Symmetry and holomorphy of the third-order ordinary differential equation satisfied by the third Painlevé Hamiltonian
aa r X i v : . [ m a t h . AG ] M a y Symmetry and holomorphy of the third-orderordinary differential equation satisfied by the thirdPainlev´e Hamiltonian
ByYusuke Sasano(The University of Tokyo, Japan)November 16, 2018
Abstract
We study symmetry and holomorphy of the third-order ordinary differen-tial equation satisfied by the third Painlev´e Hamiltonian.
Key Words and Phrases.
B¨acklund transformation, Birational transfor-mation, Holomorphy condition, Painlev´e equations.2000
Mathematics Subject Classification Numbers.
In this paper, we study symmetry and holomorphy of a 2-parameter family of third-order ordinary differential equation: d udt = 12 t (cid:0) dudt (cid:1) (cid:26)(cid:18) t d udt − (2 α − u − α (cid:19) (cid:18) t d udt + (2 α − u + α (cid:19)(cid:27) − dudt (cid:0) u + t dudt (cid:1) t + 4 t + (2 α − t dudt − t d udt , (1)where α , α are complex constants.The Hamiltonian of the third Painlev´e system (see [5, 6, 7, 9])(2) u := H III ( q, p, t ; α , α ) = q p ( p −
1) + q { (1 − α ) p − α } + tpt satisfies the equation (1). 1n [3, 4, 9], it is well-known that the Hamiltonian of the third Painlev´e systemsatisfies the second-order differential equation of binormal form. Since we studyits phase space, we need to remake a normal form. In order to consider the phasespace for the equation (1), by resolving its accessible singular locus we transformthe system of rational type into the system of polynomial type:(3) dxdt = y,dydt = yz + 2 α − t x + α t ,dzdt = − z − t x − y − t z + (2 α − α − t + 2 t . In this paper, we present symmetry and holomorphy of the system (3) (seeTheorems 4.1,4.2,4.3 and Proposition 4.1). At first, we make its phase space. Thesepatching data suggest its B¨acklund transformations. For example, it is well-knownthat the patching data(4) r : ( X, Y, Z ) = (cid:18) x , − ( xy + α ) x, z (cid:19) ( α ∈ C )suggests the symmetry condition:(5) s : ( x, y, z ; α ) → (cid:18) x + αy , y, z ; − α (cid:19) . We note that dX ∧ dY ∧ dZ = dx ∧ dy ∧ dz,ds ( x ) ∧ ds ( y ) ∧ ds ( z ) = dx ∧ dy ∧ dz. In the case of the Painlev´e systems, we can recover all birational symmetries fromthese patching data by using the correspondence (4),(5) when z = 0 (see [5, 11]).Systsm Holomorphy conditions Transition functions Symmetry P III D (1)6 -surface preserving 2-form (2 A ) (1) P II E (1)7 -surface preserving 2-form A (1)1 (3) Figure 1 preserving 3-form < s , s , π > Considering certain reductions of the soliton equations, some symmetries are closedin the Painlev´e systems. It is still an open question whether there exist some sym-metries of the soliton equations that are closed in the equations satisfied by thePainlev´e Hamiltonians. In this paper, we consider this problem from the differentviewpoints of certain reductions of the soliton equations.2he phase space (see Figure 1) for the system (3) is obtained by gluing threecopies of ( x i , y i , z i , t ) ∈ C × ( C − { } ) via the birational transformations preserving3-form: dx i ∧ dy i ∧ dz i = dx ∧ dy ∧ dz ( i = 0 , . These patching data r i ( i = 0 ,
1) are slightly different from the type (4). How-ever, for r i ( i = 0 , s , s by modifying thecorrespondence (4),(5), that is, the patching data(6) ˜ r : ( X, Y, Z ) = (cid:18) x , − ( xy + g ( z )) x, z (cid:19) ( g ( z ) ∈ C ( t )[ z ])suggests the symmetry condition:(7) ˜ s : ( x, y, z ) → (cid:18) x + g ( z ) y , y, z (cid:19) . These B¨acklund transformations s , s and π generate an infinite group (see Theorem4.3). In particular, the composition s s becomes its translation (see Proposition4.2). These properties are different from symmetry and holomorphy of the thirdPainlev´e system (see [5, 7, 8, 10, 12]).We also show that the system (3) with ( α , α ) = (cid:0) , (cid:1) admits a special solutionsolved by classical transcendental functions (see Section 5):(8) F ( a ; t ) = ∞ X k =0 a ) k t k k ! . Here the symbol ( a ) k denotes ( a ) k := a ( a + 1) · · · ( a + k − At first, we make the equation (1). After differentiating once, we obtain(9) dH III dt = − q ( p − qp + qp − α − α p ) t .
3e can express p by using the variables H III , dH III dt :(10) p = t dH III dt + H III . After differentiating again, we obtain(11) d H III dt = − p − qp + 2 qp − α − α p − H III + t dH III dt t . We can express q by using the variables H III , dH III dt , d H III dt :(12) q = 2 t dH III dt α + (2 α − H III + t (2 α + 1) dH III dt + t d H III dt . After differentiating, we obtain the equation (1).In order to consider the phase space for the equation (1), we make the rationaltransformation.
Proposition 2.1.
The rational transformations (13) x = u,y = dudt ,z = d udt − (2 α − u + α t dudt take the system of rational type to the system (3) of polynomial type. After we give the notion of accessible singularities and local index, we will proveProposition 2.1 in next section.
Let us review the notion of accessible singularity. Let B be a connected open domainin C and π : W −→ B a smooth proper holomorphic map. We assume that H ⊂ W is a normal crossing divisor which is flat over B . Let us consider a rational vectorfield ˜ v on W satisfying the condition˜ v ∈ H ( W , Θ W ( − log H )( H )) . Fixing t ∈ B and P ∈ W t , we can take a local coordinate system ( x , . . . , x n ) of W t centered at P such that H smooth can be defined by the local equation x = 0.4ince ˜ v ∈ H ( W , Θ W ( − log H )( H )), we can write down the vector field ˜ v near P = (0 , . . . , , t ) as follows:˜ v = ∂∂t + g ∂∂x + g x ∂∂x + · · · + g n x ∂∂x n . This vector field defines the following system of differential equations(14) dx dt = g ( x , . . . , x n , t ) , dx dt = g ( x , . . . , x n , t ) x , · · · , dx n dt = g n ( x , . . . , x n , t ) x . Here g i ( x , . . . , x n , t ) , i = 1 , , . . . , n, are holomorphic functions defined near P =(0 , . . . , , t ) . Definition 3.1.
With the above notation, assume that the rational vector field ˜ v on W satisfies the condition ( A ) ˜ v ∈ H ( W , Θ W ( − log H )( H )) . We say that ˜ v has an accessible singularity at P = (0 , . . . , , t ) if x = 0 and g i (0 , . . . , , t ) = 0 for every i, ≤ i ≤ n. If P ∈ H smooth is not an accessible singularity, all solutions of the ordinarydifferential equation passing through P are vertical solutions, that is, the solutionsare contained in the fiber W t over t = t . If P ∈ H smooth is an accessible singularity,there may be a solution of (14) which passes through P and goes into the interior W − H of W .Here we review the notion of local index . Let v be an algebraic vector field withan accessible singular point −→ p = (0 , . . . ,
0) and ( x , . . . , x n ) be a coordinate systemin a neighborhood centered at −→ p . Assume that the system associated with v near −→ p can be written as ddt Q x x ... x n = 1 x Q a a . . . a n Q − · Q x x ... x n + x h ( x , . . . , x n , t ) h ( x , . . . , x n , t )... h n ( x , . . . , x n , t ) , ( h i ∈ C ( t )[ x , . . . , x n ] , Q ∈ GL ( n, C ( t )) , a i ∈ C ( t ))(15)where h is a polynomial which vanishes at −→ p and h i , i = 2 , , . . . , n are polyno-mials of order at least 2 in x , x , . . . , x n , We call ordered set of the eigenvalues( a , a , · · · , a n ) local index at −→ p . 5e are interested in the case with local index(16) (1 , a /a , . . . , a n /a ) ∈ Z n . These properties suggest the possibilities that a is the residue of the formal Laurentseries:(17) y ( t ) = a ( t − t ) + b + b ( t − t ) + · · · + b n ( t − t ) n − + · · · ( b i ∈ C ) , and the ratio ( a /a , . . . , a n /a ) is resonance data of the formal Laurent series ofeach y i ( t ) ( i = 2 , . . . , n ), where ( y , . . . , y n ) is original coordinate system satisfying( x , . . . , x n ) = ( f ( y , . . . , y n ) , . . . , f n ( y , . . . , y n )) f i ( y , . . . , y n ) ∈ C ( t )( y , . . . , y n ).If each component of (1 , a /a , . . . , a n /a ) has the same sign, we may resolve theaccessible singularity by blowing-up finitely many times. However, when differentsigns appear, we may need to both blow up and blow down.The α -test,(18) t = t + αT, x i = αX i , α → , yields the following reduced system: ddT P X X ... X n = 1 X P a ( t ) a ( t ) . . . a n ( t ) P − · P X X ... X n , P ∈ GL ( n, C ) . (19)From the conditions (16), it is easy to see that this system can be solved by rationalfunctions. Proof of Proposition 2.1.
At first, setting p = u, q = dudt , r = d udt . Next, we calculate the singular points when q = 0. In this case, we solve theequation: (cid:0) t r − (2 α − p − α (cid:1) (cid:0) t r + (2 α − p + α (cid:1) = 0 . We obtain 2 solutions: r = (2 α − p + α t or − (2 α − α t . P, Q, R ) centered at the point (
P, Q, R ) =( p, , (2 α − p + α t ): P = p, Q = q, R = r − (2 α − p + α t . This system is rewritten as follows: ddt PQR = 1 Q α t
00 0 α t PQR + · · · satisfying (15). In this case, the local index is (cid:0) , α t , α t (cid:1) . This suggests the possi-bilities that we will resolve this accessible singular locus by blowing-up once in thedirection R .Blowing up along the curve { ( P, Q, R ) = ( P, , } : x = P, y = Q, z = RQ , then we obtain the system (3).In order to consider the phase spaces for the system (3), let us take the com-pactification [ z : z : z : z ] ∈ P of ( x, y, z ) ∈ C with the natural embedding( x, y, z ) = ( z /z , z /z , z /z ) . Moreover, we denote the boundary divisor in P by H . Extend the regular vectorfield on C to a rational vector field ˜ v on P . It is easy to see that P is covered byfour copies of C : U = C ∋ ( x, y, z ) ,U j = C ∋ ( X j , Y j , Z j ) ( j = 1 , , , via the following rational transformations X = 1 /x, Y = y/x, Z = z/x,X = x/y, Y = 1 /y, Z = z/y,X = x/z, Y = y/z, Z = 1 /z. Lemma 3.1.
The rational vector field ˜ v has two accessible singular loci :(20) ( C = { ( X , Y , Z ) | X = Z = 0 } ∪ { ( X , Y , Z ) | Y = Z = 0 } ∼ = P ,P = { ( X , Y , Z ) | X = Y = Z = 0 } , where C ∼ = P is multiple locus of order . P .Singular point Type of local index P ( , , )In order to do analysis for the accessible singular locus C , we need to replace asuitable coordinate system because each point has multiplicity of order 2.At first, let us do the Painlev´e test. To find the leading order behaviour of asingularity at t = t one sets x ∝ a ( t − t ) m ,y ∝ b ( t − t ) n ,z ∝ c ( t − t ) p , from which it is easily deduced that m = 1 , n = 2 , p = 1 . The order of pole ( m, n, p ) = (1 , ,
1) suggests a suitable coordinate system to doanalysis for the accessible singularities, which is explicitly given by( X (1) , Y (1) , Z (1) ) = (cid:18) xz , yz , z (cid:19) . In this coordinate, the singular point is given as follows: P = (cid:26) ( X (1) , Y (1) , Z (1) ) = (cid:18) − , − , (cid:19)(cid:27) . Let us take the coordinate system ( p, q, r ) centered at the point P : p = X (1) + 12 , q = Y (1) + 14 , r = Z (1) . The system (3) is rewritten as follows: ddt pqr = 1 r − − − − pqr + · · · satisfying (15). To the above system, we make the linear transformation XYZ = −
00 1 00 0 1 pqr
8o arrive at ddt XYZ = 1 Z − − − XYZ + · · · . In this case, the local index is ( − , − , − ).The α -test,(21) t = t + αT, X = αX , Y = αY , Z = αZ , α → , yields the following reduced system: ddt X Y Z = 1 Z − − − X Y Z . This system can be solved by rational functions:( X , Y , Z ) = (cid:18) c ( t − c ) , c ( t − c ) , −
12 ( t − c ) (cid:19) ( c i ∈ C ) . This suggests the possibilities that − is the residue of the formal Laurent series:(22) z ( t ) = − ( t − t ) + b + b ( t − t ) + · · · + b n ( t − t ) n − + · · · ( b i ∈ C ) , and the ratio ( − − , − − ) = (1 ,
4) is resonance data of the formal Laurent series of( x ( t ) , y ( t )), respectively. There exist meromorphic solutions with two free parame-ters which pass through P . Theorem 4.1.
The phase space X for the system (3) is obtained by gluing threecopies of C × C ∗ : U j × C ∗ ∼ = C × C ∗ ∋ { ( x j , y j , z j , t ) } , j = 0 , , E (1)1 E (2)1 E (1)2 E (2)2 E (3)2 E (5)2 E (4)2 y x z C Figure 1: The part surrounding bold lines coincides with ( − K ˜ X ) red (see Theorem4.1). The symbol E denotes the proper transform of boundary divisor of P × C ∗ by ϕ and E ( j ) i denote the exceptional divisors; E (1)2 is isomorphic to P , E (1)1 , E (3)2 areisomorphic to F , E (2)1 , E (4)2 , E (5)2 are isomorphic to P × P and E , E (2)2 are isomorphicto the surface which can be obtained by blowing up one point in P × P . via the following birational transformations :0) x = x, y = y, z = z, x = x, y = − (cid:18) yz + 2(2 α − x + 2 α t (cid:19) z, z = 1 z , x = x + z , y = − (( y + 2 t x − t − α − α − t ) z − α − tx + 2( α + 1) t + 4 α − α + 3 t + tz + 44 t z ) z,z = 1 z . (23) These transition functions satisfy the condition : dx i ∧ dy i ∧ dz i = dx ∧ dy ∧ dz ( i = 1 , . We note that C ∗ = C − { } . Theorem 4.2.
After nine successive blowing-ups in P × C ∗ , we obtain the smoothprojective -fold ˜ X and a morphism ϕ : ˜ X → P × C ∗ . Its canonical divisor K ˜ X is iven by K ˜ X = − E − E (1)1 − E (2)1 − E (1)2 − E (2)2 − E (3)2 − E (4)2 − E (5)2 , (24) where the symbol E denotes the proper transform of boundary divisor of P × C ∗ by ϕ and E ( j ) i denote the exceptional divisors; E (1)2 is isomorphic to P , E (1)1 , E (3)2 areisomorphic to F , E (2)1 , E (4)2 , E (5)2 are isomorphic to P × P and E , E (2)2 are isomorphicto the surface which can be obtained by blowing up one point in P × P (see Figure1) . Moreover, ˜ X − ( − K ˜ X ) red satisfies(25) ˜ X − ( − K ˜ X ) red = X . Proof of Theorems 4.1 and 4.2.
By the following steps, we can resolve the accessible singular point Q := { ( X , Y , Z ) = (0 , , } . Step 1 : We blow up at Q : p (1) = X Z , q (1) = Z , r (1) = Y Z . Step 2 : We blow up along the curve { ( p (1) , q (1) , r (1) ) = ( p (1) , , } : p (2) = p (1) , q (2) = q (1) r (1) , r (2) = r (1) . Step 3 : We make the change of variables: p (3) = p (2) , q (3) = 1 q (2) , r (3) = r (2) . Step 4 : We blow up at the point { ( p (3) , q (3) , r (3) ) = ( − , − , } : p (4) = p (3) + r (3) , q (4) = q (3) + r (3) , r (4) = r (3) . Step 5 : We blow up along the curve { ( p (4) , q (4) , r (4) ) | q (4) = r (4) = 0 } : p (5) = p (4) , q (5) = q (4) r (4) , r (5) = r (4) . Step 6 : We blow up along the curve { ( p (5) , q (5) , r (5) ) | q (5) − − α +4 α +4 t − tp (5) t = r (5) = 0 } : p (6) = p (5) , q (6) = q (5) − − α +4 α +4 t − tp (5) t r (5) , r (6) = r (5) . tep 7 : We blow up along the curve { ( p (6) , q (6) , r (6) ) | q (6) − − α +4 α +2 t +2 α t − tp (6) +4 α tp (6) t = r (6) = 0 } : p (7) = p (6) , q (7) = q (6) − − α +4 α +2 t +2 α t − tp (6) +4 α tp (6) t r (6) , r (7) = r (6) . We have resolved the accessible singular point Q . By choosing a new coordinatesystem as ( x , y , z ) = ( p (7) , − q (7) , r (7) ) , we can obtain the coordinate system ( x , y , z ) in the description of X given inTheorem 4.1.The proof on the accessible singular point P is same.Now we complete the proof of Theorems 4.1 and 4.2. Proposition 4.1.
Let us consider a system of first order ordinary differential equa-tions in the polynomial class : dxdt = f ( x, y, z ) , dydt = f ( x, y, z ) , dzdt = f ( x, y, z ) . We assume that( A deg ( f i ) = 3 with respect to x, y, z .( A
2) The right-hand side of this system becomes again a polynomial in eachcoordinate system ( x i , y i , z i ) ( i = 1 , Theorem 4.3.
The system (3) is invariant under the following transformations : s : ( x, y, z, t ; α , α ) → (cid:18) x, y, z + 2(2 α − x + 2 α t y , t ; − α , − α (cid:19) ,s : (cid:18) x, y, z, t ; α , (cid:19) → (cid:18) x + 2 G, y + 2 G − G , z − G, t ; − − α , (cid:19) ,π : (cid:18) x, y, z, t ; α , (cid:19) → (cid:18) x + z , − y − xt − z − zt + (2 α − α − t + 1 t , − z − t , t ; − α − , (cid:19) , where the symbol G denotes(26) G := t { α − x + (2 α − z + 2( α + 1) } + (2 α − α − t { t (4 y + z ) + 4 t (2 x + z − − (2 α − α − } . We note that the transformations s and π become its B¨acklund transformationsfor the system (3) only if α = , respectively. Proposition 4.2.
The transformation s s acts on α as (27) s s ( α ) = α − . Special solutions
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