One Step Degeneration of Trigonal Curves and Mixing of Solitons and Quasi-Periodic Solutions of the KP Equation
aa r X i v : . [ n li n . S I] N ov One Step Degeneration of Trigonal Curves and Mixing ofSolitons and Quasi-Periodic Solutions of the KP Equation
Atsushi Nakayashiki ∗ To the memory of Victor Enolski
Abstract
We consider certain degenerations of trigonal curves and hyperelliptic curves, whichwe call one step degeneration. We compute the limits of corresponding quasi-periodicsolutions using the Sato Grassmannian. The mixing of solitons and quasi-periodicsolutions is clearly visible in the obtained solutions.
The aim of this paper is to compute explicitly the limits of quasi-periodic solutions of theKP equation according to certain degenerations of trigonal and hyperelliptic curves, whichwe call oen step degeneration.The KP equation is the 2 + 1 dimensional equation given by3 u t t + ( − u t + 6 uu t + u t t t ) t = 0 , (1.1)where ( t , t ) and t are space and time variables respectively. It can be rewritten in theHirota bilinear form: ( D t − D t D t + 3 D t ) τ · τ = 0 , (1.2)where D t i ’s are Hirota derivatives defined by f ( t + s ) g ( t − s ) = ∞ X n =0 D nt f · g s n n ! . For a solution τ of (1.2) u = 2(log τ ) t t gives a solution of (1.1). The KP hierarchy is theinfinite system of differential equations which contains the KP equation (1.2) as its firstmember [7]. It is given by Z τ ( t − s − [ z − ]) τ ( t + s + [ z − ])e − P ∞ j =1 s j z j dz πi = 0 , (1.3) ∗ Department of Mathematics, Tsuda University, Kodaira, Tokyo 187-8577, Japan, [email protected] t = ( t , t , ... ), s = ( s , s , ... ), [ z − ] = [ z − , z − / , z − / , ... ] and the integral sig-nifies taking the coefficient of z − in the series expansion of the integrand. Expanding(1.3) by s we get differential equations for τ ( t ) in the Hirota bilinear form. A solution τ ( t ) is sometimes called a tau function. The introduction of the infinitely many variablesis indispensable to the Sato theory which we use in this paper.The KP hierarchy has a variety of solutions. Among them soliton solutions and algebro-geometric solutions are relevant to us. Soliton solutions are the solutions expressed byexponential functions given as follows (see [12] for example). Take positive integers N 3. Fix m and denote by C n the non-singular curve before taking the limit.We define some canonical tau function τ n, ( t ) (see (3.3) ) corresponding to the curve C n . Then we express the limit of τ n, ( t ) in terms of τ n − , ( t ) with the variable t being2ppropriately shifted. Then a solitonic structure can be seen clearly in the degenerationof the algebro-geometric solution τ n, ( t ). This is another crucial idea in this paper.The results are as follows. For m = 2, that is, the case of a hyperelliptic curve, wehave (Theorem 4.5),lim τ n, ( t ) = C e − P ∞ l =1 α l t l × (cid:16) e η ( α / ) τ n − , ( t − [ α − / ]) + ( − n e η ( − α / ) τ n − , ( t − [ − α − / ]) (cid:17) , (1.6)for some constant C . It is observed that the soliton factors e η ( ± α / ) pop out from τ n, ( t ).Then the solution (1.6) looks the mixture of solitons and quasi-periodic solutions. Usingthe formula repeatedly and noting that τ , ( t ) = 1 if α = 0 we get well known solitonsolutions of the KdV equation.For m = 3 we have (Theorem 3.8)lim τ n, ( t )= e − P ∞ l =1 α l t l X ≤ i Let V = C (( z )) be the vector space of Laurent series in the variable z and V φ = C [ z − ], V = z C [[ z ]] two subspaces of V . Then V is isomorphic to V φ ⊕ V . Let π : V −→ V φ bethe projection map. Then the Sato Grassmannian UGM is defined as the set of subspaces U of V such that the restriction π | U has the finite dimensional kernel and cokernel whosedimensions coincide.To an element P a n z n ∈ V we associate the infinite column vector ( a n ) n ∈ Z . Then aframe of a point U of UGM is expressed by an Z × N ≤ matrix ξ = ( ξ i,j ) i ∈ Z ,j ∈ N ≤ , wherecolumns, and therefore a basis of U , are labeled by the set of non-positive integers N ≤ .A frame ξ is written in the form ξ = ... ... · · · ξ − , − ξ − , · · · ξ , − ξ , − − − − − − − − −· · · ξ , − ξ , · · · ξ , − ξ , ... ... . (2.1)It is always possible to take a frame satisfying the following condition, there exists anegative integer l such that ξ i,j = (cid:26) j < l and i = j j < l and i < j ) or ( j ≥ l and i < l ) . (2.2)In the sequel we always take a frame which satisfies this condition, although it is notunique.A Maya diagram M = ( m j ) ∞ j =0 is a sequence of decreasing intergers such that m j = − j for all sufficiently large j . For a Maya diagram M = ( m j ) ∞ j =0 the corresponding partitionis defined by λ ( M ) = ( j + m j ) ∞ j =0 . By this correspondence the set of Maya diagrams andthe set of partitions bijectively correspond to each other.For a frame ξ and a Maya diagram M = ( m j ) ∞ j =0 define the Pl¨ucker coordinate by ξ M = det( ξ m i ,j ) − i,j ≤ Due to the condition (2.2) and the condition of the Maya diagram M this infinite deter-minant can be computed as the finite determinant det( ξ m i ,j ) k ≤− i,j ≤ for sufficiently small k . Define the elementary Schur function p n ( t ) bye P ∞ n =1 t n κ n = ∞ X n =0 p n ( t ) κ n . λ = ( λ , ..., λ l ) is defined by s λ ( t ) = det( p λ i − i + j ( t )) ≤ i,j ≤ l . Assign the weight j to the variable t j . Then it is known that s λ ( t ) is homogeneous ofweight | λ | = λ + · · · + λ l . To a point U of UGM take a frame ξ and define the taufunction by τ ( t ; ξ ) = X M ξ M s λ ( M ) ( t ) . (2.3)If we change the frame ξ τ ( t ; ξ ) is multiplied by a constant. We call τ ( t ; ξ ), for any frame ξ of U , a tau function corresponding to U . So tau functions of a point of UGM differ byconstant multiples to each other.Then Theorem 2.1. [25] The tau function τ ( t ; ξ ) is a solution of the KP-hierarchy. Converselyfor a formal power series solution τ ( t ) of the KP-hierarchy there exists a point U of UGMsuch that τ ( t ) coincides with a tau function of U .The point U of UGM corresponding to a solution τ ( t ) in Theorem 2.1 is given asfollows [25, 24, 11, 17].Let Ψ ∗ ( t ; z ) be the adjoint wave function [7] corresponding to τ ( t ) which is defined byΨ ∗ ( t ; z ) = τ ( t + [ z ]) τ ( t ) e − P ∞ i =1 t i z − i . (2.4)Define Ψ ∗ i ( z ) by the following expansion( τ ( t )Ψ ∗ ( t ; z )) | t =( x, , , ,... ) = τ (( x, , , , ... ) + [ z ])e − xz − = ∞ X i =0 Ψ ∗ i ( z ) x i . (2.5)Then U = ∞ X i =0 C Ψ ∗ i ( z ) . (2.6)By this correspondence between points of UGM and tau functions the following prop-erty follows. Let U be a point of UGM, τ ( t ) be a tau function corresponding to U and f ( z ) = e P ∞ i =1 a i zii be an invertible formal power series. Then f ( z ) U belongs to UGM andthe corresponding tau function is given bye P ∞ i =1 a i t i τ ( t ) . (2.7)It is sometimes called the gauge transformation of τ ( t ).5 .2 Embedding of algebro-geometric data to UGM In this section we recall the construction of points of UGM from alebraic curves (see[15],[20] for more details).Let C be a compact Riemann surface of genus g , p ∞ a point on it, z a local coordinatearound p ∞ . For m ≥ p i , 1 ≤ i ≤ m , on C , such that p j = ∞ for any j , wedenote by H ( C, O ( m X j =1 p j + ∗ p ∞ )) (2.8)the vector space of meromorphic functions on C which have a pole at each p j of order atmost 1 and have a pole at p ∞ of any order. By expanding functions in the local coordinate z we can consider H ( C, O ( P mj =1 p j + ∗ p ∞ )) as a subspace of V = C (( z )). Then Proposition 2.2. [20][15] The subspace z g − m H ( C, O ( P mj =1 p j + ∗ p ∞ )) belongs to UGM. Remark 2.3. This Proposition was proved in [20] from the general results [15], for m ≤ g .But the case m > g can be proved in the same way. For n ≥ { α i } ni =1 consider the compact Riemannsurface C n corresponding to the algebraic curve defined by the equation y = n +1 Y j =1 ( x − α j ) . (2.9)The genus of C n is g = 3 n and there is a unique point on C n over x = ∞ which we denoteby ∞ .Consider the space H ( C n , O ( ∗∞ )) which corresponds to m = 0 in (2.8). It is thespace of meromorphic functions on C which are regular on C n \{∞} . It can be easilyproved that it coicides with the vector space C [ x, y ] of polynomials in x, y . A basis of thisvector space is given by x i , x i y, x i y i ≥ . (2.10)We take the local coordinate z around ∞ such that x = z − , y = z − (3 n +1) F n ( z ) , F n ( z ) = n +1 Y j =1 (1 − α j z ) / . (2.11)In the following we denote by z this local coordinate unless otherwise stated. The function F n ( z ) is considered as a power series in z by the Taylor expansion at z = 0.By Proposition 2.2 z g H ( C n , O ( ∗∞ )) determines a point of UGM. Writing (2.10) interms of z and multiplying them by z g we get a basis of it, z n − i , z − − i F n ( z ) , z − n − − i F n ( z ) i ≥ . (2.12)6e define the frame ˜ ξ n from this basis as follows.For an element v ( z ) = P n ≤ i a i z i , a n = 0, define the order of v ( z ) to be − n and writeord v ( z ) = − n . Definition 2.4. Label the elements of (2.12) by ˜ v i , i ≤ 0, in such a way that ord ˜ v < ord ˜ v − < ord ˜ v − < · · · and define the frame ˜ ξ n of z g H ( C n , O ( ∗∞ )) by˜ ξ n = ( . . . , ˜ v − , ˜ v − , ˜ v ) . (2.13)By the construction of ˜ ξ n the tau function τ ( t ; ˜ ξ n ) has the following expansion (see[17]) τ ( t ; ˜ ξ n ) = s λ ( n ) ( t ) + h . w . t , (2.14)where h.w.t means the higher weight terms, λ ( n ) is the partition determined from the gapsequence w < · · · < w g at ∞ of C n and is given by λ ( n ) = ( w g − ( g − , ..., w − , w ) . Example 2.5. λ (1) = (3 , , λ (2) = (6 , , , , , λ (3) = (9 , , , , , , , , Let us take a complex number α which is different from α i , 1 ≤ i ≤ n − α n +1 , α n , α n − → α, (2.15)which means that the curve C n degenerates to y = ( x − α ) n − Y j =1 ( x − α j ) . (2.16)which we call one step degeneration of C n .In the limit F n ( z ) −→ (1 − αz ) F n − ( z ) , and the basis (2.12) tends to z n − i , z − − i (1 − αz ) F n − ( z ) , z − n − − i (1 − αz ) F n − ( z ) , i ≥ . (2.17)Let W n be the point of UGM generated by this basis. Multiply (2.17) by (1 − αz ) − wehave z n − i (1 − αz ) , z − − i (1 − αz ) F n − ( z ) , z − n − − i F n − ( z ) i ≥ . (2.18)By taking linear combinations we have 7 emma 2.6. The following set of elements gives a basis of (1 − αz ) − W n . z n − − i , z − − i F n − ( z ) , z − n − − i F n − ( z ) , i ≥ ,z n (1 − αz ) , z n − − αz , z − − αz F n − ( z ) . (2.19)We arrange the basis elements of this lemma according as their orders and define theframe ξ n as follows. Definition 2.7. Define the frame ξ n of W n by ξ n = ( . . . , v − , v − , v ) , with v = z n (1 − αz ) ,v − = z n − − αz ,v − (2+ i ) = z n − − i , ≤ i ≤ n − ,v − ( n +1) = z − − αz F n − ( z ) ,v − ( n +2+2 i ) = z − − i , ≤ i ≤ n − ,v − ( n +3+2 i ) = z − − i F n − ( z ) , ≤ i ≤ n − ,v − (3 n +2+3 i ) = z − n − − i F n − ( z ) , i ≥ ,v − (3 n +3+3 i ) = z − n − − i , i ≥ ,v − (3 n +4+3 i ) = z − n − − i F n − ( z ) , i ≥ . Since we have the expansion log(1 − αz ) − = 6 ∞ X l =1 α l z l l , the following relation holds by (2.7), τ ( t ; ξ n ) = e P ∞ l =1 α l t l lim τ ( t ; ˜ ξ n ) , (2.20)where the lim signifies taking the limit (2.15). Consider the curve C n − defined by (2.9) where n is replaced by n − 1. The genus of C n − is g ′ = 3 n − g − 3. Let Q j = ( c j , Y j ) , j = 0 , , , (2.21)8e points on C n − . We assume c j = α i for any i, j . Define ϕ j by ϕ j = y + Y j y + Y j x − c j . The pole divisor of this function is Q j + (2 g ′ − ∞ . Consider the space H ( C n − , O ( Q + Q + Q + ∗∞ )). A basis of it is given by x i , x i y, x i y , ϕ j , i ≥ , j = 0 , , . Write this basis in terms of the local coordinate z and multiply it by z g ′ − we have z n − − i , z − − i F n − ( z ) , z − n − − i F n − ( z ) , z n − ϕ j , i ≥ , j = 0 , , . (2.22)By Proposition 2.2 z g ′ − H ( C n − , O ( Q + Q + Q + ∗∞ )) is a point of UGM and the set offunctions (2.22) is a basis of it. Using this basis define the frame of z g ′ − H ( C n − , O ( Q + Q + Q + ∗∞ )) by ξ n − ( Q , Q , Q ) = ( . . . , v − ( n +3) , v − ( n +2) , v − n , ..., v − , z n − ψ , z n − ψ , z n − ψ ) , where v j is the same as that in ξ n . Corresponding to the parameter α in (2.15) let P i ( α ) = ( α, ω i y ( α )), i = 0 , , C n − , where ω = e πi/ . Take Q j = P j ( α ) in (2.21) and denote the function ϕ j by ϕ j ( α ). Then ϕ j ( α ) = y + ( ω j y ( α )) y + ( ω j y ( α )) x − α . Lemma 2.8. For 0 ≤ i ≤ y i x − α = 13 y ( α ) − i X j =0 ω ( i +1) j ϕ j ( α )The lemma can be verified by direct computation. From these relations we have v − = z n − − αz = 13 y ( α ) X i =0 ω i z n − ϕ i ( α ) (2.23) v − ( n +1) = z − F n − ( z )1 − αz = 13 y ( α ) X i =0 ω i z n − ϕ i ( α ) (2.24) v = z n (1 − αz ) = ∂∂β y ( β ) X i =0 ω i z n − ϕ i ( β ) !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β = α . (2.25)The third equation is obtained by differentiating the first equation in α .Let λ be a partition and consider the Pl¨ucker coordinate of ( ξ n ) λ . Substitute the aboveexpression to the definition of ( ξ n ) λ of ξ n . Then Equations (2.23)-(2.25) mean that each9f the column vectors of ξ n corresponding to v , v − , v − ( n +1) is a sum of vectors. So wehave ( ξ n ) λ = ( − n y n − , ( α ) X ≤ i Theorem 2.9. Consider the limit (2.15). Then the limit of the tau function of the frame˜ ξ n defined by (2.13) is given by the following formula:lim τ ( t ; ˜ ξ n ) = ( − n y n − , ( α ) e − P ∞ l =1 α l t l X ≤ i The new feature of the trigonal case compared with the hyperellipticcase studied in [3] (see Theorem 4.1 ) is the existence of a derivative in the parameter β . In [19] the degeneration to genus zero curve in the trigonal case was directly studied.The obtained solutions are not solitons but generalized solitons. The appearence of thederivative corresponds to this phenomenon. In this section we derive the analytic expression of tau functions appeared in Theorem 2.9in terms of the multivariate sigma function [4] [5][6][16][17]. The fundamental idea behindconstructing the expression is due to Krichever [13]. ( N, M ) curve We consider the general ( N, M )-curve [5] defined by f ( x, y ) = 0 with f ( x, y ) = y N − x M − X Ni + Mj 0, and define f i , i ≥ 1, to bethe i -th monomial in this order. For example f = 1, f = x . Then the set of differentials du i = − f g +1 − i dxf y , ≤ i ≤ g constitutes a basis of holomorphic one forms. We choose an algebraic fundamental form b ω ( p , p ) on C × C as in [16]. It has the decomposition of the form b ω ( p , p ) = d p Ω( p , p ) + g X i =1 du i ( p ) dr i ( p ) , where Ω( p , p ) is a certain meromorphic one form on C × C and dr i ( p ) is a certaindifferential of the second kind on C with a pole only at ∞ (see [16] for more precise formof b ω , Ω, dr i ). Taking a symplectic basis { α i , β i } gi =1 of the homology group of C we definethe period matrices ω k , η k , k = 1 , 2, Π by2 ω = Z α j du i ! , ω = Z β j du i ! , − ω = Z α j dr i ! , − ω = Z β j dr i ! , and Π = ω − ω . Define Riemann’s theta function by θ [ ǫ ]( z, Π) = X m ∈ Z g e πi t ( m + ǫ ′ )Π( m + ǫ ′ )+2 πi t ( m + ǫ ′ )( z + ǫ ′′ ) , where ǫ = t ( ǫ ′ , ǫ ′′ ) ∈ R g , ǫ ′ , ǫ ′′ ∈ R g . Let Π δ ′ + δ ′′ , δ ′ , δ ′′ ∈ (1 / Z g , be a representative ofRiemann’s constant with respect to the choice of the base point ∞ and { α i , β i } gi =1 , and δ = t ( δ ′ , δ ′′ ) ∈ (1 / Z g .Let ( w , ..., w g ), w < · · · < w g , be the gap sequence of the curve C at ∞ (see [16], [8]for example). Define the partition λ ( N,M ) by λ ( N,M ) = ( w g − ( g − , ..., w − , w ) . By the definition λ ( n ) = λ (3 , n +1) . Definition 3.1. The sigma function is defined by σ ( u ) = C e t uη ω − u θ [ − δ ]((2 ω ) − u, Π) ,u = t ( u , ..., u g )for some constant C . 11ssign the weight w i to u i . Then the constant C is specified by the condition that σ ( u ) has the expansion of the form σ ( u ) = s λ ( N,M ) ( t ) | t wi = u i + h . w . t . It is known that C is explicitly expressed by some derivatives of the Riemann’s thetafunction [21][18]. The sigma function satisfies the following quasi-periodicity property: σ ( u + X i =1 ω i m i ) =( − t m m +2( t δ ′ m − t δ ′′ m ) e t ( P i =1 η i m i )( u + P i =1 ω i m i ) σ ( u ) . (3.2) Here we derive sigma function expressions for the tau functions corresponding to the spacesin Proposition 2.2 in the case of ( N, M ) curves.We take the local coordinate z around ∞ such that x = z − N , y = z − M (1 + O ( z )) . Expand du i , b ω in z as du i = ∞ X j =1 b i,j z j − , b ω ( p , p ) = z − z ) + X i,j ≥ b q i,j z i − z j − dz dz , where z i = z ( p i ). The differential du g has a zero of order 2 g − ∞ and has the expansionof the form du g = z g − (1 + ∞ X j =2 g b g,j z j − g +1 ) dz. Define c i by the expansion log r z g − du g dz ! = ∞ X i =1 c i z i i . In [17] there is a pisprint, c i z i should be c i z i /i as above. Define g × N matrix B and thequadratic form b q by B = ( b i,j ) ≤ i ≤ g,j ≥ , b q ( t ) = ∞ X i,j =1 b q i,j t i t j . The following theorem is proved in [17].12 heorem 3.2. [17] A tau function correspodning to z g H ( C, O ( ∗∞ )) is given by τ ( t ) := e − P ∞ i =1 c i t i + b q ( t ) σ ( Bt ) . (3.3)It has the expansion of the form τ ( t ) = s λ ( N,M ) ( t ) + h . w . t . (3.4) Remark 3.3. In [17] it is proved that τ ( t ) defined by (3.3) is a solution of the N -reducedKP-hierarchy [7].More generally the tau function corresponding to the m -point space with m ≥ τ ( t ). Theorem 3.4. Let p i , 1 ≤ i ≤ m , be points on C \{∞} and z i = z ( p i ). A tau functioncorresponding to z g − m H ( C, O ( P mi =1 p i + ∗∞ )) is given by τ ( t | p , ..., p m ) := e P ∞ i =1 η ( z − i ) τ ( t − m X i =1 [ z i ]) , (3.5)where η ( κ ) = P ∞ i =1 t i κ i , [ w ] = [ w, w / , w / , ... ].By (2.7) and by that the KP-hierarchy is the system of autonomous equations, if τ ( t )is a solution of the KP-hierarchy, so is e P ∞ i =1 γ i t i τ ( t + ζ ) for any set of constants { γ i } anda constant vector ζ ∈ C g . Therefore τ ( t | p , ..., p m ) is a solution of the KP-hierarchy.Then the theorem is proved by calculating the adjoint wave function using (2.6). Tothis end we need some notation.Let E ( p , p ) be the prime form [9] (see also [11]). Define E ( z , z ), E ( q, p ) with z i = z ( p i ) and q being a fixed point on C by E ( p , p ) = E ( z , z ) √ dz √ dz , E ( q, p ) = E ( z ( q ) , z ( p )) p dz ( p ) . Define ˜ E ( q, p ) for q fixed by˜ E ( q, p ) = E ( q, p ) q du g ( p )e R pq t du ( η ω − ) R pq du ,du = t ( du , . . . , du g ) . In [16] two variables ˜ E ( p , p ) and one variable ˜ E ( ∞ , p ) were introduced and studied. Itshould be noticed that ˜ E ( q, p ) is a multiplicative function of p while E ( q, p ) is a − / E ( ∞ , p ) in [16] the following lemma can be proved. Lemma 3.5. (i) The function ˜ E ( q, p ) has the expansion in z = z ( p ) near ∞ of the form˜ E ( q, p ) = ( z − z ( q )) z g − (1 + O ( z )) . (ii) Let γ be an element of π ( C, ∞ ) and its Abelian image be P gi =1 ( m ,i α i + m ,i β i ).Then ˜ E ( q, γ ( p )) / ˜ E ( q, p ) =( − t m m +2( t δ ′ m − t δ ′′ m ) e t ( P i =1 η i m i )( R pq du + P i =1 ω i m i ) , (3.6)where m i = t ( m i, , . . . , m i,g ). 13y (i) of this lemma ˜ E ( ∞ , p ) has a zero of order g at ∞ .Let d ˜ r i be the normalized differential of the second kind with a pole only at ∞ , thatis, it satisfies Z α j d ˜ r i = 0 , ≤ j ≤ g, d ˜ r i = d ( z − i + O (1)) . Define d ˆ r i = d ˜ r i + g X j,k =1 b j,i ( η ω − ) j,k du k . By the construction their periods can be computed as (Lemma 5 in [17]) Z α j d b r i = (cid:0) t (2 η ) B (cid:1) j,i , Z β j d b r i = (cid:0) t (2 η ) B (cid:1) j,i . (3.7)In Lemma 5 of [17] there is a misprint: the right hand side is not the ( i, j ) component butthe ( j, i ) component. Proof of Theorem 3.4 The adjoint wave function (2.4) corresponding to the tau function (3.5) is computedas Ψ ∗ ( t, z ) = C ( z , ..., z m ) z g − m ˜ E ( ∞ , p ) m − σ (cid:0)R p ∞ du − P mi =1 R p i ∞ du + Bt (cid:1)Q mi =1 ˜ E ( p i , p ) σ (cid:0) − P mi =1 R p i ∞ du + Bt (cid:1) × e − P ∞ i =1 t i R p d b r i ,C ( z , ..., z m ) = ( − m ( m Y i =1 z i )e P mi =1 R pi ∞ t du ( η ω − ) R pi ∞ du . By Lemma 3.5 and (3.7) we can check that z − g + m Ψ ∗ ( t, z ) is, as a function of p ∈ C , π ( C, ∞ ) invariant. Then the same is true for any expansion coefficient of Ψ ∗ ( t, z ) in t .Expansion coefficients in t are regular except p i , 1 ≤ i ≤ m , ∞ and have at most a simplepole at p i . Therefore the point U of UGM corresponding to τ ( t | p , ..., p m ) is containedin z g − m H ( C, O ( P mi =1 p i + ∗∞ )). Since a strict inclusion relation is impossible for twopoints of UGM (Lemma 4.17 of [3]), these two points of UGM coincide. (cid:3) In this section we apply the results in the previous section to the curves C n , C n − andassociated tau functions in Theorem 2.9. So, in this section τ n, ( t ) denotes the functiondefined by (3.3) for the curve C n . Lemma 3.6. We have τ ( t ; ˜ ξ n ) = τ n, ( t ) . (3.8)14 roof. Since ˜ ξ n is a tau function corresponding to z g H ( C n , O ( ∗∞ )), we have, byTheorem 3.2, τ ( t ; ˜ ξ n ) = Cτ n, ( t ) , for some constant C . Comparing the expansions (2.14) and (3.4) we have C = 1.Next we consider tau functions appeared in the right hand side of the equation inTheorem 2.9. We need a point ( α, y ( α )) of C n − . To specify y ( α ) is equivalent tospecify one value of z such that z − = α , that is, α − / . In fact, if z = α − / is given thevalue of y ( α ) is determined by (2.11) as y ( α ) = α n − α / F n − ( α − / ) . (3.9)Since P i ( α ) = ( α, ω i y ( α )), we have z ( P i ( α )) = ω − i α − . (3.10)For simplicity we set z i ( α ) = ω − i α − . (3.11)Since, in general ξ n − ( Q , Q , Q ) is a frame of the point z g ′ − H ( C n − , O ( X i =0 P i + ∗∞ )) ∈ UGMwe have, by Theorem 3.4, τ ( t ; ξ n − ( P i ( α ) , P j ( α ) , P k ( β )))= C i,j,k ( α, β )e η ( z i ( α ) − )+ η ( z j ( α ) − )+ η ( z k ( β ) − ) × τ n − , ( t − [ z i ( α )] − [ z j ( α )] − [ z k ( β )]) , (3.12)for some constsnt C i,j,k ( α, β ). Remark 3.7. The explicit forms of the constants C i,j,k ( α, β ) are not yet determined.They should be calculated by comparing the Schur function expansions and are expectedto be expressed by some derivatives of the sigma function.Substituting (3.8), (3.12) into the relation in Theorem 2.9 we get Theorem 3.8. Let τ n, ( t ) be defined by the right hand side of (3.3) for the curve C n and z i ( α ) defined by (3.11). Then, in the limit α j → α for j = 3 n, n ± 1, we havelim τ n, ( t )= ( − n y ( α ) e − P ∞ l =1 α l t l X ≤ i 2. The curve X g has the unique point over x = ∞ whichwe also denote by ∞ . We take the local coordinate z around ∞ such that x = z − , y = z − g − F g ( z ) , F g ( z ) = g +1 Y j =1 (1 − α i z ) / . (4.3)Let µ ( g ) = ( g, g − , ..., ξ g a frame of z g H ( X g , O ( ∗∞ )) such that the corresponding taufunction has the expansion of the form τ ( t ; ˜ ξ ) = s µ ( g ) ( t ) + h . w . t . (4.4)Fix one of the square root α − / and define y by y = α g − / F g − ( α − / ) . (4.5)Then ( α, y ) is a point of X g − . Set p ± = ( α, ± y ) . (4.6)Then the values of the local coordinates of p ± are z ( p ± ) = ± α − / . Let ξ g − ( p ± ) be a frame of z g − H ( X g − , O ( p ± + ∗∞ )) such that their tau functionshave the following expansions τ ( t ; ξ g − ( p ± )) = s µ ( g − ( t ) + h . w . t . (4.7)The following theorem is proved in [3] in a similar way to Theorem 2.9.16 heorem 4.1. [3] The following relation holds.lim τ ( t ; ˜ ξ )= ( − g − (2 y ) − e − P ∞ l =1 α l t l ( τ ( t ; ξ g − ( p + )) − τ ( t ; ξ g − ( p − ))) , (4.8)where lim in the left hand side means the limit taking α g +1 , α g to α .Let τ g, ( t ) denote the function defined by the right hand side of (3.2) for X g . Lemma 4.2. (i) τ ( t ; ˜ ξ g ) = τ g, ( t ) . (ii) For some constant C ǫ ( α ) τ ( t ; ξ g − ( p ǫ )) = C ǫ ( α )e P ∞ l =1 ( ǫα − / ) − l t l τ g − , ( t − [ ǫα − / ]) , ǫ = ± . Proof. (i) Both τ ( t ; ˜ ξ g ) and τ g, ( t ) are tau functions corresponding to z g H ( X g , O ( ∗∞ )).By comparing the expansions (2.14) and (4.4) we get the result.(ii) Since the right hand side and the left hand side without C ǫ ( α ) of the equation inthe assertion are the tau functions corresponding to z g − H ( X g − , O ( p ǫ + ∗∞ )) by thedefinition of ξ g − ( p ǫ ) and Theorem 3.4, the assertion follows.This lemma is proved in [3] in a different form. The explicit form of the constant C ǫ ( α )can be extracted from there. Let us give the formula.Let m ( g ) = h g +12 i . Define the sequence A ( g ) and s ( g ) ∈ {± } by A ( g ) = ( a ( g )1 , ..., a ( g ) m ( g ) ) = (2 g − , g − , g − , ... ) ,s ( g ) = ( − ( g − m ( g ) . Example 4.3. A (1) = (1), A (2) = (3), A (3) = (5 , A (4) = (7 , s (1) = 1, s (2) = − s (3) = 1, s (4) = 1.The following property of A ( g ) is known [23][21], | A ( g ) | := m ( g ) X j =1 a ( g ) j = 12 g ( g + 1) . (4.9)Denote the sigma function of X g − by σ ( g − ( u ). Set b i = ( a ( g − i + 1) / ∈ { , , ..., g − } , ≤ i ≤ m ( g − , and define σ ( g − A ( g − ( u ) = ∂ m ( g − ∂u b · · · ∂u b m ( g − σ ( g − ( u ) . Then, by Theorem 4.14 of [3], we can deduce that C ǫ ( α ) = s ( g − σ ( g − A ( g − ( − Z p ǫ ∞ du ) − , du = t ( du , ..., du g ) . (4.10)17 emma 4.4. The following relation is valid. C − ( α ) = ( − g − C + ( α ) . (4.11) Proof. It is known that the sigma function satisfies the following relation [23][16] σ ( g − ( − u ) = ( − g ( g − σ ( g − ( u ) . By differentiating it we get σ ( g − A ( g − ( − u ) = ( − g ( g − m ( g − σ ( g − ( u ) . (4.12)We can easily verify that 12 g ( g − 1) + m ( g − = g − . . (4.13)For the hyperelliptic curve X g − the following relation holds, Z p − ∞ du = − Z p + ∞ du. (4.14)The assertion of the lemma follows from (4.10), (4.12), (4.13), (4.14).Substituting the equations of (i), (ii) in Lemma 4.2 into (4.8) and using (4.11) we get Theorem 4.5. Let τ g, ( t ) be given by the right hand side of (3.3) for the hyperellipticcurve X g defined by (4.1). Then in the limit α g +1 , α g → α we have the following formula,lim τ g, ( t ) = ( − g (2 y ) − C + ( α )e − P ∞ l =1 α l t l × (cid:16) e η ( α / ) τ g − , ( t − [ α − / ]) + ( − g e η ( − α / ) τ g − , ( t − [ − α − / ]) (cid:17) , where y , p ± , C + ( α ) are given by (4.5), (4.6), (4.10) respectively. Remark 4.6. The tau function τ g, ( t ) gives a solution of the KdV hierarchy (see remark3.3). Again it can be shown that τ , = 1 for the genus zero curve y = x which coorespondsto α = 0. Using the formula repeatedly we get the well known soliton solution [22][10].For α = 0 we can show that τ , ( t ) = e L ( t )+ Q ( t ) , where L ( t ) and Q ( t ) are certain linearand quadratic functions of t . Acknowledegements I would like to thank Samuel Grushevsky for discussions and helpful comments on thedegenerations of Z N curves and associated theta functions, and Julia Bernatska and VictorEnolski for former collaborations. This work was supported by JSPS KAKENHI GrantNumber JP19K03528. now deceased eferences [1] T. Ayano and A. Nakayashiki, On Addition Formulae for Sigma Functions of Tele-scopic Curves, Symmetry, Integrability and Geometry: Methods and Applications(SIGMA) (2013), 046, 14 pages.[2] J. Bernatska and D. Leykin, On degenerate sigma-functions in genus two, GlasgowMath. J. 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