A unified family of $P_{\mathrm{J}}$-hierarchies (J=I, II, IV, 34) with a large parameter (Algebraic analytic methods in complex partial differential equations)

全文

(1)92. 数理解析研究所講究録第2020巻 2017年 92-95. A unified. family of P_{\mathrm{J} ‐hierarchies (\mathrm{J}=\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{V}34) with a large parameter By. Yoko UMETA. *. Abstract The purpose of this note is to give a unified family of P_{\mathrm{J} ‐hierarchies This note is a short summary of papers [18] and [19].. ( \mathrm{J}=\mathrm{I} II, IV, 34) ,. with. a. large parameter.. A unified. §1.. family. of P_{\mathrm{J} ‐hierarchies. (\mathrm{J}=\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{I}\mathrm{V}34). by Kudryashov ([11],[12]), Gordoa and Pickering ([4]), Shimomura ([13],[14]), Gordoa, Joshi and Pickering ([5]), Clarkson, Joshi and Pickering ([3]). To establish connection formulas for solutions of the higher The. P_{\mathrm{I} , P_{\mathrm{J}\mathrm{I} , P_{\mathrm{J}\mathrm{V} and P_{34} ‐hierarchies. order Painlevé. equations. perturbation theory. they. introduced. a. is. one. were. studied. of important. In the series of papers. large parameter. $\eta$ to. subjects in algebraic analysis by Kawai, Koike, Nishikawa. of. P_{\mathrm{J} ‐hierarchies ( \mathrm{J}=\mathrm{I} II, IV, 34) ([6], ,. and many important results have been established from. a. view. point of the. analysis. (See [6], [7], [8], [15], [16], [17] and etc). In what follows, family of P_{\mathrm{J} ‐hierarchies ( \mathrm{J}=\mathrm{I} II, IV, 34) with a large parameter $\eta$.. we. singular. and. Takei,. [9], [10]). exact WKB. give. a. unified. ,. Let. m. be. an. arbitrary. natural number. Let. of unknown functions u_{k}, v_{k}. (k=1,2, \ldots , m). U, V and C denote generating functions and constants c_{k}. as. follows.. U( $\theta$):=\displaystyle \sum_{k=1}^{m+1}u_{k}$\theta$^{k},V( $\theta$):=\sum_{k=1}^{m+1}v_{k}$\theta$^{k},C( $\theta$):=\sum_{k=1}^{m}c_{k}$\theta$^{k}. Here $\theta$ denotes. functions of t. .. an. independent variable,. Throughout. the note, the notation A\equiv B. are. arbitrary holomorphic. means. that A-B is. zero. Subject Classification(s): Primary 34\mathrm{E}20 ; Secondary 34\mathrm{M}40. analysis, Painlevé hierarchies, instanton‐type solutions This work was supported by JSPS KAKENHI Grant \mathrm{N}\mathrm{o}.15\mathrm{K}17557. Department of Mathematical Sciences, Yamaguchi University, Yamaguchi City, Yamaguchi Pref. 753‐8512, Japan. 2010Mathematics. Key Words:. *. u_{m+1} and v_{m+1}. exact WKB.

(2) 93 YOKO UMETA. modulo $\theta$^{m+2} with. large. a. Let. .. us. consider the system of non‐linear. ordinary differential equations. parameter $\eta$ for these generating functions:. $\eta$^{-1}\displaystle\frac{d} t\left(\begin{ar y}{l U$\thea$\ V$\thea$ \end{ar y}\right)\equiv\left(\begin{ar y}{l f_{1}\ f_{2} \end{ar y}\right)\imes(1-U)+\left(\begin{ar y}{l 0-1\ 10 \end{ar y}\right)(_{\frac{} ^{\frac{\partilH}{\partilH\partilV\partilU})+\left(\begin{ar y}{l 0\ \frac{H(U,V)}{1-U} \end{ar y}\right),. (1.1). H(U, V). where. following. is. a. polynomial. in U and V with. arbitrary complex. constants p_{i} of the. form. H(U, V):=(p_{1}U^{2}+p_{2}V^{2}) $\theta$+p_{3}UV+p_{4}CU+p_{5}CV+p_{6}U+p_{7}V+p_{8}C+p_{9}, and. f_{1}, f_{2}. are. defined. by. f_{1}:=p_{7}+( $\alpha$ u_{1}+p_{5}c_{1}) $\theta$+(y_{1}+(y_{1}u_{1}+y_{2}) $\theta$)$\theta$^{m}, f_{2}:=- $\beta$-(2 $\beta$ u_{1}+ $\alpha$ v_{1}+ $\epsilon$ c_{1}) $\theta$+(z_{1}+(2z_{1}u_{1}-y_{1}v_{1}+z_{2}) $\theta$)$\theta$^{m}. Here y_{i}, z_{i}. are. arbitrary holomorphic functions of $\alpha$:=p_{3}+p_{7},. $\beta$,. t and $\alpha$,. $\beta$ :=p_{6}+p_{9}. and. follows,. (1.1). $\epsilon$ are. given by. $\epsilon$:=p_{4}+p_{8},. respectively. If p_{i} , yí, z_{i}. (P_{\mathrm{J} )_{m}. are. determined. as. then. is. same as. the. general. member. (See [19]).. of P_{\mathrm{J} ‐hierarchy with $\eta$. If p_{2}=-1, p_{8}=2, p_{9}=1, z_{2}=2t , the others. =0\Rightarrow(P_{\mathrm{I}})_{m}.. z_{1}=2 $\gamma$ t( $\gamma$\neq 0) z_{2}=4 $\gamma$ tc_{0} the others =0\Rightarrow(P_{34})_{m}. If p_{2}=1, p_{3}=2, p_{5}=2 the others =0\Rightarrow(P_{\mathrm{I}\mathrm{I}})_{m}. If p_{2}=1, p_{3}=2, p_{5}=2, y_{1}=-2 $\gamma$ t( $\gamma$\neq 0) the others =0\Rightarrow(P_{\mathrm{I}\mathrm{V}})_{m}. If p_{2}=-1, p_{8}=2, p_{9}=1,. ,. ,. ,. ,. §2. We. can. apply. The existence of. the method. given. general. in. [2]. formal solutions of. to the. Case I:. $\alpha$=p_{3}+p_{7}\neq 0,. p_{2}\neq 0.. Case II:. $\alpha$=p_{3}+p_{7}=0,. $\beta$=p_{6}+p_{9}\neq 0,. and. we. have the. following. Theorem 2.1. In the. theorem.. cases. called instanton‐type solutions. I,. for. II,. (For we. (1.1):. more. have. cases. I,. (1.1). II:. p_{2}\neq 0. precise statements,. formal. see. [19].). solutions with 2m. free parameters.

(3) 94 A. UNIFIED FAMILY OF. §3. Theorem 3.1. Let the. following. Here. us. P_{j} ‐HIERARCHIES (\mathrm{J}=\mathrm{I} II, IV, 34 ) ,. Lax. (1.1). for. pair. WITH $\eta$. determine p_{1}, u_{m+1} and v_{m+1}. of (1.1). so. that. they satisfy. conditions.. \left{\begin{ar y}{l p_{1}=0,p_{2}\neq0,\ $\gam \alph \tea$^{k-2}=$\alph$u_{7n+1}^{J$\thea$^{m+1}(y_{1}'$\thea$^{m}+y_{2}^J$\thea$^{m+1}),\ z_{1}$\thea$^{m}+(z_{1}^lu_{1}-y ^{J}v_1+z_{2}^l)$\thea$^{m+1}(2$\beta$u_{\acute{m}+1 $\alph$v_{\acute{m}+1)$\thea$^{m+1} $\gam \beta\heta$^{k-2}=0\prime. \nd{ar y}\right.. u_{\acute{m}+\mathrm{I}. denotes the derivative of u_{m+1} with respect to t, $\gamma$ is. and k is determined. by. the above conditions. Then. compatibility condition of. (I). the. our. system. a non‐zero. (1.1). is. constant,. equivalent. to the. following equations:. ($\gam a\theta$^{k}\displaystyle\frac{\partial}{\partial$\theta$}-$\eta$A)$\psi$($\theta$,t)=0. (11). ,. (\displaystyle \frac{\partial}{\partial t}- $\eta$ B) $\psi$( $\theta$, t)=0,. where. with. A:=\left(\begin{ar y}{l \triangle_{1}(\mathrm{l}-&U)$\thea$\ \triangle_{2}&-\triangle_{1} \end{ar y}\right),B:=\left(\begin{ar y}{l \square_{1}&1\ \coprod_{2}&-\fbox_{1} \end{ar y}\right) \displaystyle \triangle_{1}:\equiv-\frac{1}{2}\frac{\partial H}{\partial V}-\frac{p_{3} {2}(1-U)+\frac{1}{2}(y_{1}$\theta$^{m}+y_{2}$\theta$^{m+1})-\frac{ $\alpha$}{2}u_{m+1}$\theta$^{m+1},. \displaystyle \triangle_{2}:\equiv p_{2}\times(-\frac{\partial H}{\partial U}-\frac{H(U,V)}{1-U}-(z_{1}$\theta$^{m}+(z_{1}u_{1}-y_{1}v_{1}+z_{2})$\theta$^{m+1}) -(2 $\beta$ u_{m+1}+ $\alpha$ v_{m+1})$\theta$^{m+1}). \displaystyle \coprod_{1}:=-\frac{1}{2 $\theta$}( $\alpha$+( $\alpha$ u_{1}+p_{5}c_{1}) $\theta$). ,. \displaystyle \square _{2}:=-\frac{p_{2} { $\theta$}( $\beta$+(2 $\beta$ u_{1}+ $\alpha$ v_{1}+ $\epsilon$ c_{1}) $\theta$) The Lax pair associated with. ,. (1.1) plays. .. an. important role. in. analyzing. the Stoke. geometry of (1.1) (See [18]).. Acknowledgments The author would like to express her sincere gratitude to Professors Okada, Naofumi Honda and Hiroshi Yamazawa for giving the opportunity to. Yasunori. her to take part in the conference.. References. [1] Aoki, T., Multiple‐scale analysis Bessatsu B5 (2008), 89‐98.. for. higher‐order. Painlevé. equations, RIMS Kôkyûroku.

(4) 95 YOKO UMETA. [2] Aoki, T., Honda,. N. and Umeta, Y., On a construction of general formal solutions for equations of the first Painlevé hierarchy I, Adv. in Math., 235 (2013), 496‐524. [3] Clarkson, P. A., Joshi. N, Pickering. A, Bäcklund transformations for the second Painlevé. approach, Inverse Problems 15 (1999) 175‐187. Pickering. A, Nonisospectral scattering problems: A key to integrable hierarchies, J. Math. Phys., 40 (1999), 5749‐5786. [5] Gordoa. P. R, Joshi. N and Pickering. A, On a generalized 2+1 dispersive water wave hierarchy, Publ. RIMS, Kyoto Univ., 37 (2001), 327‐347. [6] Kawai, T., Koike, T., Nishikawa, Y. and Takei, Y., On the Stokes geometry of higher order Painlevé equations, Astérisque 297 (2004), 117‐166. [7] Kawai, T. and Takei, Y., WKB analysis of higher order Painlevé equations with a large hierarchy:. [4]. modified truncation. a. Gordoa. P. R and. parameter‐ Local reduction of ‐parameter solutions for Painlevé hierarchies (P_{\mathrm{J} )(\mathrm{J}=\mathrm{I}, II‐1 or II‐2), Adv, Math., 203 (2006), 636‐672. [8J—, WKB analysis of higher order Painlevé equations with a large parameter. II. Structure theorem for instanton‐type solutions of (P_{J})_{m} (J I, 34, II‐2 or IV) near a simple P‐turning point of the first kind, Pub. RIMS, Kyoto Univ. 47 (2011),153-219. [9] Koike, T., On the Hamiltonian structures of the second and the forth Painlevé hierarchies =. and. [10]. degenerate. —,. (2008),. On. Garnier systems, RIMS Kôkyûroku Bessatsu B2 (2007), 99‐127. expressions of the Painlevé hierarchies, RIMS Kôkyûroku Bessatsu B5. new. 153‐198.. [11] Kudryashov,. N.. A., The first and second Painlevé equations of higher order and some them, Phys. Lett. A, 224 (1997), 353‐360. N. A. and Soukharev, M. B., Uniformization and transcendence of solutions Kudryashov, [12] for the first and second Painlevé hierarchies, Phys. Lett. A, 237 (1998), 206‐216. [13] S. Shimomura, Painlevé properties of a degenerate Garnier system of (9/2)‐type and of a certain fourth order non‐linear ordinary differential equation, Ann. Scuola Norm. Sup. Pisa, 29 (2000), 1‐17. A certain expression of the first Painlevé hierarchy, Proc. Japan Acad., Ser. A, [14] relations between. —,. 80 (2004), 105‐109. [15] Takei, Y., An explicit description of the. Toward the Exact WKB. Analysis. (2000),. 271‐296.. University. [16]. —,. Press. Toward the exact WKB. connection formula for the first Painlevé. of Differential. Equations, Linear. analysis for instanton‐type. or. equation,. Non‐Linear, Kyoto. solutions of Painlevé hierar‐. Kôkyûroku (2007), Instanton‐type formal solutions for the first Painlevé hierarchy, in: Algebraic Analysis of Differential Equations, Springer‐ Verlag (2008), 307‐319. [18] Umeta. Y, On the Stokes geometry of a unified family of P_{\mathrm{J} ‐hierarchies ( \mathrm{J}=\mathrm{I} II, IV, 34), chies,. [17]. RIMS. Bessatsu B2. 247‐260.. —,. ,. in. [19]. preparation.. —,. General formal solutions for. preparation.. a. unified. family. of P_{\mathrm{J} ‐hierarchies. ( \mathrm{J}=\mathrm{I} II, IV, 34), ,. in.

(5)