Studies on the second member of the second Painlevé hierarchy
aa r X i v : . [ m a t h . AG ] N ov STUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV ´EHIERARCHY
Abstract.
In this paper, we study the second member of the second Painlev´e hierarchy P (2) II . We show that the birational transformations take this equation to the polynomialHamiltonian system in dimension four, and this Hamiltonian system can be consideredas a 1-parameter family of coupled Painlev´e systems. This Hamiltonian is new. We alsoshow that this system admits extended affine Weyl group symmetry of type A (1)1 , andcan be recovered by its holomorphy conditions. We also study a fifth-order ordinarydifferential equation satisfied by this Hamiltonian. After we transform this equation intoa system of the first-order ordinary differential equations of polynomial type in dimensionfive by birational transformations, we give its symmetry and holomorphy conditions. Introduction
In this paper, we study the second member of the second Painlev´e hierarchy [1, 2, 3]explicitly given by(1) P (2) II : d udt = 10 u (cid:18) dudt (cid:19) + 10 u d udt − u + tu + α ( α ∈ C ) . It was found by Martynov in 1973 (see [5]) and rediscovered independently by Ablowitzand Segur in 1977 (see [6]) and Flaschka and Newell in 1980 (see [7]). It is known thatthis system can be obtained by self-similar reduction of the Modified KdV5 equation(2) g = 10( g − R ( s )) g + 40 gg w g ww + 10 g w − g − R ( s )) g w + g s , where R ( s ) is an arbitrary locally analytic function of s .We note that this equation appears as the equation F-XVII in Cosgrove’s classificationof the fourth-order ordinary differential equations in the polynomial class having thePainlev´e property (see [17]).At first, we show that the birational transformations (see Section 2) take the equation(1) to a polynomial Hamiltonian system in dimension four. We make this polynomialHamiltonian from the viewpoint of accessible singularity and local index (see Section 5).This Hamiltonian system can be considered as a 1-parameter family of coupled Painlev´esystems in dimension four. This Hamiltonian is new.It is known that for the equation (2) M. Mazzocco and M. Y. Mo found a canonicalvariables ( q i , p i ) i =1 and obtained a polynomial Hamiltonian through the monodromy pre-serving deformation equation of the second-order ordinary differential equation (see [2]). Mathematics Subjet Classification . 34M55; 34M45; 58F05; 32S65.
Key words and phrases.
Affine Weyl group, birational symmetry, coupled Painlev´e system.
The degree of this Hamiltonian with respect to q , p , q , p is 4. On the other hand, oneof our Hamiltonian is 3.We also study its symmetry and holomorphy conditions. We show that this systemadmits extended affine Weyl group symmetry of type A (1)1 as the group of its B¨acklundtransformations. These B¨acklund transformations satisfy(3) s i ( g ) = g + α i f i { f i , g } + 12! (cid:18) α i f i (cid:19) { f i , { f i , g }} + · · · ( g ∈ C ( t )[ q , p , q , p ]) , where poisson bracket { , } satisfies the relations: { p , q } = { p , q } = 1 , the others are . Since these B¨acklund transformations have Lie theoretic origin, similarity reduction of aDrinfeld-Sokolov hierarchy admits such a B¨acklund symmetry.These properties of its symmetry and holomorphy conditions are new.We find a rational solution of this Hamiltonian system given by(4) ( q , p , q , p ; α ) = (cid:18) , − t , ,
0; 0 (cid:19) as a seed solution. Applying the translation operators, we can obtain an infinite series ofthe rational solutions.Finally, we also study a fifth-order ordinary differential equation satisfied by this Hamil-tonian. We show that the birational transformations take this equation to a system ofthe first-order ordinary differential equations of polynomial type in dimension five. Forthis system, we give its symmetry and holomorphy conditions.It is still the following open questions on the system (6):(1) Relation between Mazzocco’s Hamiltonian (see [2]) with (7).(2) Special polynomials.(3) Lax pair.(4) Reduction involving the A (1)1 -symmetry.(5) Bilinear form.(6) Classification of the rational solutions or special solutions.(7) Irreducibility.(8) Discrete version.(9) q-Analogue.(10) Quantum version.(11) The remained members of P ( n ) II . TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 3 Polynomial HamiltonianTheorem . The birational transformations (5) q = u,p = d udt + (cid:18) dudt (cid:19) − t (cid:18) u − u dudt − d udt (cid:19) u,q = d udt − u dudt ,p = dudt − u take the system to the Hamiltonian system (6) dq dt = ∂H∂p = q + p ,dp dt = − ∂H∂q = − q p + α − ,dq dt = ∂H∂p = − p + p + t ,dp dt = − ∂H∂q = q with the polynomial Hamiltonian (cf. [2]) H = K ( q , p ; α ) + H I ( q , p , t ) + p p = q p + (cid:18) − α (cid:19) q − p + t p − q p p . (7)The symbols K ( x, y ; α ) and H I ( z, w, t ) denote K ( x, y ; α ) = x y + (cid:18) − α (cid:19) x,H I ( z, w, t ) = − w + t w − z . (8)The system with the Hamiltonian K ( x, y ; α ) has itself as its first integral, and H I ( z, w, t )denotes the Painlev´e I Hamiltonian.This system is a 1-parameter family of coupled Painlev´e systems. This Hamiltonian isnew (cf. [2]).Before we will prove Theorem 2.1, we review the notion of accessible singularity andlocal index. 3. Accessible singularity and local index
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
STUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY normal crossing divisor which is flat over B . Let us consider a rational vector field ˜ 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. Since˜ 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(9) 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 . 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 ordinary differentialequation passing through P are vertical solutions, that is, the solutions are contained inthe fiber W t over t = t . If P ∈ H smooth is an accessible singularity, there may be asolution of (9) 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 with anaccessible singular point −→ p = (0 , . . . ,
0) and ( x , . . . , x n ) be a coordinate system in aneighborhood centered at −→ p . Assume that the system associated with v near −→ p can bewritten as ddt x x ... x n − x n = 1 x a . . . a a . . . a ( n − a ( n − . . . a ( n − n − a n a n . . . a n ( n − a nn x x ... x n − x n + x h ( x , . . . , x n , t ) h ( x , . . . , x n , t )... h n − ( x , . . . , x n , t ) h n ( x , . . . , x n , t ) , ( h i ∈ C ( t )[ x , . . . , x n ] , a ij ∈ C ( t ))(10)where h is a polynomial which vanishes at −→ p and h i , i = 2 , , . . . , n are polynomials oforder at least 2 in x , x , . . . , x n , We call ordered set of the eigenvalues ( a , a , · · · , a nn ) local index at −→ p . TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 5
We are interested in the case with local index(11) (1 , a /a , . . . , a nn /a ) ∈ Z n . These properties suggest the possibilities that a is the residue of the formal Laurentseries:(12) y ( t ) = a ( t − t ) + b + b ( t − t ) + · · · + b n ( t − t ) n − + · · · ( b i ∈ C ) , and the ratio (1 , a /a , . . . , a nn /a ) is resonance data of the formal Laurent seriesof each 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 nn /a ) has the same sign, we may resolve theaccessible singularity by blowing-up finitely many times. However, when different signsappear, we may need to both blow up and blow down.The α -test,(13) t = t + αT, x i = αX i , α → , yields the following reduced system: ddT X X ... X n − X n = 1 X a ( t ) 0 0 . . . a ( t ) a ( t ) 0 . . . a ( n − ( t ) a ( n − ( t ) . . . a ( n − n − ( t ) 0 a n ( t ) a n ( t ) . . . a n ( n − ( t ) a nn ( t ) X X ... X n − X n , (14)where a ij ( t ) ∈ C . Fixing t = t , this system is the system of the first order ordinarydifferential equation with constant coefficient. Let us solve this system. At first, we solvethe first equation:(15) X ( T ) = a ( t ) T + C ( C ∈ C ) . Substituting this into the second equation in (14), we can obtain the first order linearordinary differential equation:(16) dX dT = a ( t ) X a ( t ) T + C + a ( t ) . By variation of constant, in the case of a ( t ) = a ( t ) we can solve explicitly:(17) X ( T ) = C ( a ( t ) T + C ) a t a t + a ( t )( a ( t ) T + C ) a ( t ) − a ( t ) ( C ∈ C ) . This solution is a single-valued solution if and only if a ( t ) a ( t ) ∈ Z . STUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY
In the case of a ( t ) = a ( t ) we can solve explicitly:(18) X ( T ) = C ( a ( t ) T + C ) + a ( t )( a ( t ) T + C )Log( a ( t ) T + C ) a ( t ) ( C ∈ C ) . This solution is a single-valued solution if and only if a ( t ) = 0 . Of course, a ( t ) a ( t ) = 1 ∈ Z . In the same way, we can obtain the solutions for each variables( X , . . . , X n ). The conditions a jj ( t ) a ( t ) ∈ Z , ( j = 2 , , . . . , n ) are necessary condition in orderto have the Painlev´e property.4. The case of the second Painlev´e system
In this section, we review the case of the second Painlev´e system:(19) d udt = 2 u + tu + α ( α ∈ C ) . Let us make its polynomial Hamiltonian from the viewpoint of accessible singularityand local index.
Step 0:
We make a change of variables.(20) x = u, y = dudt . Step 1:
We make a change of variables.(21) x = 1 x , y = yx . In this coordinate system, we see that this system has two accessible singular points:(22) ( x , y ) = { (0 , , (0 , − } . Around the point ( x , y ) = (0 , Step 2:
We make a change of variables.(23) x = x , y = y − . In this coordinate system, we can rewrite the system satisfying the condition (10): ddt x y ! = 1 x ( − − ! x y ! + · · · ) , and we can obtain the local index ( − , −
4) at the point { ( x , y ) = (0 , } . The ratio ofthe local index at the point { ( x , y ) = (0 , } is a positive integer.We aim to obtain the local index ( − , −
2) by successive blowing-up procedures.
Step 3:
We blow up at the point { ( x , y ) = (0 , } .(24) x = x , y = y x . TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 7
Step 4:
We blow up at the point { ( x , y ) = (0 , } .(25) x = x , y = y x . In this coordinate system, we see that this system has the following accessible singularpoint:(26) ( x , y ) = (0 , t/ . Step 5:
We make a change of variables.(27) x = x , y = y − t/ . In this coordinate system, we can rewrite the system as follows: ddt x y ! = 1 x ( − α − / − ! x y ! + · · · ) , and we can obtain the local index ( − , − x , y ) and ( x, y )is given by x = 1 x ,y = y − x − t . Finally, we can choose canonical variables ( q, p ). Step 9:
We make a change of variables.(28) q = 1 x , p = y , and we can obtain the system dqdt = q + p + t ,dpdt = − qp + α − H II :(29) H II = q p + 12 p + t p − (cid:18) α − (cid:19) q. We remark that we can discuss the case of the accessible singular point ( x , y ) = (0 , − x , y ) = (0 , STUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY Proof of theorem 2.1
By the same way of the second Painlev´e system, we can prove Theorem 2.1.At first, we rewrite the equation (5) to the system of the first order ordinary differentialequations.
Step 0:
We make a change of variables.(30) x = u, y = dudt , z = d udt , w = d udt . Step 1:
We make a change of variables.(31) x = 1 x , y = yx , z = zx , w = wx . In this coordinate system, we see that this system has four accessible singular points:(32) ( x , y , z , w ) = (cid:26) (0 , , , , (0 , − , , − , (cid:18) , , , (cid:19) , (cid:18) , − , , − (cid:19)(cid:27) . Around the point ( x , y , z , w ) = (0 , − , , − Step 2:
We make a change of variables.(33) x = x , y = y + 1 , z = z − , w = w + 6 . In this coordinate system, we can rewrite the system satisfying the condition (10): ddt x y z w = 1 x − x y z w + · · · . To the above system, we make the linear transformation: X Y Z W = − − x y z w to arrive at ddt X Y Z W = 1 X X Y Z W + · · · , and we can obtain the local index (1 , , ,
6) at the point { ( X , Y , Z , W ) = (0 , , , } .The continued ratio of the local index at the point { ( X , Y , Z , W ) = (0 , , , } are allpositive integers(34) (cid:18) , , (cid:19) = (2 , , . TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 9
This is the reason why we choose this accessible singular point.We aim to obtain the local index (1 , , ,
2) by successive blowing-up procedures.
Step 3:
We blow up at the point { ( x , y , z , w ) = (0 , , , } .(35) x = x , y = y x , z = z x , w = w x . Step 4:
We blow up at the point { ( x , y , z , w ) = (0 , , , } .(36) x = x , y = y x , z = z x , w = w x . In this coordinate system, we see that this system has the following accessible singularlocus:(37) ( x , y , z , w ) = (0 , y , − y , y ) . Step 5:
We blow up along the curve { ( x , y , z , w ) = (0 , y , − y , y ) } .(38) x = x , y = y , z = z + 2 y x , w = w − y x . In this coordinate system, we see that this system has the following accessible singularlocus:(39) ( x , y , z , w ) = (0 , y , z , − z ) . Step 6:
We blow up along the surface { ( x , y , z , w ) = (0 , y , z , − z ) } .(40) x = x , y = y , z = z , w = w + 2 z x . In this coordinate system, we see that this system has the following accessible singularlocus:(41) ( x , y , z , w ) = (cid:18) , y , z , y − t (cid:19) . Step 7:
We make a change of variables.(42) x = x , y = y , z = z , w = w − y + t . In this coordinate system, we can rewrite the system as follows: ddt x y z w = 1 x − t α + x y z w + · · · , and we can obtain the local index (1 , , , x , y , z , w )and ( x, y, z, w ) is given by x = 1 x ,y = x + y,z = z + 2 xy,w = w + t − x − x y − y + 2 xz. Step 8:
We make a change of variables.(43) x = 1 x , y = y , z = z , w = w . In this coordinate system, we can rewrite the system as follows: dx dt = − x + y ,dy dt = z ,dz dt = 3 y + w − t ,dw dt = 2 x w + α + 12 . Finally, we can choose canonical variables ( q , p , q , p ). Step 9:
We make a change of variables.(44) q = − x , p = − w , q = − z , p = − y , and we can obtain the system (6) with the polynomial Hamiltonian (7).Thus, we have completed the proof of Theorem 2.1 (cid:3) .We note on the remaining accessible singular points.Around the point ( x , y , z , w ) = (0 , , , Step 2:
We make a change of variables.(45) x = x , y = y − , z = z − , w = w − . In this coordinate system, we can rewrite the system satisfying the condition (10): ddt x y z w = 1 x − − − − − − x y z w + · · · . and we can obtain the local index ( − , − , − , −
6) at the point { ( x , y , z , w ) = (0 , , , } .The continued ratio of the local index at the point { ( x , y , z , w ) = (0 , , , } are all TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 11 positive integers(46) (cid:18) − − , − − , − − (cid:19) = (2 , , . We remark that we can discuss this case in the same way as in the case of ( x , y , z , w ) =(0 , − , , − x , y , z , w ) = (0 , / , / , / Step 2:
We make a change of variables.(47) x = x , y = y − / , z = z − / , w = w − / . In this coordinate system, we can rewrite the system satisfying the condition (10): ddt x y z w = 1 x − / − − / − / − x y z w + · · · . and we can obtain the local index ( − / , / , − , −
4) at the point { ( x , y , z , w ) =(0 , , , } . The continued ratio of the local index at the point { ( x , y , z , w ) = (0 , , , } are(48) (cid:18) / − / , − − / , − − / (cid:19) = ( − , , . In this case, the local index involves a negative integer. So, we need to blow down.Around the point ( x , y , z , w ) = (0 , − / , / , − / Step 2:
We make a change of variables.(49) x = x , y = y + 1 / , z = z − / , w = w + 3 / . In this coordinate system, we can rewrite the system satisfying the condition (10): ddt x y z w = 1 x / − / / − x y z w + · · · . and we can obtain the local index (1 / , − / , ,
4) at the point { ( x , y , z , w ) = (0 , , , } .The continued ratio of the local index at the point { ( x , y , z , w ) = (0 , , , } are(50) (cid:18) − / / , / , / (cid:19) = ( − , , . In this case, the local index involves a negative integer. So, we need to blow down. Symmetry and holomorphy conditions
In this section, we study the symmetry and holomorphy conditions of the system (6).These symmetries, holomorphy conditions and invariant divisors are new.
Theorem . Let us consider a polynomial Hamiltonian system with Hamiltonian H ∈ C ( t )[ q , p , q , p ] . We assume that ( A deg ( H ) = 5 with respect to q , p , q , p . ( A This system becomes again a polynomial Hamiltonian system in each coordinatesystem r i ( i = 0 , r : ( x , y , z , w ) = (cid:18) q , − (cid:18) q p + 12 − α (cid:19) q , q , p (cid:19) ,r : ( x , y , z , w ) =( 1 q , − (cid:0) ( p − p + t + 4 q ( q p + q )) q + 1 / α (cid:1) q ,q + 4 q ( q + p ) , p + 2 q ) . (51) Then such a system coincides with the system (6) with the polynomial Hamiltonian (7) . We note that the condition ( A
2) should be read that r ( H ) , r ( H − q )are polynomials with respect to x i , y i , z i , w i . Theorem . The system (6) admits extended affine Weyl group symmetry of type A (1)1 as the group of its B¨acklund transformations, whose generators s , s , π defined asfollows : with the notation ( ∗ ) := ( q , p , q , p , t ; α ): s : ( ∗ ) → (cid:18) q + − α p , p , q , p , t ; 1 − α (cid:19) ,s : ( ∗ ) → ( q + 2 α + 12( p + t − p + 4 q ( q + q p )) ,p − α + 1)( q + 2 q p ) p + t − p + 4 q ( q + q p ) + (2 α + 1) ( p + 2 q )( p + t − p + 4 q ( q + q p )) ,q − α + 1)( p − q ) p + t − p + 4 q ( q + q p ) + 3(2 α + 1) q ( p + t − p + 4 q ( q + q p )) + (2 α + 1) p + t − p + 4 q ( q + q p )) ,p − α + 1) q p + t − p + 4 q ( q + q p ) − (2 α + 1) ( p + t − p + 4 q ( q + q p )) ,t ; − − α ) ,π : ( ∗ ) → ( − q , − ( p + t − p + 4 q ( q + q p )) , − ( q + 4 q ( q + p )) , − ( p + 2 q ) , t ; − α ) . TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 13
Proposition . Let us define the following translation operators (52) T := πs , T := s π : T ( ∗ ) → ( − q − α + 12( p + t − p + 4 q ( q + q p )) , − p − t + 2 p − q ( q + q p )) , − q − q ( q + p ) , − p − q , t ; α + 1) and T ( ∗ ) → ( − q + 2 α − p , − p − t + 2 p − (2 q p − α + 1)( p + 2 q p p + 2 p q − α p ) p , − q − q p + 1 − α ) p p − (2 q p + 1 − α ) p , − p − (2 q p + 1 − α ) p , t ; α − . These translation operators act on α as follows :(53) T ( α ) = α + 1 , T ( α ) = α − . Finally, we study a solution of the system (6) which is written by the use of knownfunctions.By the transformation π , the fixed solution is derived from α = − α ,q = − q , p = − ( p + t − p + 4 q ( q + q p )) ,q = − ( q + 4 q ( q + p )) , p = − ( p + 2 q ) . (54)Then we obtain(55) ( q , p , q , p ; α ) = (cid:18) , − t , ,
0; 0 (cid:19) as a seed solution.Applying the B¨acklund transformations T , T , we can obtain an infinite series of therational solutions: α q p q p -3 t +96) t ( t − − t ( t − t − t − t +2088 t +114048 t − t ( t − − t +432 t +3456) t ( t − -2 t t − t t − t -1 t − t t − t − t − t − t − t − t t − t − t +96) t ( t − − t − t t − t The system (6) with special parameter α = admits a particular solution expressedin terms of the Painlev´e I function ( q , p ):(56) p = 0 , and(57) dq dt = q + p ,dq dt = − p + t ,dp dt = q . Other polynomial Hamiltonian system
In this section, we study some Hamiltonians transformed by birational and symplectictransformations r and r (see (51)) for the Hamiltonian (7), respectively.At first, we study the following Hamiltonian system explicitly given by(58) dq dt = ∂ ˜ H∂p = − q p − ,dp dt = − ∂ ˜ H∂q = 2 q p p + (cid:18) − α (cid:19) p ,dq dt = ∂ ˜ H∂p = − p + t − q p − (cid:18) − α (cid:19) q ,dp dt = − ∂ ˜ H∂q = q with the polynomial Hamiltonian˜ H = − p + H I ( q , p , t ) − (cid:18) q p + 1 − α (cid:19) q p = − p − q − p + t p − (cid:18) q p + 1 − α (cid:19) q p , (59)where H I ( q, p, t ) := − q − p + t p is the Painlev´e I Hamiltonian. Proposition . The birational and symplectic transformation r (see (51)) takesthe system (6) into the Hamiltonian system (58). Theorem . Let us consider a polynomial Hamiltonian system with Hamiltonian H ∈ C ( t )[ q , p , q , p ] . We assume that ( B deg ( H ) = 5 with respect to q , p , q , p . TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 15 ( B This system becomes again a polynomial Hamiltonian system in each coordinatesystem ˜ r i ( i = 0 , r : ( x , y , z , w ) = (cid:18) q , − (cid:18) q p −
12 + α (cid:19) q , q , p (cid:19) , ˜ r : ( x , y , z , w ) = (cid:18) q , p + 2 α q + 2 p − tq − q q − p q , q + 4 p q + 4 q , p + 2 q (cid:19) . Then such a system coincides with the system (58) with the polynomial Hamiltonian (59) . We note that the condition ( B
2) should be read that˜ r ( H ) , ˜ r (cid:18) H − q (cid:19) are polynomials with respect to x i , y i , z i , w i . Theorem . The system (58) is invariant under the following transformations : withthe notation ( ∗ ) := ( q , p , q , p , t ; α ): s : ( ∗ ) → (cid:18) q + α − p , p , q , p , t ; 1 − α (cid:19) ,s : ( ∗ ) → (cid:18) − q , − p − α q − p − tq + 4 q q + 4 p q , − q − p q − q , − p − q , t ; − α (cid:19) . We remark that the birational and symplectic transformation r (see (51)) takes thesystem (6) into a polynomial Hamiltonian system. We can also obtain the same resultson this Hamiltonian.8. Fifth-order ordinary differential equation satisfied the Hamiltonian
In this section, we study a 1-parameter family of the fifth-order ordinary differentialequation: d udt =((1 − α ) α − dudt (cid:18) dudt (cid:19) − t ! − d udt (cid:18) dudt − t d udt (cid:19) + 8 d udt dudt t − (cid:18) dudt (cid:19) ! − (cid:18) d udt (cid:19) − dudt d udt ! + 2 d udt (cid:18) dudt d udt + 2 d udt − (cid:19) ) / (cid:18) dudt (cid:19) + 8 d udt − t ! . (60)The Hamiltonian (7) u := H satisfies the equation (60).In order to obtain the symmetry and holomorphy conditions for the equation (60),by making birational transformations we transform the system of rational type into the system of the first-order ordinary differential equations of polynomial type in dimensionfive:(61) dxdt = y,dydt = z,dzdt = − y + w + t ,dwdt = wq + 2 α − ,dqdt = − q − y. This system is a Riccati extension of the system (6) by a scale transformation.9.
Differential system of Polynomial type
At first, we make the equation (60). After differentiating once, we obtain dHdt = w . We can express w by using the variables dHdt : w = 2 dHdt . After differentiating again, we obtain d Hdt = z . We can express z by using the variables d Hdt : z = 2 d Hdt . After differentiating again, we obtain d Hdt = 14 t + 2 y − (cid:18) dHdt (cid:19) ! . We can express y by using the variables dHdt , d Hdt : y = 12 − t + 24 (cid:18) dHdt (cid:19) + 4 d Hdt ! . After differentiating again, we obtain d Hdt = 12 tx + α − x (cid:18) dHdt (cid:19) − dHdt d Hdt − x d Hdt ! . TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 17
We can express x by using the variables dHdt , d Hdt , d Hdt , d Hdt : x = 2 d Hdt + 24 dHdt d Hdt − α t − (cid:0) dHdt (cid:1) − d Hdt . After differentiating, we obtain the equation (60).Now, let us transform the equation (60) into a system of polynomial type by birationaltransformations.
Theorem . The birational transformations x = u,y = dudt ,z = d udt ,w = d udt − t − (cid:18) dudt (cid:19) ! ,q = d udt − α − dudt d udt d udt − (cid:16) t − (cid:0) dH III dt (cid:1) (cid:17) take the system of rational type into the system (61) . Proof.
At first, we rewrite the equation (60) to the system of the first-order ordinarydifferential equations.
Step 0:
We make a change of variables.(62) x = u, y = dudt , z = d udt , w = d udt , q = d udt . Step 1:
We make a change of variables.(63) x = x, y = y, z = z, w = w + 6 y − t , q = q. In this coordinate system, we see that this system has two accessible singular loci:(64)( x , y , z , w , q ) = (cid:26)(cid:18) x , y , z , w , − y z + 12 α (cid:19) , (cid:18) x , y , z , w , − y z + 1 − α (cid:19)(cid:27) . Step 2:
We make a change of variables.(65) x = x , y = y , z = z , w = w , q = q + 12 y z − α . In this coordinate system, we can rewrite the system satisfying the condition (10): ddt x y z w q = 1 w t
00 0 0 α −
00 0 0 0 α − x y z w q + · · · , and we can obtain the local index (cid:0) , , , α − , α − (cid:1) at the point { ( x , y , z , w , q ) =(0 , , , , } . The continued ratio of the local index at the point { ( x , y , z , w , q ) =(0 , , , , } are all positive integers(66) (cid:18) α − , α − , α − , α − α − (cid:19) = (0 , , , . Let us resolve this accessible singular locus.
Step 3:
We blow up along the 3-fold { ( x , y , z , w , q ) = ( x , y , z , , } .(67) x = x , y = y , z = z , w = w , q = q w , and we can obtain the system (61).Thus, we have completed the proof of Theorem 9.1. (cid:3) We remark that we can discuss the case of( x , . . . , w , q ) = (cid:18) x , . . . , w , − y z + 1 − α (cid:19) in the same way.Next, we give its symmetry and holomorphy conditions. Theorem . The system (61) admits extended affine Weyl group symmetry of type A (1)1 as the group of its B¨acklund transformations, whose generators s , s , π defined as TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 19 follows : with the notation ( ∗ ) := ( x, y, z, w, q, t ; α ): s : ( ∗ ) → (cid:18) x, y, z, w, q + α − w , t ; 1 − α (cid:19) ,s : ( ∗ ) → ( x + 2 α + 12( t + 2 w + 2 yq − y − zq ) ,y + (2 α + 1) q t + 2 w + 2 yq − y − zq ) − (2 α + 1) t + 2 w + 2 yq − y − zq ) ,z + (2 α + 1)( q − y )4( t + 2 w + 2 yq − y − zq ) − α + 1) q t + 2 w + 2 yq − y − zq ) + (2 α + 1) t + 2 w + 2 yq − y − zq ) ,w + 2(2 α + 1)( yq − z ) t + 2 w + 2 yq − y − zq + (2 α + 1) ( q + 4 y )4( t + 2 w + 2 yq − y − zq ) ,q − α + 1 t + 2 w + 2 yq − y − zq , t ; − − α ) ,π : ( ∗ ) → ( x + q , − (cid:18) y + q (cid:19) , − (cid:18) z − q ( q + 8 y ) (cid:19) , − (cid:18) w + yq − y − zq + t (cid:19) , − q, t ; − α ) . Theorem . Let us consider a system of first order ordinary differential equationsin the polynomial class : dxdt = f ( x, y, z, w, q ) , dydt = f ( x, y, z, w, q ) , . . . , dqdt = f ( x, y, z, w, q ) ,f i ∈ C ( t )[ x, y, z, w, q ] ( i = 1 , · · · , . We assume that( A deg ( f i ) = 2 with respect to x, y, z, w, q .( A
2) The right-hand side of this system becomes again a polynomial in each coordinatesystem r i : ( x i , y i , z i , w i , q i ) ( i = 1 , ,
3) : r : ( x , y , z , w , q ) = (cid:18) x, y, z, − (cid:18) wq + 12 (2 α − (cid:19) q, q (cid:19) ,r : ( x , y , z , w , q ) =( x + q , y + q , z − q ( q + 8 y ) , − (cid:18)(cid:18) w + t − y + q ( yq − z ) (cid:19) q − − α (cid:19) q, q ) , r : ( x , y , z , w , q ) =( 1 x , yx + x, − x z − y (4 t + 5 w − y ) − x (4 wq + 2 α + 1) + 3 y ( y − w ) x − x y − x − x (4 t + 15 w − y ) + ( w − y )( w − y )128 x ,wx , qx + 34 x ( x − y ) + w − y x ) . Then such a system coincides with the system dxdt = y + g ( t ) ( g ( t ) ∈ C ( t )) ,dydt = z,dzdt = − y + w + t ,dwdt = wq + 2 α − ,dqdt = − q − y. Finally, we study a solution of the system (61) which is written by the use of knownfunctions.By the transformation π , the fixed solution is derived from α = − α ,x + q x, − (cid:18) y + q (cid:19) = y, − (cid:18) z − q ( q + 8 y ) (cid:19) = z, − (cid:18) w + yq − y − zq + t (cid:19) = w, − q = q. (68)Then we obtain(69) ( x, y, z, w, q ; α ) = (cid:18) x, , , − t ,
0; 0 (cid:19) as a seed solution.The system (61) with special parameter α = admits a particular solution expressedin terms of the Painlev´e I function ( y, z ):(70) w = 0 , and(71) dxdt = y, dydt = z, dzdt = − y + t , dqdt = − q − y. TUDIES ON THE SECOND MEMBER OF THE SECOND PAINLEV´E HIERARCHY 21
References [1] P. Clarkson, N. Joshi and A. Pickering,
B¨acklund transformations for the second Painlev´e hierarchy:a modified truncation approach , Inverse Problem. (1999), 175–187.[2] M. Mazzocco and M. Y. Mo, The Hamiltonian structure of the second Painlev´e hierarchy ,http://arXiv.org/abs/nlin.SI/0610066.[3] N. Joshi,
The second Painlev´e hierarchy and the stationary KdV hierarchy , Publ. RIMS. Kyoto Univ. (2004), 1039-1061.[4] Y. Sasano, Symmetry in the Painlev´e systems and their extensions to four-dimensional systems , toappear in Funkcial. Ekvac..[5] I. P. Martynov,
Differential equations with stationary critical singularities , Differential Equations. (1973), 1368–1376.[6] M. J. Ablowitz and H. Segur, Exact linearization of a Painlev´e transcendendent , Phys. Rev. Lett. (1977), 1103–1106.[7] H. Flaschka and A. C. Newell, Monodromy and spectrum preserving deformations I , Commun. Math.Phys. (1980), 65–116.[8] P. Painlev´e, M´emoire sur les ´equations diff´erentielles dont l’int´egrale g´en´erale est uniforme , Bull.Soci´et´e Math´ematique de France. (1900), 201–261.[9] P. Painlev´e, Sur les ´equations diff´erentielles du second ordre et d’ordre sup´erieur dont l’int´egrale estuniforme , Acta Math. (1902), 1–85.[10] B. Gambier, Sur les ´equations diff´erentielles du second ordre et du premier degr´e dont l’int´egraleg´en´erale est `a points critiques fixes , Acta Math. (1910), 1–55.[11] C. M. Cosgrove and G. Scoufis, Painlev´e classification of a class of differential equations of the secondorder and second degree , Studies in Applied Mathematics. (1993), 25-87.[12] C. M. Cosgrove, All binomial-type Painlev´e equations of the second order and degree three or higher ,Studies in Applied Mathematics. (1993), 119-187.[13] F. Bureau, Integration of some nonlinear systems of ordinary differential equations , Annali di Matem-atica. (1972), 345–359.[14] J. Chazy, Sur les ´equations diff´erentielles dont l’int´egrale g´en´erale est uniforme et admet des singu-larit´es essentielles mobiles , Comptes Rendus de l’Acad´emie des Sciences, Paris. (1909), 563–565.[15] J. Chazy,
Sur les ´equations diff´erentielles dont l’int´egrale g´en´erale poss´ede une coupure essentiellemobile , Comptes Rendus de l’Acad´emie des Sciences, Paris. (1910), 456–458.[16] J. Chazy,
Sur les ´equations diff´erentielles du trousi´eme ordre et d’ordre sup´erieur dont l’int´egrale ases points critiques fixes , Acta Math. (1911), 317–385.[17] C. M. Cosgrove, Higher order Painlev´e equations in the polynomial class II, Bureau symbol P1 ,Studies in Applied Mathematics.116