Koike, Tatsuya

Nishikawa, Yukihiro

Kyoto University Research Information Repository

Title On the exact WKB analysis for the fourth Painleve hierarchy(Recent Trends in Exponential Asymptotics)Author(s)Koike, Tatsuya; Nishikawa, YukihiroCitation数理解析研究所講究録 (2005), 1424: 160-169Issue Date2005-04URL http://hdl.handle.net/2433/47225RightType Departmental Bulletin PaperTextversionpublisherKyoto University160On the exact WKB analysis for the fourthPainleve hierarchyTatsuya KOIKEDepartment of MathematicsGraduate School of ScienceKyoto UniversityYukihiro NISHIKAWAHitachi Ltd.1 IntroductionIn the exact WKB anatytic study of Painleve equations, [AKT], [KT1], [KT2]and [T] gave the explicit description of the connection formulas of Painlevetranscendents. In their analysis some properties on the relation which holdsbetween the Stokes geometry of the Painleve equations and that of theirLax pairs play the key role, After their success, [KKNT] showed that theseproperties of the Stokes geometry also hold for several Painleve hierarchies( $P_{1},$ $P_{\mathrm{I}1-1}$ and $P_{\mathrm{I}1-2}$ in the notation of [KKNT]).In this article we discuss the fourth Painleve hierarchy which was intro-duced by Gordoa-Joshi-Pickering [GJP], and report that such basic proper-ties of the Stokes geometry also hold for this hierarchy. Here we give theoutline of our results; the detailed explanation of our results including nu-merical results of the Stokes geometry and of Nishikawa Phenomena will bepublished somewhere.1424 2005 160-1691612 The fourth Painleve hierarchy and theirLax pairs with a large parameterThe fourth Painleve hierarchy we treat here was introduced in [GJP]. Thefollowing is essentially same with that given in [GJP, p.336], but we introducea large parameter $\eta$ to employ the exact WKB analysis:Definition 2.1. We call(2.1) $(P_{IV})_{m}$ : $\{$$\eta^{-1}\partial_{t}X_{m}=2Y_{m}+uX_{m}+g-2\alpha$,$\eta^{-1}X_{m}\partial_{t}Y_{m}=-vX_{m}^{2}+(Y_{m}+\frac{1}{2}g-\alpha)^{2}-\frac{1}{4}\beta^{2}$ .the fourth Painleve hierarchy for $m=1,2,$ $\cdots$ . Here $\alpha,$ $\beta$ and $g$ are arbitraryconstants, and$X_{m}=K_{m}+gt$, $Y_{m}=L_{m}$ ,where $\{K_{j}\}$ and $\{L_{j}\}$ are $polynom\iota al$ of unknotnn functions $u,$ $v$ and theirderivatives recursively defined by the follorning relation:(2.2)$\eta^{-1}\partial_{t}(\begin{array}{l}K_{j+1}L_{j+1}\end{array})=\frac{1}{2}(\begin{array}{ll}\eta^{-2}\partial_{t\eta^{-}}^{2\mathrm{l}}u’+u\eta^{-1}\partial_{t}- 2\eta^{-1}\partial_{t}2\eta^{-\mathrm{l}}v\partial_{t}+\eta^{-1}v_{t} u\eta^{-1}\partial_{t}+\eta^{-2}\partial_{t}^{2}\end{array}) (\begin{array}{l}K_{j}L_{j}\end{array})$$(j\geq 0)$ utith $K_{0}=2$ and $L_{0}=0$ . Here, and in what follouts, ’ denotes thedifferentiation with respect to $t$ .Remark 2.1. Precisely speaking, this recursion relation (2.2) ioes not define$\{K_{j}\}$ and $\{L_{j}\}$ uniquely; there remain ambiguities of integral constants foreach step. In the following we take these integral constants as zero at eachstep. See [N1], [N2] for the precise definition of $\{K_{j}\}$ and $\{L_{j}\}$ , and for theproof that this recursion relation (2.2) really defines $\{K_{j}\}$ and $\{L_{j}\}$ as poly-nomials of $u$ and $v$ artd their derivatives. Here we list up first few membersof $\{K_{j}\}$ and $\{L_{j}\}$ in a vector notatzon:(2.3) $(\begin{array}{l}K_{1}L_{1}\end{array})$ $=$ $(\begin{array}{l}uv\end{array})$ ,(2.4) $(\begin{array}{l}K_{2}L_{2}\end{array})$ $=$ $\frac{1}{2}(\begin{array}{l}u^{2}+2v-\eta^{-1}u’2uv+\eta^{-1}v\end{array})$ ,(2.5) $(\begin{array}{l}K_{3}L_{3}\end{array})$ $=$ $( \frac{1}{2})^{2}(\begin{array}{l}-u36uv\eta^{-1}uu’+\eta^{-23}+u^{l\prime}+3v^{2}+3\eta^{-1}uv’+3uv\eta^{-22}v’’\end{array})$ .182Remark 2.2. $(P_{1\mathrm{V}})_{1}$ becomes(2.6) $\{$$\eta^{-1}u’$ $=$ $2v+u(u+gt)+g-2\alpha$ ,$\eta^{-1}(u+gt)v$ $=$ $-v(u+gt)^{2}+(v+ \frac{1}{2}\alpha-g)^{2}-\frac{1}{4}\beta^{2}$.We can solve the first equation utith respect to $v$ . By substituting this $v$ intothe second equation, we obtain(2.7) $y”= \frac{(y’)^{2}}{2y}+\eta^{2}\{\frac{3}{2}y^{3}-2gty^{2}+(\frac{g^{2}}{2}t^{2}+g-2\alpha-\eta^{-1}g)y-\frac{\beta^{2}}{2y}\}$ ,where $y=u+gt$ . This is the rraditional fourth Painleve’ equation (thestandard form of the fourth Painleve equation is obtained by setting $g=-2$and $\eta=1$ in (2.7) $)$ . This is the reason whey toe call (2.1) the fourth Fainlevehierarchy.Remark 2.3. As is expiained in [GJP] we can add more terms with arbitraryconstants in $(P_{\mathrm{I}\mathrm{V}})_{m}$ . Our analysis in the following can be applied to suchextenied $(P_{1\mathrm{V}})_{m}$ .The underlying Lax pair of $(P_{1\mathrm{V}})_{m}$ is also introduced in [GJP]:(2.8) $(L_{1\mathrm{V}})_{m}$ : $\{$$( \frac{\partial}{\partial x}-\eta A)\psi=0$, (2.8.a)$( \frac{\partial}{\partial t}-\eta B)\psi=0$ , (2.8.b)where(2.9) A $= \frac{1}{gx}(\begin{array}{ll}-\frac{1}{2}\eta^{-1}\frac{\partial S_{m}}{\partial x} S_{m}\frac{1}{2}\eta^{-2}\frac{\partial^{2}S_{m}}{\partial x^{2}}qS_{m}- \frac{1}{2}\eta^{-1}\frac{\partial S_{m}}{\partial x}\end{array}),$ $B=(\begin{array}{l}010q\end{array})$and(2.10) $q=(x- \frac{1}{2}u)^{2}-v+\frac{1}{2}\eta^{-1}u_{t}$, $S_{m}= \frac{1}{2}(tg+\sum_{j=0}^{m}x^{m-j}K_{j})$Actually the compatibility condition(2.11) $\frac{\partial A}{\partial t}-\frac{\partial B}{\partial x}+\eta(AB-BA)=0$of $(L_{\mathrm{I}\mathrm{V}})_{m}$ is reduced to $(P_{\mathrm{I}\mathrm{V}})_{m}$ .1833 Relations of the Stokes geometry of the $P_{\mathrm{I}\mathrm{V}}$hierarchy and its underlying Lax pairIn this section we considered the Stokes geometry of $(P_{1\mathrm{V}})_{m}$ and $(L_{\mathrm{I}\mathrm{V}})_{m}$ (wewill fix $m$ in the following). To this purpose we first construct a formalsolution of $(P_{\mathrm{I}\mathrm{V}})_{m}$ of the following form(3.1) $\{$ $u=$\^u $(t, \eta)$ $=$ $u_{0}(t)+\eta^{-[perp]}u_{1}(t)+\cdots$ ,$v$ $=\hat{v}(t, \eta)$ $=$ $v_{0}(t)+\eta^{-1}v_{1}(t)+\cdots$By substituting this expansion into $(P_{\mathrm{I}\mathrm{V}})_{m}$ , we find that $u_{0}$ and $v_{0}$ satisfy(3.2) $\{$$2Y_{m}^{(0)}+uX_{m}^{(0)}+g-2\alpha=0$ ,$-v(X_{m}^{(0)})^{2}+(Y_{m}^{(0)}+ \frac{1}{2}g-\alpha)^{2}-\frac{1}{4}\beta^{2}=0$,or, equivalently,(3.3) $\{$$2L_{m_{1}}^{(0)}+uK_{m}^{(0)}+g-2\alpha=0$ ,$-v(K_{m}^{(0)}+gt)^{2}+(L_{m}^{(0)}+ \frac{1}{2}g-\alpha)^{2}-\frac{1}{4}\beta^{2}=0$ ,where $X_{m}^{(0)},$ $Y_{m}^{(0)},$ $K_{m}^{(0)}$ and $L_{m}^{(0)}$ are top order terms of $X_{m},$ $Y_{m},$ $K_{m}$ and $L_{m}$$\mathrm{W}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{r}\mathrm{e}\mathrm{S}\mathrm{a}\mathrm{n}\mathrm{d}K_{m}^{(0}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{o}1\mathrm{f}^{\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}1\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\eta,\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{l}\mathrm{y}.\mathrm{W}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}L_{m}^{(0\rangle}}\mathrm{y}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{a}1\mathrm{s}\mathrm{o}\mathrm{f}u\mathrm{a}\mathrm{n}\mathrm{d}v(\mathrm{S}\mathrm{e}\mathrm{e}[\mathrm{N}\mathrm{l}],[\mathrm{N}2].),\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}(3.3)\mathrm{i}\mathrm{s}$algebraic equation with respect to $u$ and $v$ . Once $u_{0}$ and $v_{0}$ are chosen andfixed so that they satisfy (3.3), higher order term $\mathrm{s}u_{j}$ and $v_{I}(j\geq 1)$ aredetermined uniquely and recursively. A formal soiution (3.1) constructed inthis way is cailed a zero parameter solution in the following.Once a zero parameter solution u=\^u, $v=\hat{v}$ is constructed, we substitutethis zero parameter solution into our quantities $A,$ $B,$ $K_{m},$ $L_{m},$ $S_{m}$ and $q$ .As a result, we obtain expansions of $A$ and $B$ etc. with respect to the largeparameter $\eta$ like(3.4) $A_{m}|_{u=\hat{u},v=\hat{v}}$ $=$ $A_{m,0}(x, t)+\eta^{-1}A_{m,1}(x, t)+\cdots$ ,(3.5) $B_{m}|_{u=\hat{u},v=\hat{v}}$ $=$ $B_{m,0}(x, t)+\eta^{-1}B_{m,1}(x, t)+\cdots$Explicitly we obtain(3.6) $A_{m,0}= \frac{1}{gx}(\begin{array}{ll}0 S_{m_{\prime}0}q_{0}S_{m,0} 0\end{array})$ , $B_{m,0}=(\begin{array}{ll}0 1q_{0} 0\end{array})$ ,164where(3.7) $q_{0}$ $=$ $(x- \frac{1}{2}u_{0})^{2}-v_{0}$ ,(3.8) $S_{m,0}$ $=$ $\frac{1}{2}(tg+\sum_{j=0}^{m}x^{m-i}K_{i,0})$ .Now we consider the Stokes geometry of (2.8.a) which is defined with thecharacteristic polynomial(3.9) $\det$ (A $-A_{m,0}$ ( $x$ , $?)$ ) $=\lambda^{2}+\det A_{m,0}(x, t)$ .First we find $A_{m,0}=(2S_{m,0}/gx)B_{m,0}$ by (3.6), which implies(3.10) $\det A_{m,0}=(\frac{2S_{m,0}}{gx})^{2}\det B_{m,0}$ .Hence we obtain the following proposition on turning points of (2.8.a):Proposition 3.1. (i) Eq(2.8.a) has m double turning points$\{b_{1}(t), b_{2}(t), \ldots, b_{m}(t)\}$$and_{J}$ each $b_{\gamma}(t)$ is a root of $S_{m,0}(x, t)=0$ .(ii) $Eq(\mathit{2}.\mathit{8}.a)$ has two simple turning points $\{a_{1}(t), a_{2}(t)\}$ and each $a_{j}(t)$ isa root of$\det B_{m,0}=(x-\frac{1}{2}u_{0})^{2}-v_{0}=0$ .We also obtain the relation between the eigenvalues of $A_{0}$ and that of $B_{0}$ .The following proposition can be shown easily (See [KKNT]):Proposition 3.2. Let $\lambda_{+}$ and A-be eigenvalues of $A_{0_{J}}\mu_{+}$ and $\mu_{-}$ eigenval-ues of $B_{0}$ . Then we obtain(3.11) $\frac{\partial}{\partial t}\lambda_{+}=\frac{\partial}{\partial x}\mu_{+}$ , $\frac{\partial}{\partial t}\lambda_{-}=\frac{\partial}{\partial x}\mu_{-}$ ,or(3.12) $\frac{\partial}{\partial t}\lambda_{+}=\frac{\partial}{\partial x}\mu-$ , $\frac{\partial}{\partial t}\lambda_{=}=\frac{\partial}{\partial x}\mu_{+}$ .165Next we consider the Stokes geometry for the fourth Painleve hierarchy.The Stokes geometry of $(P_{\mathrm{I}\mathrm{V}})_{m}$ is defined as that of linearized equation of$(P_{\mathrm{I}\mathrm{V}})_{m}$ at a zero parameter solutions u=\^u, $v=\hat{v}$ . Hence we first determinethe (characteristic polynomial of) linearized equation. To this purpose weset u=\^u+\triangle u and $v=\hat{v}+$ Av in $(P_{\mathrm{I}\mathrm{V}})_{m}$ , and consider the linear part in$(\triangle u, \triangle v)$ . Let $\triangle X_{m}$ and $\triangle Y_{m}$ be linear parts of $X_{m}$ and $Y_{m}$ in ( $\triangle u$ , Au),respectively. In terms of this $\triangle X_{m}$ and $\triangle Y_{m}$ , the linearized equation of$(P_{\mathrm{I}\mathrm{V}})_{m}$ can be written as(3.13) $\{$$\eta^{-1}\frac{d}{dt}\triangle X_{m}$ $=$ $2\triangle Y_{m}+\hat{u}\triangle X_{m}+\hat{X}_{m}\triangle u$ ,$\hat{X}_{m}\eta^{-1}\frac{d}{dt}\triangle Y_{m}$ $=$$- \eta^{-1}\frac{d\hat{Y}_{m}}{dt}\triangle X_{m}-(\hat{X}_{m})^{2}\triangle v-2\hat{v}\hat{X}_{m}\triangle X_{m}$+2 $( \hat{Y}_{m}+\frac{1}{2}g-\alpha)\triangle Y_{m}$ ,where(3.14) $\hat{X}_{m}=X_{m}|_{u=\dot{u},v=\hat{v}}$ , $\hat{Y}_{m}=Y_{m}|_{u=\hat{u},v=\hat{v}}$ .This linearized equation can be expressed in the following form:(3.15) $\eta^{-1}\frac{d}{dt}(\begin{array}{l}\triangle X_{m}\triangle Y_{m}\end{array})$ $=$ $(\begin{array}{lll}\hat{u} 2-2\hat{v}-\eta^{-1}\frac{1}{\hat{X}_{m}} \frac{d\hat{Y}_{m}}{dt} -\hat{u}\end{array})(\begin{array}{l}\triangle X_{m}\triangle Y_{m}\end{array})$$+\hat{X}_{m}(\begin{array}{ll}1 0\mathrm{O} -1\end{array})(\begin{array}{l}\triangle u\triangle v\end{array})$where we have used(3.16) $2\hat{Y}_{m}+g-2\alpha=-u\hat{X}_{m}$ .We will now determine $\triangle X_{m}$ and $\triangle Y_{m}$ . By its definition we easily findthat $\triangle X_{m}=\triangle K_{m}$ and $\triangle Y_{m}=\triangle Y_{m}$ . Since $K_{m}$ and $L_{m}$ are polynomials in$u,$ $v$ and their derivatives, there exits a2 $\mathrm{x}2$ matrix valued linear differentialoperator(3.17) $P^{(m)}(t,\eta^{-1}\partial_{t};\eta)=P_{0}^{(m)}(t,\eta^{-1}\partial_{t})+\eta^{-1}P_{1}^{(m\rangle}(t,\eta^{-1}\partial_{t})+\cdots$for which(3.18) $(\begin{array}{l}\triangle K_{m}\triangle L_{m}\end{array})=P^{(m)}(t, \eta^{-1}\partial_{t\mathrm{i}}\eta)(\begin{array}{l}\triangle u\triangle v\end{array})$166hoids. In terms of this operator $P^{(m)}(t, \eta^{-1}\partial_{t};\eta)$ , the characteristic polyno-mial $C(t, \nu)$ (i.e., the top order part with respect to $\eta$ of the symbol obtainedby replacing $\eta^{-1}\partial_{t}$ by $\nu$ ) of (3.15) is expressed as(3.19)$C(t, \nu)$ $=$ $\det[(\begin{array}{ll}-1 00 1\end{array})\{$ $(\begin{array}{ll}\nu-u_{0} -22v_{0} \nu+u_{0}\end{array})P_{0}^{(m)}(t, \nu)+\hat{X}_{m,0}(\begin{array}{ll}-1 00 1\end{array})\}]$$=$ $\det[(\begin{array}{ll}-\nu+u_{0} 22v_{0} \nu+u_{0}\end{array})P_{0}^{(m)}(t, \nu)+\hat{X}_{m,0}(\begin{array}{ll}1 00 1\end{array})\ovalbox{\tt\small REJECT}\cdot$(Here we multiply a matrix $(\begin{array}{ll}-\mathrm{l} 00 1\end{array})$ for the later convenience).Proposition 3.3. We have(3.20) $C(t, \nu)=(-1)^{m}4\prod_{j=1}^{m}\det(\mu-B_{0})|_{\mu=\nu/2,x=b_{\mathrm{j}}(t)})$where $\{b_{j}(t)\}$ are double turning points defined in Proposition 3.1.Proof. To determine $P_{0}^{(m)}(t, \nu)$ we use the recursion relation (2.2). Considerthe linear part in (Au, $\triangle v$ ) of both side of (2.2), we find that $\{P_{0}^{(j)}(t, \nu)\}$should satisfy the following recursion relation:(3.21) $P_{0}^{(j+1)}(t, \nu)=\mathcal{R}P_{0}^{(j)}(t, \nu)+\frac{K_{j,0}}{2}I_{2}$where(3.22) $\mathcal{R}=\frac{1}{2}(\begin{array}{ll}-\nu+u_{0} 22v_{0} \nu+u_{0}\end{array})$ , $I_{2}=(\begin{array}{ll}\mathrm{l} 00 1\end{array})\mathrm{I}$By solving this recursion relation with the initial condition $P_{0}^{(1)}(t, \iota/)=I_{2}$ ,we have(3.23) $P_{0}^{(m)}(t, \nu)=\frac{1}{2}\sum_{j=0}^{m-1}’ \mathcal{R}^{m-j-1}\hat{K}_{j,0}$.167Hence we obtain(3.24) $C(t, \nu)$ $=$ $\det[2\mathcal{R}P_{0}^{(m)}(t, \nu)+\hat{X}_{m,0}]$(3.25) $=$ $\det\ovalbox{\tt\small REJECT}\sum_{j=0}^{m}\mathcal{R}^{m-j}K_{j,0}+gtI_{2}]$(3.26) $=$ 4 $\det\ovalbox{\tt\small REJECT}\frac{1}{2}(gtI_{2}+\sum_{j=0}^{m}\mathcal{R}^{m-j}K_{j,0})\ovalbox{\tt\small REJECT}$ ,Here we note that $S_{m,0}$ can be factorized as(3.27) $S_{m,0}= \frac{1}{2}(tg+\sum_{j=0}^{m}x^{m-j}K_{j_{1}0})=\prod_{j=1}^{m}(x-b_{j}(t))$(See Proposition 3.1 (i)). Substituting 72 into $x$ in this equation (3.27) wefind(3.28) $C(t, \nu)=4\prod_{j=1}^{m}(\mathcal{R}-b_{j}I_{2})$ .Hence we conclude that(3.29) $C(t, \nu)$ $=$ $4 \det\prod_{j=1}^{m}(\mathcal{R}-b_{j}(t)I_{2})$(3.30) $=$ 4 $\prod_{j=1}^{m}\{\frac{1}{4}(\begin{array}{llll}-\nu+u_{0} -2b_{j} 2 2v_{0} \nu+u_{\mathrm{O}} -2b_{j}\end{array}) \}$(3.31) $=$ 4 $\prod_{j=1}^{m}\{-(\frac{1}{2}\nu)^{2}+(bj-\frac{1}{2}u_{0})^{2}-v_{0}\}$(3.32) $=$ $(-1)^{m}4 \prod_{\mathrm{i}=1}^{m}\det(\mu-B\mathrm{o})|_{\mu=\nu/2,x=b_{j}(t)}$This completes the proof of the proposition. ElAs is explained already, Stokes geometry for $(P_{1\mathrm{V}})_{m}$ is defined in termsof $C(t, \nu)$ . Note that it follows from the Proposition 3.3 that $C(t, \nu)$ has the168form $f(\nu^{2}, t)$ with some polynomial $f$ of degree $m$ . This impties that thereare two kind of turning points for $(P_{1\mathrm{V}})_{m}$ :(i) A turning points where thedegree 0 part of $f$ vanishes (“a turning point of the first kind”), and (ii) aturning point where the discriminant of $f$ vanishes (“$\mathrm{a}$ turning point of thesecond kind” ). The following theorems are obtained by using Proposition 3.2and 3.3:Theorem 3.1. (i) Let $t=\tau^{1}$ be a turning point of the first kind of $(P_{IV})_{m}$ .Assume (a) $t=\tau$ is not a turning point of second kind, (b) $\partial C/\partial t$ loes notvanish at $t=\tau_{f}\nu=0$ . Then at $t=\tau^{1}$ a double tterning point $x=b_{j}(t)$merges with a simple turning point $x=a_{k}(t)$ in the Stokes geometry of(2.8.a) , Consequently the two eigenvalues $\nu_{J},\pm$ of $C$ merge and vanish at$t=\tau^{1}$ . Furthermore the following relation holds:(3.33) $\frac{1}{2}\mathit{1}_{\tau^{1}}^{t}(\nu \mathrm{j},+-\nu j)-)dt=f_{a_{h}(t)}^{b_{j}(t)}(\lambda_{+}-\lambda_{-})dx$.(ii) Let $t=\tau^{11}$ be a turning point of the second kind of $(P_{1\mathrm{V}})_{m}$ . Assume(a) $t=\tau$ is not a turning point of the fist kind, (b) $t=\tau$ is a simplezero of discriminant of $f(t, z)$ . Then at $1=\tau^{1\mathrm{I}}$ a double turning point $x=$$b_{j}(t)$ merges with another double turning point $x=b_{j},(t)$ . Consequentig twoeigenvalues $\nu_{j,+}$ and $\nu_{j’,+}$ of $C$ rnerge at $t=\tau^{\mathrm{I}\mathrm{I}}$ , and so do $\nu_{J},-and$ $\nu_{j’,-}$ .Furthermore the following relation holds:(3.34) $\int_{\tau^{\mathrm{I}\mathrm{I}}}^{t}(\nu_{j,+}-\nu_{j’,+})dt=-\oint_{\tau^{\mathrm{I}\mathrm{I}}}^{t}(\nu_{j,-}-\nu_{j’,-})dt=\int_{b_{j},(t)}^{b_{j}(t)}.(\lambda_{+}-\lambda_{-})dx$ .References[AKT] T. Aoki, T. Kawai and Y. Takei: WKB analysis of Painleve’ tran-scendents with a large parameter II, in Structure of Solutions ofDifferential Equations, World Scientific, 1996, p. 1-49.[GJP] P.R. Gordoa, N. Joshi and A. Pickering: On ageneralized $2+1$ dis-persive water wave hierarchy, Publ. RIMS, Kyoto Univ,, 37(2001),327-347.[KKNT] T. Kawai, T. Koike, Y. Nishikawa and Y. Takei: On the Stokes ge-ometry of Higher order Painleve equations, To appear in Aste’risque.169[KT1] T. Kawai and Y. Takei: WKB analysis of Painleve transcendentswith a large parameter. I, Adv. Math., 118(1996), 1-33.[KT2] –: WKB analysis of Painleve transcendents with a large pa-rameter. III, Adv. Math., 134(1998), 178-218.[N1] Y. Nishikawa: WKB analysis of $P_{\mathrm{I}\mathrm{I}}$ - $P_{1\mathrm{V}}$ hierarchies, Master Thesis,Kyoto Univ., 2003. (In Japanese.)[N2] Y. Nishikawa: Towards the exact WKB analysis of $P_{\mathrm{I}1}$ - $P_{1\mathrm{V}}$ hierar-chies, In preparation.[T] Y. Takei: An explicit description of the connection formula for thefirst Painleve equation, in Towards the Exact WKB Analysis ofDifferential Equations, Lirtear or Non-Linear, Kyoto Univ, Press,2000, p. 271-296.

On the exact WKB analysis for the fourth Painleve hierarchy (Recent Trends in Exponential Asymptotics)

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On the exact WKB analysis for the fourth Painleve hierarchy (Recent Trends in Exponential Asymptotics)

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