数理解析研究所講究録1088 巻 1999 年 1-21

1Birkhoff normal form of Hamiltonian systems
and
WKB-type formal solutions
Yoshitsugu TAKEIRIMS, Kyoto University
Kyoto, 606-8502, Japan(京大数理研1

竹井義次)

Introduction

Fuchsian
As is illustrated by the computation of monodromy groups of second-ordertool
equations (cf. [AKTI]), the exact WKB analysis provides us with a powerfulToequations.
for studying global behavior of solutions of linear ordinary differential
Univ.),
generalize such an analysis to nonlinear equations, T. Kawai (RIMS, Kyoto
Painlev\’efor
T. Aoki (Kinki Univ.) and the author have developed the WKB theory[KT2]).
[AKT2],
equations with a large parameter in our series of articles ([KT1],
behavior(See [T1], [T2] also.) Although we have almost succeeded in analyzing theturning pointsof 2-parameter formal solutions constructed in [AKT2] near simple
for the(cf. [KT2]), their behavior near fixed singular points, which is also important
reportglobal study of Painlev\’e equations, has not been clarified yet. The aim of thisnearbehaveis thus to consider the following problem: How do our formal solutions
fixed regular-type singular points for Painlev\’e equations?
In the case of second-order linear equations, the corresponding formal solutions
are given by the WKB solutions and two typical methods are known for their conand
struction: One is to transform equations in question into Riccati equations,
equations. Bethe other is to solve the so-called eiconal equation and transport
the behavior
tween these two methods the first one is more effective to determine
forof WKB solutions near regular singular points. Now, to construct 2-parametertheemployed
mal solutions of Painlev\’e equations with a large parameter, we havemultiple-scale analysis in [AKT2], that is, we have constructed them by solving somedifferential equations degree by degree. In this sense this construction corresponds
is not effito the second method for WKB solutions mentioned above and hence
cient to discuss the present problem. In this report we propose a new construction

2of 2-parameter formal solutions of Painlev\’e equations with a large parameter toinvestigate their behavior near fixed regular-type singular points.
The new construction of formal solutions we propose here is based on the workof Kimura [K] and its improvement by Takano [Tkal] (see [Tka2] also), where they
respectively constructed a 2-parameter family of analytic solutions at each regulartype singular point of (ordinary) Painlev\’e equations. Making use of the well-known
fact that Painlev\’e equations can be written in the form of Hamiltonian systems(which we call Painlev\’e Hamiltonian systems in this report), Kimura first establishedsome reduction theorem for Hamiltonian systems to construct analytic solutions
and later Takano modified his method to enlarge the domain of convergence of
these analytic solutions. Their reduction theorem is closely related to the following“Birkhoff normal form” of Hamiltonian systems (cf. [B], [SM]).

Birkhoff normal form
(1)

Consider a Hamiltonian system$dq/dt.=\partial H/\partial p$,$dp/dt=\backslash -\partial H/\partial q$with a Hamiltonian $H=H(t, q,p)$ . If we can find a canonical transformation
which transforms the original system (1) to$(q,p)arrow(\overline{q},\overline{p})$(2)$d\overline{q}/dt=\partial\overline{H}/\partial\overline{p}$

with
(3)
(

,

$\overline{H}(t,\overline{q},\overline{p})=\sum_{n\geq 0}\overline{h}_{n}(t)(\overline{q}\overline{p})^{n+}1$

is a function of and the productBirkhoff normal form of (1).
$i.e.,\overline{H}$$d\overline{p}/dt=-\partial\overline{H}/\partial\overline{q}$$t$$\overline{q}\overline{p}$only), then the new system (2) is called

Roughly speaking, to construct 2-parameter formal solutions, we will revise their
reduction theorem so that it may be adapted to Hamiltonian systems of singular perturbations and prove the existence of a canonical transformation which reduces the Painlev\’e Hamiltonian system to its “Birkhoff normal form” in a singularperturbative manner. The existence of singular-perturbative reduction will be discussed in Section 3 and the behavior near fixed regular-type singular points of our
2-parameter formal solutions thus constructed will be investigated in Section 4. Before considering Painlev\’e Hamiltonian systems, in Section 2 we will study the relationship between this viewpoint and WKB solutions of second-order linear ordinary
differential equations.
The author would like to express his gratitude to Professors T. Kawai and T.
Aoki for the stimulating discussions with them. He also thanks to Professor M.
Yoshino for his valuable comment on Birkhoff normal form. This work is supported
by Grant-in-Aid for Scientific Research for Encouragement of Young Scientists (No.
09740101), the Japanese Ministry of Education, Science, Sports and Culture.

3

2Birkhoff normal form and WKB solutions of
Schr\"odinger equations
In this section we discuss the construction of WKB solutions of l-dimensional
Schr\"odinger equations(4)$(- \frac{d^{2}}{dx^{2}}+\eta^{2}Q(X))\psi=0$($\eta$: large parameter)from the viewpoint of reduction of Hamiltonian systems to their Birkhoff normal
form. Let us begin by reviewing two well-known methods for the construction of
WKB solutions.
The first method is to transform the unknown function of (4) into definedby$\psi=\exp\int^{x}Sdx$ .
(5)$S$$\psi$

Then we readily verify that$S$

must satisfy the so-called Riccati equation:(6)$S^{2}+ \frac{dS}{dx}=\eta^{2}Q(x)$This equation (6) has the following two formal.

$\mathrm{p}\mathrm{o}\mathrm{W}\mathrm{e}\mathrm{r}\backslash$series solutions denoted by$S_{\pm}:$(7)$S_{\pm}$

$=$

$=$

$\pm\eta S_{-1}(X)+s0(_{X)\pm}\eta^{-}1S_{1}(X)+\cdots$,$\pm S_{\mathrm{o}\mathrm{d}\mathrm{d}}+s_{\mathrm{e}\mathrm{V}\mathrm{e}}\mathrm{n}$where $S_{-1}(x)=\sqrt{Q(x)}$ and the other $S_{j}(x)(j\geq 0)$ are determined recursively.
Note that the comparison of odd order terms (with respect to the power of ) of
both sides of (6) entails$\eta$.

(8)$s_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}=- \frac{1}{2}\frac{d}{dx}\log s_{\mathrm{o}\mathrm{d}\mathrm{d}}$Substituting (7) and (8) into (5), we obtain the WKB solutions of (4) of the form

(9)$\psi_{\pm}=\frac{1}{\sqrt{S_{\mathrm{o}\mathrm{d}\mathrm{d}}}}\exp\int^{x}(\pm S_{\mathrm{O}}\mathrm{d}\mathrm{d}dx)$.

On the other hand, in the second method we seek for a solution of (4) in thefollowing form:(10)$\psi_{=}\exp(\eta p(x))A(x)$

,

where

$A(x)=a_{0}(x)+\eta^{-1}a_{1}(X)+\cdots$ .

4In order thatsatisfy$\psi$of the form (10) may be a solution of (4) $p(x)$ and $A(x)$ should$( \frac{dp}{dx})^{2}=Q(X)$

(11)

,

$\{$

$\frac{d^{2}p}{dx^{2}}A+2\frac{dp}{dx}\frac{dA}{dx}+\eta^{-1}\frac{d^{2}A}{dx^{2}}=0$.

equation”) and each coefficient
Hence $dp/dx=\pm\sqrt{Q(x)}$ (“should be determined by the following differential equations (“in a recurslve manner:$\mathrm{e}\mathrm{i}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}$(12)$(j\geq 0)$$\{4Q(x)\frac{d}{dx}+Q’(x)\}a_{j}(X)=\mp 2\sqrt{Q(x)}a_{j-1}^{\prime/}(X)$of $A(x)$equations”)$a_{j}(x)$.

(Here and in what follows
the differentiation with respect to and we$a_{-1}(x)\equiv0.)$
In this way the eiconal equation and transport
conventionally define
equations also determine the WKB solutions of (4) of the form (10). The solutions
thus obtained are essentially the same with (9).
Let us now reconsider the construction of WKB solutions from the viewpoint of
reduction of Hamiltonian systems. To do so, by putting $\varphi=\eta^{-1}d\psi/dx$ we rewrite
the equation (4) in the following Hamiltonian form:$/\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{S}$

(13)

$x$

$d\psi/dx=\eta\partial H/\partial\varphi$

where
(14)

,

$d\varphi/dx=-\eta\partial H/\partial\psi$$H=H(_{X}, \psi, \varphi)=\frac{1}{2}\varphi^{2}-\frac{1}{2}Q(x)\psi 2$.

This system (13) is a Hamiltonian system of singular perturbations. What we want
to do is to transform
into its Birkhoff normal form by some canonical trans. In this case such a canonical transformation should beformation
linear, i.e.,$(\vee 13)$$(\psi, \varphi)arrow(\psi,\overline{\varphi})$$\psi$(15)$=$$a(x)\tilde{\psi}+b(x)\tilde{\varphi}$$=$$c(x)\overline{\psi}+d(_{X})\overline{\varphi}$$\{$$\varphi$,and the Birkhoff normal form should be of the form
(16)

$d\tilde{\psi}/dx=\eta\partial\overline{H}/\partial\tilde{\varphi}$

where
(17)

,

$d\overline{\varphi}/dx=-\eta\partial\overline{H}/\partial\overline{\psi}$$\overline{H}=f(x)\overline{\psi}\overline{\varphi}$.

(Here $a(x),$, $f(x)$ may depend on also). If we successfully find such a canonicaltransformation and a normal form, we automatically obtain solutions of the original$\ldots$$\eta$

5equation (4) in the following way: The reduced system (16) are easily solved and$\overline{\psi}$

(18)

$=$

$\alpha\exp\eta\int^{x}f(_{X})dx$$=$$- \beta\exp(-\eta\int^{x}f(x)dX)$$\{$$\overline{\varphi}$

gives a solution of it. (Here and denote free parameters and we have added
minus sign (-) in front of for the sake of convention.) Then, substitution of (18)
into (15) produces the following solution of (4):$\beta$$\alpha$$\beta$(19)$\psi=\alpha a(x)\exp(\eta\int^{x}f(x)dx)-\beta b(X)\exp(-\eta\int^{x}f(x)d_{X})$ .

Our problem is thus to find such a linear canonical transformation (15). Roughly) tospeaking, we employ an inductive argument (with respect to the power of
construct a canonical transformation. To illustrate our inductive argument, let us
first consider the top degree part of the problem. Since the original Hamiltonian isgiven by (14), as the top degree part of the transformation we choose$\eta^{-1}$$\sqrt{Q(x)}\psi+\varphi)$

(20)

$==$

$\{\overline{\psi\overline{\varphi}}$$2^{-1/}2Q(_{X}2^{-1/}2Q(x)^{-})^{-1/4}1/4\{$$-\sqrt{Q(x)}\psi+\varphi)$,that is,$\psi$(21)$=$$2^{-1/2}Q(x)^{-1/4}(\overline{\psi}-\overline{\varphi})$$=$$2^{-1/2}Q(_{X})^{1/4}(\overline{\psi}+\overline{\varphi})$$\{$$\varphi$.

Note that the factors $2^{-1/2}Q(x)^{-1/4}$ etc. are added so that the transformation becomes canonical. Then, by straightforward computations, we find that the system(13) is transformed into another Hamiltonian system with the Hamiltonian(22)

For the top degree part (22) is now of the required form, that is, its top degreepart has the same structure with the Hamiltonian (17) of the Birkhoff normal form.
Similarly, by adding appropriate degree $(-1)$ terms to the transformation (20) or(21) we could obtain a Hamiltonian system which is the Birkhoff normal form up
to the degree $(-1)$ , and this procedure could further be continued up to arbitrarily. However, to construct a canonical transformation
higher orders with respect toin all orders, we here employ the following argument, which is conciser than thenaive inductive argument explained above.
$\eta^{-1}$6

Let us assume that a transformation we are seeking for has the following form:$\psi$(23)$=$$a(x, \eta)\overline{\psi}+b(x, \eta)\overline{\varphi}$$=$$c(_{X}, \eta)\tilde{\psi}+d(x, \eta)\overline{\varphi}$$\{$$\varphi$

where $a(x, \eta)$ etc. are formal power series of$\eta^{-1}$

we suppose(24),. To guarantee that (23) is canonical,$a(x, \eta)d(x, \eta)-b(x, \eta)c(x, \eta)=1$.

The transformation (23) is obtained also by using the following generating function:$W(x,\overline{\psi}, \varphi)$(25)$W(x, \overline{\psi}, \varphi)=-\frac{b}{2d}\varphi^{2}+\frac{c}{2d}\overline{\psi}2-\frac{1}{d}\overline{\psi}\varphi$,

in other words, (23) is equivalent to(26)$\psi=-\partial W/\partial\varphi$,.

$\overline{\varphi}=-\partial W/\partial\overline{\psi}$

The relation between the original Hamiltonian and the transformed one is described
also in terms of the generating function $W$ as follows:(27)$\overline{H}=$$=$$H(x, \psi(\overline{\psi},\overline{\varphi}), \varphi(\overline{\psi},\overline{\varphi}))+\eta-1\frac{\partial W}{\partial x}(x,\overline{\psi}, \varphi(\overline{\psi},\overline{\varphi}))$$\frac{1}{2}(c\overline{\psi}_{+}d\overline{\varphi})^{2}-\frac{1}{2}Q(x)(a\overline{\psi}+b\overline{\varphi})2$$+ \eta^{-1}(-(\frac{b}{2d})’(C\overline{\psi}+d\overline{\varphi})^{2}+(\frac{c}{2d}\mathrm{I}^{\overline{\psi}^{2}}/-(\frac{1}{d})’\overline{\psi}(C\overline{\psi}+d\overline{\varphi}))$$=$$\{(1-\eta^{-1}(\frac{b}{d}\mathrm{I}’)cd-Q(x)ab-\eta-1(\frac{1}{d})’d\}\overline{\psi}\overline{\varphi}$$+ \frac{1}{2}\{(1-\eta^{-1}(\frac{b}{d})’)C^{2}-Q(x)a^{2}+\eta-1(\frac{c}{d})’-2\eta^{-1}(\frac{1}{d})’c\}\overline{\psi}^{2}$$+ \frac{1}{2}\{(1-\eta^{-1}(\frac{b}{d})’\mathrm{I}^{d^{2}}-Q(X)b^{2\}\overline{\varphi}^{2}}$.

In order that
may be Birkhoff normal form, it is sufficient that the following
equalities should be satisfied:$\overline{H}$$0\dot{\mathrm{f}}$(28)$(1- \eta^{-1}(\frac{b}{d})’)c^{2}-Q(x)a^{2}+\eta-1(\frac{c}{d})’-2\eta^{-1}(\frac{1}{d})’c=0$(29)$(1- \eta^{-1}(\frac{b}{d}\mathrm{I}’)d^{2}-Q(x)b^{2}=0$

.

,

7

In particular, since(29)$\Leftrightarrow$$( \frac{d}{b})^{2}-\eta-1(\frac{b}{d})’(\frac{d}{b})^{2}-Q(x)=0$$\Leftrightarrow$$( \frac{d}{b})^{2}+\eta-1(\frac{d}{b})’=Q(X)$,satisfies the Riccati equation (6). Furthermore, since (24) implies $(b/d)’=$$(a/c)’-(1/cd)’$ , we have$\eta d/b$

(28)$\Leftrightarrow$$(1- \eta^{-1}(\frac{a}{c})’)c2-Q(X)a^{2}+\eta^{-1}\{(\frac{1}{cd})C^{2}+/(\frac{c}{d})’-2(\frac{1}{d})’c\}=0$$\Leftrightarrow$$(1- \eta^{-1}(\frac{a}{c})’)C2-Q(_{X)a^{2}}=0$$\Leftrightarrow$$( \frac{c}{a})^{2}+\eta^{-}1(\frac{c}{a})’=Q(_{X)}$.

also satisfies the Riccati equation (6). Note that the Riccati equation (6)Hencecan be solved in a singular-perturbative manner and we obtain two formal solutions
must be different solutions since it
andgiven by (7). In this situation
follows from (24) that$\eta c/a$$\eta d/b$$\eta c/a$

$S_{\pm}$$( \frac{d}{b})-(\frac{c}{a})=\frac{1}{ab}$.

Thus we may assume

(30)

$\{$

$\eta\frac{d}{b}=$

and
(31)

$=S_{+}$$S_{\mathrm{o}\mathrm{d}\mathrm{d}}+s_{\mathrm{e}\mathrm{V}\mathrm{e}}\mathrm{n}$$\eta\frac{c}{a}=$$=S_{-}$$-s_{\mathrm{o}\mathrm{d}\mathrm{d}}+s_{\mathrm{e}\mathrm{V}\mathrm{e}}\mathrm{n}$$\frac{1}{ab}=-2\eta^{-1}s_{\mathrm{O}}\mathrm{d}\mathrm{d}$.

These relations (30) and (31) are describing the condition that the transformation(23) is canonical and reduces the original Hamiltonian system (13) $-(14)$ into its
Birkhoff normal form. By (27) and the identity$-( \frac{b}{d})’cd-(\frac{1}{d})’d$$=$$( \frac{d}{b})’\frac{b^{2}c}{d}+\frac{d’}{d}$$=$$( \frac{d}{b})’\frac{b}{d}(ad-1)+\frac{d’}{d}$..

8we find also that the coefficient offollowing:(32)$\tilde{\psi}\overline{\varphi}$$=$$( \frac{d}{b})’ab-(\frac{d}{b})’\frac{b}{d}+\frac{d’}{d}$$=$$( \frac{d}{b})’ab+\frac{b’}{b}$,in the Birkhoff normal form is given by the$(1- \eta^{-1}(\frac{b}{d})’\mathrm{I}^{C}d-Q(x)ab-\eta-1(\frac{1}{d})’d$$=cd-Q(x)ab+ \eta^{-1}(\frac{d}{b})’ab+\eta^{-1}\frac{b’}{b}$$=cd-( \frac{d}{b})^{2}ab+\eta^{-}1_{\frac{b’}{b}}$$=- \frac{d}{b}+\eta^{-1}\frac{b’}{b}$.

However, it is obvious that (30) and (31) cannot determine the transformation
uniquely. Concerning the determination ofand
we have the following(typical) options:
$a,$

$b,$

$c$

$d$

Idea A: We determine
and in such a way that the coefficient (32) ofin the Birkhoff normal form may become as simple as possible. For that purpose
we should define by solving
$a,$

$b,$

$d$

$c$

$\overline{\psi}\overline{\varphi}$

$b$

(33)

$\frac{db}{dx}-(S_{-}+\eta\sqrt{Q(x)})b=0$

in view of (30) and (32). Consequently the Hamiltonian of the Birkhoff normal form
becomes(34).
$_{\overline{H}=\sqrt{Q(x)}\overline{\psi}\overline{\varphi}}$Note that due to the assumption that is a formal power series of
we cannot
eliminate the coefficient ofcompletely and the top degree part $\sqrt{Q(x)}$ remains.
The differential equation (33) for together with (30) and (31) determines
and modulo constants of integration.
In this determination of the transformation we have to solve the differential
equation (33) and the transformation itself inevitably contains some constants of integration. In that sense this approach is closer to the construction of WKB solutions
via eiconal and transport equations.
$b$$\eta^{-1}$$\overline{\psi}\overline{\varphi}$

$b$

$a,$

$b,$

$c$

$d$

Idea
ment:
(35)

$\mathrm{B}$: To determine

$a,$

$b,$

$c$

and

$d$

we make the following additional require-$a=-b$ .

9this requirement is to pick out the odd part of solutions as the
The meaning
and the even part as the canonical transformation and (cf.
coefficient of(19) . As a matter of fact, (35) together with (31) entails$\underline{\mathrm{o}}\mathrm{f}$$a$$\psi\tilde{\varphi}$

$b$

$)$

(36)

$a=-b=(2\eta^{-1)^{-1}}S_{\mathrm{o}\mathrm{d}}\mathrm{d}/2$

and further the coefficient of(37)$- \frac{d}{b}+\eta^{-1}\frac{b’}{b}$

$=$

$=$

$=$

$\tilde{\psi}\overline{\varphi}$,

becomes$- \eta^{-1}(-S_{\circ}\mathrm{d}\mathrm{d}+s\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{n})+\eta^{-}\frac{d}{dx}1\log(2\eta-1s_{\mathrm{o}\mathrm{d}\mathrm{d}})-1/2$$- \eta^{-1}(-S_{\mathrm{o}\mathrm{d}\mathrm{d}}+S\mathrm{e}\mathrm{V}\mathrm{e}\mathrm{n})+\eta^{-1}(-\frac{1}{2})\frac{d}{dx}\log$Sodd$\eta^{-1}S\mathrm{o}\mathrm{d}\mathrm{d}$thanks to the relation (8). We thus obtain solutions of (4) of the form
(38)$\psi=(2\eta^{-1}s_{\mathrm{o}\mathrm{d}}\mathrm{d})-1/2\{\alpha\exp(\int^{x}s_{\mathrm{o}\mathrm{d}}\mathrm{d}dx)+\beta\exp(-\int^{x}S_{\mathrm{o}\mathrm{d}}\mathrm{d}dX\mathrm{I}\}\cdot$

The requirement (35) enables us to determine the transformation uniquely. Thisapproach is closer to the construction of WKB solutions via the Riccati equation.
In this way the WKB solutions of Schr\"odinger equations can be constructed also
by using reduction of Hamiltonian systems to Birkhoff normal form. In Section
3 we employ this idea to construct formal solutions of Painlev\’e equations. Again
there we will encounter a similar problem of unique determination of canonicaltransformations as above. Throughout this report we follow mainly the line ofeven in the case of Painlev\’e equations.
“Idea$\mathrm{B}$3”

Construction of formal solutions of Painlev\’eequations via reduction to Birkhoff normal form

In this section we consider the construction of 2-parameter formal solutions ofwith a large parameter , which are tabuPainlev\’e equations
lated in Table 1 below.
$(P_{J})(J=\mathrm{I}, \ldots, \mathrm{V}\mathrm{I})$

Table 1$(P_{\mathrm{I}\mathrm{I}})$.

$\frac{d^{2}\lambda}{dt^{2}}$$=$$\eta^{2}(6\lambda^{2}+t)$$\frac{d^{2}\lambda}{dt^{2}}$$=$$\eta^{2}(2\lambda^{3}+t\lambda+C)$$(P_{\mathrm{I}})$.

$\eta$

10$\frac{d^{2}\lambda}{dt^{2}}$$(P_{\mathrm{I}\mathrm{I}\mathrm{I}})$$=$$\frac{1}{\lambda}(\frac{d\lambda}{dt}\mathrm{I}^{2}-\frac{1}{t}\frac{d\lambda}{dt}+\eta^{2}[16c\lambda\infty+\frac{8c_{\infty}’\lambda^{2}}{t}3-\frac{8c_{0}’}{t}-\frac{16c_{0}}{\lambda}]\cdot$.

$(P_{\mathrm{I}\mathrm{V}})$$(P_{\mathrm{V}})$$\frac{d^{2}\lambda}{dt^{2}}$$=$$\frac{d^{2}\lambda}{dt^{2}}$$=$$\frac{1}{2\lambda}(\frac{d\lambda}{dt})^{2}.-\cdot\frac{2}{\lambda}+\eta^{2}[\frac{3}{2}\dot{\lambda}^{3}+4t\lambda 2(+2t^{2}+8c1)\lambda-\frac{8c_{0}}{\lambda}]$.

$( \frac{1}{2\lambda}+\frac{1}{\lambda-1})(\frac{d\lambda}{dt})^{2}-\frac{1}{t}\frac{d\lambda}{dt}+\frac{(\lambda-1)^{2}}{t^{2}}(2\lambda-\frac{1}{2\lambda})$.

$+ \eta^{2}\frac{2\lambda(\lambda-1)2}{t^{2}}[(c_{0}+c_{\infty})-\frac{c_{0}}{\lambda^{2}}-\frac{c_{2}t}{(\lambda-1)^{2}}-\frac{c_{1}t^{2}(\lambda+1)}{(\lambda-1)^{3}}]$.
$\frac{d^{2}\lambda}{dt^{2}}$$(P_{\mathrm{V}\mathrm{I}})$$=$$\frac{1}{2}(\frac{1}{\lambda}$$+ \frac{1}{\lambda-1}+\frac{1}{\lambda-t}\mathrm{I}(\frac{d\lambda}{dt})^{2}-(\frac{1}{t}+\frac{1}{t-1}+\frac{1}{\lambda-t})\frac{d\lambda}{dt}$$+ \frac{2\lambda(\lambda-1)(\lambda-t)}{t^{2}(t-1)^{2}}[1-\frac{\dot{\lambda}^{2}-2t\lambda+t}{4\lambda^{2}(\lambda-1)2}$$+ \eta^{2}\{(c_{0}+c_{1}+C_{t}+C\infty)-\frac{c_{0}t}{\lambda^{2}}+\frac{c_{1}(t-1)}{(\lambda-1)^{2}}-\frac{c_{t}t(t-1)}{(\lambda-t)^{2}}\}]$.

As is well known, Painlev\’e equations can also be represented in the form of Hamiltonian systems$(H_{J})$$d\lambda/dt=\eta\partial K_{J}/\partial_{l^{\text{ノ}}}$,$d\nu/dt=-\eta\partial K_{J}/\partial\lambda$(cf., e.g., [O]). One explicit choice of Hamiltonians$K_{J}(t, \lambda, \nu, \eta)$

is the following:

Table 2.

$=$$\frac{1}{2}[I^{\text{ノ^{}2}-}(4\lambda^{3}+2t\lambda)]$$K_{\mathrm{I}\mathrm{I}}$$=$$\frac{1}{2}[\nu^{2}-(\lambda^{4}+t\lambda^{2}+2_{C\lambda)]}$$K_{\mathrm{I}\mathrm{I}\mathrm{I}}$$=$$\frac{2\lambda^{2}}{t}[\nu^{2}-\eta^{-1_{\frac{3\nu}{2\lambda}-}}(\frac{c_{0}t^{2}}{\lambda^{4}}+\frac{c_{0}’t}{\lambda^{3}}+\frac{c_{\infty}’t}{\lambda}+c\infty^{t})2]$$K_{\mathrm{I}\mathrm{V}}$$=$$2 \lambda[\nu 2-\eta^{-}-1_{\frac{\nu}{\lambda}}(\frac{c_{0}}{\lambda^{2}}+C_{1}+(\frac{\lambda+2t}{4})^{2})]$$K_{\mathrm{V}}$$=$$\frac{\lambda(\lambda-1)^{2}}{t}$$K_{\mathrm{I}}$

.

.

.

$\cross[\nu^{2}-\eta^{-1}(\frac{1}{\lambda}+\frac{1}{\lambda-1})\nu-(\frac{c_{0}}{\lambda^{2}}+\frac{c_{1}t^{2}}{(\lambda-1)^{4}}+\frac{c_{2}t}{(\lambda-1)^{3}}+\frac{c_{\infty}}{(\lambda-1)^{2}}\mathrm{I}]\cdot$$K_{\mathrm{V}\mathrm{I}}$$=$$\frac{\lambda(\lambda-1)(\lambda-t)}{t(t-1)}$$\mathrm{x}[I\text{ノ}-2\eta^{-}1(\frac{1}{\lambda}+\frac{1}{\lambda-1})\nu-(\frac{c_{0}}{\lambda^{2}}+\frac{c_{1}}{(\lambda-1)^{2}}+\frac{c_{\infty}}{\lambda(\lambda-1)}+\frac{c_{t}}{(\lambda-t)^{2}}\mathrm{I}]\cdot$

11

by using reduction of
In what follows we try to construct formal solutions of
to its Birkhoff normal form.
this Hamiltonian system
Let us first note that each Painlev\’e equation has the following structure in com$(P_{J})$$(H_{J})$

mon:
(39)

$\frac{d\lambda}{dt},$$\frac{d^{2}\lambda}{dt^{2}}=G_{J}(\lambda,$$t)+\eta^{2}F_{J}(\lambda, t)$,

are rational functions. In view of (39) we easily find that
and
where
:
the following formal power series solutions denoted by$F_{J}$$(P_{J})$$G_{J}$

has$\lambda_{J}^{(0)}(t)$(40)$\lambda_{J}^{(0)}(t)=\lambda 0(t)+\eta^{-}\lambda 2(2t)+\eta^{-4}\lambda 4(t)+\cdots$

where the top term$\lambda_{0}(t)$,satisfies$F_{J}(\lambda_{0}(t), t)=0$

and the other $\lambda_{2j}(t)(j\geq 1)$ are determined in a recursive manner. Corresponding
to these solutions (40), there exist formal power series solutions called O-parameter:
solutions of$(H_{J})$$\lambda_{J}^{(0)}(t)$$=$$\lambda_{0}(t)+\eta-2\lambda_{2}(t)+\eta-4\lambda_{4}(t)+\cdots$$\nu_{J}^{(0)}(t)$$=$$\eta^{-1_{U_{1}}}(t)+\eta-3\nu 3(t)+\eta-5\nu_{5}(t)+\cdots$$\{$, Proposition 1.1]). Let us next consider the following localization of(cf.
at this -parameter solution:$(H_{J})$$[\mathrm{K}\mathrm{T}1$

$0$

(41)

$\lambda=\lambda_{J}^{(}0)(t)+\eta-1/2U$,$\nu=\nu^{(0}j)(t)+\eta^{-1/2}V$,

to $(U, V)$ . Then we
fromthat is, we transform the unknown function ofreadily verify that $(U, V)$ must obey another Hamiltonian system$(H_{J})$

,

(42)
where

(43)

$dU/dt=\eta\partial \mathcal{K}J/\partial V$$\mathcal{K}_{J}$$(\lambda, \nu)$$dV/dt=-\eta\partial \mathcal{K}J/\partial U$,

is given by the following:$\mathcal{K}_{J}=\sum_{j+k\geq 2}\eta)-(j+k-2/2\frac{1}{j!k!}\frac{\partial^{j+k}K_{J}}{\partial\lambda^{j}\partial\nu^{k}}(t, \lambda_{J}(0)(t),$$\nu_{J}(0)(t),$ $\eta)U^{j}V^{k}$.

Now the main result of this report is the following:
Theorem 1 There exists a formal canonicalthe form
(44)

$U$

$=$

$V$

$=$

$transf_{\mathit{0}}rmation(U, V)\mapsto(\overline{U},\overline{V})$,$u_{0}(\overline{U},\overline{V})+\eta^{-1/}u_{1}(2\overline{U},\overline{V})+\cdots$$\{$$v_{0}(\overline{U},\overline{V})+\eta^{-1/}v_{1}(2\overline{U},\overline{V})+\cdots$

,

of

12

and
are homogeneous polynomials of degree $(j+1)$ in(whose
power
are
with
being
coefficients
formalofcoefficientsfunctions of ),so that the Hamiltonian system (42) may be taken into the following normal form:
where$u_{j}$$(\tilde{U},\tilde{V})$$v_{j}$$serie\mathit{8}$(45)$\eta^{-1/2}$,$d\overline{U}/dt=\eta\partial\tilde{\mathcal{K}}_{J}/\partial\tilde{V}$

where
(46)

$t$

$d\tilde{V}/dt=-\eta\partial\overline{\mathcal{K}}_{J}/\partial\overline{U}$,$\overline{\mathcal{K}}_{J}=\sum_{=l0}^{\infty}\eta^{-l}f(l)(t, \eta)(\overline{U}\tilde{V}\mathrm{I}^{l1}+$

and eachcients being functions of .

$f^{(l)}(t, \eta)=\Sigma_{j\geq 0}\eta^{-j}f/2(l)(j/2t)$

is a formal power series of$\eta^{-1/2}$

with coeffi-$t$Remark The concrete form of the first few terms offollowing:$f^{(l)}$in the case of$f^{(0)}$$=$$(12\lambda_{0)^{1}-}/2\eta-2_{\frac{3^{2}\cdot 5^{2}}{2}}(12\lambda_{0})^{-9/}2+\cdots$$f^{(1)}$$=$15 $(12\lambda 0)^{-2}+\eta^{-2}33.5^{2}\cdot 31(12\lambda 0)^{-7}+\cdots$$f^{(2)}$$=$$-3\cdot 5\cdot 47(12\lambda_{0})^{-9/}2+\cdots$

For the top degree partis the.

we also have the following equalities for any :$f_{0}^{(0)}(t)$(47)$J=\mathrm{I}$$J$$f_{0}^{(0)}(t)=\sqrt{\frac{\partial F_{J}}{\partial\lambda}(\lambda_{0}(t),t)}$.

Theorem 1 claims that the Hamiltonian system (42) can be transformed into
its Birkhoff normal form. Since the reduced Hamiltonian
depends only on the
product, the system (45) is easily solved; taking account of the fact that the
product
is independent of , we find$\overline{\mathcal{K}}_{J}$$\overline{U}\overline{V}$$\overline{U}\overline{V}$$t$$\overline{U}$

(48)

$=$

$\alpha\exp(\eta\int^{t}\sum\eta^{-l}(l+1)f^{(}l)(s, \eta)(-\alpha\beta)ld_{S})$$=$$- \beta\exp(-\eta\int^{t}\sum\eta^{-l}(l+1)f^{(l)}(s, \eta)(-\alpha\beta)ld_{S})$$\{$$\tilde{V}$gives a solution of (45). Substituting (48) into (44) and then into (41), we obtain
2-parameter formal solutions ofand.
Let us now sketch the proof of Theorem 1. The proof consists of the following two
steps; reduction of the linear part and that of the nonlinear part. We first consider
reduction of the linear part, that is, we seek for a linear canonical transformation$(H_{J})$(49)$(P_{J})$

$U$

$=$

$a(t, \eta)\overline{U}+b(t, \eta)\overline{V}$

$V$

$=$

$c(t, \eta)\tilde{U}+d(t, \eta)\overline{V}$

$\{$

13

with the generating function(50)$W(t, \overline{U}, V)=-\frac{b}{2d}V^{2}+\frac{c}{2d}\overline{U}2-\frac{1}{d}\overline{U}V$

which transforms the Hamiltonian system (42) into its Birkhoff normal form up tois transformed into
quadratic terms. By (49) the Hamiltonian$\mathcal{K}_{J}$(51)$-1\partial W$$\overline{\mathcal{K}}_{J}$$=$$\mathcal{K}_{J}+\eta$$\overline{\partial t}$$=$$\frac{1}{2}\frac{\partial^{2}K}{\partial\lambda^{2}}(a\overline{U}+b\overline{V})^{2}+\frac{\partial^{2}K}{\partial\lambda\partial\nu}(a\overline{U}+b\overline{V})(c\overline{U}+d\overline{V})+\frac{1}{2}\frac{\partial^{2}K}{\partial\nu^{2}}(C\tilde{U}+d\overline{V})^{2}$$+ \eta^{-1}\{-(\frac{b}{2d})’(c\overline{U}+d\overline{V})^{2}+(\frac{c}{2d})’\overline{U}^{2}-(\frac{1}{d})’\overline{U}(C\overline{U}+d\overline{V}\mathrm{I}\}$$+$(terms of degree greater than 2 in$(\overline{U},\overline{V})$).

(Here and in what follows we often omit the suffix for simplicity and abbreviate
etc. if there is no fear of confusions.)toNamely$J$$(\partial^{2}K_{J}/\partial\lambda^{2})(t, \lambda_{J}^{(}0)(t),$ $\nu^{(0}(J)t),$(52)(coeff. of$\overline{U}\overline{V}$)$=$$\partial^{2}K/\partial\lambda^{2}$$\eta)$$\frac{\partial^{2}K}{\partial\lambda^{2}}ab+\frac{\partial^{2}K}{\partial\lambda\partial\nu}(ad+bc)+\frac{\partial^{2}K}{\partial\nu^{2}}cd$$- \eta^{-1}((\frac{b}{d})’Cd+(\frac{1}{d})d)/$(53)(coeff. of$\overline{U}^{2}$

)

$=$

,

$\frac{1}{2}\frac{\partial^{2}K}{\partial\lambda^{2}}a^{2}+\frac{\partial^{2}K}{\partial\lambda\partial\nu}ac+\frac{1}{2}\frac{\partial^{2}K}{\partial\nu^{2}}c^{2}$$+ \eta^{-1}(-(\frac{b}{2d})’c+2(\frac{c}{2d})’-(\frac{1}{d})’c)$(54)(coeff. of$\overline{V}^{2}$)$=$$\frac{1}{2}\frac{\partial^{2}K}{\partial\lambda^{2}}b^{2}+\frac{\partial^{2}K}{\partial\lambda\partial\nu}bd+\frac{1}{2}\frac{\partial^{2}K}{\partial\nu^{2}}d^{2}-\eta^{-1}(\frac{b}{2d})’d^{2}$

andWe are thus required to choose
really possible, that is, we can prove

Proposition 1 There exrst a,
(55)

,

$a,$

$b,$

$b,$

$c$

$c$

and

$d$

$d$

so that (53) and (54) may vanish. It is

which satisfy

ad–bc $=1$ ,(56)(coeff. of(57)(coeff. of.

$\overline{U}^{2}$) $=0$ ,$\overline{V}^{2}$) $=0$

14

together with the additional requirement(58)$a=-b$ .

These conditions (55)determine a,(55)
entail the following:$-(\mathit{5}\mathit{8})$

$b,$

$c$

and

$d$

(almost) uniquely. Furthermore$-(\mathit{5}s)$(coeff. of(59)$\overline{U}\overline{V}$)$=\eta^{-1}S_{\mathrm{o}\mathrm{d}\mathrm{d}}$

,

where
denotes the odd part (in the sense of [AKT2, Definition 2.1]) of solutions of the Riccati equation associated with the Fr\’echet derivative ($i.e.$ , linearized
equation) of
along the -parameter.
$S_{\mathrm{o}\mathrm{d}\mathrm{d}}$$(H_{J})$$soluti_{on}.(\lambda_{J’ J}^{(0)(0)}\nu)$$\mathit{0}$Before mentioning some comments on Proposition 1, let us recall here the definition of the Riccati equation associated with the Fr\’echet derivative of.
Substituting
andinto, we find that the Fr\’echet
derivative ofis given by the following:$(H_{J})$$\nu=\nu_{J}^{(0)}+\varphi$$\lambda=\lambda_{J}^{(0)}+\psi$$(H_{J})$$(H_{J})$(60),$\psi’$$=$$\eta(\frac{\partial^{2}K}{\partial\lambda\partial\nu}\psi+\frac{\partial^{2}K}{\partial\nu^{2}}\varphi)$$\varphi’$$=$$- \eta(\frac{\partial^{2}K}{\partial\lambda^{2}}\psi+\frac{\partial^{2}K}{\partial\lambda\partial\nu}\varphi \mathrm{I}\cdot$$\{$

We consider WKB solutions of (60), which is of the form$\psi=\exp\int^{t}Sdt$

Then

$S$

and

$T$

,

$\varphi=\exp\int^{t}Tdt$.

must satisfy(61)$(S- \eta\frac{\partial^{2}K}{\partial\lambda\partial\nu})\exp\int^{t}sdt-\eta\frac{\partial^{2}K}{\partial\nu^{2}}\exp\int^{t}\tau dt=0$

,

(62)

$\eta\frac{\partial^{2}K}{\partial\lambda^{2}}\exp\int^{t}sdt+(T+\eta\frac{\partial^{2}K}{\partial\lambda\partial\nu})\exp\int tTdt=0$.

Let us take the logarithmic derivative of (61).
(63)$\frac{d}{dt}\log(S-\eta\frac{\partial^{2}K}{\partial\lambda\partial\nu})+S=\frac{d}{dt}\log\frac{\partial^{2}K}{\partial\nu^{2}}+T$.

Furthermore, since neither $\exp\int^{t}Sdt$ nor $\exp\int^{t}Tdt$ is equal to zero, (61) and (62)entail(64)$(S- \eta\frac{\partial^{2}K}{\partial\lambda\partial\nu})(T+\eta\frac{\partial^{2}K}{\partial\lambda\partial\nu})+\eta^{2}\frac{\partial^{2}K}{\partial\lambda^{2}}\frac{\partial^{2}K}{\partial\nu^{2}}=0$.

15

A single equation which determines
fact, putting$S$

can be easily obtained from (63) and (64). In$T^{\uparrow}=T+ \eta\frac{\partial^{2}K}{\partial\lambda\partial\nu}$$s \uparrow_{=S-}\frac{\partial^{2}K}{\partial\lambda\partial\nu}\eta$,

we have$T^{\uparrow}=S \dagger+\frac{d}{dt}\log s\uparrow+2\eta\frac{\partial^{2}K}{\partial\lambda\partial\nu}-\frac{d}{dt}\log\frac{\partial^{2}K}{\partial\nu^{2}}$$s \uparrow T\dagger\eta+\frac{\partial^{2}K}{\partial\lambda^{2}}2\frac{\partial^{2}K}{\partial\nu^{2}}=0$

,

.

Hence
$(S^{\uparrow})^{2}+ \frac{dS^{\uparrow}}{dt}+(2\eta\frac{\partial^{2}K}{\partial\lambda\partial\nu}-\frac{d}{dt}\log^{\frac{\partial^{2}K}{\partial\nu^{2}}})S^{\uparrow\frac{\partial^{2}K}{\partial\nu^{2}}}+\eta^{2}\frac{\partial^{2}K}{\partial\lambda^{2}}=0$

or, in terms of the original
(65)

$S$

instead of$s\dagger$,,$S^{2}+ \frac{dS}{dt}-\frac{d}{dt}\log^{\frac{\partial^{2}K}{\partial\nu^{2}}+}\eta^{2}(\frac{\partial^{2}K}{\partial\lambda^{2}}\frac{\partial^{2}K}{\partial\nu^{2}}-(\frac{\partial^{2}K}{\partial\lambda\partial\nu})^{2})$$+ \eta(\frac{\partial^{2}K}{\partial\lambda\partial\nu}\frac{d}{dt}\log\frac{\partial^{2}K}{\partial\nu^{2}}-\frac{d}{dt}\frac{\partial^{2}K}{\partial\lambda\partial\nu})=0$

Fr\’echet derivashould be satisfied. This is the Riccati equation associated with the.
tive oftwo
Just like (6) we can solve (65) in a singular-perturbative manner to obtainformal power series solutions$(H_{J})$(66)$s_{\pm}$

$=$

$=$

$\pm\eta S_{-1}(t)+S\pm,\mathrm{o}(t)+\eta-1S_{\pm,1}(t)+\cdots$$\pm S_{\mathrm{o}\mathrm{d}\mathrm{d}}+s_{\mathrm{e}\mathrm{V}\mathrm{e}}\mathrm{n}$,.

(65) we findBy comparing the odd part (in the sense of [AKT2, Definition2.1]) of
that, instead of (8), the following relation holds in the present situation:(67)$s_{\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}}= \frac{1}{2}\frac{d}{dt}(\log\frac{\partial^{2}K}{\partial\nu^{2}}-\log S_{\mathrm{O}}\mathrm{d}\mathrm{d})$part (in ) ofward computations we can show the following:
Furthermore, since the degree

(68)

$0$

$(\partial^{2}K/\partial\lambda\partial\nu)$$\eta$$S_{-1}(t)=( \frac{\partial F_{J}}{\partial\lambda}(\lambda_{0}, t))^{1/2}$.
vanishes, by straightfor-

16(cf. (1.11) in [KT1], (1.35) and (1.41) in [KT2]). The relation (47) is an immediate
consequence of (59) and (68).
Proposition 1 can be proved by a similar argument as that in Section 2 which
verified the Hamiltonian system (13) $-(14)$ is reduced into its Birkhoff normal form.
See [T3] for the details of the proof of Proposition 1. Here we only note thatthe additional requirement (58) again determines a canonical transformation (49)(almost) uniquely. Its coefficientsand are, as a matter of fact, given by thefollowing:
$a,$

(69)

$=$

$a$

(70)
(71)

$b,$

$c$

$d$

$-b=( \frac{\partial^{2}K}{\partial\nu^{2}}\frac{1}{2\eta^{-1}S\mathrm{o}\mathrm{d}\mathrm{d}}\mathrm{I}^{1/2}$

,
,

$c=$

$(2 \eta^{-1}S_{\mathrm{o}\mathrm{d}}\mathrm{d}\frac{\partial^{2}K}{\partial\nu^{2}})^{-1/2}(\eta^{-1}S_{+}-\frac{\partial^{2}K}{\partial\lambda\partial\nu})$$d$$-(2 \eta^{-1}S_{\mathrm{o}\mathrm{d}}\mathrm{d}\frac{\partial^{2}K}{\partial\nu^{2}})^{-1/2}(\eta^{-1}S_{-\frac{\partial^{2}K}{\partial\lambda\partial\nu}}-)$$=$.

By this reduction of the linear part we thus obtain the following reduced Hamiltonian:(72)$\mathcal{K}_{J}$

$=$

$=$

$\eta^{-1}S_{\circ \mathrm{d}\mathrm{d}}UV+\sum_{+jk\geq 3}\eta^{-(}j+k-2)/2_{\frac{1}{j!k!}\frac{\partial^{j+k}K}{\partial\lambda^{j}\partial\nu^{k}}}(aU+bV)j(_{CU}+dV)k$$\eta^{-1}$

sodd

,

$UV+|j+ arrow\sum_{\geq\vec{k}|3}\eta-2/2\frac{1}{j!\vec{k}!arrow}-(|^{\sim}j+\vec{k}|)\frac{\partial^{|j+\vec{k}|}Karrow}{\partial\lambda^{1^{arrow}}j|\partial_{\mathcal{U}}|\vec{k}|}abj1?2k_{1}d^{kj_{1}}C2U+k_{1}Vj2+k_{2}$where $jarrow=(j_{1}, j_{2})$ and. (We have omitted tildes
for the sake ofsimplicity.) The Hamiltonian (72) is written also in the following form:$\vec{k}=(k_{1}, k_{2})$(73)$(^{-})$$\mathcal{K}_{J}=\eta^{-1}s_{\circ \mathrm{d}\mathrm{d}}UV+$$\sum_{+,p,qp\geq-q\geq 11}\eta^{-}K_{pq}(p+q)/2(t, \eta)Up+1Vq+1$

where
(74)

$K_{pq}(t, \eta)=j_{1,2^{+}}k_{1p+}=j+k=1\sum_{2q+1}\frac{1}{j!\vec{k}!arrow}\frac{\partial^{|j+\vec{k}|}Karrow}{\partial\lambda|j|\partial\nuarrow|\vec{k}|}ab^{j_{2}}j_{1}Cdk_{1}k2$.

What we next have to do is to find reduction of the nonlinear part, that is, acanonical transformation with the trivial linear terms
(75)

$U$

$=$

$\overline{U}+\eta^{-1/}u_{1}(2t,\tilde{U},\tilde{V}, \eta)+\eta^{-}u_{2}(1t,\overline{U},\tilde{V}, \eta)+\cdots$

$V$

$=$

$\overline{V}+\eta^{-1/}v_{1}(2t,\overline{U},\tilde{V}, \eta)+\eta^{-}v_{2}(1t,\overline{U},\overline{V}, \eta)+\cdots$

$\{$

,

17

where
(76)

$=$

$u_{j}(t,\overline{U},\overline{V}, \eta)$$p+q=jp,q \geq\sum_{1 ,-1-}u_{p}q(t, \eta)\overline{U}^{p}+1\overline{V}q+1$

(77)

$=$

$v_{j}(t,\overline{U},\tilde{V}, \eta)$$p+q=p,q \geq^{j}\sum_{1 ,-1-}v(pqt, \eta)\overline{U}^{p}+1\overline{V}q+1$, which transforms the Hamilbeing formal power series of
andwithinto its Birkhoff normal form. For that purpose we again make use of atoniangenerating function of the following form:$\eta^{-1/2}$$u_{pq}$$v_{pq}$$\mathcal{K}_{J}$

(78)

$W$

$W(t,\overline{U}, V)$$=$.

$-\overline{U}V+$$=$$\sum_{p+q\geq,p,q\geq-11}\eta^{-}a_{pq}((p+q)/2t, \eta)\overline{U}^{p+1q+}V1$

Roughly speaking, by introducing more additional requirements, we can uniquely
in a recursive manner so that the associated canonical transformadeterminetion$\{a_{pq}\}$$U=- \frac{\partial W}{\partial V}$$\overline{U}-$$=$$\sum_{p+,p,q\geq^{\geq}q-11}\eta(q+1)a_{p}\overline{U}p+1V^{q}-(p+q)/2q$

(79)

$\{$

$\overline{V}=-\frac{\partial W}{\partial\overline{U}}$

$V-$

$=$

$\sum_{\geq p+q1,p,q\geq-1}\eta^{-}((p+q)/2p+1)a\overline{U}^{p}V^{q+}pq1$

reduces the Hamiltonian (73) into its Birkhoff normal form. Note that, if we sucare explicitly given by
and, thencessfully find suchthe following:$u_{j}(t,\overline{U},\overline{V}, \eta)$$\{a_{pq}\}$

(80)

(81)

$u_{j}$$v_{j}(t,\tilde{U},\overline{V}, \eta)$$=$$-p+q \geq p+q+\mu 11,p,q++\sum_{0\geq-1,\iota}\cdot\mu k--\mu\geq j(q+1)a\overline{U}^{p+1}v\cdots vpq\mu 1\mu_{k}$$v_{j}$$=$$p+q+ \mu 1p+q\geq 1,p,q.\geq-1,\mu_{l}\geq+\sum_{0}(p++\mu_{k+}1^{-_{j}}-)1a_{p}\overline{U}pv_{\mu_{1\mu k+1}}\cdots vq$

respectively denoteandandand (81) recursively determine
the argument here and only refer the reader to [T3].
$(j=1,2,3, \ldots)$ where$u_{0}$$v_{0}$$\{u_{j}\}$$\overline{U}$and$\{v_{j}\}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}.\{a_{pq}\}.)$. (The relations (80)
We omit the details of$\overline{V}$18

4Local behavior of formal solutions of Painlev\’eequations near regular-type singular points
We have seen in the preceding section that singular-perturbative reduction of(more precisely, its localization at the -parameter solution) to Birkhoff normal form
produces 2-parameter formal solutions of. In this section we study their local
behavior at fixed regular-type singular points. As a typical example of fixed regulartype singular points of Painlev\’e equations we pick up the origin $t=0$ of the sixth
Painlev\’e equationand discuss the problem only for this typical example inthis report.
As is shown in Theorem 2 below, the regular-type singularness of fixed singularpoints of( $t=0$ ofhere) should entail the simpleness of poles which thecoefficientsof the Birkhoff normal form may possess there. Furthermore,in the global study of
the residues ofat regular-type singular points
would play an important role. Hence it is desirable to be able to compute such
residues explicitly. However, our choice of Hamiltonians
which is listed
up in Table 2 is not convenient for that purpose; if we work with, we can show
the simpleness of poles, but the computation of the residues becomes quite difficult.
To overcome this difficulty we use the following “polynomial Hamiltonian$(H_{J})$$0$$(P_{J})$$(P_{\mathrm{V}\mathrm{I}})$$(P_{J})$$(P_{\mathrm{V}\mathrm{I}})$$f^{(l)}(t, \eta)$$(P_{J})$$f^{(l)}(t, \eta)$$K_{J}(t, \lambda, \nu, \eta)$$K_{J}$$H_{\mathrm{V}\mathrm{I}}$(82)$d\lambda/dt=\eta\partial H\mathrm{v}\mathrm{I}/\partial\mu$,”$d\mu/dt=-\eta\partial H_{\mathrm{v}}\mathrm{I}/\partial\lambda$

where
(83)
$H_{\mathrm{V}\mathrm{I}}$$=$$\frac{1}{t(t-1)}[\lambda(\lambda-1)(\lambda-t)\mu^{2}-\eta^{-}\{\kappa 0(\lambda-1)(\lambda-1t)+\kappa 1\lambda(\lambda-t)$$+( \kappa_{t}-1)\lambda(\lambda-1)\}\mu+\frac{1}{4}\eta-2\{(\kappa_{0}+\kappa 1+\kappa_{t}-1)^{2}-\kappa^{2}\}\infty(\lambda-t)]$which is first discovered by Okamoto
relations between
and$H_{\mathrm{V}\mathrm{I}},$$\mu,$$\kappa_{*}$$\frac{1}{4}(\kappa_{*}^{2}-1)=C_{*}\eta^{2}$$K_{\mathrm{V}\mathrm{I}},$$([\mathrm{O}])$$\nu,$$c_{*}$, instead ofin this report.
are given by the following:$K_{\mathrm{V}\mathrm{I}}$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}*=0,1,$ $t$$\frac{1}{4}(\kappa_{\infty}^{2}-\kappa^{2}0-\kappa^{2}1-\kappa^{2}t-1)=c_{\infty}\eta 2$,,$\mu+\frac{1}{2}\eta^{-1}(\frac{1-\kappa_{0}}{\lambda}+\frac{1-\kappa_{1}}{\lambda-1}+\frac{1-\kappa_{t}}{\lambda-t})=\nu$,$H_{\mathrm{V}\mathrm{I}}+ \frac{1}{2}\eta^{-}2(1-\kappa_{t})(\frac{1-\kappa_{0}}{t}+\frac{1-\kappa_{1}}{t-1}+\frac{1}{\lambda-t})=K_{\mathrm{V}\mathrm{I}}$

Note that every $\kappa_{*}(*=0,1, t, \infty)$ is a quantity of degree 1 in .
$\eta$

.

,

The

19

Let us now state our results. The top degree part
, i.e.,
by the equationis$\lambda_{0}(t)$of our formal solutions$F_{\mathrm{V}\mathrm{I}}(\lambda_{0}(t), t)=0$$\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{C}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{z}\dot{\mathrm{e}}\mathrm{d}$.

$(_{C_{0}++}c_{1}Ct+c_{\infty})-C0^{\frac{t}{\lambda_{0}^{2}}+\frac{t-1}{(\lambda_{0^{-}}1)^{2}}C_{t^{\frac{t(t-1)}{(\lambda_{0^{-}}t)^{2}}=0}}}C_{1}-$following behavior
This algebraic equation has six solutions, one of which shows theat $t=0$ :.
$\lambda_{0}(t)=at+bt^{2}+\cdots$
with
(84)
$a= \frac{\sqrt{c_{0}}}{\sqrt{c_{0}}+\sqrt{c_{t}}}$in this report. (The other cases
We restrict ourselves to this special choice of
the above
will be discussed elsewhere.) Then, for 2-parameter formal solutions with, we can verify the following:
top degree part$\lambda_{0}(t)$$\lambda_{0}(t)$be the coefficients of the Birkhoff normal form obtained
Theorem 2 Letat the -parameter solution with the top
in Theorem 1 from the localization ofhas a simple pole at $t=0$ and. Then each
degree part$f^{(l)}(t, \eta)$$\mathit{0}$$(H_{\mathrm{V}\mathrm{I}})$$f^{(l)}(t, \eta)$$\lambda_{0}\mathit{8}ati_{S}fying(\mathit{8}\mathit{4})$(85)${\rm Res}_{t=0}f^{(0)}(t, \eta)$$=$$\eta^{-1}(\kappa_{0}+\kappa_{t})$(86)${\rm Res}_{t=0}f(1)(t, \eta)$

$=$

1,

(87)

${\rm Res}_{t=0}f(l)(t, \eta)$

$–0$

,

$(l\geq 2)$.

Furthermore, concerning the local behavior of the canonical transformation obtained
reduces the Hamiltonian system (82) to its Birkhoff normal form,in Section 3 which
andand bywe can also verify the following: (Note that, if we replacerespectively, all formulas in Section 3 hold even for the polynomial Hamiltonian$K_{\mathrm{V}\mathrm{I}}$$\nu$$H_{\mathrm{V}\mathrm{I}}$$\mu$$H_{\mathrm{V}\mathrm{I}}.)$and of the linear part of the canonical
Proposition 2 (i) The coefficients a,following local
transformation obtained in Proposition 1 (cf. (69) (71) also) show the
behavior at $t=0$ :
$b,$

(88)

$a$

$=$

(89)

$c$

$=$

$d$

$=$

(90)

$c$

$d$

$-b=( \frac{-\kappa_{0}\kappa_{t}}{\eta^{-1}(\kappa_{0+}\kappa_{t})^{3}})^{1/2}t+\cdots$$( \frac{\eta^{-1}(\kappa_{0}+\kappa t)^{3}}{-\kappa_{0}\kappa_{t}}\mathrm{I}^{1/2}\frac{1}{t}+\cdots$$0 \cdot\frac{1}{t}+\cdots$

,

,

.

and
of the nonlinear part of the canonical trans(ii) The coefficientsofformation given respectively by (76) and (77) as well as the coefficients$\{u_{pq}\}$$\{v_{pq}\}$$\{a_{pq}\}$

20

the generating function (78) are holomorphic (more precisely, formal power series
with holomorphic coefficients) at $t=0$ for anyofwith $q\geq-1$ and$p+q\geq 1$ (cf. (80) and (81)).
$\eta^{-1/2}$$p,$ $q$$p,$

For the proof of Theorem 2 and Proposition 2 see [T3].

References[AKTI] T. Aoki, T. Kawai and Y. Takei: Algebraic analysis of singular perturbations. S\={u}gaku Expositions, 8(1995), 217-240. (Originally appeared inJapanese in S\={u}gaku, 45(1993), 299-315.)[AKT2] –: WKB analysis of Painlev\’e transcendents with a large parameter. II. Structure of Solutions of Differential Equations, World Scientific,
1996, pp. 1-49.
[B]

G.D. Birkhoff: Dynamical Systems. Amer. Math. Soc., 1927, revised edition, 1966.

[K]

H. Kimura: The construction of a general solution of a Hamiltonian systemwith regular type singularity and its application to Painlev\’e equations. Ann.
Mat. Pura Appl., 134(1983), 363-392.

[KT1]

T. Kawai and Y. Takei: WKB analysis of Painlev\’e transcendents with alarge parameter. I. Adv. in Math., 118(1996), 1-33.

[KT2]–: WKB analysis of Painlev\’e transcendents with a large parameter. III. Adv. in Math., 134(1998), 178-218.

[O]

K. Okamoto: Isomonodromic deformation and Painlev\’e equations, and theGarnier systems. J. Fac. Sci. Univ. Tokyo, Sect. IA, 33(1986), 575-618.

[SM]

C.L. Siegel and J.K. Moser: Lectures on Celestial Mechanics, SpringerVerlag, 1971.

[T1]

Y. Takei: On a WKB-theoretic approach to the Painlev\’e transcendents.
XIth International Congress of Mathematical Physics, International Press,
1995, pp. 533-542.

[T2]–: On a WKB-theoretic approach to the Painlev\’e transcendents. II.
RIMS K\^oky\^uroku, No. 1058, 1998, pp. 114-128.

[T3]–: Singular-perturbative reduction to Birkhoff normal form andinstanton-type formal solutions of Hamiltonian systems. To appear in Publ.
RIMS, Kyoto Univ.

21[Tkal][Tka2]

singular points
K. Takano: Reduction for Painlev\’e equations at the fixedof the first kind. Funkcial. Ekvac., 29(1986), 99-119.
points of: Reduction for Painlev\’e equations at the fixed singular
the second kind. J. Math. Soc. Japan, $42(1990),$ 423-443