Rational KdV potentials and Differential Galois Theory

In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schr\"odinger equation for rational time dependent KdV potentials. In particular, we compute the fundamental matrices of the linear systems associated to the Schr\"odinger equation. Furthermore we prove the invariance of the Galois groups with respect to time, to generic values of the spectral parameter and to Darboux transformations.

One of the goals of the paper is to study the invariance of the Galois group of the linear system associated to the KdV hierarchy, with respect to the Darboux transformations and respect to the KdV flow (ie, to time). In fact as a by-product we have obtained more than that: the Galois group is also invariant with respect to generic values of the spectral parameters (see section 7). Thus, in some sense this paper can be considered as a continuation of our previous paper [15], where we studied the invariance of the Galois group of the AKNS systems with respect to the Darboux transformations. But one of the essential differences here is that in general we can not use the Darboux invariance result in [15], because the Darboux transformation here is not a well-defined gauge transformation, ie, it is not inversible. Thus we must use the classical Darboux tranformation of the Schrödinger equation, we call it the Darboux-Crum transform; and then to verify the compatibility of this transform with the complete linear system (1.2).
In Section 3 we study the action of the Darboux transformations over the recursive relations (2.1) inside the KdV hierarchy. We point out that the results in Section 3 hold not only for rational KdV potentials but also for any arbitrary KdV potential.
Also, in Section 6 we study the action over the spectral curve of the Darboux transformations for stationary KdV arbitrary potentials.
Brezhnev in three papers [4,5,6] also consider the Galois groups associated to spectral problem for some KdV potentials. More specifically the so-called finite-gap potentials, where the spectral curve is non-singular. Here we study a completely different situation, where the spectral curves are cuspidal curves, corresponding to Adler-Moser rational type solutions.
However, the general results obtained in Sections 3 and 6 open the door to study more general families of KdV potentials, such as Rosen-Morse potentials or elliptic KdV potentials.

Basic facts on KdV hierarchy
Let K be a differential field with compatible derivations ∂ x , ∂ t 1 , ∂ t 2 , . . . , ∂ t m with respect to the variables x and t = (t 1 , . . . , t m ). Let us assume that its field of constants is the field of complex numbers C. Let E ∈ C be a complex parameter and u ∈ K be a fixed element of K.
2 Let us consider the differential recursive relations: see [12], where the authors also provided an algorithm to compute ∂ −1 x ( f j,x ). Functions f j are differential polynomials in u, see [12,18]. For the first terms one finds for some integration constants c i . It is well known that the time dependent KdV hierarchy can be constructed as zero curvature condition of the family of integrable systems (see [13] chapter 1, section 2): where F r , G r and H r are differential polynomials of the potential u defined by

5)
H r = (E − u)F r − G r,x = (E − u)F r + F r,xx 2 . (2.6) Observe that the degree in E of the matrices V r and functions H r is r + 1. We point out that the first equation of (2.3) is equivalent to the Schrödinger equation with L = −∂ xx + u. Now, fix a positive integer r and consider the corresponding system (2.3). Its zero curvature condition U t r − V r,x + [U, V r ] = 0, (2.8) yields to the KdV r equation Using expressions (2.1) and (2.4), this equation can be rewritten as: KdV r : u t r = 2 f r+1,x . (2.10) We recall that the equation (2.10) is called the level r equation of the KdV hierarchy. Whenever we want to specify the dependence on the potential u, we will write f j (u), F j (u), G j (u) and H j (u) to emphasize this fact.

Adler-Moser rational potentials
In this section we review the KdV r rational potentials that Adler and Moser constructed in [1]. These are a family of rational potentials u n for Schrödinger operator −∂ xx + u of the form u n = −2(log θ n ) xx , where θ n are functions in the variables x, t defined by the differential recursion: θ 0 = 1, θ 1 = x, θ n+1,x θ n−1 − θ n+1 θ n−1,x = (2n + 1)θ 2 n . (2.11) The solutions of this recursion are polynomials in x with coefficients in the field F = C(t). This is an easy consequence of the next result, which is an easy extension of the proof of Lemma 2 in [1]. 3 Lemma 2.1. Let be F = C(t), and a ∈ C * , b ∈ C. Let (F [x], ∂ x ) be the ring of polynomials with derivative ∂ x , whose field of constants is F. Let consider the sequence defined recursively by: P 0 = 1, P 1 = ax + b, P n+1,x P n−1 − P n+1 P n−1,x = (2n + 1)P 2 n . (2.12) Then P n ∈ F[x] for all n.
Now, applying Lemma 2.1 for a = 1 and b = 0, we obtain that functions θ n are polynomials of x with coefficients in C(t) for all n. We call these polynomials Adler-Moser polynomials.
Remark 2.5. Theorem 2.4 shows that for each level r the formula (2.13) for θ n is a solution of the KdV r equation. Hence the constants τ 2 , . . . , τ j must be adapted to get a solution of the KdV r equation. When this is the case, we will denote adjusted polynomials as θ r,n and adjusted potentials as u r,n to stress this fact.  2 uu x − 1 4 u xxx .. The computations were made using SAGE. We have n u 1,n (τ 2 , · · · , τ n ) 0 0 We notice that the adjustment of τ i is not linear in t 1 .

Spectral curves for KdV hierarchy
Now, we consider the stationary KdV hierarchy.
s-KdV r : 2 f r+1,x = 0. (2.16) We have the following result for Adler-Moser potentials u r,n in the stationary case [1]: Lemma 2.8. For τ j = 0, j = 2, . . . , n, we have θ n (x, 0) = θ (0) n (x) = x n(n+1)/2 and u (0) r,n (x) = u r,n (x, t r = 0) = n(n + 1)x −2 . (2.17) The first level of the stationary KdV hierarchy for which potentials u (0) r,n (x) = n(n + 1)x −2 defined in the aforementioned Lemma are solutions of is level n, which implies that in the stationary case we will have r = n. We will denote them just by u (0) n (x). Therefore, the associated system will be (2.18) To simplify the notation, from now on we write F (0) n , G (0) n and H (0) n instead of F n (u (0) n ), G n (u (0) n ) and H n (u (0) n ). The zero curvature condition of this system is now the stationary KdV n equation: After applying expressions (2.1) and (2.4), this equation can be rewritten as: Of course, this coincides with equation (2.16) for these potentials for r = n.
When the potential u (0) is a solution of the zero curvature condition (2.19) we will say that it is a s-KdV n potential. Under this assumption, the spectral curve of system (2.18) for this potential is the characteristic polynomial of matrix iV (0) n : (2.21) (see for instance [13]). We denote by p n (E, µ) = µ 2 − R 2n+1 (E) the equation that defines the spectral curve. We will use the following notation where C i are differential polynomials in u (0) with constant coefficients.
Lemma 2.9. We have the following equality ∂ Derivating with respecto to x and using formula (2.1) we arrive to the desire expression.
With this matrix presentation it is easy to prove the following result: Proposition 2.10 (Burchnall and Chaundy, [7]). Let u = u(x) be solution of equation This proposition together with Lemma 2.9 and formula (2.1) easily implies the following result.
It is well known that the spectral curve associated to system (2.18) for stationary potential (2.17) is (2.23) Therefore, these are the spectral curves associated to system (2.3) for Adler-Moser potentials u r,n .
Remark 2.12. Take potential u r,n solution of KdV r equation, then potential u (0) n (x) is solution of s-KdV n equation. Thus, we can link level r of the time-dependent KdV hierarchy with level n of the stationary KdV hierarchy.

Darboux transformations for f j
In this section we will present the behavior of Darboux transformations acting on the differential polynomials f j . Therefore we will consider the Schrödinger equation where E 0 is a fixed energy level. Let φ 0 be a solution of such equation. Recall that a Darboux transformation of a function φ by φ 0 is defined by the formula Then the transformed function φ = DT (φ 0 )φ is a solution of the Schrödinger equation for potential u = u − 2(log φ 0 ) xx , whenever φ is a solution of Schrödinger equation for potential u and energy level E E 0 ( [8,9,10,16]). We will denote by DT (φ 0 )u the potential u to point out the fact that it depends on the choice of φ 0 . Next we can observe that the Riccati equation has σ 0 = (log φ 0 ) x as solution, and then In this way, we retrieve a Riccati equation for u as we have Moreover, whenever we have a solution φ of the Schrödinger equation (2.7), the formula σ = (log φ) x gives a solution of the Riccati equation (3.2). Hence, σ satisfies the nonlinear differential equation Next, we consider the matrix differential system (2.3). Then we perform a Darboux tramsformation, DT (φ 0 ), on it obtaing a new differential system, say Φ x = UΦ , Φ t r = V r Φ, whose zero curvature condition is still equation (2.9). Let F r ( u), G r ( u) and H r ( u) be the corresponding entries of the matrix V r . These differential polynomials are given by expressions (2.4), (2.5) and (2.6) in terms on f j ( u). We will establish the relation between f j ( u) and f j (u) in the next theorem.
where A j is a differential polynomial in u and σ. Moreover, A j satisfies the recursive differential relations Proof. We will proceed by induction on n. First, we prove by induction that f j ( u) = f j (u) + A j . For j = 0 we have f 0 ( u) = 1 = f 0 (u) + A 0 , where A 0 = 0. We suppose it true for j and prove it for j + 1. Applying equation (2.1) and induction hypothesis we find: Thus, f j+1 ( u) = f j+1 (u) + A j+1 as we wanted to prove. Now, we prove statements 1 and 2. We do it by induction and simultaneously. Since A 0 = 0 and f 0 (u) = f 0 ( u) = 1, the case j = 0 is the trivial one. So, we start the induction process in j = 1. For this, using recursion formula (2.1) we have: For j = 1 statements 1 and 2 read: by equation (3.5). Now, we suppose both statements true for j and prove them for j + 1. Derivation with respect to x in the right hand side of statement 1 yields to: Applying equality (3.5) to the term σ xx A j /2 we get: Applying induction hypothesis for statement 2 we have: which is exactly expression (3.6) for A j+1,x . So, we can assume that Thus, statement 1 is proved. Finally, by equations (2.1), (3.6), (3.5) and induction hypothesis we find for statement 2: by statement 1. Therefore, statement 2 is also proved. This completes the proof.
Example 3.2. To illustrate the previous theorem we will consider the following KdV 2 potentials in the system (2.3). Take u = 6(2x 10 +270x 5 t+675t 2 ) Observe that: and also Hence, in this case By direct computation we can verify that the A j satisfy the retations 1 and 2 of 3.1.
Theorem 3.1 has several interesting consequences. The main ones are the relations that the transformed potential u produce for functions F r (u). Next we stablish some of them, which will be used in the following sections. In particular, Proposition 3.5 is specially interesting since it gives a relation between σ x and σ t r .
Proof. It is an immediate consequence of Theorem 3.1.
Proposition 3.5. Let u be a solution of KdV r equation. Let φ be a solution of Schrödinger equation (2.7) for potential u and energy E 0 . Let be σ = (log φ) x . Consider A r+1 as defined in 3.1 and P r as defined in 3.4. Then, we have: Proof. We compare the zero curvature conditions for u and u: We prove the first equality. For this, we have u t r = (u − 2σ x ) t r = u t r − 2σ x,t r and 2 f r+1,x ( u) = 2 f r+1,x (u) + 2A r+1,x by Theorem 3.1. Then: Thus, σ t r = −A r+1 . Now, we prove the second equality. Using expression (3.3) for u and applying 3.4 (1), we obtain Since 2σ x,t r = u t r − u t r , we have 2σ x,t r = 1 2 P r,xxx + 2EP r,x − 2uP r,x + 4σ x F r,x (u) + 4σ x P r,x − u x P r + 2σ xx F r (u) + 2σ xx P r .
Applying 3.4 (2) to the expresion σ x P r,x , we find: − u x P r + 2σ xx F r (u) + 2σ xx P r Moreover, for the coefficient of P r we have: Hence we have proved the statement.
We finish this section with the following technical result. It makes a connection between differential polynomials f r (u) and some differential polynomials g r (σ) defined by Proposition 3.6. We have the following relations: Proof. Statement 1 is just statement 2 of Theorem 3.1 rewritten. For statement 2 we have: by statement 1 and equation (3.10).

Fundamental matrices for KdV r rational Schrödinger operators
In this section we give a fundamental matrix for the system (2.3) depending on the energy level E. The spectral curve is the tool that will allow us to understand why fundamental matrices present different behaviors according to the values of the energy.
For stationary rational potentials u (0) n = n(n +1)x −2 , it is well known that the spectral curve associated to system is the algebraic plane curve in C 2 given by Whenever an Adler-Moser potential u r,n (x, t) is time dependent, we will consider Γ n as the spectral curve associated to its corresponding linear differential system (2.3). Observe that (E, µ) = (0, 0) is the unique affine singular point of Γ n . It turns out that for E 0 the behavior of the fundamental matrix associated to the system presents a similar behavior since the point P = (E, µ) is a regular point of Γ n . A fundamental matrix for E = 0 can be also computed. However, it is not obtained by a specialization process from the fundamental matrix obtained for a regular point. We include some examples in this section.

Fundamental matrices for E = 0
In this section, we compute explicitly fundamental matrices of system (2.3) when the potential u is u r,n = −2(log θ r,n ) xx and E = 0. Recall that u r,n is a solution of KdV r (see Remark 2.5). Hence, we study the system It is obvious that the zero curvature condition of this system is the KdV r equation for c i = 0, i = 1, . . . , r: From now on we will denote u r,n,t r = ∂ t r (u r,n ).
Proof. We prove it by induction on n. For n = 0 the definition θ r,0 = 1 gives u r,0 = 0. So, system (4.4) reads Thus, φ 1,r,0 = 1 and φ 2,r,0 = x generate B (r) 0,0 . Since θ r,1 = x we have that φ 1,r,0 = θ r,−1 θ r,0 and φ 2,r,0 = θ r,1 θ r,0 . Now, we suppose it true for n and prove it for n + 1. For n we know that φ 1,r,n = θ r,n−1 θ r,n and φ 2,r,n = θ r,n+1 θ r,n generate B (r) n,0 . Therefore, φ 1,r,n and φ 2,r,n are solutions of Schrödinger equation φ xx = u r,n φ. We apply a Darboux transformation with φ 2,r,n to this Schrödinger equation and we obtain: So, φ 1,r,n+1 = θ r,n θ r,n+1 is a solution of φ xx = u r,n+1 φ and, obviously, (φ 1,r,n+1 , φ 1,r,n+1,x ) t is a column solution of the first equation of the system for u r,n+1 . Now we verify that this column matrix is also a solution of the second equation: We notice that the second row is just the partial derivative with respect to x of the first one. Hence, we just have to verify that expressions (4.8) and (4.9) satisfy the equation Applying expression (4.9) and induction hypothesis we obtain for the left hand side of this equation: and for the right hand side: (4.12) Now, we prove that both expressions are equal. Applying Theorem 3.1 statement 2 for σ = φ 2,r,n,x φ 2,r,n to expression (4.12) leads to: which is equal to expression (4.11). Therefore, both sides of expression (4.10) coincide.
Adler and Moser proved in [1] that matrix B (r) n,0 is a fundamental matrix for the Schrödinger equation (2.7) for E = 0. But they did not prove there that this matrix is also a fundamental matrix for the second equation of the system (4.4). To do that, it is necessary to control the action of the Darboux transformations over the differential polynomials f j , as we did in Section 3.

Fundamental matrices for E 0
In this section, we compute explicitly fundamental matrices of system (2.3) when u = u r,n = −2(log θ r,n ) xx and E 0. In this case, the system is: The zero curvature condition of this system is still the KdV r equation for c i = 0, i = 1, . . . , r: u r,n,t r = 2 f r+1,x (u r,n ). (4.16) When E 0, we take λ ∈ C a parameter over K such that E + λ 2 = 0. 13 Next, we consider the differential systems: We have the following relations for solutions of the differential systems (4.17)-(4.18) and (4.19)-(4.20).
Finally, using relation (4.17) for Q + n and statements 1 and 2 of Corollary 3.4, the right hand side of equation Therefore, both expressions coincide and Q + n+1 is a solution of equation (4.18). The proof for Q − n+1 is analogous. As a consequence, we have the following result: Theorem 4.6. Let n be a non negative integer, then, for E = −λ 2 0 and u = u r,n , a fundamental matrix for system (4.15) is:
As far as we know, a general expression for fundamental matrices for system (4.15) has never been computed when E 0. As in Theorem 4.1, the key to do that is to control the action of the Darboux transformations over the differential polynomials f j , as we showed in Section 3. In Section 5 we will give some examples of these fundamental solutions both in the general framework of unadjusted functions τ i and in the particular case r = 1, in the same line as in Example 4.4.  Proof. We notice that φ + r,n (x, t, −λ) = e −λx−(−1) r λ 2r+1 t r Q + r,n (x, t, −λ) θ r,n , since θ r,n does not depend on λ. So, both relations are equivalent and it suffices to prove that Q + r,n (x, t, −λ) = (−1) n Q − r,n (x, t, λ). We prove it by induction on n. For n = 0, we have that Q + r,0 = 1 = Q − r,0 . Hence, Q + r,0 (x, t, −λ) = (−1) 0 Q − r,0 (x, t, λ). Using expresions (4.21) and (4.22), we obtain , as we wanted to prove.
Remark 4.9. Theorem 4.8 implies that matrix B (r) n,λ is not a fundamental matrix of system (2.3) for λ = E = 0, since it is not invertible for that value of E. The reason of this is that, by Proposition 4.7, when λ = 0 we have φ + r,n (x, t, 0) = (−1) n φ − r,n (x, t, 0), so, both column solutions are linearly dependent. We will detail this phenomenon in Section 6. In fact, we will show that it is not the same to set E = 0 in (2.3) and then solve the system, than to solve the system for a generic E and then replace E = 0 in the solution obtained, i.e., there is not specialization process in this sense.
In next section we will show a method to compute functions Q + r,n and Q − r,n more efficient than solving explicitly equations (4.17), (4.18), (4.19) and (4.20) which will allow us to obtain fundamental matrices B (r) n,λ . In particular φ + r,1 and φ − r,1 are linearly independent solutions for the Schrödinger operator −∂ 2 + u r,1 − E = 0 where u r,1 = 2/x 2 is the constructed rational KdV r potential, as long as E 0.

Examples of fundamental matrices for the case E 0
Along this section we prove that funtions Q ± r,n defined in Theorem 4.6 satisfy the recursion formula (2.11). This implies in particular that they are polynomials of x with coefficients in C(λ, t). Thus, they generalized the family of Adler-Moser polynomials θ n .
For the following computations we do not suppose that functions θ n and Q ± n and potentials u n are adjusted to any level of the KdV hierarchy.
Finally, expression (4.17) for Q + n,xx yields to: Analogously, the second recursion formula can be proved. So we have established our result Remark 5.2. By Lemma 2.1 for F = C(λ, t) and a = λ, b = −1, we can conclude from this theorem that the functions Q ± n (x, t, λ) are polynomials of x with coefficients in C(λ, t) for all n. Indeed, their degree as functions of λ is n. Thus, Theorems 4.6 and 5.1 determine the algebraic structure of φ + r,n and φ − r,n . Since polynomials Q ± n are not adjusted to any level of the KdV hierarchy, when we iterate recurrences (5.6) and (5.7) we will obtain integration constants of x which may depend on λ and τ 2 , . . . , τ n . We will denote such integration constants by τ ± 2 , . . . , τ ± n .
Example 5.3. For the first polynomials we find (5.11)

Examples of fundamental matrices for the case E 0
We can compute fundamental matrices for system (4.15) for any n using recursion formulas (5.6) and (5.7).
Example 5.4. We present explicit computations using SAGE for the fundamental solutions of the system (4.15) when E = −λ 2 0 for same potentials as in Example 4.4.

We first expose examples of unadjusted fundamental solutions:
where Q + 3 and Q − 3 are the ones given in (5.11).
2. Next, we expose fundamental solutions for potentials which are solutions of the first level of the KdV hierarchy, KdV 1 equation: u t 1 = 3 2 uu x − 1 4 u xxx . We also show the explicit choice of the functions τ ± i . The choice of functions τ i is the same as in Example 4.4.

Spectral curves and Darboux-Crum transformations
Let Γ n ⊂ C 2 be the spectral curve associated to the stationary Schrödinger operator −∂ xx + u − E where u is a s-KdV n potential. Next we consider the Zariski closure of Γ n , say Γ n , in the complex projective plane P 2 . Let be p(E, µ) = µ 2 − R 2n+1 (E) = µ 2 − 2n+1 j=0 C j E j = 0 an equation for Γ n . Then an equation for Γ n is

Extended Green's function
Following [13], we define the Green's function on Γ n × C as where φ 1 and φ 2 are two independent solutions of Schrödinger equation for the same value of E and W(φ 1 , φ 2 ) stands for their wronskian. Let be functions defined over the spectral curve. We recall the following result.
Lemma 6.1 (Lemma 1.8 of [13]). Let u be solution of s-KdV n equation (2.10). Let φ 1 and φ 2 be solutions of Schrödinger equation (6.5) for this potential and with corresponding functions over the spectral curve σ + and σ − defined by (6.6). Then σ + and σ − are solutions of the Riccati type equation: Moreover, the following identities are satisfied: We remark that this lemma is essentially a reformulation of a classic result that goes back to Hermite when he was studying closed form solutions for Lamé equation ([14]). In [19] call this approach the Lindeman-Stieljes theory but, as far as we know, this approach was used for the first time by Hermite, and then by others: Halphen, Brioschi, Crawford, Stieljes.... The method used that the product of solutions X = φ 1 φ 2 is a solution of the second symmetric power of the Schrödinger equation Then the relations (6.8) connect the solutions of the Riccati equation with that of the second symmetric power. The fact that there is a connection between the solutions of the second symmetric product and the Riccati equation of the Schrödinger equation is relevant for the differential Galois theory, although we will not use explicitely this connection in this paper. Furthermore it is interesting to point out that the solutions of the Lamé equation obtained by Hermite in [14], are associated to other algebro-geometric solutions of KdV, finite-gap solutions with regular spectral curves, see [17] and references therein. As far as we know, the relevance of the equation (6.9) for the KdV equation was considered for the first time by Gel'fand and Dikii in their fundamental paper about the asymptotic behaviour of the resolvent of the Schrödinger equation associated to the KdV equation [12].
By Lemma 6.1, the Green's function can be rewritten as Observe that g is well defined whenever µ 0, i. e. for energy levels such that R 2n+1 (E) 0. Next, let define a extension of g on Γ n × C x as We call g h the homogenized Green's function. Next we will show that g h is well defined and also that it extends g, To do that, observe that for any a ∈ C, a 0. Moreover, we have that is an homogeneous polynomial in E of degree n and then Also, we get the following formula: , (6.14) 21 where F n,x = ν n−1 F n,x (E/ν) and F n,xx = ν n−1 F n,xx (E/ν) (6.15) are homogeneous polynomials in E and ν of degree n − 1. Now, take equation (2.19): after multiplication by F n and integration, this equation reads where c is a integration constant. By (6.10) we have the following differential relation for the function g: since g x = (σ + + σ − )g and g xx = 2(u − E + σ + σ − )g. Now let define the extensions of σ + and σ − on Γ n × C x as where we have used previous notations. Notice that the functions (σ + ) h and (σ − ) h are solutions of the Riccati type equation

Transformed Green's functions
Now, we analyze how Darboux-Crum transformations change Green's functions g and g h . For that, we will use solutions of the Riccati type equation (6.7) as a esential tool.
Let u be solution of s-KdV n equation (2.10). Let φ 1 and φ 2 be solutions of Schrödinger equation (6.5) for this potential and energy level E. Next we consider φ 0 a solution of Schrödinger equation for u and E 0 , with E 0 E and choose as corresponding point of the spectral curve (E 0 , µ 0 ). Recall that after applying a Darboux-Crum transformation with φ 0 to u, φ 1 and φ 2 , we get where σ 0 = (log φ 0 ) x is a solution of the Riccati equation σ 2 + σ x = u − E 0 . By Lemma 6.1, the function σ 0 equals where F 0 n = F n (E 0 ), is a solution of the same Riccati equation for E = E 0 . Thus, we conclude that we can perform a Darboux transformation using σ 0 instead of σ 0 . The transformed functions are solutions of the Schrödinger equation for potential Now, we take the functions σ 1 = (log φ 1 ) x and σ 2 = (log φ 2 ) x , which are solutions of the Riccati equation (6.7) for E E 0 . Then, by equations (6.8), we get the equalities Next we define the transformed Green's function The relations (6.21)-(6.23) link the Green's functions as follows: Hence we obtain a rational presentation of g as a consequence of the formulas (6.20) and (6.6). We write this formula in (6.25).
Proposition 6.4 (Lemma G.1 in [13]). Let u be solution of s-KdV n equation, let (E 0 , µ 0 ) and (E, µ) be two different points of Γ n . Then the transformed Green's function explicitly reads: where F n is a polynomial in E of degree n and µ is such that Γ n : µ 2 − R 2 n+1 = 0 for some polynomial R 2 n+1 (E) of degree 2 n + 1, with 0 ≤ n ≤ n + 1.
Using above notation we have the following results.
Remark 6.6. Formula is an homogeneous polynomial in E and ν of degree n + 1.
Remark 6.8. Formula ν 2 F n,xx 2 + (E − νu) F n is an homogeneous polynomial in E and ν of degree n + 1.

Darboux-Crum transformations for the Spectral curve
In this subsection we present how Darboux-Crum transformations affect the spectral curve Γ n . We observe that the action of the tranformation DT (φ 0 ) strongly depend on the type of point P in the spectral curve we use to construct φ 0 . In fact, if P is a regular point, the curve associated with the transformed potential is the same; in the other cases the new curve is a blowing-down or a blowing-up of Γ n .
Futhermore, the spectral curve associated to u is Γ n : µ 2 − R 2 n+1 = 0, with The idea of the proof is to compute Green's function (6.25) associated to u and interpret the result by means of Lemma 6.4.
Proof. First, we suppose that (E 0 , µ 0 ) is a regular point and µ 0 0. In this case, we compute We use Corollaries Appendix A.1 and Appendix A.2 to rewrite the expressions F 0 n F n,x −F 0 n,x F n and µ 2 (F 0 n ) 2 −µ 2 0 F 2 n . This yields to the equality Finally, we replace this expression in Green's fuction (6.25): Since F n = F n + P n,x 2F 0 n − P n σ 0 F 0 n is a polynomial in E of degree n, by means of Lemma 6.4, we conclude that n = n and µ = µ. Thus, R 2 n+1 = R 2n+1 .
Now, we suppose that (E 0 , µ 0 ) is a regular point and µ 0 = 0. In this case, we have that R 0 2n+1 = R 2n+1 (E 0 ) = 0 and R 0 2n+1,E = ∂ E (R 2n+1 )(E 0 ) 0, thus, where M 2n (E) is a polynomial in E of degree 2n such that M 2n (E 0 ) 0. Hence for µ 0 = 0, µ 2 = (E − E 0 )M 2n and Corollary Appendix A.1, the equality (6.25) becomes is a polynomial in E of degree n. By Lemma 6.4, we obtain that n = n, µ = µ and R 2 n+1 = R 2n+1 . Therefore, for regular points R 2 n+1 is a polynomial of degree 2n + 1 in E. By Corollary 2.11, we conclude that u is solution of a s-KdV n equation. Thus, a Darboux-Crum transformation with a regular point preserves the spectral curve and the level of the s-KdV hierarchy.
Next, we suppose that (E 0 , µ 0 ) is a singular point of Γ n , i.e., where Z 2n−1 (E) is a polynomial in E of degree 2n − 1. Hence for µ 0 = 0, µ 2 = (E − E 0 ) 2 Z 2n−1 and Appendix A.1, the equality (6.25) becomes is a polynomial in E of degree n. By Lemma 6.4, we obtain that n = n − 1 and µ = (E − E 0 ) −1 µ. Therefore, Next, we will proceed to establish the situation at the point of infinity P ∞ = [0 : 1 : 0] of the spectral curve. For that, we will need to work with the Zariski closure in P 2 of the spectral curve to understand its behavior under Darboux transformations for the energy level E 0 = 0 . In addition, we will use the blowing-up map in P 2 to control the KdV level of the transformed potential u.
Let π : P 2 → P 2 be the blowing-up of P 2 with center [0 : 0 : 1]. Hence, if [E : µ : ν] are homegeneous coordinates in P 2 , then the new ones are denoted by [ E : µ : ν], and π is given by Futhermore, the spectral curve associated to u is Γ n+1 : Proof. First, consider the homogeneized Green's function associated to transformed Green's function g. Then, by Propositions 6.5 and 6.7, ( g) h is a well defined rational function on Γ n . But also we have: ( g) h = G h • π on the spectral curve.
Moreover G h is a Green function for the curve defined by µ 2 − R 2n+3 ( E) = 0, where R 2n+3 ( E) = E 2 R 2n+1 (E); that is, for Γ n+1 , the strict transform of Γ n . Observe that R 2n+3 = E 2 R 2n+1 is a polynomial of degree 2n + 3 in E. Then, by Corollary 2.11, we conclude that u is solution of a s-KdV n+1 equation. 26 Finally we can rewrite 6.9 and 6.10 to establish how the spectral curve Γ n behaves under Darboux-Crum transformations. Futhermore, the spectral curve associated to u is Γ n : µ 2 − R 2 n+1 = 0, with if P is a regular point of Γ n , (E − E 0 ) −2 R 2n+1 if P is an affine singular point of Γ n . Example 6.12. Next we apply the previous theorem to a rational s-KdV 2 potential.
Take the s-KdV 2 potential u = 6 x 2 in the Schrödinger equation (6.5). The spectral curve associated to this potential is Γ 2 : µ 2 − E 5 = 0. When E = 0, we have the fundamental solutions φ 1 = x −2 and φ 2 = x 3 . We consider the Darboux transformations of u with these solutions: We have that potential u 1 is a solution of s-KdV 1 equation. It is well known that the spectral curve associated to this potential is Γ 1 : µ 2 − E 3 = 0, the blowing-up of Γ 2 at (0, 0). Furthermore, potential u 3 is a solution of s-KdV 3 equation, and its associated spectral curve Γ 3 is the blowing-down of Γ 2 , that is Γ 3 : µ 2 − E 7 = 0. Now, we take a regular value of E in Γ 2 , for instance, E = −1. Then, a solution of the Schrödinger equation (6.5) for this value of E is φ + = e x (x 2 −3x+3) x 2 . The Darboux transformation of u with this solution reads: Then this transformed potential is a solution of s-KdV 2 equation and the spectral curve associated to this potential is still Γ 2 : µ 2 − E 5 = 0.
We sum up this example in the following diagram: therefore, φ + is a solution for a regular point, ( u 1 , Γ 1 ) therefore, φ 1 is a solution for the affine singular point (0, 0).
Remark 6.13. The importance of Theorem 6.11 lies in the fact that we need to introduce the homogenized Green's function to state it. This new function is the essential tool that allows us to include in our study the point of infinity P ∞ of the affine curve Γ n . As far as we know, this is a new approach to the understanding of the spectral curve under Darboux transformations. Similar problems to our result 6.11 were treated by several authors, see [11,Thm 5] and [13,Thm G.2]. In [11], F. Ehlers and H. Knörrer studied the action of the Darboux transformations on the spectral curves by means of the eigenfunctions of the centralizer of the Schrödinger operator.

Spectral curves and KdV hierarchy in 1 + 1 dimensions
In this section we will show how the points of the spectral curves in the stationary setting are related with the solutions of the Schrödinger operator with rational potential in the 1 + 1 KdV hierarchy.
Recall that the rational soliton u r,n restricted to t = 0 is the well known n-soliton u (0) n (x) = n(n + 1)x −2 . Let Γ n be its affine spectral curve. This complex plane curve has a defining equation Our goal was to obtain the algebraic structure of a fundamental matrix of the Schrödinger operator −∂ 2 x + u r,n − E by means of the system (4.3). For this purpose we needed to use a parametric representation of the spectral curve Γ n . Observe that Γ n is a rational singular plane curve, nevertheless we can have a global parametrization in the sense given in [3]. In fact, we have taken the parametrization: and then E = −λ 2 as was taken since Section 4. Observe that the unique affine singular point of the spectral curve is reached for λ = 0. Hence, whenever λ 0 we obtain regular points on Γ n and we can obtain the desired description of the fundamental matrix B (r) n,λ as is given in Theorem 4.6. On the other hand, at the singular point χ(0) = (0, 0) the fundamental matrix for the system (4.3) must be obtained in a specific way, see Theorem 4.1.
The fundamental solutions φ 1,r,n (x, t), φ 2,r,n (x, t) obtained in Theorem 4.1 were used as source to perform Darboux transformations. In particular, for t = 0, we get the functions: and the corresponding potentials are transformed as is suggested in the following diagram: This situation is a particular case of a more general one that has been obtained in Theorem 6.11. The diagram (6.32) has its time dependent counterpart (see (4.8) and ( / / u r,n+1 (6.33) The fundamental matrix B (r) n,0 associated to the functions φ 1,r,n and φ 2,r,n can not be changed by the same Darboux transformations used for the potentials since there is a loss of independent solutions; in fact we have the following diagram On the other hand, whenever the point on the spectral curve is a regular point, that is λ 0, we have obtained the behavior of the fundamental matrices B (r) j,λ , for j = n − 1, n, n + 1, as it is encoded in the following diagram: All these situations are reflected in the time dependent frame coming from the stationary one, as we have seen. In particular, in the lack of specialization process from B (r) n,λ to B (r) n,0 . According to Theorem 4.8, we have that det B (r) n,λ = −2λ 2n+1 , whereas we have det B (r) n,0 = 2n + 1. 28 Remark 6.14. We notice then that, despite functions φ (0) 1,n and φ (0) 2,n are fundamental solutions of the Schrödinger equation for E = 0, they are not solutions for the same point of the spectral curve. Therefore, for each singular point of this spectral curve we can only compute one fundamental solution by means of Darboux transformations.
Next we have computed an explicit example to illustrate the relationship between spectral curves and KdV hierarchy in 1 + 1 dimensions for rational solitons.
Next, we perform the Darboux transformations by means of φ 1,1,2 and φ 2,1,2 to our initial potential u 1,2 . In these cases we have obtained / / u 1,3 = 6x(2x 9 + 675x 3 t 2 1 + 1350t 3 1 ) (x 6 + 15x 3 t 1 − 45t 2 1 ) 2 . (6.40) Then, we must consider the Schrödinger operators Their solutions φ + 1, j and φ − 1, j were given in Example 5.4. It should be noted that if the energy is not zero, these solutions inherit the same behavior as their corresponding potentials when the Darboux transformations DT (φ 1,1,2 ) and DT (φ 2,1,2 ) act on them. Hence we obtain the following diagram The zero energy case is essentially different from the point of view of the Darboux transformations. We only can partially obtain the previous diagram, To compute fundamental matrices associated to u 1,1 and u 1,3 we have to use Theorem 4.1 (see Example 4.4).

Differential Galois groups
In this section we study the Picard-Vessiot extensions of the differential systems (4.4) and (4.15), obtained for energy levels E = 0 and E 0 respectively. We denote the base differential field by K r = C(x, t r ) with constants field C.
We point out that the behavior that they present depend strongly on the affine point P = (E, µ) of the corresponding spectral curve. They present a similar behavior when the point P = (E, µ) is a regular point of Γ n .
A fundamental matrix for E = 0 can be also computed. However, it is not obtained by a specialization process from the fundamental matrix obtained for a regular point.
We obtain the Picard-Vessiot extensions given by B (r) n,0 and B (r) n,λ and compute their corresponding differential Galois group, say G (r) n,0 and G (r) n,λ respectively.
To compute the differential Galois group G (r) n,λ 0 in this case, we just have to compute the action of G (r) n,λ 0 on η r . For this, let σ in G (r) n,λ 0 be an automorphism of the differential Galois group, then Therefore σ(η r ) η r is a constant in K r . Hence σ(η r ) = c · η r for some c ∈ C. As a consequence we get that, for each λ 0 and every n, the differential Galois group is isomorphic to the multiplicative group, say Remark 7.1. Since the Galois groups G (r) n,λ 0 are obtained for a particular value of λ by especialization process, they do not depend on λ. For a spectral study of the Picard-Vessiot extensions see [17].

Global behavior of the differential Galois groups
Let us consider the family of linear algebraic groups G (r) n,λ λ∈C . Then for each point in Γ n we have found a linear algebraic group. As a result of our constructions we have a sheave structure of groups on the regular points of Γ n Γ n \ Sing( Γ n ) ∋ (−λ 2 , iλ 2n+1 ) −→ G (r) n,λ For each λ ∈ C, the situation is encoded in the following diagram We observe the invariance of the Galois groups with respect to: • Time (ie, it is invariant by the flow of the KdV equation).
• Generic values of the spectral parameter, ie, moving along the regular points of the spectral curve.

• Darboux transformations.
Although this invariant behaviour of the Galois group is proved here for the rational solutions of Adler-Moser type, we conjectured that it is also true for arbitrary algebro geometric solutions of the KdV, ie, for solutions associated to spectral curves different of µ 2 − E 2n+1 = 0.
for P n a polynomial in E of degree at most n − 1,as it is stated.
Corollary Appendix A.2. We have µ 2 (F 0 n ) 2 − µ 2 0 F 2 n = (E − E 0 )       F n F 0 n P n,x 2 + F 2 n (F 0 n ) 2 − P n (F n F 0 n,x + F n,x F 0 n ) 4 where P n is the polynomial obtained in Corollary Appendix A.1. In particular for E 0 = 0 we obtain µ 2 f 2 n − µ 2 0 F 2 n = E F n f n P n,x 2 + F 2 n ( f n ) 2 − P n (F n f n,x + F n,x f n ) 4 .
Corollary Appendix A.3. Let (E 0 , µ 0 ) be a regular point of Γ n and µ 0 = 0. We have that is a polynomial in E of degree n, with P n the polynomial obtained in Corollary Appendix A.1 and M 2n the polynomial defined in (A.3).
Proof. We have We replace these expressions in the formula and we get: The numerator of this function is a polynomial in E of degree n + 1 and has a root in E = E 0 as can be easily verified replacing E by E 0 : 2(F 0 n ) 2 F 0 n,xx − 4(u − E 0 )(F 0 n ) 3 − (F 0 n,x ) 2 F 0 n = 4F 0 n µ 2 0 = 0.
So, we get that 2(F 0 n ) 2 F n,xx − 4(u − E)(F 0 n ) 2 F n + (F 0 n,x ) 2 F n − 2F 0 n F 0 n,x F n,x = (E − E 0 )Q n , where Q n denotes a polynomial in E of degree n. Hence M 2n F n + (E − E 0 )P 2 n 4F n (F 0 n ) 2 = Q n 4(F 0 n ) 2 and then the result follows.
Corollary Appendix A.4. Let (E 0 , µ 0 ) be a singular point of Γ n . We have that Z 2n−1 F n + P 2 n 4F n (F 0 n ) 2 is a polynomial in E of degree n − 1, with P n the polynomial obtained in Corollary Appendix A.1 and Z 2n−1 the polynomial defined in (A.4).
Proof. It follows by an analogous computation to that of Corollary Appendix A.3.