2Organization of the paper 5Acknowledgments 5Notation 51. The Dubrovin–Frobenius manifold 61.1. The Dubrovin–Frobenius manifold for Catalan numbers 61.2. The canonical coordinates 61.3. The normalised canonical frame 72. The deformed flat connection and the principal hierarchy 72.1. The deformed flat connection 72.2. The superpotential and the deformed flat coordinates 82.3. The calibration 92.4. The principal hierarchy 92.5. The R matrix 103. Givental quantization formalism and potentials 103.1. Symplectic loop space and quantization 103.2. Symplectic transformations and potentials 123.3. Higher genera Catalan numbers 123.4. The descendent potential and the KP hierarchy 164. The period vectors 174.1. Definition and main properties 174.2. Monodromy 184.3. At a special point 194.4. Asymptotics of the period vectors for λ ∼ u i λ ∼ ∞ λ ∼ u i W a,b ( t, λ ) 235.4. The functions c a ( t, λ ) 235.5. Splitting 235.6. Conjugation by R λ ∼ ∞ S Introduction
This paper is devoted to the study of the integrable hierarchy associated with theDubrovin–Frobenius manifold of rank 2 given in the flat coordinates t , t bythe metric η αβ = δ α + β, , (0.1)the prepotential F ( t , t ) = ( t ) t + ( t ) log t , (0.2)and the Euler vector field E = t ∂ t + 2 t ∂ t , (0.3)first introduced in [17, Example 1.1, Equation (1.24b)]. Despite the fact that it isone of the first non-trivial examples of semi-simple Dubrovin–Frobenius manifolds, itwas until recently not well-studied in the literature, since the enumerative meaningof its genus expansion was unclear. Digression . In many examples, a Dubrovin–Frobenius manifold capturesthe primary genus 0 part of the Gromov-Witten partition function of some target variety, or, moregenerally, the partition function of some naturally constructed cohomological field theory. In thesecases, the enumerative meaning of the genus expansion is encoded in the all-genera descendentpartition function. For instance, the rank 2 Dubrovin–Frobenius manifolds given in flat coordinatesby the same metric as above and the prepotentials ( t ) t + ( t ) and ( t ) t + e t (seeagain [17, Example 1.1]) are related to the Witten 3-spin class and the Gromov-Witten theory of CP , respectively, and there is an extensive literature studying these examples.In general, there is a universal reconstruction procedure for the genus expansion of a semi-simple Dubrovin–Frobenius manifold. It can be given either by the universal Givental formula [31],or, alternatively, as the tau-function that linearizes a special system of symmetries called theVirasoro constraints [22]. Equivalence of these two approaches is proved in [22]. This tau-function IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 3 determines the tau-structure of a bi-Hamiltonian dispersive deformation of an integrable hierarchyof hydrodynamic type associated with the initial semi-simple Dubrovin–Frobenius manifold.Note that the construction of this hierarchy given in [22] doesn’t guarantee the regular (poly-nomial) dependence of the Poisson brackets and the densities of the Hamiltonians. The regularityof the first Poisson bracket and the densities of the Hamiltonians is proved in [8, 9], while thepolynomiality of the second Poisson bracket is still an important open problem.
Quite recently it was proved in [24, 5] that there exist a specialization of the loga-rithm of the partition function D = D ( { t id } i =1 , d > ) associated with the Dubrovin–Frobenius manifold (0.1)-(0.3) (say, consider the partition function given by theGivental formula) that is the generating function of the generalized Catalan numbers (see a table of their values in [4, Part III, Section 1.1]) weighted by combinatorialfactors. The same numbers (in some instances, up to some small combinatorialrescaling) are also studied under the names of strictly monotone Hurwitz numbers,enumerations of ribbon graphs, (rooted) maps on surfaces, Grothendieck’s dessinsd’enfants for strict Belyi functions, lattice points in the moduli spaces of curves, etcetera, see e.g. [30, 45, 23] for some references to the vast literature on this sub-ject. This motivates us to have a closer look at this example of Dubrovin–Frobeniusmanifold. Note that though it is known that there exists a specialization of D to agenerating function of generalized Catalan numbers, its explicit form is not availablein the literature, so we give it below, see Theorem 8. Digression . The generalized Catalan numbers enumerategraphs with n > g >
0, where this graph can beembedded such that its complement is a union of open disks. We call g the genus of the graph.By C g ; k ,...,k n we denote the number of such graphs of genus g with n vertices of indices k , . . . , k n .In a dual language we can say that C g ; k ,...,k n counts the number of ways (up to orientationpreserving homeomorphisms) to glue a genus g surface out of n ordered polygons with k , . . . , k n sides, respectively, by identifying the pairs of sides, where each polygon has one distinguished side(these are the rooted maps). Obviously, for g = 0 and n = 1 C ,k is not equal to 0 if and only if k = 2 m is even, and in this case it is equal to the m -th Catalan number.A closely related concept is D g ; k ,...,k n := ( k · · · k n ) − · C g ; k ,...,k n . These numbers can be definedvia enumeration of ribbon graphs, where each graph is counted with the weight equal to the inverseorder of its automorphism group, or (not rooted) maps on surfaces, lattice points in the modulispaces, and strictly monotone Hurwitz numbers / Grothendieck’s dessins d’enfants for strict Belyifunctions.Recently both C g ; k ,...,k n and D g ; k ,...,k n got a lot of attention since their generating functionsserve as the basic examples for the Chekhov–Eynard–Orantin topological recursion and hypergeo-metric tau-function of the KP hierarchy. In particular, their relation to the Dubrovin–Frobeniusmanifold (0.1)-(0.3) is a byproduct of their study in the context of topological recursion [24, 5]. Consider the partition function D . The goal of this paper is to construct anintegrable hierarchy for which this partition function would be a tau-function corre-sponding to the string solution. We prove that D is a tau-function of the extendednon-linear Schr¨odinger or AKNS [1] hierarchy defined in [10, Section 5], which canalso be considered as an extension of a particular rationally reduced KP or con-strained KP hierarchy, see [7, 14, 15, 38, 46, 39, 36, 41] and references therein. Digression . It is now well known that the exponential of the gener-ating function of the numbers C g ; k ,...,k n is a tau function of the KP hierarchy [30]. It is anexample of the so-called hypergeometric tau-function, let us denote it by Z = Z ( { t d } d > ) = G. CARLET, J. VAN DE LEUR, H. POSTHUMA, AND S. SHADRIN D| t d =( d +1)! t d +1 ,t d =0 , d > . It is a natural general open question for the hypergeometric tau-functionswhat kind of further reduction of KP or lattice KP they would still satisfy. To this end, in the caseof lattice KP a number of interesting examples is systematically studied in [49].From that point of view, we do here one more step. Namely, we first start with Z , which has avery clear combinatorial enumerative meaning, and identify the reduction of KP for Z as a specialrational reduction of the KP hierarchy also known as the non-linear Schr¨odinger hierarchy [10] orAKNS hierarchy [1].It is known from [10] that in addition to the standard set of Hamiltonians generating the flowsof ∂/∂t d , there is a an additional set of commuting Hamiltonians. The corresponding flows extendthe tau-function Z , and this extension is identified with with D , where ∂/∂t d are the flows of thisadditional set of Hamiltonians. Our construction consists of three big steps loaned from the existing literature andmodified to fit our needs. First, we use the techniques of Givental, Milanov, Tseng,et al. [33, 34, 42, 43] to construct the Hirota quadratic equations for the partitionfunction Z . In fact, we modify their argument to avoid using the superpotential ofthe Dubrovin–Frobenius manifold (0.1)-(0.3). The reason to do it this way is ourintention to use this example as a departure point for the development of generalstructures producing Hirota equations and intrinsically existing for a Dubrovin–Frobenius manifold. Second, we use the well-known method developed e.g. in [42, 12]to pass from the Hirota equations to the Lax representation. Notice that the Hirotaequations coincide with those of the extended Toda hierarchy, but with the primarytimes interchanged. The natural approach would be to follow the usual approachfor KP reductions, but it is not clear how to apply the fundamental lemma in termsof pseudo-differential operators to obtain the Sato equations for the additional setof times. We therefore first recall (a particular case) of the computation in [12]obtaining the Lax equations of the extended Toda hierarchy in the “unnatural”spatial variable. In the third step we revisit in terms of the dressing operators theconstruction of [10] that allows to perform a change of the time corresponding to thespatial x -variable in order to re-organize the resulting hierarchy into the extendednonlinear Schr¨odinger hierarchy. In the last subsection we finally propose a directbut somehow not standard derivation of the pseudo-differential Sato equations fromthe Hirota quadratic equations.This result, though quite non-trivial, is very much expected. Indeed, on the onehand it is already mentioned in [10] that on the level of the underlying Dubrovin–Frobenius manifolds the change of the time shifted by x that turns the extended Todahierarchy into the extended nonlinear Schr¨odinger hierarchy is reduced to a Legendre-type transformation that turns the Dubrovin–Frobenius manifold structure of theGromov-Witten invariants of CP into the one given by Equations (0.1)-(0.3). Onthe other hand, this result can be considered as the very first example that indirectlyaffirms a much more general conjecture of Liu, Zhang, and Zhou in [41], which theyposed for a different reason. This paper studies their extended 1-constrained KPcase. The n -constrained case will be addressed in a forthcoming publication. Digression . The way Liu, Zhang, and Zhou arrive to theirconjecture is quite different and very interesting. They study the so-called central invariants [40, 20,11] of the bi-Hamiltonian structures of a special class of the rationally constrained KP hierarchiesand they show that all central invariants are constants equal to 1 /
24. This is exactly the propertythat the bi-Hamiltonian structures of the Dubrovin–Zhang hierarchies associated to Dubrovin–Frobenius manifolds must have (as we mentioned above, the existence of the second bracket isan open conjecture, but for the definition of the central invariants one needs only its existence
IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 5 up to order 2 in the dispersion parameter ǫ , which is proved in [21]). There is a direct relationbetween the dispersionless limits of this special class of hierarchies and the principal hierarchiesof a particular family of Dubrovin–Frobenius manifolds (one for each rank r > In conclusion, let us mention that we believe that a detailed study of this exampleof Dubrovin–Frobenius manifold in the way we performed it is quite helpful in theview of its possible generalizations (and a revision of similar results available in theliterature). Indeed, we paid a special attention to specifying the convergence issuesand emerging choices (for instance, calibration, choices of roots, etc.), which arequite often jammed in the literature though their effect on the resulting formulas isquite essential, as one can see from the detailed analysis in this paper.
Organization of the paper.
In Section 1 we introduce the Dubrovin–Frobeniusmanifold that we study in this paper and recall all essential structures related toit. In Section 2 we study the structure of the principle hierarchy associated tothis Dubrovin–Frobenius manifold, with a special attention to the possible choice ofcalibration. In Section 3 we recall the Givental quantization formalism and define theall-genera partition functions (the ancestor potential and the descendent potential)associated to our Dubrovin–Frobenius manifold. In Section 4 we study in detail theperiod vectors of our Dubrovin–Frobenius manifold, their values at a special pointand their asymptotics. In Section 5 we introduce the associated vertex operators andstudy their structural properties. In Section 6 (in Section 7, respectively) we provethe Hirota quadratic equations for the ancestor (descendent, respectively) potentialof this Dubrovin–Frobenius manifold. We derive in Section 8 the Lax formulation ofthe obtained integrable system, which we then identify with the extended nonlinearSchr¨odinger hierarchy.
Acknowledgments.
G. C., H. P., and S. S. were supported by the Netherlands Or-ganization for Scientific Research. G. C. is supported by the ANER grant “FROBE-NIUS” of the Region Bourgogne-Franche-Comt´e. The IMB receives support fromthe EIPHI Graduate School (contract ANR-17-EURE-0002).
Notation. • i : the imaginary unit. • h ( n ) := P nk =1 k − : the n -th harmonic number, h (0) = 0. • ( a ) n := Γ( a + n ) / Γ( a ) : the Pochhammer symbol. • R + : the non-negative real axis as a subset of C . • R − : the non-positive real axis as a subset of C . • e , . . . , e n : the canonical basis in C n . G. CARLET, J. VAN DE LEUR, H. POSTHUMA, AND S. SHADRIN
We will often use, for n > (cid:18) (cid:19) n = Γ(1 / n ) √ π = (2 n − n = (2 n )!4 n n ! , (0.4) (cid:18) (cid:19) − n = Γ(1 / − n ) √ π = ( − n (2 n − . (0.5)1. The Dubrovin–Frobenius manifold
The Dubrovin–Frobenius manifold for Catalan numbers.
Let M = C × C ∗ with coordinates ( t , t ). Let us define a charge d = − M with potential F ( t ) = 12 ( t ) t + 12 ( t ) log t . (1.1)For a general introduction to the theory of Dubrovin–Frobenius manifolds referto [17, 18, 19], or the more recent review in the first part of [16].The unit and Euler vector fields, the metric, and the product on the tangent spaceare given by e = ∂∂t , E = t ∂∂t + 2 t ∂∂t , η = (cid:18) (cid:19) , ∂∂t • ∂∂t = 1 t ∂∂t . (1.2)The intersection form g ij = E k c ijk , where c kij are the structure constants of theproduct and the indexes are raised and lowered by the metric η , is equal to g = (cid:18) t t t (cid:19) . (1.3)The discriminant ∆ ⊂ M is the locus where the intersection form g degenerates,that is ∆ = { t ∈ M | t = ( t ) } . (1.4)We denote by ∆ λ ⊂ M × C the locus where the pencil g − λη degenerates, which is∆ λ = { ( t, λ ) ∈ M × C | t = ( t − λ ) } . (1.5)We have the following two standard endomorphisms on the tangent space, which inthe flat trivialization read µ = (cid:18) / − / (cid:19) , U = (cid:18) t t t (cid:19) (1.6)respectively defined by µ = (2 − d ) / − ∇ E and U = E • , with ηµη = − µ, η U η = U T . (1.7)1.2. The canonical coordinates.
Let us define two canonical coordinates chartson M . Let V = { ( t , t ) ∈ M | t R − } ⊂ M and U = { ( u , u ) ∈ C | Re( u − u ) > } . The map V → U given by u = t + 2 √ t , u = t − √ t (1.8)is a diffeomorphism with inverse t = u + u , t = (cid:18) u − u (cid:19) , (1.9) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 7 where √ t is the principal branch of the square root on the cut plane C \ R − . Letlog( t ) denote the principal branch of the logarithm. We always denote ( t ) α =exp( α log t ). The same formulas give a diffeomorphism V → U , where V = { ( t , t ) ∈ M | t R + } and U = { ( u , u ) ∈ C | Im( u − u ) > } . To define theroots in this case we choose the branch of logarithm on C \ R + such that log( −
1) = π i .Without further notice we will systematically work on V , the extension of ourformulas to V being obvious.The canonical coordinate vectors given by (over both coordinate charts) ∂∂u = 12 ∂∂t + 12 √ t ∂∂t , ∂∂u = 12 ∂∂t − √ t ∂∂t , (1.10)are idempotents for the multiplication on the tangent spaces, and e = ∂∂u + ∂∂u , E = u ∂∂u + u ∂∂u . (1.11)1.3. The normalised canonical frame.
Let use denote∆ − i = ( ∂∂u i , ∂∂u i ) (1.12)so that ∆ − = √ t , ∆ − = − √ t . The normalised canonical frame is defined as e i = ∆ / i ∂∂u i , (1.13)where we fix ∆ / = √ t ) − / , ∆ / = i √ t ) − / , (1.14)and the transition matrix Ψ from the normalised canonical frame to the flat frame,defined by ∂∂t α = X j e j Ψ j,α , (1.15)is given byΨ = (Ψ − ) T η = 1 √ (cid:18) ( t ) / ( t ) − / − i ( t ) / i ( t ) − / (cid:19) , Ψ − = 1 √ (cid:18) ( t ) − / i ( t ) − / ( t ) / − i ( t ) / (cid:19) . (1.16)In the following we will represent a tangent vector by the column matrix of itscomponents in the relevant frame. The coefficients V i,can = Ψ iα V αflat of a vector inthe normalised canonical frame are obtained from the coefficients V iflat in the basisof flat coordinated by left multiplication by the matrix Ψ. Note that the row indexis i in Ψ iα , and α in (Ψ − ) αi . Remark also that in the formulas above the branch ofthe roots depends on the chart used.2. The deformed flat connection and the principal hierarchy
The deformed flat connection.
Let Y ( t, z ) be a two by two matrix valuedfunction on M × C which solves the deformed flatness equations − z ∂Y∂z = ( µ + U z ) Y, z ∂Y∂t α = C α Y, (2.1)with C = and C = (cid:18) t ) − (cid:19) . (2.2) G. CARLET, J. VAN DE LEUR, H. POSTHUMA, AND S. SHADRIN
The columns of the fundamental matrix Y are the gradients of a system of deformedflat coordinates ˜ t α ( t, z ): Y βγ = η βγ ∂ ˜ t α ∂t γ . (2.3)After fixing a branch of log z near z = 0, we look for solutions of the form Y ( t, z ) = S ( t, z ) z − µ z − R (2.4)where S = P k > S k z − k is a matrix valued power series which converges in a smallneighbourhood of z = 0, with S = and R = (cid:18) (cid:19) . (2.5)The matrix S is uniquely fixed by setting ( S ) , = ψ + log t , see below. Thecolumns of the matrix S are the gradients of analytic functions θ α on M × C suchthat (˜ t , ˜ t ) = ( θ , θ ) z − µ z − R , (2.6)and the ˜ t α are said to form a Levelt system of deformed flat coordinates on M .2.2. The superpotential and the deformed flat coordinates.
The Dubrovin–Frobenius manifold structure on M can be given in terms of the following superpo-tential f : M × C ∗ → C f ( t, ζ ) = ζ + t + t ζ − , (2.7)see [17]. Notice that ∆ λ coincides with the set of points ( t, λ ) ∈ M × C where thepreimage of λ via f ( t, · ) degenerates, namely is not given by two distinct points in C ∗ .We define the formal logarithm f log f as the formal Laurent series P k ∈ Z a k ζ k , withcoefficients given by X k > a k ζ k = 12 log( f ζ ) , ζ ∼ , (2.8) X k< a k ζ k = 12 log( f /ζ ) , ζ ∼ ∞ . (2.9) Proposition 5.
The Levelt system ˜ t α ( t, z ) of deformed flat coordinates for the Cata-lan Dubrovin–Frobenius manifold M is given by ˜ t = θ z − , ˜ t = θ z − θ z − log z, (2.10) where θ α = Res g α dζ (2.11) and g α ( t, ζ , z ) are defined as g = z (cid:0) e f/z − (cid:1) , g = 2 e f/z (cid:18) f log f + ψ − Ein( f /z ) (cid:19) . (2.12) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 9
The calibration.
The S matrix is uniquely determined by the recursive for-mula kS k + S k µ − µS k = U S k − − S k − R, for k > S = and ( S ) , = ψ + log t , and R = (cid:18) (cid:19) . (2.14)The first two terms are S = (cid:18) t ψ + log t t t (cid:19) , S = (cid:18) ( t ) + t t ( ψ + log t ) t t ( t ) + t ( ψ + log t − (cid:19) . (2.15)We can write the S matrix in terms of the superpotential as follows S k = Res f k k ! ζ − f k − ( k − (cid:16) f log f + ψ − h ( k − (cid:17) ζ − f k k ! 2 f k − ( k − (cid:16) f log f + ψ − h ( k − (cid:17) dζ . (2.16)This expression can be derived from (2.10) or checked by substitution in the recursiveformula for S k .For example, at the point given by t = 0 and t = 1 (we often consider special-ization to this point below and call this the special point of M denoted by t sp ) wecan explicitly compute the S matrix coefficients, which are given by S k = k !) ψ + k − − h ( k ) k !( k − ! , S k +1 = ψ − h ( k )( k !) k +1)! k ! ! . (2.17)The first few terms are S = (cid:18) ψ (cid:19) , S = (cid:18) ψ − (cid:19) , S = (cid:18) ψ − (cid:19) , (2.18) S = (cid:18) ψ − (cid:19) , S = (cid:18) ψ − (cid:19) . The principal hierarchy.
We can easily obtain the following explicit Hamil-tonian form of the principal hierarchy.
Proposition 6.
The principal hierarchy of the Catalan Dubrovin–Frobenius mani-fold is given in Hamiltonian form by ∂t i ∂t α,p = { t i ( x ) , H α,p } (2.19) where the Hamiltonians and their densities are given by H α,p = Z h α,p dx, h α,p = Res g α,p dζ , (2.20) the functions g α,p ( t, ζ ) are defined as g ,p = f p +2 ( p + 2)! , g ,p = 2 f p +1 ( p + 1)! (cid:18) f log f − h ( p + 1) + ψ (cid:19) , (2.21) and the Poisson structure is { t i ( x ) , t j ( y ) } = η ij δ ′ ( x − y ) . (2.22) The R matrix. Let us represent the fundamental matrix Y in the normalisedcanonical frame ˜ Y = Ψ Y. (2.23)The z part of the deformed flatness equations becomes − z ∂ ˜ Y∂z = ( V + Uz ) ˜ Y , (2.24)where U := Ψ U Ψ − = (cid:18) u u (cid:19) , V := Ψ µ Ψ − = (cid:18) i / − i / (cid:19) . (2.25)There exists a unique formal solution of this equation of the form˜ Y ( t, z ) = R ( t, z ) e U/z , (2.26)where R = X k > R k z k , R = . (2.27)Indeed the coefficients R k are uniquely determined by the recursion relation[ R k +1 , U ] = ( V + k ) R k . (2.28)Explicitly we have R k = (cid:0) (cid:1) k − (cid:0) (cid:1) k ( k )! ( − k +1 k i ( − k +1 k i − ! ( u − u ) − k . (2.29)3. Givental quantization formalism and potentials
In this section we introduce the quantization formalism of Givental [31, 32] (seealso an exposition in [6]) and explain his definitions of the so-called ancestor and de-scendent potentials associated to a Dubrovin–Frobenius manifold. We use the Given-tal formula for the descendent potential in order to link the Dubrovin–Frobeniusmanifold (0.1)-(0.3) to the higher genera Catalan numbers.3.1.
Symplectic loop space and quantization.
Let V be a C -vector space witha non-degenerate symmetric bilinear form ( , ) V , with basis { φ i } and dual basis { φ i } . Let V := V (( z )) be the loop space of formal Laurent series with values in V ,equipped with the symplectic formΩ( f , g ) = Res z ( f ( − z ) , g ( z )) dz, f , g ∈ V . (3.1) Remark . In the following V will be identified with the tangent space at a pointof M either via the flat trivialisation or via the normalised canonical frame. In thefirst case the basis is that of flat coordinate vector fields ∂∂t i and the bilinear form isthe flat metric η at the point, in the second case the V is identified via the canonicalbasis with the Euclidean space together with the standard inner product.Darboux coordinates on V can be defined as q ik = Ω(( − z ) − k − φ i , · ) , p i,k = Ω( · , φ i z k ) , k > , (3.2)by which we can express any element f ∈ V as f = X k > (cid:0) q ik φ i z k + p i,k φ i ( − z ) − k − (cid:1) . (3.3) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 11
Let V + := V [[ z ]] be the space of formal Taylor series with values in V . Then wehave a natural isomorphism V ∼ = T ∗ V + . Let ǫ be a formal parameter. Consider theFock space of functions on V + given by formal power series in the variables q ik + δ i δ k with the coefficients in formal Laurent series in ǫ .We consider linear and quadratic Hamiltonians on V . The standard (Weyl) quan-tization associates to them differential operators of order q iℓ )ˆ = 1 ǫ q iℓ , ( p i,ℓ )ˆ = ǫ ∂∂q iℓ , (3.4)( q ik q jℓ )ˆ = 1 ǫ q ik q jℓ , ( q ik p j,ℓ )ˆ = q ik ∂∂q jℓ , ( p i,k p j,ℓ )ˆ = ǫ ∂ ∂q iℓ ∂q jℓ . (3.5)For instance, the linear Hamiltonian h f ( · ) = Ω( f , · ) associated with the constantvector field f = P l I l ( − z ) l ∈ V is given by h f = X l > (cid:2) ( − l +1 ( I l , φ i ) p i,l + ( I − ( l +1) , φ i ) q il (cid:3) . (3.6)It is convenient to denote ( I k , φ i ) (respectively, ( I k , φ i )) by ( I k ) i (respectively, ( I k ) i ).The quantization of h f readsˆ f := ( h f )ˆ = X l > (cid:20) ǫ ( − l +1 ( I l ) i ∂∂q il + 1 ǫ ( I − ( l +1) ) i q il (cid:21) . (3.7)Note that in particular [ˆ f , ˆ f ] = Ω( f , f ) . (3.8)We consider two Lie algebras, of purely positive and purely negative series in z ,that is, either m = P ℓ > m ℓ z ℓ or m = P ℓ − m ℓ z ℓ , m ℓ ∈ End( V ), representing linearvector fields m commuting with z . These vector fields are infinitesimally symplecticif m ij ( z ) + η ik m lk ( − z ) η lj = 0. The Hamiltonian of m is defined as h m ( f ) := Ω( mf , f ),and ˆ m denotes its quantization, ˆ m := ( h m )ˆ . (3.9)For the operator M = exp( m ) the symbol ˆ M denotes the operator exp( ˆ m ).For instance, the quadratic Hamiltonian associated with s = P ℓ > s ℓ z − ℓ is givenby h s ( f ) = 12 X a,b > ( − b +1 q ia q jb ( s a + b +1 ) ki η kj − X a > ,ℓ > q ia + ℓ p j,a ( s ℓ ) ji , (3.10)where ( s a + b +1 ) ki η kj = ( s a + b +1 φ i , φ j ) and ( s a + b +1 ) ji = ( s a + b +1 φ i , φ j ). Its quantizationreads ( h s )ˆ = 12 ǫ X a,b > ( − b +1 q ia q jb ( s a + b +1 ) ki η kj − X a > ,ℓ > q ia + ℓ ∂∂q ja ( s ℓ ) ji . (3.11)The quadratic Hamiltonian of the element r := P ℓ > r ℓ z ℓ is given by h r ( f ) = 12 X a,b > ( − a p i,a p j,b ( r a + b +1 ) ik η kj − X a > ,ℓ > q ia p j,a + ℓ ( r ℓ ) ji , (3.12) leading to the quantization( h r )ˆ = ǫ X a,b > ( − a ∂∂q ia ∂∂q jb ( r a + b +1 ) ik η kj − X a > ,ℓ > q ia ∂∂q ja + ℓ ( r ℓ ) ji . (3.13)3.2. Symplectic transformations and potentials.
Recall the series S = S ( t, z )defined by Equation (2.4) and discussed in detail in Section 2.3. It is a symplecticoperator on V for φ i = ∂∂t i , ( , ) V = η that commutes with multiplication by z , sowe can apply the quantization procedure described above and define ˆ S .Recall the series R = R ( t, z ) defined in Section 2.5. Consider its action on thesame V in a different basis given by e j = X α ∂∂t α (Ψ − ) αj (3.14)(note that ( e i , e i ) V = δ ij ). It is a symplectic operator on V commuting with multi-plication above as well, and the quantization procedure above defines the operatorˆ R .Note that since the matrix R is given in a different basis, then in order to applyEquation (3.13) one has to consider the operator Ψ − R Ψ. A better alternative is touse the basis e i and the more natural variables Q ia := P α Ψ iα q αa .We can now define the ancestor potential as A ( { q a , q a } a > ) := ˆΨ − ˆ R Y i =1 τ KdV ( { Q ia } a > ) , (3.15)where τ KdV is the Witten–Kontsevich τ -function for the KdV hierarchy in the vari-ables with the dilaton shift (that is, with respect to the standard descendent variables t a , a >
0, we have Q ia = t a − δ ,a ), and the operator ˆΨ − recomputes the functionˆ R Q i =1 τ KdV ( { Q ia } a > ) in the variables q ia .Finally, the total descendent potential is defined as D := C ˆ S − A , (3.16)where the extra factor C is set to be (up to a multiplicative constant)log C ( t , t ) := Z ( t ,t ) ( R ) du + ( R ) du = −
116 log t . (3.17)Note that the factor C and operators ˆ S , ˆΨ, and ˆ R do depend on the point ( t , t ) ofthe Dubrovin–Frobenius manifold. The coefficients of the ancestor potential A alsodepend on the point ( t , t ) of the Dubrovin–Frobenius manifold, however, Giventalproved in [31] that the coefficients of D are constant in ( t , t ), that is, the descendentpotential D does not depend on the point of the Dubrovin–Frobenius manifold.3.3. Higher genera Catalan numbers.
The variables q ia , i = 1 , a > D are related to the variables t ia by the dilaton shift: q ia = t ia − δ i, δ a, . Recall the definition of the numbers C g,k ,...,k n given in Digression 2. Theorem 8.
We have: log D| t a =0 ,a > = ∞ X g =0 ∞ X n =1 ǫ g − n ! X k ,...,k n > C g,k +1 ,...,k n +1 n Y i =1 t k i ( k i + 1)! (3.18) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 13
Proof.
The first step of the proof is to rewrite the formula for the descendent poten-tial in terms of the variables { t a , t a } a > . To this end, we have to shift the variables: (cid:18) e − ∂∂q C ˆ S − e ∂∂q (cid:19) (cid:12)(cid:12)(cid:12) q ia → t ia ˆΨ − (cid:18) e − Ψ ∂∂Q − Ψ ∂∂Q ˆ Re ∂∂Q + ∂∂Q (cid:19) (cid:12)(cid:12)(cid:12) Q ia → T ia Y i =1 τ KdV ( { T ia } a > , ǫ ) , (3.19)and in this formula the Witten–Kontsevich τ -function τ KdV are considered in thestandard descendent variables. We can further rewrite this formula as (cid:18) e − ∂∂q C ˆ S − e ∂∂q (cid:19) (cid:12)(cid:12)(cid:12) q ia → t ia ˆΨ − (cid:18) e − Ψ ∂∂Q − Ψ ∂∂Q ˆ Re Ψ ∂∂Q +Ψ ∂∂Q (cid:19) (cid:12)(cid:12)(cid:12) Q ia → T ia ·· e − Ψ ∂∂T − Ψ ∂∂T + ∂∂T + ∂∂T Y i =1 τ KdV ( { T ia } a > , ǫ ) , (3.20)Recall that Ψ i = ∆ − / i , i = 1 ,
2. The dilaton equation implies that e − ∆ − / i ∂∂Ti + ∂∂Ti τ KdV ( { T ia } a > , ǫ ) = τ KdV ( { ∆ / i T ia } a > , ∆ i ǫ ) , i = 1 , . (3.21)Thus the resulting formula for the descendent potential that is used in applicationsin the variables { t ia } is given by C t ˆ S − ˆΨ − t ˆ R Y i =1 τ KdV ( { ∆ / i T ia } a > , ∆ i ǫ ) , (3.22)where t ˆ S := (cid:18) e − ∂∂q ˆ Se ∂∂q (cid:19) (cid:12)(cid:12)(cid:12) q ia → t ia ; (3.23) t ˆ R := (cid:18) e − Ψ ∂∂Q − Ψ ∂∂Q ˆ Re Ψ ∂∂Q +Ψ ∂∂Q (cid:19) (cid:12)(cid:12)(cid:12) Q ia → T ia . (3.24)Equation (3.22) is the standard expression for the Givental formula for the totaldescendent potential in the coordinates { t ia } used in application, see e.g. [26, 25].Its advantage is that at all steps of the computation of its coefficients one has towork with the formal power series.At the second step we have to use some results from the theory of the Chekhov–Eynard–Orantin topological recursion [28]. It is a recursive procedure that producessymmetric differentials from the small set of input data that consists of a Riemannsurface Σ, two functions x and y defined on it, and a symmetric bi-differential B on Σ × Σ, (all these pieces of data are subjects to some extra conditions). Thisdata is related to a choice of Dubrovin’s superpotential for the Dubrovin–Frobeniusmanifolds [24]. We don’t use the explicit formulation of the topological recursionitself, but we have to recall two results for the data given by the Riemann sphere C P with a global coordinate z on it, functions x = z + z − , y = z , and the bi-differential B ( z , z ) = dz dz / ( z − z ) from [45, 23] and from [24, 5].It is proved in [45, 23] that the topological recursion applied to this input datareturns the symmetric n -differentials ω g,n ( z , . . . , z n ), 2 g − n >
0, that expand near z = · · · = z n = 0 in the variables x i = x ( z i ), i = 1 , . . . , n , as ω g,n = X k ,...,k n > C g,k +1 ,...,k n +1 n Y i =1 dx i x k i +2 i = X k ,...,k n > C g,k +1 ,...,k n +1 Q ni =1 ( k i + 1)! n Y i =1 ( k i + 1)! dx i x k i +2 i . (3.25)On the other hand, it is proved in [24, 5] that ω n := P ∞ g =0 ǫ g − ω g,n is given by ω n = X i ,...,i n =1 , a ,...,a n > n Y j =1 ∂∂T i j a j log t ˆ R Y i =1 τ KdV ( { ∆ / i T ia } a > , ∆ i ǫ ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T ia =0 n Y j =1 d (cid:16) − ddx j (cid:17) a j ξ i j ( z j ) , (3.26)where ξ j ( z ) := dzd p x ( z ) − x ( p j )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = p j · p j − z (3.27)for j = 1 , p = 1 , p = − x , and the wholeexpression (3.26) is considered at the special point ( t , t ) = (0 ,
1) of the underlyingDubrovin–Frobenius manifold. The choice of the square roots is required to bealigned with the choices made for ∆ / , ∆ / . In particular, in the case y = z wehave: dzd p x ( z ) − x ( p j )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = p j = ∆ − / j , j = 1 , . (3.28)An equivalent form of Equation (3.26) in the flat frame is ω n = X α ,...,α n =1 , a ,...,a n > n Y j =1 ∂∂t α j a j log ˆΨ ˆ R Y i =1 τ KdV ( { ∆ / i T ia } a > , ∆ i ǫ ) ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t ia =0 n Y j =1 d (cid:16) − ddx j (cid:17) a j ˜ ξ α j ( z j ) , (3.29)where ˜ ξ α = (Ψ − ) αi ξ i , α = 1 , S − for 2 g − n > n Y j =1 ∂∂t α j a j log ˆ S − ˆΨ ˆ R Y i =1 τ KdV ( { ∆ / i T ia } a > , ∆ i ǫ ) ! (cid:12)(cid:12)(cid:12) t ia =0 == X ℓ j a j ,j =1 ,...,n ( S ℓ ) β j α j n Y j =1 ∂∂t β j a j − ℓ j log ˆΨ ˆ R Y i =1 τ KdV ( { ∆ / i T ia } a > , ∆ i ǫ ) ! (cid:12)(cid:12)(cid:12) t ia =0 . (3.30)Note that at the special point ( t , t ) = (0 ,
1) we have C ( t , t ) = 0. Therefore, thestatement of the theorem is equivalent to the following identity:Res z =0 x k +1 ( k + 1)! d (cid:16) − ddx (cid:17) a ˜ ξ α ( z ) = ( , a > k > − S k − a ) α , k > a > . (3.31)To this end, we just explicitly compute ξ ( z ) = 1 √ − z ; ξ ( z ) = 1 √ i z ; ˜ ξ ( z ) = z − z ; ˜ ξ ( z ) = 11 − z ; (3.32) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 15 (recall (1.14) and (1.16)), and then we see that indeedRes z =0 x k +1 ( k + 1)! d (cid:16) − ddx (cid:17) a z − z = Res z =0 (cid:16) ddx (cid:17) a +1 x k +1 ( k + 1)! zd ( z + z − ) z − , a > k > − m !) − , k > a > , k − a = 2 m ;0 , k > a > , k − a = 2 m + 1 = ( , a > k > − S k − a ) , k > a > z =0 x k +1 ( k + 1)! d (cid:16) − ddx (cid:17) a − z = Res z =0 (cid:16) ddx (cid:17) a +1 x k +1 ( k + 1)! d ( z + z − ) z − , a > k > − , k > a > , k − a = 2 m ;( m !( m + 1)!) − , k > a > , k − a = 2 m + 1 = ( , a > k > − S k − a ) , k > a > . This completes the proof of the theorem for 2 g − n > g, n ) = (0 , g, n ) = (0 , g, n ) = (0 , ǫ − t a in log D| t a =0 ,a > is given by[ ǫ − t a ] log D| t a =0 ,a > = η α ( S a +2 ) α . (3.35)Then, using Equation (2.17) we see indeed that η α ( S a +2 ) α = ( , a = 2 m ; m +1)!( m +2)! , a = 2 m + 1 = (3.36)= ( , a = 2 m ; m +2)! (2 m +2)!( m +1)!( m +2)! , a = 2 m + 1 = C ,a +1 ( a + 1)! . (3.37)Consider ( g, n ) = (0 , ǫ − t a t b inlog D| t a =0 ,a > is given by[ ǫ − t a t b ] log D| t a =0 ,a > = [ z a w b ] − η + P ∞ m,n =0 ( S m ) µ z m ( S n ) ν w n η µν z + w . (3.38)Therefore, using explicit formulas (2.17), we have( z + w ) ∞ X a,b =0 [ ǫ − t a t b ] log D| t a =0 ,a > = ∞ X p,q =0 (cid:18) z p w q +1 ( p !) q !( q + 1)! + z p +1 w q p !( p + 1)!( q !) (cid:19) . (3.39)On the other hand, it is proven in [23] that X k ,k > C ,k +1 ,k +1 ( k + 1)!( k + 1)! Y i =1 ( k i + 1)! dx i x k i +2 i == d d log (cid:18) z − − z − x − x (cid:19) = − d d log(1 − z z ) . (3.40) Therefore,( z + w ) X k ,k > C ,k +1 ,k +1 ( k + 1)!( k + 1)! z k w k (3.41)= X k ,k > ( z k +1 w k + z k w k +1 ) Res z =0 Res z =0 x k +11 ( k + 1)! x k +12 ( k + 1)! · − d d log(1 − z z )= X k ,k > k + k > z k w k Res z =0 Res z =0 x k ( k )! x k ( k )! · (cid:0) dx d log(1 − z z ) + dx d log(1 − z z ) (cid:1) = X k ,k > k + k > z k w k Res z =0 Res z =0 x k ( k )! x k ( k )! · (cid:16) z + 1 z (cid:17) dz dz = X p.q > (cid:18) z p w q +1 p !) q !( q + 1)! + z p +1 w q p !( p + 1)!( q !) (cid:19) . The latter expression coincides with the right hand side of Equation (3.39), whichproves the ( g, n ) = (0 ,
2) case of the theorem. (cid:3)
Remark . Note that the computations done above for 2 g − n > C P mentioned in Digression 1. This is explained by the following twofacts. First, note that the S -matrix at the special point ( t , t ) = (0 ,
1) with ψ = 0,see Equation (2.17), conjugated by η , is equal to the S -matrix of the Dubrovin–Frobenius manifold given by the prepotential ( t ) t + e t . Second, note that the ˜ ξ i -functions in the above computation are obtained from the corresponding ˜ ξ i -functionsin the computations in [25] by the interchanging of the superscripts.3.4. The descendent potential and the KP hierarchy.
Consider the generat-ing function Z for the higher genera Catalan numbers Z := exp X g > ǫ g − X n > n ! X k ,...,k n > C g,k ,...,k n n Y i =1 t i ! (3.42)as a formal power series in the variables t , t , . . . . It is proved in [30, Theorem 5.2]that Z is a tau-function of the KP-hierarchy (more precisely, one should speak of ~ -KP hierarchy in the sense of [48, 44]for ~ = ǫ , see [2]). In particular, it is provedin [30, Theorem 5.2] that Z takes the following form: Z = X λ s λ ( { p i } i > ) s λ ( { ˜ p i } i > ) Y ( i,j ) ∈ λ (1 + ǫ ( i − j )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p i = i t i /ǫ, i > p i = δ i /ǫ, i > . (3.43)Here the sum is taken over all Young diagrams λ including the empty one, and s λ are the Schur functions considered in the two copies of the power sums variables p , p , . . . and ˜ p , ˜ p , . . . . This equation makes Z a special case of the so-calledhypergeometric family of KP tau-functions introduced and considered in [37, 47].So, Theorem 8 has the following corollary. Corollary 10. D| t d =( d +1)! , t d +1 ,t d =0 ,d > is a KP tau-function. IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 17
Our main result can be considered as a refinement of this Corollary, cf. Digression 3in the Introduction. Namely, one of the ways to interpret the results that we provein Sections 7 and 8 is that D| t d =( d +1)! t d +1 ,t d =0 ,d > appears to satisfy a particularrational reduction of the KP hierarchy [7, 14, 15, 38, 39, 36, 41] that possessesextra symmetries, which, once added explicitly form a second series of times in D ( { t id } i =1 , d > ). Remark . Note that though our approach to the construction of the rationallycontrained KP hierarchy in the Hirota form does not use the results mentioned inthis section, the change of variables t d = ( d + 1)! t d +1 , d > q d , ¯ q d are shifted by ∓ ǫd ! /λ d +1 , which matches precisely that in the standard formulation ofthe Hirota bilinear equations for KP hierarchy the corresponding shifts for t d +1 , ¯ t d +1 would be ∓ ǫ/ (( d + 1) λ d +1 ). Remark . The step from the standard KP hierarchy for Z to its rational reductionfits very well into the context of a recent paper of Takasaki [49], where he studies thepossible reductions of the lattice KP hierarchy for several families of hypergeometrictau-functions.
Remark . The function Z in variables p i , i > Z := exp X g > ǫ g − X n > n ! X k ,...,k n > D g,k ,...,k n n Y i =1 p i ! (3.44)(recall the definition of D g,k ,...,k n in Digression 2 in the Introduction). The obviousinterpretations of the numbers D g,k ,...,k n that follow directly from the definition ofthe higher genera Catalan numbers is via the (weighted) enumeration of ribbongraphs, non-rooted maps, or Grothendieck dessins d’enfants for the strict Belyifunctions, see e.g. [30, 45]. It is proved in [35] that these numbers have also aninterpretation in the framework of the theory of weighted Hurwitz numbers as theso-called 2-orbifold strictly monotone Hurwitz numbers (see also [3] for a differentproof). 4. The period vectors
Definition and main properties.
We denote by I ( l ) e i ( t, λ ) for l ∈ Z and i = 1 , λ ∼ u i . Let usfix two cuts L i = u i + e i π/ R + in the λ -plane, and the roots of λ − u i determinedby the principal branch of the logarithm near each u i in the cut plane C \ ∪ i L i . Weuniquely define the period vectors as follows. Proposition 14.
For each l ∈ Z and i = 1 , there exists a unique multivaluedholomorphic solution I ( l ) e i ( t, λ ) defined on ( M × C ) \ ∆ λ with values in C of theequation ( U − λ ) ∂I ( l ) ∂λ = ( µ + l + 12 ) I ( l ) (4.1) such that I ( l ) e i = ( − l √ π Γ( l + 1 / λ − u i ) − l − (Ψ − e i + O ( λ − u i )) (4.2) for λ ∈ C \ ∪ i L i , and such that the analytic continuation of I ( l ) e i along a small path γ i surrounding u i is equal to − I ( l ) e i . A general proof of this statement can be easily adapted from the case l = 0 shownin [19, lemma 5.3].The period vectors thus defined satisfy also the following equation ∂I ( l ) ∂t i = − ∂∂t i • ∂I ( l ) ∂λ , (4.3)and one can easily check that, in general I ( l +1) e i = ∂I ( l ) e i ∂λ , l ∈ Z , (4.4) I ( l − e i ( t, λ ) = Z λu i I ( l ) e i ( t, ρ ) dρ, l . (4.5)In the following, for a = a e + a e ∈ C we denote I ( l ) a = a I ( l ) e + a I ( l ) e thecorresponding solution of (4.1). Remark
15 (To be used in Section 6) . With this setup it is straightforward to checkthat the vector I ( − e − I ( − e is constant on M , with the values of the components( I ( − e ) − ( I ( − e ) = − π i , ( I ( − e ) − ( I ( − e ) = 0.4.2. Monodromy.
Let π = π ( C \ { u , u } ) be the fundamental group of thepointed λ -plane with base point λ . Denote by γ i , i = 1 ,
2, the generators of π corresponding to the two small loops around the points u i in counterclockwise di-rection connected to λ by paths in the cut λ -plane. Moreover let us denote by γ ∗ I the analytic continuation of a multivalued analytic function I on the pointed plane C \ { u , u } along the loop γ ∈ π .Denoting by ( , ) λ the flat pencil g − λη on the cotangent to M , let us define thesymmetric bilinear form <, > on C by < a, b > := ( η ∗ I (0) a , η ∗ I b ) λ , which is well-known to be independent of t and λ . Here η ∗ is the isomorphism from the tangentto the cotangent spaces to M induced by the metric. In our case we have that thecorresponding matrix G ij = < e i , e j > is G = − (cid:18) (cid:19) . (4.6)We call reflection along w ∈ C the involution of C given by v v − < v, w >< w, w > w. (4.7)Let us denote by γ i the reflection along e i , and define the group homomorphism π → GL ( C ) by sending the loop γ i to the reflection γ i . We have that: Proposition 16.
For each γ ∈ π we have γ ∗ I ( l ) a = I ( l ) γa . This is just a reformulation of a general result, see [19, lemma 5.3].Explicitly the monodromy action is given by
IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 19
Proposition 17.
The action of the generators of π on the canonical basis of C isgiven by γ e = − e , γ e = e − e , (4.8) γ e = e − e , γ e = − e . (4.9)The action of γ , resp. γ , corresponds to the horizontal, resp. vertical, reflectionw.r.t. the invariant subspace V spanned by e − e . Note that γ γ sends a = a e + a e to a + 2( a + a )( e − e ), therefore acts on the affine line a + a = b bytranslation by − b along e − e . We also denote π i the projections to the invariantssubspace, π a = a ( e − e ) and π a = a ( e − e ).We have therefore two types of orbits: the trivial orbits are given by the pointsof the invariant subspace, and the infinite orbits { a, γ a } + Z a + a )( e − e ) for a = ( a , a ) ∈ C not invariant, i.e., a + a = 0. Remark . The representation of π on the space C should not be confused withthe action on the space of solutions of (4.1). Indeed we have that I (0) e = I (0) e andconsequently I ( l ) e = I ( l ) e for l >
0, so the period vectors span a one-dimensionalsubspace of the solution space. For negative l the two period vectors differ bypolynomials in λ , so they are indeed a basis of the solution space of (4.1).4.3. At a special point.
In the rest of this section we work at a special point t sp of M given by t = 0 and t = 1. This corresponds to canonical coordinates u = − u = 2. As can be easily checked, the two solutions of the system (4.1) atthe special point t sp are given by I (0) e ( t sp , λ ) = I (0) e ( t sp , λ ) = (cid:18) λ/ (cid:19) ( λ − − . (4.10)This expression defines a holomorphic C -valued function on the cutted given by λ -plane C \ { (2 + i R + ) ∪ ( − i R + ) } , where √ λ − √ λ − √ λ + 2 and √ λ ∓ ± Asymptotics of the period vectors for λ ∼ u i . By asymptotic series near u i we mean a formal Laurent series in ( λ − u i ) / . By asymptotic expansion of amultivalued function g ( λ ) for λ ∼ u i we mean an asymptotic series near u i whichsatisfies the asymptotic condition on a cut neighbourhood of u i for a choice of branchof g ( λ ) and of √ λ − u i .At the special point t sp we can easily compute the asymptotic expansions of theperiod vectors. Lemma 19.
The asymptotic expansions of the period vector I ( l ) e i for λ ∼ u i at thespecial point t sp are given by I ( l ) e ( t sp , λ ) ∼ X k > ( − − k k ! (cid:18) (cid:0) (cid:1) k (cid:0) (cid:1) k − k − (cid:0) (cid:1) k +1 (cid:0) (cid:1) k (cid:19) ∂ l − k +1 λ √ λ − , λ ∼ u = 2 (4.11) I ( l ) e ( t sp , λ ) ∼ i X k > − k k ! (cid:18) (cid:0) (cid:1) k (cid:0) (cid:1) k k − (cid:0) (cid:1) k +1 (cid:0) (cid:1) k (cid:19) ∂ l − k +1 λ √ λ + 2 , λ ∼ u = − on the cut λ -plane C \ ((2 + i R + ) ∪ ( − i R + )) with principal branches of the roots √ λ ± . Proof.
The formulas for the case l = 0 are obtained by a simple expansion of (4.10).For other values of l they follow by integration or derivation. (cid:3) Asymptotics of the period vectors for λ ∼ ∞ . At the special point t sp of M the period vectors I (0) e = I (0) e have the following asymptotic expansion for | λ | ∼ ∞ and arg λ = π/ I (0) e i ( t sp , λ ) ∼ I (0) asy ( λ ) (4.13)where I (0) asy := 12 X s > (2 s )! s ! s ! λ − s − (cid:18) λ (cid:19) = (cid:18) λ + λ + · · · + λ + λ + · · · (cid:19) . (4.14)By taking derivatives we obtain the asymptotic expansions of the period vectors I ( l ) e i with l > I ( l ) e i ( t sp , λ ) ∼ I ( l ) asy := ∂ lλ I (0) asy (4.15)where we have explicitly I ( l ) asy = ( − l X s > (2 s + l )! s !( s + 1)! (cid:18) s + 1(2 s + l + 1) λ − (cid:19) λ − s − l − . (4.16)Let us define I ( − l ) formal for l > λ (withoutintegration constants) of the asymptotic expansion I (0) asy , i.e., I ( − l ) formal := ∂ − lλ I (0) asy , l > , (4.17)where we have set ∂ − λ λ − = log λ , ∂ − λ ( λ p log λ ) = λ p +1 p +1 (cid:16) log λ − p +1 (cid:17) for p >
0, and ∂ − λ λ k = λ k +1 / ( k + 1) for k = −
1. Explicitly we have that the two components of I ( − l ) formal are equal to( I ( − l ) formal ) = X s l − λ l − s − (log λ − h ( l − s − s ! s !( l − s − X s > l (2 s − l )! λ l − s − s ! s !( − l , (4.18)( I ( − l ) formal ) = 12 λ l l ! − X s l λ l − s (log λ − h ( l − s )) s !( s − l − s )! + X s > l +12 (2 s − l − λ l − s s !( s − − l . (4.19)In these expressions log λ denotes the principal branch of the logarithm on thecomplex plane C \ i R + cut along the positive imaginary axis.Finally we can give the asymptotic expansion of all the period vectors for λ ∼ ∞ . Proposition 20.
The period vectors I ( l ) e i ( t sp , λ ) , i = 1 , , at the special point t sp of M have the following asymptotic expansions for λ ∼ ∞ and arg λ = π/ : I ( l ) e i ( t sp , λ ) ∼ I ( l ) asy ( λ ) , for l > , (4.20) and I ( l ) e i ( t sp , λ ) ∼ I ( l ) formal ( λ ) + P ( l ) i ( λ ) , for l < , (4.21) To see this easily note that, once having fixed arg λ = π/
2, for | λ | big enough one has( λ − − / ( λ + 2) − / = λ − (1 − λ − ) − / . However, for arg λ = π/ IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 21 where P ( l )1 ( λ ) := P i,j > i + j = − l − h ( i ) i ! λ j j ! − P i,j > i +1+ j = − l − h ( i )+ h ( i +1) i !( i +1)! λ j j ! , (4.22) P ( l )2 ( λ ) := P i,j > i + j = − l − h ( i )+ π i i ! λ j j ! − P i,j > i +1+ j = − l − h ( i )+ h ( i +1)+2 π i i !( i +1)! λ j j ! . (4.23) Proof.
Let us first compute the asymptotic expansion of I ( − e i at the special point,by observing that I ( − e i ( t sp , λ ) = Z λu i I (0) e i ( t sp , ρ ) dρ (4.24)= Z λu i (cid:18) ρ (cid:19) dρ + Z ∞ u i (cid:18) I (0) e i − (cid:18) ρ (cid:19)(cid:19) dρ + Z λ ∞ (cid:18) I (0) e i − (cid:18) ρ (cid:19)(cid:19) dρ. (4.25)In this expression the first integral contributes the linear and logarithmic termsappearing in the formal asymptotics plus some constants, while the last integralexactly reproduces the negative powers of λ . Therefore I ( − e i ( t sp , λ ) is asymptotic to I ( − formal plus a constant equal to (cid:18) − log u i − u i (cid:19) + Z ∞ u i (cid:18) I (0) e i − (cid:18) ρ (cid:19)(cid:19) dρ. (4.26)Evaluating the integral one obtains exactly the constants given by P ( − = (cid:18) (cid:19) , P ( − = (cid:18) π i (cid:19) , (4.27)hence the formula (4.21) is valid for l = − l −
2. In this case we havethat µ + l + 1 / t sp that I ( l ) e i = ( µ + l + 12 ) − ( U − λ ) I ( l +1) e i , (4.28)which is asymptotic to( µ + l + 12 ) − ( U − λ ) (cid:16) I ( l +1) formal + P ( l +1) i (cid:17) (4.29)by inductive assumption. Proving that this asymptotic expansion is equal to I ( l ) formal + P ( l ) i is equivalent to prove that( U − λ ) ∂ λ I ( l ) formal − ( µ + l + 12 ) I ( l ) formal + ( U − λ ) P ( l +1) i − ( µ + l + 12 ) P ( l ) i (4.30)is zero. Notice that ∂ λ P ( l ) i = P ( l +1) i , therefore deriving this expression with respectto λ amounts to send l to l + 1 in which case we know that it is zero by inductiveassumption, by substituting (4.21) in (4.1). We conclude that (4.30) is a constant.By induction it is also easy to see that ( U − λ ) ∂ λ I ( l ) formal − ( µ + l + ) I ( l ) formal is apolynomial, so to evaluate (4.30) it is sufficient to set λ = 0. Since we have defined I ( l ) formal by formal integration without constant coefficient,we have that (cid:20) ( U − λ ) ∂ λ I ( l ) formal − ( µ + l + 12 ) I ( l ) formal (cid:21) λ =0 (4.31)is given by minus the coefficient of λ − in I ( l +1) formal which can be read off (4.18)and (4.19), and which cancels with (cid:20) U P ( l +1) i − ( µ + l + 12 ) P ( l ) i (cid:21) λ =0 = − l − )! ( l odd) − − l − − l )! ( l even) ! . (4.32) (cid:3) The vertex operators
In this section we use the period vectors of Section 4 to define certain vertexoperators and we compute the action by conjugation of the R and S Givental groupelements on them.5.1.
Vertex loop space elements.
For a = ( a , a ) ∈ C let us define f a ( t, λ, z ) = X l ∈ Z I ( l ) a ( t, λ )( − z ) l . (5.1)The associated vertex operator is defined as the exponential of the quantisation oflinear Hamiltonian h f a of f a , i.e., Γ a = e b f a = e \ ( f a ) − e \ ( f a ) + .As a consequence of equations (4.1), (4.3) and (4.4) we have that Corollary 21.
The functions f a for a ∈ C satisfy − z ∂ f a ∂λ = f a , z ∂ f a ∂t i = ∂∂t i • f a , ( z ∂∂z + λ ∂∂λ + E ) f a = ( − µ −
12 ) f a . (5.2)5.2. Asymptotics for λ ∼ u i . Let us define the following formal C -valued Laurentseries in z with coefficients which are multivalued functions on C ∗ corresponding tothe vertex operator of the KdV hierarchy f KdV ( λ, z ) = X l ∈ Z I ( l ) KdV ( λ )( − z ) l , I ( l ) KdV ( λ ) = ∂ lλ (2 λ ) − / , (5.3)where ∂ ± λ is the formal differentiation/integration in λ as above. As above we con-sider the principal branches of the roots on the cut λ -plane C \ i R + . Proposition 22.
For i = 1 , we have the equality of asymptotic series at λ ∼ u i f e i ( t, λ, z ) = Ψ − ( t ) R ( t, z ) f KdV ( λ − u i , z ) e i (5.4) Proof.
By substitution it is easy to check that the formula holds at the special point t sp . It is then sufficient to observe that the two sides of the equality satisfy the sameequation in u i . (cid:3) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 23
The functions W a,b ( t, λ ) . Given a, b ∈ C let us define the function W a,b ( t, λ ) = ( I (0) a ( t, λ ) , I (0) b ( t, λ )) , (5.5)which is clearly symmetric and linear in a and b . In our case we have in particular W a,a = ( a + a ) W e i ,e i (5.6)for a = a e + a e ∈ C , and at the special point W e i ,e i ( t sp , λ ) = λ ( λ − − ( λ + 2) − .Notice that W π i a,b is always vanishing, since I (0) π i a = 0.The generators of the fundamental group π of the pointed plane acts on theintegral of W a,a as follows. Lemma 23.
For a ∈ C we have that γ ∗ i (cid:18)Z λλ W a,a dρ (cid:19) = Z λλ W a,a dρ + π i ( a + a ) . (5.7) Proof.
By deforming the path of integration we can write γ i Z λλ W a,a dρ − Z λλ W γ i a,γ i a dρ = Z u i λ ( W a,a − W γ i a,γ i a ) dρ + lim r → + Z C ( u i ,r ) W a,a dρ (5.8)where C ( u i , r ) is the circle with center at u i and radius r . In our case W γ i a,γ i a = W a,a so we just need to evaluate the last integral, which is equal to 2 π i times the residueof ( a + a ) W e i ,e i at λ = u i , which equals ( a + a ) by the normalisation (4.2). (cid:3) The functions c a ( t, λ ) . Let us consider an orbit of the action of the funda-mental group π of the pointed λ -plane on C . The elements in such orbit can beparametrised as (cid:18) r − r (cid:19) + b (cid:18) ( − k + 2 k ( − k − k (cid:19) , k ∈ Z . (5.9)We define c a ( t, λ ) = d ( a ) exp (cid:20) − Z λλ W a,a ( t, ρ ) dρ (cid:21) . (5.10)where d ( a ) is a function defined on the chosen orbit such that c a is covariant underthe action of π , i.e., γ ∗ c a = c γa for any γ ∈ π . This is equivalent to the followingcondition on the function d ( a ) d ( γ i a ) = e π i ( a + a ) d ( a ) . (5.11)The function d ( a ) is therefore fixed, up to multiplication by a constant, to be d ( a k ) = e kπ i b , (5.12)where a k is the element of the orbit parametrised as above.5.5. Splitting.Proposition 24.
Let a = ( a , − a ) ∈ C , then for λ ∼ u i we have the equality ofasymptotic series in √ λ − u i Γ a + ce i = Γ a Γ ce i (5.13) Proof.
Simply observe that ( f a ) + = 0, soΓ a + ce i = e ( f a ) − e ( f cei ) − e ( f cei ) + = Γ a Γ ce i . (5.14) (cid:3) Conjugation by R . Let us now consider the conjugation of the vertex operatorby the R action of the Givental group. Proposition 25.
For λ ∼ u i we have ˆ R − e \ Ψ f cei ˆ R = e c R λui (cid:16) W ei,ei −
12 1 ρ − ui (cid:17) dρ e \ c f KdV ( λ − u i ,z ) e i (5.15) Proof.
We follow the proof given in [43]. Let us first recall the following consequenceof BCH formula: for f = P l I ( l ) ( − z ) l ∈ V and R ( z ) = P k > R k z k in the twistedloop group one has ˆ R − b e f ˆ R = e V f − [ e R − f , (5.16)where the phase factor is given by12 V f − = 12 X k,l > ( V k,l I ( − l − , I ( − k − ) (5.17)where the matrices V k,l ∈ End( V ) are defined by X k,l > V k,l w k z l = 1 − R ( w ) R ∗ ( z ) w + z . (5.18)It follows that V k − ,l + V k,l − = − R k R ∗ l + δ k, δ l, , (5.19)assuming that V − ,l = V k, − = 0.We need to compute the asymptotic expansion for λ ∼ u i of the phase factor thatin our case is given by V (Ψ f ce i ) − = c X k,l > ( V k,l Ψ I ( − l − e i , Ψ I ( − k − e i ) . (5.20)Recall that each entry in Ψ I ( l ) e i is asymptotic to a formal Laurent series in √ λ − u i for λ ∼ u i , with leading term of degree (at worst) − l −
1. This implies that theright-hand side of (5.20) converges to a formal series in √ λ − u i , which in particularvanishes at u i .Deriving (5.20) by λ and changing the indexes in the sums we obtain c X k,l > (cid:0) ( V k − ,l + V k,l − )Ψ I ( − l ) e i , Ψ I ( − k ) e i (cid:1) , (5.21)which, by substituting (5.19), equals c times(Ψ I (0) e i , Ψ I (0) e i ) − X l > R ∗ l Ψ I ( − l ) e i , X k > R ∗ k Ψ I ( − k ) e i ! . (5.22)Note that (5.4) gives X l > R ∗ l Ψ I ( − l ) e i = I (0) KdV ( λ − u i ) e i , (5.23)so the second term in (5.22) is equal to − (2( λ − u i )) − . By integration we obtain12 V (Ψ f ce i ) − = c Z λu i (cid:20) W e i ,e i ( t, ρ ) −
12 1 ρ − u i (cid:21) dρ. (5.24) (cid:3) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 25
Asymptotics at λ ∼ ∞ . Let us define the following formal Laurent series in z with coefficients which are C -valued functions over C \ i R + : f e , ∞ ( λ, z ) := X l ∈ Z ∂ lλ (cid:18) ∂ λ (log λ + ψ ) (cid:19) ( − z ) l , (5.25) f e , ∞ ( λ, z ) := f e , ∞ ( λ, z ) + X l ∈ Z ∂ l +1 λ (cid:18) π i (cid:19) ( − z ) l , (5.26)and for a = ( a , a ) ∈ C let f a, ∞ = a f e , ∞ + a f e , ∞ . As above log λ denotes the prin-cipal branch of the logarithm over C \ i R + , ∂ ± λ is formal differentiation/integrationin λ , and ψ is the constant parametrising the calibration, defined in § Proposition 26.
For any a ∈ C , the asymptotic behaviour of f a for | λ | ∼ ∞ , arg λ = π/ is given by f a ( t, λ, z ) ∼ S ( t, z ) f a, ∞ ( λ, z ) . (5.27) Proof.
We need to prove that I ( l ) e i ( t, λ ) ∼ ∞ X k =0 ( − k S k I ( l + k ) e i , ∞ (5.28)where f e i , ∞ = P l I ( l ) e i , ∞ ( − z ) l . It is easy to check that this equality of asymptoticseries holds at t sp . The case i = 1 follows by direct substitution, while in the case i = 2 one can observe that the difference between the asymptotic expansions of thetwo period vectors is a polynomial in λ which is exactly given by shifting log λ by π i as in the second term on the right-hand side of (5.26).Because of (2.1), both sides of (5.28) satisfy the same equation in t i , thereforethe must coincide on the whole M . (cid:3) Conjugation by S . Let us now consider the conjugation of the vertex operatorby the S action of the Givental group. We define W ∞ a,b = (( I (0) a, ∞ ( t, λ ) , I (0) b, ∞ ( t, λ )).Notice that W ∞ e i ,e i = λ − for i = 1 , Proposition 27.
For a ∈ C we have ˆ S Γ a ∞ ˆ S − = e c ( t )2 + R ∞ λ ( W a,a −W ∞ a,a ) dρ Γ a , (5.29) with c ( t ) = ( a + a ) t + ψ ) . (5.30) Proof.
It follows from the Baker-Campbell-Hausdorff formula that, for f = P l f l z l in the loop space V and S ( z ) in the twisted loop group, we haveˆ S b e f ˆ S − = e W ( f + , f + ) c e S f , (5.31)where W ( f + , f + ) = X k,l > ( W k,l f l , f k ) , (5.32)and the coefficients W k,l ∈ End( V ) are defined by the generating formula X k,l > W k,l w − k z − l = S ∗ ( w ) S ( z ) − w − + z − . (5.33) We need therefore to evaluate the phase factor W (( f a, ∞ ) + , ( f a, ∞ ) + ). We followhere the argument of [43, § a, b ∈ C W (( f a, ∞ ) + , ( f b, ∞ ) + ) = X k,l > ( − k + l ( W k,l I ( l ) a, ∞ , I ( k ) b, ∞ ) , (5.34)therefore ddλ W (( f a, ∞ ) + , ( f b, ∞ ) + ) = − X k,l > ( − k + l (( W k − ,l + W k,l − ) I ( l ) a, ∞ , I ( k ) b, ∞ ) , (5.35)where we assumed that W − ,l = W k, − = 0 and we used the fact that ∂ λ I ( l ) a, ∞ = I ( l +1) a, ∞ .From (5.33) we get W k − ,l + W k,l − = S ∗ k S l − δ k, δ l, , (5.36)so (5.35) equals − ( X l > S l ( − l I ( l ) a, ∞ , X k > S k ( − k I ( k ) b, ∞ ) + ( I (0) a, ∞ , I (0) b, ∞ ) . (5.37)By (5.27), we have ddλ W (( f a, ∞ ) + , ( f b, ∞ ) + ) = −W a,b + W ∞ a,b . (5.38)From (5.28) it follows that the right hand-side has leading term of order λ − . There-fore we can formally integrate the asymptotic series and the integration constant isgiven by the leading term ( a + a )( b + b )4 ( S ) , in the expansion of (5.34). We haveshown that for a, b ∈ C we have the equality of asymptotic series for λ ∼ ∞ W (( f a, ∞ ) + , ( f b, ∞ ) + ) = ( a + a )( b + b )4 (log t + ψ )+ Z ∞ λ (cid:2) W a,b ( t, ρ ) − W ∞ a,b ( t, ρ ) (cid:3) dρ. (5.39)The Proposition is proved. (cid:3) The Hirota quadratic equations for the ancestor potential
In this section we define the ancestor Hirota quadratic equations and prove thatthe ancestor potential A satisfies them.6.1. Definition of Hirota quadratic equations for the ancestor potential.
Recall that the total ancestor potential A can be considered as a formal powerseries in the variables q iℓ + δ i δ ℓ for i = 1 , ℓ > ǫ , and it analytically depends on the point of M .Recall the discussion of the action of the monodromy group on C in Section 4.2,in particular Proposition 17. We choose a finite subset A in an infinite orbit ofthe monodromy group defined as A = { a + , a − } , where a + = a e + a e and a − =( − a − a ) e + a e . Note that γ a ± = a ∓ ; (6.1) γ a ± = a ∓ ± a + a )( e − e ) . (6.2)In terms of the parametrization of the full orbit generated by A given in Section 5.4,we have b = a + a and r = ( a − a ) /
2. Recall that we associate in Section 5.4 to
IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 27 each element a of this full orbit a function c a ( t, λ ) given on a ± , up to a multiplicationby a non-zero constant, by c a + = exp (cid:20) − Z λλ W a + ,a + ( t, ρ ) dρ (cid:21) ; (6.3) c a − = exp (cid:20) − π i ( a + a ) − Z λλ W a − ,a − ( t, ρ ) dρ (cid:21) ; (6.4)Recall also that to each point a ∈ C we associate in Section 5.1 a vertex operatorΓ a .Let o = e − e . Recall that ( I ( − o ) = 0, ( I ( − o ) = 0 = − π i (see Remark 15, weuse the scalar product to lower the indices). Let us define N = exp − X ( j,ℓ ) =(2 , ( I ( − ℓ − o ) j q jℓ ( I ( − o ) ∂∂q = exp X ℓ > ,j =1 , ( I ( − ℓ − o ) j q jℓ π i ∂∂q ! . (6.5) Lemma 28. If b ∈ Z and bǫ ( q − ¯ q ) ∈ Z , the following expression is a single-valuedfunction of λ : N ⊗ N (cid:16) c a + Γ a + ⊗ Γ − a + + c a − Γ a − ⊗ Γ − a − (cid:17) ( A ⊗ A ) dλ. (6.6) Here the two copies of A depend on the variables q iℓ and ¯ q iℓ respectively.Proof. We need to prove that it is invariant under the action of the two generators γ , γ of the fundamental group of the pointed complex plane. Note that the coefficientsof N are single-valued functions in λ . Indeed, since I (0) o = 0, all I ( − ℓ − o , ℓ >
0, arepolynomials in λ . Recall also the action of the fundamental group of the Γ a -operators(Proposition 16) and the coefficients c a (Section 5.4).For γ we have: γ ∗ N ⊗ N (cid:16) c a + Γ a + ⊗ Γ − a + + c a − Γ a − ⊗ Γ − a − (cid:17) ( A ⊗ A ) dλ (6.7)= N ⊗ N (cid:16) c a − Γ a − ⊗ Γ − a − + c a + Γ a + ⊗ Γ − a + (cid:17) ( A ⊗ A ) dλ, so this expression is invariant under the action of γ .For γ we first observe that the action of γ on the vertex operator is given by γ ∗ Γ a ± = Γ a ∓ ± bo = e ± b ˆ f o Γ a ∓ , (6.8)where we use the Baker-Campbell-Hausdorff formula and the fact that ( f o ) + = 0 forthe second equality (see Proposition 24). Note also (cf. Section 5.4) that γ ∗ c a ± = c γ a ± = e ± π i b c a ∓ = c a ∓ (6.9)(here we use the condition b ∈ Z for the last equation). Therefore, the action of γ on (6.6) is γ ∗ N ⊗ N (cid:16) c a + Γ a + ⊗ Γ − a + + c a − Γ a − ⊗ Γ − a − (cid:17) ( A ⊗ A ) dλ (6.10)= N ⊗ N (cid:16) e b ˆ f o ⊗ e − b ˆ f o c a − Γ a − ⊗ Γ − a − + e − b ˆ f o ⊗ e b ˆ f o c a + Γ a + ⊗ Γ − a + (cid:17) ( A ⊗ A ) dλ. Now note that ˆ f o = ǫ − P ℓ > ( I ( − ℓ − o ) i q iℓ (cf. Equation (3.7)), and, therefore, N e ± b ˆ f o = e ± bǫ − ( I ( − o ) q N . Therefore, (6.10) is equal to (cid:16) e bǫ ( I ( − o ) ( q − ¯ q ) N ⊗ N c a − Γ a − ⊗ Γ − a − ++ e − bǫ ( I ( − o ) ( q − ¯ q ) N ⊗ N c a + Γ a + ⊗ Γ − a + (cid:17) ( A ⊗ A ) dλ. (6.11)Under the condition that bǫ ( I ( − o ) ( q − ¯ q ) ∈ π i Z (which can be simplified using( I − o ) = − π i to bǫ ( q − ¯ q ) ∈ Z ) this expression is equal to (6.6), which proves theinvariance under the action of γ . (cid:3) Lemma 28 implies that under the condition b ∈ Z expression (6.6) can be consid-ered as a formal power series in the variables q iℓ + δ i δ ℓ and ¯ q iℓ + δ i δ ℓ for i = 1 , l > ǫ , the coefficients of whose restriction to bǫ ( q − ¯ q ) ∈ Z are rational functions of λ with possible poles at the points λ = u , u , ∞ . Definition 29.
We say that the ancestor potential A satisfies the ancestor Hirotaquadratic equations for the set A = { a + , a − } if the aforementioned dependence on λ is polynomial, that is, if there are no poles at λ = u , u .6.2. Proof of the ancestor HQE.Theorem 30.
Let b = 1 . Then the ancestor potential A satisfies the ancestorHirota quadratic equation.Proof. Let us prove that (6.6) is regular at λ = u (the case of λ = u is completelyanalogous). Note that a ± = − a o ± be . According to Proposition 24, expres-sion (6.6) is equal to( N ⊗ N ) (cid:0) Γ − a o ⊗ Γ a o (cid:1) (cid:0) c a + Γ be ⊗ Γ − be + c a − Γ − be ⊗ Γ be (cid:1) ( A ⊗ A ) dλ. (6.12)Here the first two factors do not change the regularity in λ at λ = u (recall that I (0) o = 0), so we have to show that (cid:0) c a + Γ be ⊗ Γ − be + c a − Γ − be ⊗ Γ be (cid:1) ( A ⊗ A ) dλ. (6.13)is regular in λ at λ = u .Recall that A = ˆΨ − ˆ R τ
KdV τ KdV , where the KdV tau-functions τ KdV i depend onthe coordinates that correspond to the normalized canonical frame. In particular,the KdV hierarchy for these KdV tau-functions can be written as the regularity at λ = u i , i = 1 ,
2, of the expression( λ − u i ) − (cid:16) e \ f KdV ( λ − u i ,z ) e i ⊗ e − \ f KdV ( λ − u i ,z ) e i −− e − \ f KdV ( λ − u i ,z ) e i ⊗ e \ f KdV ( λ − u i ,z ) e i (cid:17) τ KdV i ⊗ τ KdV i dλ. (6.14)Proposition 25 implies that expression (6.13) is equal to ˆΨ − ˆ R ⊗ ˆΨ − ˆ R applied to (cid:18) c a + e b R λu (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ e \ b f KdV ( λ − u ,z ) e ⊗ e − \ b f KdV ( λ − u ,z ) e ++ c a − e b R λu (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ e \ − b f KdV ( λ − u ,z ) e ⊗ e \ b f KdV ( λ − u ,z ) e (cid:19) ·· ( τ KdV τ KdV ⊗ τ KdV τ KdV ) dλ. (6.15) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 29
Let us compute the coefficients. We have: c a + e b R λu (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ = e − b R λλ W e ,e dρ + b R λu (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ (6.16)= e b log( λ − u )+ b R λ u (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ ( λ − u ) − b ; c a − e b R λu (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ = e − π i b − b R λλ W e ,e dρ + b R λu (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ (6.17)= e b log( λ − u )+ b R λ u (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ e − π i b ( λ − u ) − b . Under the assumption b = 1 the coefficients, up to a common invertible factor thatdoes not depend on λ , are equal to ± ( λ − u ) − .Thus, under the assumption b = 1 expression (6.15) is equal up to an invertiblefactor e log( λ − u )+ R λ u (cid:16) W e ,e −
12 1 ρ − u (cid:17) dρ τ KdV ⊗ τ KdV (6.18)that does not depend on λ to the expression (6.14) with i = 1, whose regularity in λ at λ = u is part of the definition of A . This implies the regularity at λ = u of (6.12). (cid:3) Remark . The condition b = 1 implies that b = 1 or b = −
1. Since a ± = − a o ± be , the set A does not depend on this choice, and in the construction of theancestor Hirota quadratic this choice only affects a common non-vanishing coefficientthat we can ignore, cf. Equation (6.3). Remark . With b = 1 the restriction bǫ ( q − ¯ q ) ∈ Z reduces to ( q − ¯ q ) ∈ ǫ Z .7. The Hirota quadratic equations for the descendent potential
In this section we define the descendent Hirota quadratic equations and provethat the de scendant potential D satisfies them. Note that we assumefrom now onthat b = 1.7.1. The Hirota equation.
Recall the definitions of f e i , ∞ = P l I ( l ) e i , ∞ ( − z ) l , i = 1 , f a, ∞ and I ( l ) a, ∞ defined for any a ∈ C .Recall also that for any a, b ∈ C we defined in Section 5.8 the function W ∞ a,b ( λ ) =( I (0) a, ∞ ( λ ) , I (0) b, ∞ ( λ )).Recall that o = e − e . Let us also define N ∞ = exp − X ( j,ℓ ) =(2 , ( I ( − ℓ − o, ∞ ) j q jℓ ( I ( − o, ∞ ) ∂∂q = exp − X ℓ > λ ℓ ℓ ! q ℓ ∂∂q ! . (7.1)(The second equality here follows from Equations (5.25)-(5.26).)Let A = { a + , a − } be the same set of points in C as in Section 6, a ± = − a o ± e (that is, for a + = a e + a e we assume a + a = 1). Let c ∞ a + = exp (cid:20) − Z λλ W ∞ a + ,a + ( ρ ) dρ (cid:21) ; (7.2) c ∞ a − = exp (cid:20) − π i − Z λλ W ∞ a − ,a − ( ρ ) dρ (cid:21) (7.3) (cf. the analogous definitions in Sections 5.4 and 6.1). Since I (0) e , ∞ = I (0) e , ∞ = ( λ , ) t ,we have W ∞ a + ,a + = W ∞ a − ,a − = λ − and, therefore, c ∞ a ± = ± λ λ .Consider the following expression: N ∞ ⊗ N ∞ (cid:16) c ∞ a + Γ a + ∞ ⊗ Γ − a + ∞ + c ∞ a − Γ a − ∞ ⊗ Γ − a − ∞ (cid:17) ( D ⊗ D ) dλ , (7.4)where the two copies of D depend on two different sets of variables, q iℓ and ¯ q iℓ ,respectively. Lemma 33.
For ( q − ¯ q ) ∈ ǫ Z expression (7.4) is a single-valued function of λ .Proof. Since the whole expression is only defined as an asymptotic series for | λ | ∼ ∞ ,we have to check that the action of the monodromy γ ∞ along the big circle is trivial.Note that N ∞ and c ∞ a ± are single-valued in λ . For Γ ± a ± ∞ the action of the mon-odromy γ ∞ changes the branch of the logarithm in the definition of f e i , ∞ . We have: γ ∞ f e i , ∞ = f e i , ∞ + f ∞ , (7.5)where f ∞ ( λ, z ) = 2 π i X l ∈ Z ∂ l +1 λ (cid:18) (cid:19) ( − z ) l = 2 π i (cid:18) (cid:19) X l > λ l l ! ( − z ) − l − . (7.6)Thus γ ∞ Γ e i ∞ = e ˆ f ∞ Γ e i ∞ , and, therefore, for any constant cγ ∞ Γ ca ± ∞ = e ± ( c ˆ f ∞ ) Γ ca ± ∞ , (7.7)where ˆ f ∞ = 2 π i ǫ X k > ( ∂ − kλ q k = 2 π i ǫ X k > λ k k ! q k . (7.8)Using this computation, we apply γ ∞ to expression (7.4), and we obtain: N ∞ ⊗ N ∞ (cid:16) e ˆ f ∞ ⊗ e − ˆ f ∞ c ∞ a + Γ a + ∞ ⊗ Γ − a + ∞ + e − ˆ f ∞ ⊗ e ˆ f ∞ c ∞ a − Γ a − ∞ ⊗ Γ − a − ∞ (cid:17) ( D ⊗ D ) dλ . (7.9)Note that N ∞ e ± ˆ f ∞ = e ± π i ǫ q N ∞ . Therefore, (7.9) is equal to (cid:16) e π i ǫ ( q − ¯ q ) N ∞ ⊗ N ∞ c ∞ a + Γ a + ∞ ⊗ Γ − a + ∞ ++ e − π i ǫ ( q − ¯ q ) N ∞ ⊗ N ∞ c ∞ a − Γ a − ∞ ⊗ Γ − a − ∞ (cid:17) ( D ⊗ D ) dλ , (7.10)which coincides with (7.4) in the case ( q − ¯ q ) ∈ ǫ Z . (cid:3) Definition 34.
We say that the descendent potential D satisfies the descendentHirota quadratic equation if the coefficients of expression (7.4) (expanded in q iℓ + δ i δ ℓ ,¯ q iℓ + δ i δ ℓ , and ǫ ) are polynomial in λ .7.2. Hirota equations for the descendent potential.Theorem 35.
The descendent potential D satisfies the descendent Hirota quadraticequation.Proof. Recall that D = C ˆ S − A , where all three factors on the right hand sidedepend on the point of M , but their product is independent. Note that the factor C does not depend on λ and is constant in q iℓ , ¯ q iℓ (in particular, it commutes with alloperators involved in expression (7.4)). For the operator ˆ S − we use the followingtwo lemmata: IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 31
Lemma 36.
We have the equality of the asymptotic series for λ ∼ ∞ c ∞ a ± ( ˆ S ⊗ ˆ S )(Γ a ± ∞ ⊗ Γ − a ± ∞ )( ˆ S − ⊗ ˆ S − ) = F · c a ± (Γ a ± ⊗ Γ − a ± ) , (7.11) where F = exp (cid:16) (log t + ψ ) + R ∞ λ (cid:16) W a ± ,a ± − W ∞ a ± ,a ± (cid:17) dρ (cid:17) .Proof. Recall Proposition 27. Since the sum of coordinates of a ± is equal to ±
1, itimplies that c ∞ a ± ( ˆ S ⊗ ˆ S )(Γ a ± ∞ ⊗ Γ − a ± ∞ )( ˆ S − ⊗ ˆ S − ) = c ∞ a ± e (log t + ψ )+ R ∞ λ (cid:16) W a ± ,a ± −W ∞ a ± ,a ± (cid:17) dρ (Γ a ± ⊗ Γ − a ± ) . (7.12)Now observe that c ∞ a ± e (log t + ψ )+ R ∞ λ (cid:16) W a ± ,a ± −W ∞ a ± ,a ± (cid:17) dρ == ± e (log t + ψ )+ R ∞ λ (cid:16) W a ± ,a ± −W ∞ a ± ,a ± (cid:17) dρ − R λλ W ∞ a ± ,a ± ( ρ ) dρ = ± e (log t + ψ )+ R ∞ λ (cid:16) W a ± ,a ± −W ∞ a ± ,a ± (cid:17) dρ − R λλ W a ± ,a ± ( ρ ) dρ = F · c a ± (7.13)(for the last equality here recall the definition of c a ± given in Equations (6.3)-(6.4)). (cid:3) Note that the factor F does not depend on λ and is constant in q iℓ , ¯ q iℓ (in particular,it commutes with N ∞ ). Note also that the factor F doesn’t depend on the choiceof the sign ± in a ± since W ∞ a + ,a + = W ∞ a − ,a − and W a + ,a + = W a − ,a − . Lemma 37.
We have the equality of the asymptotic series for λ ∼ ∞N ∞ ˆ S − = QO N , (7.14) where Q is an exponential of a linear combination of terms ǫ − q iℓ q jm , whose coeffi-cients are polynomial in λ and depend on the point of M , and O is the exponentialof a linear vector field in q iℓ that does not contain differentiation ∂/∂q and whosecoefficients depend on the point of M .Proof. Typically, we commute the operators using the quantisation rules. However, N and N ∞ are not obtained by quantization, so we have to go into a detailed analysisof the commutation of these operators with ˆ S .Recall the structure of log ˆ S (see Equation (3.11)). It can be split into two sum-mands, a linear combination of all terms of the type ǫ − q ik q jℓ and a linear combi-nation of the terms q ik ∂/∂q jℓ , with the coefficients depending on the point of M .Consider N ∞ ˆ S − . Using the Baker-Campbell-Hausdorff formula we extract all qua-dratic terms ǫ − q ik q jℓ in ˆ S − to a separate exponential and commute it through N ∞ to the left. This gives the coefficient Q .Now we have to compute N ∞ exp X ℓ > a > q ia + ℓ ∂∂q ja ( s ℓ ) ji = (7.15)exp − X ( j,ℓ ) =(2 , ( I ( − ℓ − o, ∞ ) j q jℓ ( I ( − o, ∞ ) ∂∂q exp X ℓ > a > q ia + ℓ ∂∂q ja ( s ℓ ) ji . To this end note that (cf. Equation (5.31))exp − X ℓ > a > q ia + ℓ ∂∂q ja ( s ℓ ) ji exp − X ( j,ℓ ) =(2 , ( I ( − ℓ − o, ∞ ) j q jℓ ( I ( − o, ∞ ) ∂∂q exp X ℓ > a > q ia + ℓ ∂∂q ja ( s ℓ ) ji (7.16)= exp − X ( k,ℓ ) =(2 , X p > ( I ( − ℓ − o, ∞ ) k ( S − p ) kj q jℓ + p ( I ( − o, ∞ ) ∂∂q = exp − X ( k,ℓ ) X p > ( I ( − ℓ − o, ∞ ) k ( S − p ) kj q jℓ + p ( I ( − o, ∞ ) ∂∂q + X p > ( I ( − o, ∞ ) ( S − p ) j q jp ( I ( − o, ∞ ) ∂∂q = exp − X ( j,ℓ ) P p > ( − p ( S p I ( − ( ℓ + p ) − p ) o, ∞ ) j q jℓ + p ( I ( − o, ∞ ) ∂∂q + X p > ( I ( − o, ∞ ) ( S − p ) j q jp ( I ( − o, ∞ ) ∂∂q . Using Equation (5.28) and the observations that I ( ℓ ) o, ∞ = 0 and I ( ℓ ) o = 0 for ℓ > X p > ( S − p ) j q jp ∂∂q ! N . (7.17)Now note that the BCH formula impliesexp X ℓ > a > q ia + ℓ ∂∂q ja ( s ℓ ) ji = exp X ℓ > , a > j,a ) =(2 , q ia + ℓ ∂∂q ja ( s ℓ ) ji exp − X p > ( S − p ) j q jp ∂∂q ! . (7.18)Let O denote the first factor on the right hand side of Equation (7.18). ThenEquations (7.15), (7.16), (7.17), and (7.18), collected together, imply that N ∞ exp X ℓ > a > q ia + ℓ ∂∂q ja ( s ℓ ) ji = O N . (7.19) (cid:3) Using these two lemmata, we can rewrite Equation (7.4) as the asymptotic seriesexpansion for λ ∼ ∞ of the following expression: C F · ( Q ⊗ Q )( O ⊗ O )( N ⊗ N ) (cid:16) c a + Γ a + ⊗ Γ − a + + c a − Γ a − ⊗ Γ − a − (cid:17) ( A ⊗ A ) dλ . (7.20)Note that the operator C F · ( Q ⊗ Q )( O ⊗ O ) does not contain derivatives withrespect to q and ¯ q . Therefore, the restriction ( q − ¯ q ) ∈ ǫ Z can be applied to thisoperator and to rest of the formula simultaneously. Note also that it is an invertibleoperator that preserves the polynomiality property in λ , that is, the coefficients ofthis expression restricted to ( q − ¯ q ) ∈ ǫ Z (and expanded in q iℓ + δ i δ ℓ , ¯ q iℓ + δ i δ ℓ , IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 33 and ǫ ) are polynomial in λ if and only if the same property holds for the asymptoticexpansion of ( N ⊗ N ) (cid:16) c a + Γ a + ⊗ Γ − a + + c a − Γ a − ⊗ Γ − a − (cid:17) ( A ⊗ A ) dλ , (7.21)which is indeed the case by Theorem 30. (cid:3) Explicit form of the Hirota equations.
The goal of this Section is to workout an explicit form of the Hirota quadratic equations for the descendent potential(which is the equations of the polynomiality of the 1-form given by Equation (7.4)at q − ¯ q = ǫk , k ∈ Z ) Proposition 38.
For any value of the calibration parameter ψ the descendent po-tential D satisfies the following equations: λ = ∞ λ n − dλ h λ k exp( kψ (cid:0) ǫ P ℓ > λ ℓ +1 ( ℓ +1)! ( q ℓ − ¯ q ℓ ) − ǫ P ℓ > λ ℓ ℓ ! h ( ℓ )( q ℓ − ¯ q ℓ ) (cid:1) ×D (cid:0)(cid:8) q ℓ − ǫ ℓ ! λ ℓ +1 (cid:9) ℓ > , q − ǫ − P ℓ > λ ℓ ℓ ! q ℓ , (cid:8) q ℓ (cid:9) ℓ > (cid:1) × (7.22) D (cid:0)(cid:8) ¯ q ℓ + ǫ ℓ ! λ ℓ +1 (cid:9) ℓ > , ¯ q + ǫ − P ℓ > λ ℓ ℓ ! ¯ q ℓ , (cid:8) ¯ q ℓ (cid:9) ℓ > (cid:1) − λ − k exp( − kψ (cid:0) − ǫ P ℓ > λ ℓ +1 ( ℓ +1)! ( q ℓ − ¯ q ℓ ) + ǫ P ℓ > λ ℓ ℓ ! h ( ℓ )( q ℓ − ¯ q ℓ ) (cid:1) ×D (cid:0)(cid:8) q ℓ + ǫ ℓ ! λ ℓ +1 (cid:9) ℓ > , q + ǫ − P ℓ > λ ℓ ℓ ! q ℓ , (cid:8) q ℓ (cid:9) ℓ > (cid:1) ×D (cid:0)(cid:8) ¯ q ℓ − ǫ ℓ ! λ ℓ +1 (cid:9) ℓ > , ¯ q − ǫ − P ℓ > λ ℓ ℓ ! ¯ q ℓ , (cid:8) ¯ q ℓ (cid:9) ℓ > (cid:1)i(cid:12)(cid:12)(cid:12) q − ¯ q = kǫ . for any k ∈ Z and for any n > .Proof. Recall Equations (5.25) and (5.26). For a ± = ± e − a o they imply that f a ± , ∞ = ± X ℓ ∈ Z ∂ ℓλ (cid:18) ∂ λ (log λ + ψ ) (cid:19) ( − z ) ℓ + a X ℓ ∈ Z ∂ ℓ +1 λ (cid:18) π i (cid:19) ( − z ) ℓ (7.23)= X ℓ > ( − z ) − − ℓ λ ℓ ℓ ! (cid:16) a π i ± (log λ − h ( ℓ ) + ψ ) (cid:17) ± λ ℓ +1 ( ℓ +1)! ! ± X ℓ > ( − z ) ℓ ( − ℓ ℓ ! λ ℓ +1 (cid:18) (cid:19) ± ( − z ) (cid:18) (cid:19) . Therefore,ˆ f a ± , ∞ = ± ǫ X ℓ > λ ℓ +1 ( ℓ + 1)! q ℓ ± ǫ X ℓ > λ ℓ ℓ ! (cid:16) log λ − h ( ℓ ) + ψ (cid:17) q ℓ + 1 ǫ a π i X ℓ > λ ℓ ℓ ! q ℓ (7.24) ∓ ǫ X ℓ > ℓ ! λ ℓ +1 ∂∂q ℓ ∓ ǫ ∂∂q . Recall also Equation (7.1): N ∞ = exp − X ℓ > λ ℓ ℓ ! q ℓ ∂∂q ! . (7.25) We have: N ∞ ˆ f a ± , ∞ = " ± ǫ X ℓ > λ ℓ +1 ( ℓ + 1)! q ℓ ∓ ǫ X ℓ > λ ℓ ℓ ! h ( ℓ ) q ℓ ± ǫ (cid:16) log λ + ψ (cid:17) q + 1 ǫ a π i q (7.26) ∓ ǫ X ℓ > ℓ ! λ ℓ +1 ∂∂q ℓ ∓ ǫ ∂∂q N ∞ Note that for q − ¯ q = ǫk , k ∈ Z , we have:exp (cid:16) ± ǫ (cid:16) log λ + ψ (cid:17) ( q − ¯ q )+ 1 ǫ a π i ( q − ¯ q ) (cid:17) = exp( kπ i a ) exp (cid:16) ± kψ (cid:17) λ ± k (7.27)Since that the factor exp( kπ i a ) doesn’t depend on the choice of the point a ± anddoesn’t depend on λ , it doesn’t affect the polynomiality in λ and can be omitted.For the same reason we can replace in (7.4) the coefficients c ∞ a ± by ± λ − . Modulothese not important factors, the full expression (7.4) can be rewritten as dλλ h λ k exp( kψ ) exp (cid:0) ǫ P ℓ > λ ℓ +1 ( ℓ +1)! ( q ℓ − ¯ q ℓ ) − ǫ P ℓ > λ ℓ ℓ ! h ( ℓ )( q ℓ − ¯ q ℓ ) (cid:1) × (7.28) D (cid:0)(cid:8) q ℓ − ǫ ℓ ! λ ℓ +1 (cid:9) ℓ > , q − ǫ − P ℓ > λ ℓ ℓ ! q ℓ , (cid:8) q ℓ (cid:9) ℓ > (cid:1) ×D (cid:0)(cid:8) ¯ q ℓ + ǫ ℓ ! λ ℓ +1 (cid:9) ℓ > , ¯ q + ǫ − P ℓ > λ ℓ ℓ ! ¯ q ℓ , (cid:8) ¯ q ℓ (cid:9) ℓ > (cid:1) − λ − k exp( − kψ ) exp (cid:0) − ǫ P ℓ > λ ℓ +1 ( ℓ +1)! ( q ℓ − ¯ q ℓ ) + ǫ P ℓ > λ ℓ ℓ ! h ( ℓ )( q ℓ − ¯ q ℓ ) (cid:1) ×D (cid:0)(cid:8) q ℓ + ǫ ℓ ! λ ℓ +1 (cid:9) ℓ > , q + ǫ − P ℓ > λ ℓ ℓ ! q ℓ , (cid:8) q ℓ (cid:9) ℓ > (cid:1) ×D (cid:0)(cid:8) ¯ q ℓ − ǫ ℓ ! λ ℓ +1 (cid:9) ℓ > , ¯ q − ǫ − P ℓ > λ ℓ ℓ ! ¯ q ℓ , (cid:8) ¯ q ℓ (cid:9) ℓ > (cid:1)i(cid:12)(cid:12)(cid:12) q − ¯ q = kǫ . This expression is polynomial in λ if and only if the residues of its products with λ n , n >
0, at λ = ∞ are equal to zero, which completes the proof of the proposition. (cid:3) The Lax formulation of the Catalan hierarchy
In this section we would like to find a Lax representation for the integrable hier-archy associated with the descendent HQE. Such hierarchy has as natural spatialvariable the time X = q . Proceeding directly as in the rational reduction of KPis quite straightforward as long as we don’t consider the equations for the “loga-rithmic” times q ℓ . Noticing that the descendent HQEs are identical to those of theExtended Toda hierarchy [10, 12] under exchange of the two sets of times, we firstproceed in deriving the Lax form of the equations as in [12] using as space variablethe time q = x , and then obtain indirectly a Lax representation in terms of pseudo-differential operators in the proper space variable X = q using the approach insection 5 of [10]. Finally we reconsider the Hirota equations and directly derive theLax equations in pseudo-differential operator form.8.1. Lax representation with difference operators.
Here we quickly repeatwith slight modifications the derivation of the Lax representation of the extendedToda hierarchy following [12] in the case N = M = 1.Let us consider the multivalued one form on the λ plane ω ∞ := (cid:16) c ∞ a + Γ a + ∞ ⊗ Γ − a + ∞ + c ∞ a − Γ a − ∞ ⊗ Γ − a − ∞ (cid:17) ( D ⊗ D ) dλ . (8.1) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 35
The Hirota equation (7.4) can be equivalently formulated by saying that( N ∞ ⊗ N ∞ ) ω ∞ is regular at λ ∼ ∞ . To obtain the Lax representation we have to switch to anequivalent HQE which is ∂ x -operator valued, as in [42]. We therefore introduce theoperators Γ δ ∞ = e x ∂∂q e P l> λll ! q l ∂ x , Γ δ ∞ = e − P l> λll ! q l ∂ x e x ∂∂q . (8.2)Since we have thatΓ δ ∞ = e P l> λll ! q l ∂ x e x ∂∂q N ∞ , Γ δ ∞ = e x ∂∂q e − P l> λll ! q l ∂ x N ∞ , (8.3)it follows that ( N ∞ ⊗ N ∞ ) ω ∞ regular ⇐⇒ (Γ δ ∞ ⊗ Γ δ ∞ ) ω ∞ regular . (8.4)Along the lines of [12] by carefully commuting the operators we can prove thatΓ δ ∞ Γ a ± ∞ D = D ′ W ± λ ± q xǫ e ( ± ψ ǫ − ( a − π i ǫ )( q + x ) (8.5)and Γ δ ∞ Γ − a ± ∞ D = λ ∓ q xǫ e ( ∓ ψ ǫ +( a − π i ǫ )( q + x ) W ∗± D ′ , (8.6)where we define W ± = P ± exp ± ǫ X l > λ l +1 ( l + 1)! q l + X l> λ l l ! ( ∂ x ∓ h l ǫ ) q l ! (8.7) W ∗± = exp ∓ ǫ X l > λ l +1 ( l + 1)! q l − X l> λ l l ! ( ∂ x ∓ h l ǫ ) q l ! P ∗± (8.8)and P ± = e \ ( f a ±∞ ) + D ′ D ′ ( x − ǫ ) , P ∗± = e \ ( f − a ±∞ ) + D ′ D ′ ( x + ǫ ) . (8.9)Here D ′ ( q, x ) = D ( q ) | q → q + x .Substituting, we find that the HQE are equivalent to the regularity of (cid:2) W + ( q ) W ∗ + (¯ q ) λ k e ψk/ − W − ( q ) W ∗− (¯ q ) λ − k e − ψk/ (cid:3) dλλ (8.10)where ( q − ¯ q ) = kǫ . In residue form we haveRes λ (cid:2) W + ( q ) W ∗ + (¯ q ) λ k e ψk/ − W − ( q ) W ∗− (¯ q ) λ − k e − ψk/ (cid:3) λ n − dλ = 0 , n > . (8.11)We now convert this expression in a bilinear equation for difference operators.Given a difference operator A = P s a s Λ s = P s Λ s ˜ a s the left and right symbols arerespectively defined as σ l ( A ) = P s a s λ s and σ r ( A ) = P s ˜ a s λ s . Recall thatRes λ σ l ( A ) σ r ( B ) dλλ = Res Λ AB, where Res Λ A := a , for a proof see § W ± and W ∗± by σ l ( W + ) = W + , σ r ( W ∗ + ) = W ∗ + , (8.12) σ l ( W − ) = W − | λ → λ − e − ψ , σ r ( W ∗− ) = W ∗− | λ → λ − e − ψ (8.13) which implies W + = P + exp ǫ X l > Λ l +1 ( l + 1)! q l + X l> Λ l l ! ( ∂ x − h l ǫ ) q l ! , (8.14) W ∗ + = exp − ǫ X l > Λ l +1 ( l + 1)! q l − X l> Λ l l ! ( ∂ x − h l ǫ ) q l ! P ∗ + , (8.15) W − = P − exp − ǫ X l > Λ − l − e − ( l +1) ψ ( l + 1)! q l + X l> Λ − l e − lψ l ! ( ∂ x + h l ǫ ) q l ! , (8.16) W ∗− = exp + 12 ǫ X l > Λ − l − e − ( l +1) ψ ( l + 1)! q l − X l> Λ − l e − lψ l ! ( ∂ x + h l ǫ ) q l ! P ∗− . (8.17)Here the operators P ± and P ∗± have been defined by σ l ( P + ) = P + , σ r ( P ∗ + ) = P ∗ + (8.18) σ l ( P − ) = P − | λ → λ − e − ψ , σ r ( P ∗− ) = P ∗− | λ → λ − e − ψ . (8.19)Note that P + and P ∗ + are power series in negative powers of Λ with leading termequal to 1, while P − and P ∗− are power series in positive powers of Λ. We havethat Res Λ [ W + ( q )Λ n W ∗ + (¯ q )Λ − k ] = Res Λ [ W − ( q )Λ − n e − nψ W ∗− (¯ q )Λ − k ] . (8.20)Notice that here q = ¯ q . Since this holds for k ∈ Z and there is no k dependencein the square bracket we finally find W + ( q )Λ n W ∗ + (¯ q ) = W − ( q )Λ − n e − nψ W ∗− (¯ q ) . (8.21)For q = ¯ q we get P + Λ n P ∗ + = P − Λ − n e − nψ P ∗− , (8.22)which implies for n = 0 that P ∗± = ( P ± ) − , consequently for n = 1 we obtain theconstraint P + Λ( P + ) − = P − Λ − e − ψ ( P − ) − =: L (8.23)where L is a differential operator of the form L = Λ + v + e u Λ − .We can easily express the coefficients in the Lax operator in terms of the totaldescendent potential as v = (Λ / − Λ − / ) ǫ ∂ log D ′ ∂q , (8.24) u = Λ − / (Λ + Λ − −
2) log D ′ − ψ. (8.25)Let us now obtain the Sato equations by differentiating with respect to q il thebilinear equation (8.21) and setting ¯ q = q . We obtain ǫ ∂P ± ∂q il = ∓ ( A il ) ∓ P ± (8.26)where A l = L l +1 ( l + 1)! , A l = 2 L l l ! (log L − h ( l )) . (8.27) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 37
The logarithm of L is defined, following [10], aslog L = X k ∈ Z w k Λ k = ǫ (cid:0) P − x ( P − ) − − P + x ( P + ) − (cid:1) . (8.28) Remark . The dressing (8.23) of L by P + defines an injective map C [ v, e u ][ v k , u k ; k > ǫ ]] → C [ p i ; i > p i,k ; i, k > ǫ ]] (8.29)by the substitutions v p − p ( x + ǫ ) and e u p − p ( x + ǫ ) − p ( p − p ( x + ǫ )), where p i are the coefficients in P + . The subscript denotes the degree zerohomogeneous part of the formal power series ring, where the degree is k for u k , v k and p i,k , is − ǫ and zero for the remaining generators. It is clear thatthe coefficients in L p = P + Λ p ( P + ) − are elements of C [ v, e u ][ v k , u k ; k > ǫ ]] , orequivalently in its image via the above injection. It was proved in [10] that thecoefficients w k defined by P + ǫ∂ x ( P + ) − = ǫ∂ x − ǫP + x ( P + ) − = ǫ∂ x + 2 X k − w k Λ k (8.30)are also in the image of such injection, so they define unique elements w k in C [ v, e u ][ v k , u k ; k > ǫ ]] . One can prove that the coefficients w k for k > − P − ǫ∂ x ( P − ) − = − ǫ∂ x + ǫP − x ( P − ) − = − ǫ∂ x + 2 X k > w k Λ k (8.31)as elements of C [ q , q − , q i ; i > q i,k ; i > , k > ǫ ]] are given by w = ψ ǫ − − ∂u∂x = ψ u . . . , w k = e − u ( x + ǫ ) · · · e − u ( x + ǫk ) w − k ( x + ǫk )(8.32)for k >
1. We have therefore that the coefficients w k of log L are uniquely differentialpolynomials in A := C [ u, v, e ± u ][ v k , u k ; k > ǫ ]] . (8.33) Remark . To prove (8.32) we define, as in [42], a linear anti-involution σ on thespace of difference operators by σ ( a ( x )Λ k ) = Q ( e ψ Λ) − k a ( x ) Q − , where Q = D ′ ( x + ǫ ) D ′ ( x − ǫ ) , (8.34)is the coefficient of Λ in P − . We extend this anti-involution on the space of formaloperators in both Λ and ǫ∂ x , by σ ( ǫ∂ x ) = Q ( − ǫ∂ x − ψ ) Q − = − ǫ∂ x − ψ + ǫQ − ∂Q∂x . (8.35)Note that from (8.9) it follows that P − = Q P ∗ + , P + = Q P ∗− , (8.36)from which it is easy to derive that σ ( P ± ) = Q ( P ∓ ) − , σ ( L ) = L (8.37)and σ ( P ± ǫ∂ x ( P ± ) − ) = − P ∓ ǫ∂ x ( P ∓ ) − − ψ. (8.38) Thus σ (log L ) = log L and therefore w − k = e − kψ Q ( x ) w k ( x − ǫk ) Q ( x − ǫk ) . (8.39)From the relations w = ǫ Q − ∂Q∂x , e u + ψ = QQ ( x − ǫ ) , (8.40)the equations (8.32) easily follow. For example we have w − = ǫ (cid:18) (Λ − − ∂v∂x (cid:19) , w = ǫ (cid:18) Λ e − u (Λ − − ∂v∂x (cid:19) . (8.41)Finally, it follows from Sato equations and from the commutativity of L and A il that L satisfies the Lax equations: ǫ ∂L∂q il = [( A il ) + , L ] . (8.42)They define commuting derivations of A .8.2. Lax representation via change of variable.
Let us introduce a variable X by the shift q q + X . The Lax equations for the time q read: ǫv X = (Λ − e u , ǫu X = (1 − Λ − ) v. (8.43)Denoting φ = v ( x − ǫ ) and ρ = e u , notice that we can express the x derivatives of u and v as X differential polynomials in u , v or equivalently in ρ , φ . By substitutionin the evolutionary equations implied by (8.42) we obtain the desired hierarchy ofequations having X as space variable. A Lax representation for this hierarchy, calledextended NLS, was outlined in [10]. Here we give an equivalent presentation usingdressing operators.Denote by ˜ P the pseudo-differential operator in the variable X obtained by sub-stituting Λ by ǫ∂ X in the dressing operator P + in the previous section, namely P + = X k = −∞ p k Λ k , ˜ P = X k = −∞ p k ( ǫ∂ X ) k . (8.44)Let L = ˜ P ǫ∂ X ˜ P − , S = ˜ P ǫ∂ X Λ − ˜ P − , T = ˜ P ( ǫ∂ X ) Λ − ˜ P − (8.45)where it should be noted that the last two operators above are both pseudo-differentialin the variable X and difference in the variable x .We define log L as in [10] bylog L = X k ∈ Z ˜ w k Λ k (( ǫ∂ X − φ )Λ − ) k (8.46)where ˜ w k are differential polynomials in the variables ρ , φ and their X derivativesobtained from w k via the substitutions mentioned above. Of course also the x derivatives of φ appearing as a result of the shifts have to be expressed in terms of X derivatives of the variables ρ and φ . IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 39
Remark . In the formula for log L notice we haveΛ k (( ǫ∂ X − φ )Λ − ) k = ( ǫ∂ X − φ ( x + ǫk )) · · · ( ǫ∂ X − φ ( x + ǫ )) , k > , (8.47)Λ − k (( ǫ∂ X − φ )Λ − ) − k = ( ǫ∂ X − φ ( x + ǫ ( − k + 1))) − · · · ( ǫ∂ X − φ ( x )) − , k > , (8.48)where all x derivatives have to be expressed as differential polynomials in the X -derivatives by the substitutions above. The operator ( ǫ∂ X − φ ) − is the inverse of ǫ∂ X − φ in the algebra of usual pseudo-differential operators, explicitly( ǫ∂ X − φ ) − = X l > ( ǫ∂ X ) − ( φ ( ǫ∂ X ) − ) l . (8.49)Notice however that the infinite sum of positive powers of k in (8.46) leads to non-convergent infinite sums in front of every positive power of ǫ∂ X .To give a well-defined meaning to the formal operator (8.46) we consider it as apseudo-differential operator in ǫ∂ X − φ . Notice that the product rule for pseudo-differential operators gives the formally equivalent rule( ǫ∂ X − φ ) k f = X l > (cid:18) kl (cid:19) f ( l ) ( ǫ∂ X − φ ) k − l . (8.50)Because when written in terms of ǫ∂ X − φ the substitutions (8.47) now involvein each factor corrections of strictly positive degree in ǫ , it follows that log L is awell-defined operator of the formlog L = X k ∈ Z a k ( ǫ∂ X − φ ) k (8.51)where a k are differential polynomials in ρ , φ and their X -derivatives. Moreover themultiplication of such operator, which involves both infinite positive and negativepowers of ǫ∂ X − φ , by an operator with only finite number of arbitrary powers of ǫ∂ X − φ (see the form of L below) is also well-defined. We conclude that the Laxequations below provide derivations in the ring of differential polynomials in ρ , φ and their X -derivatives. Proposition 42.
The operator ˜ P satisfies the following Sato equations ǫ ∂ ˜ P∂q il = − ( ˜ A il ) − ˜ P (8.52) where ˜ A ℓ = L ℓ +1 ( ℓ + 1)! , ˜ A ℓ = 2 ℓ ! (log L − h ( ℓ )) L ℓ . (8.53) Moreover the operators S and T are given by S = ( ǫ∂ X − φ )Λ − , T = (( ǫ∂ X ) − ǫ∂ X φ + ρ )Λ − (8.54) so, in particular they do not contain negative powers of ǫ∂ X , and the Lax operatoris equal to their ratio L = T S − = ǫ∂ X + ρ ( ǫ∂ X − φ ) − . (8.55) Proof.
The Sato equation (8.26) for q and the dressing (8.23) of Λ can be writtenrespectively as ǫ ∂P + ∂X = − L − P + = ( − e u Λ − ) P + , (Λ + v + e u Λ − ) P + = P + Λ . (8.56) Moving all shifts Λ on the right the first equation becomes ǫ ∂P + ∂X = − e u P + ( x − ǫ )Λ − . (8.57)Notice that in this expression all power of Λ appear on the right of the remainingcoefficients, therefore we can replace them with powers of ǫ∂ X , obtaining ǫ ∂ ˜ P∂X = − e u ˜ P ( x − ǫ )( ǫ∂ X ) − . (8.58)Similarly we can rewrite the second equation above in terms of pseudo-differentialoperators as ˜ P ( x + ǫ ) ǫ∂ X + v ˜ P + e u ˜ P ( x − ǫ )( ǫ∂ ) − = ˜ P ǫ∂ X . (8.59)Substituting (8.58) in (8.59) we get˜ P ǫ∂ X Λ − ˜ P − = Λ − ( ǫ∂ X − v ) = ( ǫ∂ X − φ )Λ − (8.60)and similarly from (8.59) we find˜ P ( ǫ∂ X ) Λ − ˜ P − = ( e u + ǫ∂ X ( ǫ∂ X − φ ))Λ − , (8.61)thus proving (8.54). A similar computation shows that Sato equations (8.26) canbe rewritten in terms of pseudo-differential operators as (8.52) where the operators˜ A iℓ are obtained by substituting Λ k with Λ k S k as follows A iℓ = X k a iℓ,k Λ k ˜ A iℓ = X k a iℓ,k Λ k S k . (8.62)This follows from the observation that Sato equations are written as ǫ ∂P + ∂q iℓ = − X k< a iℓ,k Λ k P + = − X k< a iℓ,k P + ( x + ǫk )Λ k (8.63)so imply ǫ ∂ ˜ P∂q iℓ = − X k< a iℓ,k Λ k ˜ P ( x − ǫk )( ǫ∂ X ) k = − X k< a iℓ,k Λ k ˜ P Λ − k ( ǫ∂ X ) k . (8.64)Multiplication on the right by ˜ P − gives ǫ ∂ ˜ P∂q iℓ ˜ P − = − X k< a iℓ,k Λ k ˜ P Λ − k ( ǫ∂ X ) k ˜ P − = − X k< a iℓ,k Λ k S k , (8.65)which proves our assertion. Notice moreover that the projection ( · ) − commutes withthe substitution Λ k Λ k S k .To complete the proof we need to show that the operators ˜ A iℓ defined by thesubstitution (8.62) coincide with those give by formulas (8.53). Let us first considerthe case i = 1. From the dressing we know L ℓ P + = P + Λ ℓ . (8.66)Denoting L ℓ = X k b ℓk Λ k (8.67)we have X k b ℓk Λ k P + = P + Λ ℓ , (8.68) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 41 which, by the same argument above, gives X k b ℓk Λ k ˜ P Λ − k ( ǫ∂ X ) k = ˜ P ( ǫ∂ X ) ℓ (8.69)and finally X k b ℓk Λ k S k = L ℓ . (8.70)For the case i = 2 we have to use a slightly different argument because we don’thave a definition of log L in terms of dressing operators in the pseudo-differentialoperator case. The definition (8.46) of log L amounts to substituting Λ k with Λ k S k in log L for ETH as given in (8.28). We have that the product log L · L ℓ after suchsubstitution is equal to X j w j Λ j X k b ℓk Λ k S k + j = X j w j Λ j X k b ℓk Λ k S k ! S j = X j w j Λ j L ℓ S j . (8.71)Because S and L commute this expression equals log L · L ℓ which completes theproof. (cid:3) Corollary 43.
The Lax operator L satisfies the following equations ǫ ∂ L ∂q iℓ = [ − ( ˜ A iℓ ) − , L ] (8.72) for i = 1 , , ℓ > .Remark . Clearly the operators S and T satisfy the same Lax equations as L .8.3. Lax representation from the Hirota equations.
In this section we want toapproach the problem of deriving the Lax equations from the HQE directly, namelyusing a fundamental lemma to convert it into equations for pseudo-differential oper-ators, like in the usual derivation of rational reductions of KP. Obtaining the Satoequations for the new times q ℓ is not straightforward.8.3.1. Preliminaries.
Let us define P ( X, x, q, λ ) := P + (cid:12)(cid:12)(cid:12) x → x + ǫ q → q + X , ˜ P ( X, x, q, λ ) := P ∗ + (cid:12)(cid:12)(cid:12) x → x − ǫ q → q + X , (8.73)or, more explicitly P ( X, x, q, λ ) = e − ǫ P ℓ > ℓ ! λℓ +1 ∂∂q ℓ D ′′ D ′′ , ˜ P ( X, x, q, λ ) = e ǫ P ℓ > ℓ ! λℓ +1 ∂∂q ℓ D ′′ D ′′ (8.74)where D ′′ equals the total descendent potential D with dependences on X and x introduced via the shifts q q + X , q q + x . Notice that for consistency withthe usual KP reductions here the symbols P ( λ ) resp. ˜ P ( λ ) are shifted by ± ǫ/ P + resp. P ∗ + . Let us also define Q ( X, x, q ) = D ′′| x → x + ǫ D ′′ , R ( X, x, q ) = Q ( X, x − ǫ, q ) − , (8.75) φ ( X, x, q ) = − ǫR ( X, x, q ) − ∂R ( X, x, q ) ∂X , (8.76) ρ ( X, x, q ) = e − ψ Q ( X, x, q ) R ( X, x, q ) . (8.77) We now rewrite the HQE (7.28) in an equivalent form which is more convenientfor the use of the fundamental lemma for pseudo-differential operators.
Lemma 45.
The HQE (7.28) is equivalent to h exp (cid:16) ǫ X ℓ > λ ℓ +1 ( ℓ + 1)! ( q ℓ − ¯ q ℓ ) − ǫ X ℓ > λ ℓ ℓ ! h ( ℓ )( q ℓ − ¯ q ℓ ) (cid:17) ×× P ( X, q, λ ) exp (cid:16) X ℓ > λ ℓ ℓ ! ( q ℓ − ¯ q ℓ ) ∂ x (cid:17) e − ( k − ǫ∂ x λ k ˜ P ( ¯ X, ¯ q, λ ) e − ǫ∂ x e λǫ ( X − ¯ X ) i == e − kψ h Q ( X, q ) e ǫ∂ x ˜ P ( X, q, λ ) exp (cid:16) X ℓ > λ ℓ ℓ ! ( q ℓ − ¯ q ℓ ) ∂ x (cid:17) ×× e − ( k +1) ǫ∂ x λ − k ˜ P ( ¯ X, ¯ q, λ ) Q − ( ¯ X, ¯ q ) i . (8.78) Remark . Note that in this formula: (a) we have q = ¯ q and P , ˜ P and Q dependon x via the shift x → x + ǫ in (8.74) and (8.75); (b) the projections refer to thepowers of λ , so we have the equality of the coefficients of λ n in the right-hand sideand left-hand side for n
0; (c) the expression has values in power series in ∂ x ,however the dependence on ∂ x can be removed by right multiplication byexp (cid:16) − X ℓ > λ ℓ ℓ ! ( q ℓ − ¯ q ℓ ) ∂ x (cid:17) e kǫ∂ x (8.79)which depends only on nonnegative powers of λ so preserves the equality. Proof.
Starting from (7.28), notice that we can multiply it byexp( − kψ (cid:0) ǫ X ℓ > λ ℓ +1 ( ℓ + 1)! ( q ℓ − ¯ q ℓ ) − ǫ X ℓ > λ ℓ ℓ ! h ( ℓ )( q ℓ − ¯ q ℓ ) (cid:1) (8.80)since it contains only nonnegative powers of λ . We can then introduce the depen-dence on X , x by the shifts q → q + X , ¯ q → ¯ q + ¯ X , q → q + x . Rewriting theshifts of the variables appearing in D as shift operators gives the desired formulaabove. (cid:3) The Fundamental Lemma.
In the following let P ( X, ǫ∂ X ), Q ( X, ǫ∂ X ) be pseudo-differential operators and P ( X, λ ), Q ( X, λ ) the corresponding symbols, e.g. P ( X, ǫ∂ X ) = X k p k ( X )( ǫ∂ X ) k , P ( X, λ ) = X k p k ( X ) λ k . (8.81)Recall that the residue Res ∂ X of a pseudo-differential operator coincides with theresidue Res λ = ∞ dλ of its symbol and the adjoint is defined by P ( X, ǫ∂ X ) ∗ = X k ( − ǫ∂ X ) k p k ( X ) . (8.82) Lemma 47.
The equality holds:
Res λ = ∞ P ( X, λ ) Q ( ¯ X, − λ ) e ( X − ¯ X ) λǫ dλ = ǫ Res ∂ X P ( X, ǫ∂ X ) Q ∗ ( X, ǫ∂ X ) e u∂ X (cid:12)(cid:12) u = X − ¯ X . (8.83) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 43
Proof.
It suffices to show that (8.83) holds for P ( X, ǫ∂ X ) = ( ǫ∂ X ) k and Q ( X, ǫ∂ X ) = B ( X )( − ǫ∂ X ) ℓ for k + ℓ <
0. In that case we use Taylor’s formula for the left-handside, which is equal toRes λ = ∞ dλ & λ k + ℓ ∞ X m =0 ( ¯ X − X ) m m ! ∂ m B ( X ) ∂X m e ( X − ¯ X ) λǫ = ǫ k + ℓ +1 ∞ X m =0 ( − m m !( − k − ℓ − ∂ m B ( X ) ∂X m ( X − ¯ X ) − k − ℓ + m − and the right-hand side is equal to ǫ k + ℓ +1 Res ∂ X ∂ k + ℓX B ( X ) ∞ X n =0 ( u∂ X ) n n ! (cid:12)(cid:12) u = X − ¯ X == ǫ k + ℓ +1 ∞ X m (cid:18) k + ℓm (cid:19) − k − ℓ + m − ∂ mX B ( X ) ∂ X X m ( X − ¯ X ) − k − ℓ + m − . Since ( − m m !( − k − ℓ − (cid:18) k + ℓm (cid:19) − k − ℓ + m − , we obtain the desired result. (cid:3) Lemma 48.
If the following equality holds:
Res λ = ∞ dλ P ( X, λ ) Q ( ¯ X, − λ ) e ( X − ¯ X ) λǫ = X j R j ( X ) S j ( ¯ X ) , (8.84) then ( P ( X, ǫ∂ X ) Q ∗ ( X, ǫ∂ X )) − = X j R j ( X )( ǫ∂ X ) − S j ( X ) . (8.85) Proof.
We use Taylor’s formula again and the above lemma: X j R j ( X ) S j ( ¯ X ) = X j R j ( X ) ∞ X k =0 ( − k ∂ k S j ( X ) ∂X k ( X − ¯ X ) k k ! == ǫ Res ∂ X j R j ( X )( ǫ∂ ) − S j ( X ) e u∂ X (cid:12)(cid:12) u = X − ¯ X . (8.86)Thus formula (8.85) holds. (cid:3) Corollary 49.
The equality h P ( X, λ ) Q ( ¯ X, − λ ) e ( X − ¯ X ) λǫ i = h ˜ P ( X, λ ) ˜ Q ( ¯ X, λ ) i (8.87) implies the following identity of pseudo-differential operators (cid:2) P ( X, ǫ∂ X )( ǫ∂ X ) n − Q ∗ ( X, ǫ∂ X ) (cid:3) − = Res λ ˜ P ( X, λ ) λ n − ( ǫ∂ X ) − ˜ Q ( X, λ ) dλ, n > . (8.88) The Hirota equation in pseudo-differential operator form.
Using the previousCorollary we find here a version of the HQE in terms of ∂ x -valued pseudo-differentialoperators in the variable X . Proposition 50.
The HQE (7.28) is equivalent to h P ( q, ǫ∂ X ) exp (cid:16) ǫ X ℓ > ( ǫ∂ X ) ℓ +1 ( ℓ + 1)! ( q ℓ − ¯ q ℓ ) − ǫ X ℓ > ( ǫ∂ X ) ℓ ℓ ! h ( ℓ )( q ℓ − ¯ q ℓ ) (cid:17) ×× exp (cid:16) X ℓ > ( ǫ∂ X ) ℓ ℓ ! ( q ℓ − ¯ q ℓ ) ∂ x (cid:17) e − ( k − ǫ∂ x ( ǫ∂ X ) n + k − ˜ P (¯ q, − ǫ∂ X ) ∗ i − == e − kψ Res λ h Q ( q ) e ǫ∂ x ˜ P ( q, λ ) exp (cid:16) X ℓ > λ ℓ ℓ ! ( q ℓ − ¯ q ℓ ) ∂ x (cid:17) e − ( k +1) ǫ∂ x λ n − k − ×× ( ǫ∂ X ) − P (¯ q, λ ) Q − (¯ q ) e ǫ∂ x i dλ (8.89) for n > and k ∈ Z .Remark . Note that here P , ˜ P , Q depend on x and X = ¯ X , which we havesuppressed for simplicity. Moreover recall that the variables q and ¯ q are identified.8.3.4. First consequences.
For q = ¯ q the HQE becomes h P ( ǫ∂ X ) e − ( k − ǫ∂ x ( ǫ∂ X ) n + k − ˜ P ( − ǫ∂ X ) ∗ i − == e − kψ Res λ h λ n − k − Q ˜ P ( x + ǫ, λ ) e − ( k − ǫ∂ x ( ǫ∂ X ) − Q − ( x − ǫ ) P ( x − ǫ, λ ) i dλ. (8.90)For notational simplicity in this equation we have suppressed the dependence on q variables and we explicitly indicated the dependence on x only when this variableis shifted.Let us consider some consequences of this equation for small values of k and n .Notice that the right-hand side vanishes for k > n . n = 0 , k = 1 : we have h P ( ǫ∂ X ) ˜ P ( − ǫ∂ X ) ∗ i − = 0 , (8.91)hence ˜ P ( − ǫ∂ X ) ∗ = P ( ǫ∂ X ) − . n = k > h P ( ǫ∂ X ) e − ( k − ǫ∂ x ( ǫ∂ X ) k − P ( ǫ∂ X ) − i − = e − kψ Q e − ( k − ǫ∂ x ( ǫ∂ X ) − Q − ( x − ǫ ) . (8.92) n = 0 , k = 0 : let us define S ( ǫ∂ X , ǫ∂ x ) := P ( ǫ∂ X ) e ǫ∂ x ( ǫ∂ X ) − P ( ǫ∂ X ) − . The previ-ous equation for k = 0 gives S ( ǫ∂ X , ǫ∂ x ) − = Q e ǫ∂ x ( ǫ∂ X ) − Q − ( x − ǫ ) , (8.93)hence we have that S ( ǫ∂ X , ǫ∂ x ) = R − ǫ∂ X Re − ǫ∂ x . (8.94)Notice that S ( ǫ∂ X , ǫ∂ x ) = ˜ S ( ǫ∂ X ) e − ǫ∂ x , where˜ S ( ǫ∂ X ) = R − ǫ∂ X R = ǫ∂ X − φ. (8.95) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 45 n = 1 , k = 1 : defining L ( ǫ∂ X ) = P ( ǫ∂ X ) ǫ∂ X P ( ǫ∂ X ) − , we get from Equation (8.92)that h L ( ǫ∂ X ) i − = e − ψ Q ( ǫ∂ X ) − Q − ( x − ǫ ) , (8.96)therefore L ( ǫ∂ X ) = ǫ∂ X + Q ( e ψ ǫ∂ X ) − R (8.97)= ǫ∂ X + ρ ( ǫ∂ X − φ ) − (8.98)= R − (cid:0) ǫ∂ X + φ + ρ ( ǫ∂ X ) − (cid:1) R. (8.99) n = 1 , k = 2 : Let us define T ( ǫ∂ X , ǫ∂ x ) := P ( ǫ∂ X ) e − ǫ∂ x ( ǫ∂ X ) P ( ǫ∂ X ) − . Equa-tion (8.90) for n = 1, k = 2 gives T ( ǫ∂ X , ǫ∂ x ) − = 0. We have that T ( ǫ∂ X , ǫ∂ x ) =˜ T ( ǫ∂ X ) e − ǫ∂ x , where ˜ T ( ǫ∂ X ) is a differential operator in the variable X . By definitionwe have L ( ǫ∂ X ) = T ( ǫ∂ X , ǫ∂ x ) S ( ǫ∂ X , ǫ∂ x ) − = S ( ǫ∂ X , ǫ∂ x ) − T ( ǫ∂ X , ǫ∂ x ) (8.100)= ˜ T ( ǫ∂ X ) ˜ S ( ǫ∂ X ) − = ˜ S ( x + ǫ, ǫ∂ X ) − ˜ T ( x + ǫ, ǫ∂ X ) . (8.101)Notice that S ( ǫ∂ X , ǫ∂ x ) and T ( ǫ∂ X , ǫ∂ x ) commute, therefore commute with L , while˜ S ( ǫ∂ X ) and ˜ T ( ǫ∂ X ) don’t. In particular we have˜ T ( ǫ∂ X ) = R − (cid:0) ( ǫ∂ X ) + φǫ∂ X + ρ (cid:1) R (8.102)= ( ǫ∂ X − φ ) + φ ( ǫ∂ X − φ ) + ρ. (8.103) n ≥ , k = 1 : from (8.90) we deduce that h L ( ǫ∂ X ) n i − = e − ψ Res λ h λ n Q ˜ P ( x + ǫ, λ )( ǫ∂ X ) − Q − ( x − ǫ ) P ( x − ǫ, λ ) i dλ. (8.104) Remark . Notice that all remaining constraints from (8.90) for k > n are auto-matically satisfied since the left-hand side of that equations is equal to [ T n S k − n − ] − which vanishes since T and S are differential in X .8.3.5. Sato equations.
Differentiating (8.89) with respect to q ℓ for ℓ > q = ¯ q and k = 1, n = 0 we get the Sato equations for the q ℓ flows ǫ ∂P ( ǫ∂ X ) ∂q ℓ = − (cid:18) L ( ǫ∂ X ) ℓ +1 ( ℓ + 1)! (cid:19) − P ( ǫ∂ X ) . (8.105)Differentiating (8.89) with respect to q ℓ for ℓ > q = ¯ q and k = 1, n = 0 we get ǫ ∂P ( ǫ∂ X ) ∂q ℓ P ( ǫ∂ X ) − = (cid:18) L ( ǫ∂ X ) ℓ ℓ ! (cid:18) ǫ ∂P ( ǫ∂ ) ∂x P ( ǫ∂ ) − + 2 h ( ℓ ) (cid:19)(cid:19) − ++ e − ψ Res λ h λ ℓ − ℓ ! Q ˜ P ( x + ǫ, λ )( ǫ∂ X ) − ǫ ∂Q ( x − ǫ ) − P ( x − ǫ, λ ) ∂x i dλ . (8.106)Notice that two terms proportional to ∂ x cancelled in this expression thanks to (8.90).Let us define an operator log + L by dressing ǫ∂ x log + L ( ǫ∂ X ) := P ( ǫ∂ X ) ǫ∂ x P ( ǫ∂ X ) − = ǫ∂ x − ǫ ∂P ( ǫ∂ X ) ∂x P ( ǫ∂ X ) − . (8.107) Notice that log + L is given by the sum of ǫ∂ x and a pseudo-differential operator − ǫ ∂P ( ǫ∂ X ) ∂x P ( ǫ∂ X ) − = X k − w k ( ǫ∂ X ) k = X k − w k e kǫ∂ x S ( ǫ∂ X ) k (8.108)Notice that in this case we don’t have a second dressing operator so we cannotdirectly define a second logarithm of L ( ǫ∂ X ). To avoid this problem we proceedto define log L ( ǫ∂ X ) directly from the coefficients of log + L ( ǫ∂ X ). We define thecoefficients w k for k ≥ w = ǫ Q − ∂Q∂x , w − k = e − kψ Qe − kǫ∂ x w k Q − e kǫ∂ x , (8.109)and definelog L ( ǫ∂ X ) := X k − w k e kǫ∂ x S ( ǫ∂ X ) k + X k > S ( ǫ∂ X ) k w k ( x − ǫ ) e kǫ∂ x . (8.110) Remark . Note that the first part of log L ( ǫ∂ X ) coincides log + L − ǫ∂ x and in par-ticular with the first part of the operator log L defined in the previous section. Thesecond part is reminiscent of the second part of (8.46) but the explicit equivalenceof the two expressions could not be proved. Proposition 54.
The operator P satisfies the Sato equations ǫ ∂P ( ǫ∂ X ) ∂q iℓ = − ( A iℓ ) − P ( ǫ∂ X ) (8.111) A ℓ = L ( ǫ∂ X ) ℓ +1 ( ℓ + 1)! , A ℓ = 2 ℓ ! L ( ǫ∂ X ) ℓ (log L ( ǫ∂ X ) − h ( ℓ )) . (8.112) Proof.
We just need to consider the case i = 2. For ℓ = 0 it simply followsfrom (8.108). For ℓ > ǫ ∂Q − P ( ǫ∂ X ) ∂x = ǫ ∂Q − ∂x P ( ǫ∂ X ) − X k − Q − w k e kǫ∂ x S ( ǫ∂ X ) k P ( ǫ∂ X ) , (8.113)therefore ǫ ∂Q − P ( ǫ∂ X ) ∂x = − X k > e − kψ e − kǫ∂ x Q − w k S ( ǫ∂ X ) − k P ( ǫ∂ X ) (8.114)= − X k > e − kψ e − kǫ∂ x Q − w k P ( ǫ∂ X ) e kǫ∂ x ( ǫ∂ X ) − k , (8.115)which implies ǫ ∂Q − P ( λ ) ∂x = − X k > e − kψ e − kǫ∂ x Q − w k P ( λ ) e kǫ∂ x λ − k . (8.116) IROTA EQUATIONS FOR EXTENDED NONLINEAR SCHR ¨ODINGER HIERARCHY 47
Substituting this in equality (8.106) we get that the second term on the right hand-side is equal to − ℓ ! X k > e − ( k +1) ψ Res λ (cid:2) λ ℓ − k − Q ˜ P ( x + ǫ, λ )( ǫ∂ X ) − e − kǫ∂ x ·· Q ( x − ǫ ) − P ( x − ǫ, λ ) i dλ w k ( x − ǫ ) e kǫ∂ x , (8.117)which can be written using pseudo-differential operators, using the Hirota equationin the form (8.90), obtaining − (cid:2) L ℓ ℓ ! X k > S ( ǫ∂ X ) k w k ( x − ǫ ) e kǫ∂ x (cid:3) − . (8.118)The result is proved. (cid:3) The Lax equations follow as usual.
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