On Reducible Degeneration of Hyperelliptic Curves and Soliton Solutions

In this paper we consider a reducible degeneration of a hyperelliptic curve of genus $g$. Using the Sato Grassmannian we show that the limits of hyperelliptic solutions of the KP-hierarchy exist and become soliton solutions of various types. We recover some results of Abenda who studied regular soliton solutions corresponding to a reducible rational curve obtained as a degeneration of a hyperelliptic curve. We study singular soliton solutions as well and clarify how the singularity structure of solutions is reflected in the matrices which determine soliton solutions.


Introduction
By the study of [6,7,14,16] soliton solutions of the KP equation acquire a new aspect. Namely it is discovered that the shapes of soliton solutions are more various than what is known before and those shapes are classified by points of totally positive Grassmannians. This study relates soliton solutions to other areas of mathematics such as cluster algebras.
Then it is natural to ask what happens for quasi-periodic solutions. From this point of view it is important to study the connection of quasi-periodic solutions and soliton solutions, in other words, the degenerations of quasi-periodic solutions to soliton solutions. In papers [2,3,4,5,1] Abenda and Grinevich studied this problem. They constructed a singular rational curve and some divisor on it to each regular soliton solution studied in [6,7,14,16]. It is noteworthy that their rational curves are reducible in general. It means that we need to consider reducible degenerations of algebraic curves in order to obtain a variety of soliton solutions.
In [1] Abenda studied a reducible rational curve which is obtained as a degeneration of a hyperelliptic curve and the corresponding soliton solutions as a concrete example of their theory. It should be noticed that in papers [2,3,4,5,1] soliton solutions and rational curves are directly related and that the limits of quasi-periodic solutions are not actually computed.
We began the study of degenerations of quasi-periodic solutions of the KP-hierarchy by the method of the Sato Grassmannian in [21]. In this approach it is possible to calculate the limits of quasi-periodic solutions without knowing the limits of periods of a Riemann surface.
In this paper we continue this study. We compute the limit of the τ -function of the KP-hierarchy corresponding to a hyperelliptic curve when it degenerates to a reducible rational curve. From the view point of taking a limit of a solution there is no reason to restrict ourselves to regular solutions. So we consider singular solutions as well. We can see how the singularity structure of the solution is reflected in the matrix A = (a i,j ) (see §3) which determines a soliton solution.
Consider the hyperelliptic curve X of genus g = n − 1 given by We assume that λ i 's are real and ordered as λ 1 < · · · < λ n .
There are two points over x = ∞ on X which are denoted by ∞ ± . The solution corresponding to X is well known. It is constructed by the method of Baker-Akhiezer function of Krichever [15]. To construct the Baker-Akhiezer function we need to specify a base point p ∞ , a local coordinate z around p ∞ and a general divisor of degree g. We take p ∞ = ∞ + , z = x −1 . For each 0 ≤ m 0 ≤ g we consider a general divisor of the form D g = p 1 + · · · + p m 0 + (g − m 0 )∞ + , p j = ∞ + ∀j.
The number m 0 specifies the partition of the Schur function which appears as the first term in the Schur function expansion of the τ -function corresponding to D g . Let k be an integer such that 0 ≤ k ≤ m 0 . We assume that p 1 , ..., p k is in a small neiborhood of ∞ − and the remaining points are in a small neighborhood of ∞ + . The number k specifies the type of soliton solutions in the limit.
We consider the degeneration of X to the reducible curve given by To take the limit of the corresponding solution of the KP-hierarchy we use the Sato Grassmannian. Using the Sato Grassmannian it is possible to write down the solution corresponding to X as a series with the coefficients in the polynomials of {λ j }. Therefore the limit of the solution exists. By making an appropriate gauge transformation we identify this limit with a soliton solution. The paper is organized as follows. In section 2 we review the correspondence between solutions (τ -functions) of the KP-hierarchy and points of the Sato Grassmannian. We recall (n, k) solitons and the corresponding points of the Sato Grassmannian in section 3. In section 4 we review how the data of algebraic curves are embedded in the Sato Grassmannian. In order to embed the data of X to the Sato Grassmannian we need an explicit description of meromorphic functions on X with a pole only at ∞ + . It is given in section 5. We also compute the gap sequence at ∞ + of the holomorphic line bundle of degree 0 corresponding to the divisor D g − g∞ + . The top term of the Schur function expansion of the solution is determined by using it. In section 6 we recall the description of the tau function corresponding to D g in terms of Riemann's theta function. The limit of the frame of the Sato Grassmannian corresponding to D g is determined in section 7. We show that it is gauge equivalent to the frame of an (n, k + 1) soliton. Finally we give the explicit formula of the limits of the tau function and the adjoint wave function (Baker-Akhiezer function ) in section 8. We also show that our result is consistent with Dubrovin-Natanzon's theorem on the regularity of quasi-periodic solutions.

KP-hierarchy
We set In this paper the KP-hierarchy signifies the following equation [8] for the function τ (x) of x = t (x 1 , x 2 , x 3 , ...): where y = t (y 1 , y 2 , y 3 , ...) and the integral means taking the coefficient of λ −1 in the series expansion of the integrand in λ.

Sato Grassmanian
The set of formal power series solutions of the KP-hierarchy is parametrized by the Sato Grassmannian which we denote by UGM [22,23] (see also [17] [13]). Let us briefly recall the definition and the fundamental properties of UGM.
Let V = C((z)) be the vector space of formal Laurent series in the variable z and Let π : V → V φ be the projection map. Then UGM is the set of subspaces U of V which satisfy dim Ker(π| U ) = dim Coker(π| U ) < ∞.
A basis of U is called a frame of U. We express a frame of U by an infinite matrix as follows. Set and write an element f of V as We associate the infinite column vector (ξ i ) i∈Z to f . Then a frame of U is given by a matrix ξ = (ξ i,j ) i∈Z,j∈N which is written as For a point U of UGM there exists a frame ξ = (ξ i,j ) i∈Z,j∈N satisfying the following conditions: there exists a non-negative integer l such that ξ i,j = 1 if j > l and i = −j 0 if (j > l and i < −j) or (j ≤ l and i < −l).
It means that X is of the form where B is an ∞ × l matrix of rank l and its first row is placed at the −lth row of ξ.
Conversely a matrix of this form becomes a frame of a point of UGM. In the following a frame of a point of UGM is always assumed to satisfy the condition (5) unless otherwise stated.
Here we introduce the notion of Maya diagram. A Maya diagram of charge p is a sequence of integers M = (m 1 , m 2 , ...) such that m 1 > m 2 > · · · and, for some l, m i = −i + p, i ≥ l holds. In this paper we consider only a Maya diagram of charge 0 and call them simply a Maya diagram.
With each Maya diagram M we can associated the partition λ by This gives a one to one correspondence between the set of Maya diagrams and the set of partitions. Let λ = (λ 1 , ..., λ l ) be an arbitrary partition and M = (m 1 , m 2 , m 3 , ...) the corresponding Maya diagram. The Plücker coordinate ξ λ or ξ M of a frame ξ is defined by We introduce the Schur function s λ (x) of the variable x = t (x 1 , x 2 , ...) by Then we define the tau function corresponding to a frame ξ of a point of UGM by where the summation is taken over all partitions. For a given point of UGM a frame ξ of it satisfying the condition (5) is not unique. If ξ is replaced by another frame the tau function is multiplied by a non-zero constant.

Theorem 1 [24]
For a frame ξ of a point of UGM τ (x; ξ) is a solution of the KPhierarchy. Conversely for any formal power series solution τ (x) of the KP-hierarchy there exists a unique point U of UGM and a frame ξ of U such that τ (x) = τ (x; ξ).

(n, k) solitons
In this section we recall the results on (n, k) solitons (see [14] for more details).
For a positive integer N and a nonnegative integer N ′ we use the following notation: Let n, k be positive integers which satisfy n ≥ k, A = (a ij ) be an n × k matrix of rank k and λ 1 , ..., λ n non-zero complex numbers. For k we set Then becomes a solution of the KP-hierarchy [12,25]. It is called the (n, k) soliton associated with the data (A, {λ j }) or the (n, k) soliton associated with A if {λ j } are fixed. The (n, k) soliton (9) can be written in the form of Wronskian. Let Then It is a trivial solution of (1) which is obtained from the constant solution by a gauge transformation. We include this case for the sake of convenience to describe the limits of the quasi-periodic solutions later.
The point of UGM corresponding to an (n, k) soliton is determined by Sato [22]. We consider the function 1/(1 − λ i z) as a power series in z by

Then
Theorem 2 [22] The point of UGM corresponding to the (n, k) soliton associated with (A, {λ j }) is given by the following frame:

Algebraic curves and UGM
It is possible to embed certain set of data of algebraic curves to the Sato Grassmannian (see [13,17,26] and the references therein). We restrict ourselves to the sepecial case which is relevant to us. Let X be a compact Riemann surface of genus g, p ∞ a point of X, z a local coordinate of X around p ∞ , L a holomorphic line bundle of degree g − 1 and φ a local trivialization of L around p ∞ . We define a map as follows. Take an element s of H 0 (X, L( * p ∞ )). Using φ the section s can be considered as a meromorphic function on some neighborhood of p ∞ . Therefore it is possible to expand it in z as Then Theorem 3 [13,17,26] The image of ι belongs to UGM.
Let us interpret this theorem in terms of dvisors and meromorphic functions. Let m 0 be an integer satisfying 0 ≤ m 0 ≤ g, p j , j ∈ [m 0 ], points of X such that p j = p ∞ for any j, D = p 1 + · · · + p m 0 + (g − 1 − m 0 )p ∞ the divisor of degree g − 1 and L the holomorphic line bundle corresponding to D. Then L ≃ O(D) as a sheaf of O-modules. Using this isomorphism and the local coordinate z we can consider a local section of L near p ∞ as a meromorphic function on some neighborhood of p ∞ . It gives a local trivialization of L around p ∞ . So let us examine how this isomorphism looks like.
Let I be a finite index set which contains the symbol ∞, {W i |i ∈ I} an open covering of X such that each W i is a domain of a local coordinate system of X and contains at most one p j and d i a meromorphic function on W i whose divisor is D in W i . We assume that W ∞ contains p ∞ . We can take defines a meromorphic function on W whose divisor (f ) satisfies (f ) + D ≥ 0. This is the map from L to O(D).
Let us look at the neighborhood W ∞ of p ∞ . A local section s of L on W ∞ is mapped to the meromorphic function s/z g−1−m 0 on W ∞ . Conversely a local meromorphic function f on W ∞ which belongs to O(D) corresponds to the local holomorphic section We have the composition of maps: where the first map is that induced from O(D) ≃ L and the second map is ι. Using the description of the isomorphism O(D) ≃ L explained aboveι is given as follows.
Let us take a meromorphic function f ∈ H 0 (X, O(D + * p ∞ )) and expand it in z around p ∞ as

Hyperelliptic curves and functions on them
Let X be the hyperelliptic curve of genus g = n − 1 defined by where {λ j } are mutually distinct non-zero complex numbers. It can be compactified by adding two points over x = ∞ which we denote by ∞ ± . We take z = 1/x as a local coordinate around ∞ ± . We distinguish ∞ + and ∞ − by the expansion of y: We denote by σ the involution of X defined by σ(x, y) = (x, −y). Let be a general divisor. It is known that (12) is a general divisor if and only if p i = σ(p j ) for any i = j (see [11] for example). Let D = p 1 + · · · + p g − ∞ + the divisor of degree g − 1.
It can be written as for some 0 ≤ m 0 ≤ g. Since (13) is a general divisor, For simplicity we assume that p 1 , ..., p m 0 are mutually distinct and different from ∞ − .
Let us find a basis of H 0 (X, O(D + * ∞ + )). To this end we first study the case of m 0 = 0, that is, the case D = (g − 1)∞ + . In this case where the right hand side is the space of meromorphic functions on X which are holomorphic on X\{∞ + }. A basis of this space can be given as follows.
It can be easily proved that the space of meromorphic functions on X which are holomorphic on X\{∞ + , ∞ − } is equal to the space of polynomials in x and y. Let us write the expansion of y at ∞ ± as For m ≥ n define polynomials g m (x) and f m (x, y) by Since, at ∞ ± , This means that, for m ≥ n, f m is a meromorphic function on X with a pole only at ∞ + and the order of a pole is m.
Here we recall the notion of gaps. Let M be a holomorphic line bundle of degree zero, p a point of X and m a non-negative integer. If there is no meromorphic section of M with a pole of order m at p and with no other poles, then m is called a gap of M at p. If m is not a gap then it is called a non-gap of M at p. There are exactly g gaps for any M and p by Lemma 1 of [20]. If the set of gaps of the trivial line bundle at p is not [g], then p is called a Weierstrass point. It is known that the Weierstrass points of the hyperelliptic curve X are branch points (λ j , 0), j ∈ [2g]. In particular ∞ ± are not Weierstrass points.
Next we consider the general case (13) with m 0 not necessarily equal to zero. Since p i = ∞ ± , we can write for some c i ∈ C. We assume that c i does not depend on {λ j } for any i. In particular c i = λ j , i, j ∈ [2n]. Since {p i } are mutually distinct and satisfy (14), {c i } are mutually distinct. In the following we assume further It is a meromorphic function on X with the pole divisor p j + (n − 1)∞ + .
Proof. Let M be the holomorphic line bundle of degree zero corresponding to the divisor Then we have We identify the left hand side of (20) with the right hand side of (20). Then 1 and h j , j ∈ [m 0 ], belong to H 0 (X, M((m 0 + g)∞ + )). Since c 1 , ..., c m 0 are mutually distinct, the set of functions {1, h j , j ∈ [m 0 ]} is linearly independent and it spans an m 0 + 1 dimensional subspace of H 0 (X, M((m 0 + g)∞ + )). Since the degree of M is zero Notice that, for m ≥ g + 1, Therefore there are at most m 0 + g + 1 − (m 0 + 1) = g gaps in H 0 (X, M( * ∞ + )). Since there are exactly g gaps by Lemma 1 of [20], we can conclude that {1, h j , j ∈ [m 0 ]} is a basis of H 0 (X, M((m 0 + g)∞ + )). It then shows that {1, h j , j ∈ [m 0 ], f m , m ≥ g + 1} is a basis of H 0 (X, M( * ∞ + )). Let us determine the gap sequence of M defined by (18) at ∞ + . By (19) a meromorphic function from H 0 (X, O(D + * ∞ + )) with a pole of order r at ∞ + is identified with a meromorphic section of M with a pole of order r + m 0 at ∞ + . We prove Proposition 1 The gap sequence of M at ∞ + is (0, 1, ..., m 0 − 1, m 0 + 1, ..., g).
Let K = (k ij ) 1≤i,j≤m 0 be the m 0 × m 0 matrix defined by Thenh Proof. (i) It can be proved just by computation using the properties of determinants. So we leave the details to the reader.
(ii) By expanding h i (z) in z we have The assertion (ii) follows from this. By the lemma we have Proof of Proposition 1.

Theta function solution
By Corollary 1, Lemma 2 and Corollary 2 (i) it is possible to give the following definition.
By Lemma 3 the tau functions corresponding to ξ(D) andξ(D) are related by By Krichever's construction [15] the tau function τ (x;ξ(D)) is expressed in terms of Riemann's theta function as follows.
} be a canonical homology basis, {dv j | j ∈ [g]} the normalized holomorphic one forms, Ω = δ j dv i the period matrix, θ(z|Ω) Riemann's theta function and K ∞ + Riemann's constant corresponding to the point ∞ + . For i ≥ 1 we denote by dr i the normalized differential of the second kind with a pole only at ∞ + of order i + 1. Namely it satisfies Since p 1 + · · · + p m 0 + (g − m 0 )∞ + is a general divisor, θ( p ∞ + dv + e) has a zero of order g − m 0 at ∞ + by Riemann's theorem [11]. Therefore does not vanish. By Krichever's theory [15] the following function Ψ(x; z) defines an adjoint wave function [8], where p dr i is the indefinite integral without the constant term. Let By [8] there exists, up to a constant multiple, a function τ (x) which satisfies the following equation near ∞ + , Therefore τ (x) coincides with τ (x;ξ(D)) up to a constant multiple (see [19]). The function τ (x) satisfying the relation (30) can be constructed in the following way. Let E(p 1 , p 2 ) be the prime form [11]. Write By a similar computation to [19] we have Proposition 2 There exists a non-zero constant c such that By Proposition 1 the top term of the Schur function expansion of τ (x;ξ(D))) is determined. Let λ be the partition defined by λ = (g, g − 1, ..., m 0 + 1, m 0 − 1, ..., 1, 0) − (g − 1, ..., 1, 0) = (1 g−m 0 ). By Corollary 1 and Corollary 2 of [20] the partition corresponding to the Schur function which appears in the top term of the expansion of τ (x;ξ(D))) is given by the conjugate partition of λ, t λ = (g − m 0 ). Taking the conjugate of λ is due to the minus sign in the definition (28) of e. By the form of the frameξ(D) the Schur function expansion of τ (x;ξ(D)) begins from s (g−m 0 ) (x) = p g−m 0 (x). Thus
In this section we show that ξ 0 (D) can be transformed to a frame of the form (10) by a gauge transformation. The hyperelliptic curve (11) tends to (1 − λ j z).
Then the Taylor series y(z) of y around ∞ + tends to where we use the same symbol y(z) for the limit of y(z). Let g 0 m and f 0 m be the limits of g m and f m respectively. Then To determine the limit of h i we need to specify the limit of the point p i = (c i , y i ). We do this in the following way.
Since c i does not depend on {λ j }, p i goes to where ε i = ±1. Let k be an integer such that We assume, in (34), that For simplicity set Then This condition is satisfied if p 1 , ..., p k are in a small neighborhood of ∞ − and p k+1 ,...,p m 0 are in a small neighborhood of ∞ + . The limits of the quantities in the numerator of (17) are Then the frame ξ 0 (D) is given by Definition 2 We denote the point of UGM corresponding to the frame (35) by U 0 (D).
In order to identify the solution corresponding to U 0 (D) with a soliton solution we change a basis and make a gauge transformation. To this end let Consider the gauge transformation ϕ(z)U 0 (D) of U 0 (D). Then the following set of functions is a basis of ϕ(z)U 0 (D), Lemma 4 The following set of functions is a basis of ϕ(z)U 0 (D): Proof. Since d i 's are mutually distinct and deg ϕ i (z) = l − 1, where Span C { * } denotes the vector space spanned by { * }. Therefore, noticing that deg ϕ(z) = l, we have which shows the lemma.
It is possible to erase the terms of degree less than l in ϕ(z) n−1 j=0 b i,j z j by subtracting an appropriate linear combination of z r f (z), 0 ≤ r ≤ l − 1. It means that there exist constants η r , 0 ≤ r ≤ l − 1 and a unique polynomial G i (z) of degree at most n − 1 satisfying the following equation: Proposition 4 A basis of ϕ(z)U 0 (D) is given by Proof. Multiplying (38) by z −m 0 we have Then the lemma follows from Lemma 4, (37), (39). Define Then Theorem 4 The following set of functions is a basis of U 0 (D): where A = (a i,j ) i∈[n],j∈[k+1] is given by Then Theorem 4 tells that τ (x; ξ 0 (D)) is an (n, k + 1) soliton.

Remark 2
In [1] only the case m 0 = n − 1, which corresponds to regular solutions, is studied. If m 0 = n − 1, k = 0, l = n − 1, the n × 1 matrix A coicides with that in [1]. For k > 0 the matrix A is apparently different from that in [1].
The theorem is proved by expanding elements of the basis in Proposition 4 into partial fractions by using the following lemma which easily follows from the definition (38) of G i (z).
8 The tau function corresponding toξ 0 (D) In this section we examine the properties of the tau function τ (x;ξ 0 (D)). By taking the limit of (27) we have The tau function τ (x; ξ 0 (D)) can be expressed by τ (x; ξ 0 (D)) as follows.
The tau function corresponding to ξ 0 (D) can be computed by (9) with the matrix A given in Theorem 4. Let us compute A I .

Lemma 6 For I ∈ [n]
k+1 we have The lemma can be proved using the following Cauchy like formula which is easily proved: .
We assign weight i to x i . Then , and Ξ is given by (45).
(ii) It follows from Proposition 3.
Next we study the condition of the regularity of τ (x;ξ 0 (D)). Hereafter we assume that λ i , c i , d i , x i are real for all possible i and that λ 1 < · · · < λ n .
Proof. Notice that Ξ is independent of I, ∆ I is positive and the sign of i,j∈I,i<j (λ i −λ j ) is (−1) (1/2)k(k+1) is independent of I. Let us compute the sign of the remaining part of A I in Lemma 6. The sign of is the same as that of and is equal to (−1) n−i . The sign of n r =i (λ i − λ r ) is also (−1) n−i . Therefore the sign of is 1 and the proposition is proved.
Remark 3 Proposition 5 is consistent with the Dubrovin-Natanzon theorem on the positivity of algebro geometric solutions of the KP-hierarchy [9] (see also [1]). In fact the condition (47) means that, before taking the limit, each fixed point oval with respect to the anti-holomorphic involution of X except that containing ∞ + , contains one divisor point of the poles of the Baker-Akhiezer function (the adjoint wave function).
Finally we give the adjoint wave function, which we denote by Ψ 0 (x; z), corresponding to the tau function in Corollary 3. The result is Ψ 0 (x; z) = 1 Φ(x) where I c denotes the complement of I in [n], ∆ I = ∆ I (λ 1 , ..., λ n ), A I is given by (44) and Φ(x) is the part of τ (x;ξ 0 (D)) which is obtained by removing the part in front of the sum symbol (the constant and the exponential function). The function Ψ 0 (x; z) is the expression of the limit of Ψ(x; z) near ∞ + . Notice that the poles at p 1 , ..., p k of Ψ(x; z) disappear in the limit. This is possible because we consider the reducible degeneration of the curve X.