伊藤, 利明

蔡, 東生

Kyoto University Research Information Repository

Title1次分数型点変換,離散リッカチ方程式と離散パンルヴェ方程式(非線形可積分系による応用解析)Author(s)伊藤, 利明; 蔡, 東生Citation数理解析研究所講究録 (1994), 889: 49-69Issue Date1994-11URL http://hdl.handle.net/2433/84363RightType Departmental Bulletin PaperTextversionpublisherKyoto University1 ,(Toshiaki Itoh )(DongSheng Cai)1., 5$[4][8][11]$ .., ,., ..( ). ...2., 11. QRT2. (SCM) , ?3.. QRT SCM ( )4. QRTQRT5.,SCM6. WTC , ?KP WTC ,889 1994 49-69 49( )1$\#$$[$$-\backslash \prime x\Leftrightarrow\alpha_{\wedge}^{ffl1}ae$$\ni F$u$\Psi$,$\varpi$n$\Leftrightarrow\pi$ WTC$*$$\triangleright\sigma_{x}$ . $\overline{\tau}iX\vdash$1.3. QRT$[9][10]$ ,(3.1) $oRT\#_{\backslash }:y^{1}yg3(x^{t})-(y^{I}+y)g2(x^{1})+g1(x^{1})=0x^{1}xf_{3}(y)-(x^{1}+x)f_{2}(y)+f_{1}(y)=0,$ .,(3.2) $(x^{1}y^{1}|=$ ( $(g$ 5-Xyf3g(3y()X)$\iota$)) $|$ .$x^{t}=x((n+1)\Delta),$ $x=x(n \triangle)$ , ,(3.4) .$f(x)=(Ao(x))x(A_{1}(x))$$g(x)=(A_{0}^{T}(x))x(A_{1}^{T}(x))$ ,(3.3)$x=(1, x, x^{2})^{T}$$A_{i=0,1}=(\begin{array}{l}\alpha_{i}\beta_{iYi}6_{i}\epsilon_{i}\zeta_{i}\kappa_{i}\lambda_{i}\mu i\end{array})$ .(3.4) $\frac{(x^{1}\cdot A0y^{1})}{(x^{1}\cdot A_{1}y^{1})}=\frac{(x^{I}\cdot A0y)}{(x^{t}\cdot A_{1}y)}=\frac{(x\cdot A0y)}{(x\cdot A_{1}y)}$ .QRT $\emptyset$ ,i $)$ 18 - $\grave\circ$7- $p$ , ii) $(x, x^{\mathfrak{s}})$ , (y,y $\dagger$ )iii) $x=x((n- 1)\Delta)=x_{n- 1},$ $x^{\uparrow}=x((n+1)\Delta)=x_{n+1},$ $y=x(n\Delta)=x_{n},$ $y^{t}=x((n+2)\Delta)=x_{n+2}$, $f_{j}=g_{j}$ , 2 ,(3.5) $x_{n+1}x_{n- 1}f_{3}(x_{n})-(x_{n+1}+x_{n- 1})f_{2}(x_{n})+f_{1}(x_{n})=0$ .50iv) (2.11) , ,,v $)$ , ,vi) $(x Aiy)^{- 1}$ .4.4 . 1 (SCM)(35) , Ramani $[4][8][9][11]$. Ramani ,Conjecture:The movable singularities of integrable mapping are confined, i. e., they arecanceled out after a finite number of steps. Moreover, the memory of the initialcondition is not lost whenever a singularity is crossed.Implimentation:Step 1. For given a mapping, one must first find all possible ways a singularitycan emerge$<-->$ Analogy to the first step of algorithm for $ODE^{t}s$ where one looksfor all possible leading singular behaviors. (Find all singular manifold.)Step 2. Test the system wheather this divergence propagete in discrete timeor remains (confined). $<-->Reminisoent$ of the $\uparrow\uparrow$search for resonances $\dagger\uparrow$ .Step 3. Verify that indeed the singularity does not propagete beyond thedivergence. $<-->$ Resonance condition., $–$, ,. ,,. Ramani . .( ).[12].514. 2 QRT SCMA. Ramani [1] .QRT (35) 18 $\nearrow\grave\circ$ 7- , $n$.$($4.1 $)$ $\Delta PI$ : $x_{n+1}+x_{n- 1}=\frac{- x_{n}^{2}+bx_{n}+(\alpha n+\beta)}{x_{n}}$,(4.2) $\Delta$PII$:x_{n+1}+$ Xn$- 1= \frac{x_{n}(\alpha n+\beta)+\gamma}{l- x_{n}^{2}}$ ,$($4.3 $)$ $\Delta$PIII $: x_{n+1}x_{n- 1}=\frac{\gamma x_{n}^{2}+\zeta_{0}\lambda^{n/2}x_{n}+\mu 0\lambda^{n}}{\lambda^{n}x_{n}^{2}+\beta\lambda^{n/2}x_{n}+\gamma}$ ,$\Delta$PIV: Xn$+$ lXn-l $+$ Xn $($Xn$+$ l $+$ Xn-l$)=\underline{-(\alpha n+\beta)x^{3_{n}}+[\epsilon o_{4}^{1}--(\alpha n+\beta)^{2}]x_{n}^{2}+\mu}$(4.4)$x3+(\alpha n+\beta)x_{n}+[Yo+\frac{1}{4}(\alpha n+\beta)^{2}]$’(4.5) $\Delta PV:(2x_{n}- 1)x_{n+1}x_{n- 1}- x_{n}(x_{n+1}+x_{n- 1})$$= \frac{1_{-(0-\alpha_{0}\lambda^{2n})x_{n}^{3}+[\mathfrak{g}_{+}1_{(o+\alpha_{0}\lambda^{2n}- 2p_{0}\lambda^{n})]x_{n}^{2}- 2\mu x_{n}+\mu}}24}{\alpha_{0}\lambda^{2n}p0-\alpha 0)+1=(o+\alpha_{0}\lambda^{2n}- 2p0\lambda^{n})}$$x_{n}=x(n),$ $x_{n\pm 1}=x(n\pm 1)$ , $-\backslash ^{o_{\overline{7}}}$ $t=n\Delta+t_{0}$ $\triangle=1$. $\alpha$ , $\beta$ $\int\backslash \circ$ 7- $\trianglearrow 0$, . ,. $\Delta$ PIV$[6][19]$ .5. QRT, ,.52$\frac{dw}{d_{7_{\wedge}}}=po(z)+pl(z)w+p2w^{2}$$\downarrow$$f_{1}(z)u(z+1)u(z- 1)$$+f_{2}(z)u(z+1)+f_{3}(z)u(z- 1)+t_{4}(z)$$=0$$\downarrow$2 2$P2(Z)\frac{d^{2}u}{dz^{2}}-\{\frac{d_{P2}(z)}{dz}+p1(z)p2(z)\}_{dz}Lu$ $u(z+1)-\{\frac{f_{2}(z)}{f_{1}(z)}-\frac{f_{3}(z- 1)}{f_{1}(z- 1)}\}$$+po(z)p2(z)u=0$ –11$))$ 11)$)$ –ll)$)\}$ u(z-l) $=0$$(Y-\alpha)(Y-\beta)\zeta F(\alpha, \beta, Y+1;\zeta)$$+Y[(Y- 1)+(\alpha+\beta- 2_{Y}+1)\zeta]F(\alpha, \beta, Y;\zeta)$$-\gamma(Y- 1)(1-\zeta)F(\alpha, \beta, \gamma- 1;\zeta)=0$$f:zarrow zl=^{az\pm h}C7_{9}+d’|\begin{array}{ll}a bc d\end{array}|\neq 0$$P\{\lambda_{1}^{I}\lambda_{1}a1$$\lambda_{2}^{1}\lambda_{2}a2$ $\lambda_{3}^{t}\lambda_{3}a3,$$z\{=P\{\begin{array}{ll}b_{1} b_{2}. b_{3} m+|\lambda_{1}^{1}\lambda_{1} \lambda_{2}^{\dagger}\lambda_{2} \lambda_{3}^{I}\lambda_{3} Cz+D\int\end{array}$$( U_{1}(z)\frac{d}{d_{7}}+U_{2}(z))(D_{1}(z)_{d_{7}}^{\Lambda}+$ $(z))u(z,$ $\alpha)- u(z,$ $\alpha)=0$$U_{Z}u(\alpha$ ; $z)=(U_{1}(z)\frac{d}{d_{7}}+U_{2(Z))u(\alpha}$ ; $z)=u(\alpha+1,z)$$\frac{dx}{dt}=\frac{\partial H(x,y,t)}{\partial y}$ ,$- \frac{dy}{dt}=\frac{\partial H(x,y,t)}{\partial x}=0$$x^{I}xf_{3}(y)-(x^{1}+x)f_{2}(y)+f_{1}(y)=0$ ,$y^{t}yg3(x^{I})-(y^{1}+y)g2(x^{I})+g1(x^{1})=0$ .,5. 1,(5.1) $\frac{dw}{d_{7}}=po(z)+pl(z)w+p2w^{2}$$w_{1},$ $w_{2},$ $w_{3}$ .53(5 [ww—-ww –33–ww2l] $=0,$ $w^{1}w^{1}w^{1}$$w^{I}$1 $w$$w^{2}$1 $W1$ $w_{1,2}^{2}$$=0$$1$$W2$ $w_{2}$lw3 $w_{3}^{2}$.5. 2$[1][7]$ .(5.3) $f_{1}(z)u(z+1)u(z- 1)+f_{2}(z)u(z+1)+f_{3(z)u(Z- 1)}+f_{4}(z)=0$ ,$u(0)=u0$–1$\grave$R , $u(z+1)=-\frac{f_{3}(z)u(z- 1)+f_{4}(z)}{f_{1}u(z- 1)+f_{2}(z)}$ .3 .$f_{1(z)}$ ul $(z+1)u1(z- 1)+f_{2(z)u1(Z+1)}+f_{3(z)u1(Z- 1)}+r_{4(z)}=0$ ul(O) $=uO_{1}$fl(z) $u2(z+1)u2(z- 1)+r_{2(z)u2(z+1)}+f_{3(z)u2(Z- 1)}+f_{4(z)}=0$ $u2(0)=u02$fl(z) $u3(z+1)u3(z- 1)+f_{2(z)u3(Z+1)}+f_{3(z)u3(Z- 1)}+f_{4(z)}=0$ $u3(0)=u03$(5.3) ,1 $u(z+1)$ $u$ (z-1)1 $u_{1}(z+1)$ $ul(z- 1)$$1$ $u2(z+1)$ $u2(z- 1)$$1$ $u3(z+1)$ $u3(z- 1)$$u(z+1)u$ (z-1)ul(z$+$ l)ul(z-l)$u2(z+1)u2(z- 1)$$u3(z+1)u3(z- 1)$$u11^{LT\perp}/$ $-1\backslash --J$ $-1\backslash -.-J^{-1}v-\sim y$ $=0$(5.4) 1 lI, (5.1) (53)$w^{\uparrow}=u(z+1)- u(z- 1),$ $w^{2}=u(z+1)u(z- 1)$ ..., .P ,,,. ,54.$w(z)=- u^{1}(z)/(p2(z)u(z))$ ,(5.5) $P2(Z)\frac{d^{2}u}{d_{7,}^{2}}-\{\frac{dp2(Z)}{dz}+p1(z)p2(z)\}\frac{du}{dz}+po(z)p2(z)u=0$ ,2 . 2 $u_{1},$ $u_{2}$ ,$u=C_{1}u_{1}(z)+C_{2}u_{2}(z)$ , $w=- u^{1}/(P2(z)u)$,(5.6) $w(z)=\frac{C_{1}u^{1}1(z)+C_{2}u^{t}2(z)}{P2(z)\{C_{1}u1(z)+C_{2}u2(z)\}}$,(5.7) $po(z)=\frac{1}{p^{2}2(z)}\{\frac{\lambda_{1}\lambda_{1(a1^{- a2)(a1^{-}a3)}}^{1}}{z- a1}+\frac{\lambda_{2}\lambda_{2(a2}^{1}- a3)(a2^{-}a1)}{z- a2}+\frac{\lambda_{3}\lambda_{3(a3}^{I}- a1)(a3^{-}a2)}{z- a3}\}$ ,(5.8) $p1(Z)=-\frac{1}{P2(Z)}\{\begin{array}{l}(\lambda_{1}+\lambda_{1}^{1})(z- a2)(z- a3)|+(\lambda_{2}+\lambda_{2}^{I})(z- a1)(z- a3)+(\lambda_{3}+\lambda_{3}^{t})(z- a1)(z- a2)/’\end{array}$(5.9) $p2(Z)=(z- al)(z- a2)(z- a3)$ ,(5.5) $P$ . , 1)$w=- u^{1}/(P2(z)u)$ $P$ $\emptyset$. . 2) $p_{j}(z),$ $j=0,1,2$, 2. 2, . $P$.(5.3) $u(z+1),$ $u(z- 1)$ $w(z+1)_{2}w(z)$(5.10) $f_{1}(z)w(z+1)w(z)+f_{2}(z)w(z+1)+f_{3}(z)w(z)+f_{4}(z)=0$ ,$w(z)=\{f_{1}(z)u(z+1)- f_{2}(z)u(z)\}/(f_{1}(z)u(z))$(5.11) $u(z+1)-\{$ $\text{ _{}Z- 1)}^{Z- 1)}\}u(z)-\{\text{ _{}-}^{-}\text{ _{}-1)}^{-1)}- \text{ _{}-}^{-}B\}u(z- 1)=0$ ,2 $v_{1}(z),$ $v_{2}(z)$ , $u(z)=C_{1}v_{1}(z)+C_{2}$$v_{2}(z)$ , $w(z)=\{f_{1}(z)u(z+1)- f_{2}(z)u(z)\}/(f_{1}(z)u(z))$55(5.12) $w(z)=\frac{c_{1\{f_{1}(z)v1(z+1)- f_{2}(z)v1(z)\}+C_{2}\{f_{1}(z)v2(z+1)- f_{2}(z)v2(z)\}}}{f_{1}(z)\{c_{1^{V}1(Z)+C_{2^{V}2}(z)\}}}$ ,(5.12) (5.6) ..$(5.7)-(5.9)$ , 2(5.5) $p_{j}(z),$ $j=0,1,2$ .(5.5) .(5.13) $U_{Z}D_{z}u(z, \alpha)- u(z, \alpha)=(U_{1}(z)\frac{d}{d_{7,}}+U_{2}(z))(D_{1}(z)\frac{d}{d_{7_{r}}}+Dz(z))u(z, \alpha)- u(z, \alpha)=0$(UJ, (DJ .(5.14) $U_{Z}=U_{1}(z)\frac{d}{d_{7_{\wedge}}}+U_{2}(z),$ $D_{Z}=D_{1}(z)\frac{d}{d_{7}}+$,.$U_{z}:v(\alpha;z)->v(\alpha+1;z),$ $D_{z}:v(\alpha;z)->v(\alpha- 1;z)$.$z$ , $z$$\alpha$ , $:\dot{F}$ $z$ .(5.15) $U_{Z}u(\alpha;z)=(U_{1}(z)\frac{d}{d_{7}}+U_{2}(z))u(\alpha;z)=u(\alpha+1,z)$ ,,(5.16) $\frac{du(\alpha;z)}{dz}=\frac{1}{U_{1}(z)}(u(\alpha+1,z)- U_{2}(z)u(\alpha;z))$ ,.$\int\backslash$o-7 , $z$$\alpha$ ,. $p_{j}(z),$ $j=0,1,2,3$ $f_{j}(z),$ $j=1,2,3,4$.2. $f_{j}$2 .QRT56.,(5.18) $u(\alpha+1;z)=-\frac{f_{2}(\alpha;z)u(\alpha- 1;z)+f_{1}(\alpha;z)}{f_{4}(\alpha;z)u(\alpha- 1;z)+f_{3}(\alpha;z)}$ ,(5.16) , $u(\alpha;z)=C_{1}u_{1}(\alpha;z)+C_{2}u_{1}(\alpha;z)$,(5.19) $\frac{u^{1}(\alpha+1;z)}{p2(\alpha;z)u(\alpha+1;z)}=-\frac{f_{2}(\alpha;z)u^{1}(\alpha- 1;z)+f_{1}(\alpha;z)p2(\alpha- 1;z)u(\alpha- 1;z)}{f_{4}(\alpha;z)u^{1}(\alpha- 1;z)+f_{3}(\alpha;z)p2(\alpha- 1;z)u(\alpha- 1;z)}$$u^{\mathfrak{s}}(\alpha- 1;z)->u^{t}(\alpha+1;z)$ # .$P$ . $P$,(5.20) $\xi=\frac{a2^{-a3Z- a}1}{a2^{-}a_{1}z- a3}$ ,(5.21)$w=\frac{(z- a3)^{(\lambda_{1}+\lambda_{2})}}{(z- a_{1})^{\lambda_{1}}(z- a2)^{\lambda_{2}}}u(\zeta)$ ,$\alpha=\lambda_{1}+\lambda_{2}+\lambda_{3},$ $\beta=\lambda_{1}+\lambda_{2}+\lambda_{3,Y}^{I}=1+\lambda_{1}-\lambda_{1Y}^{I}-\alpha-\beta=\lambda_{2}^{1}+\lambda_{2}$ ,,(5.22) $\xi(1-\xi)\frac{d^{2_{W}}}{d\zeta^{2}}+[Y-(\alpha+\beta+1)\zeta]\frac{dw}{d\zeta}-\alpha\beta w=0$ ,(5.23)$F(\alpha, \beta, Y;\zeta)\equiv w=P\{\begin{array}{l}0 1 000 0 \alpha 1-\gamma Y-\alpha-\beta \beta\end{array}$ $\zeta$. $F(\alpha, \beta, \gamma;\zeta)$ .$(Y-\alpha)(Y-\beta)\zeta F(\alpha, \beta, Y+1;\zeta)$$($5.24$)$ $+Y[(Y- 1)+(\alpha+\beta- 2_{Y}+1)\zeta]F(\alpha, .\beta, Y;\zeta)$- $Y$ $(Y- 1)(1-\zeta)F(\alpha, \beta, Y- l; \zeta)=0$2 $u(z)rightarrow u(Y)rightarrow F(\alpha, \beta, Y;\zeta)$,57$(5.11)^{1}$$F(\alpha, \beta, \gamma+1;\zeta)-\{\frac{f_{2}(Y.\zeta)}{f_{1}(Y)\zeta)}-\frac{f_{3}(Y- 1;\zeta)}{f_{1}(Y- 1,\zeta)}\}F(\alpha, \beta, Y;\xi)$$- \{\frac{f_{2}(Y- 1,\zeta)}{f_{1}(Y- 1,\zeta)}\frac{f_{3}(Y- 1,\zeta)}{f_{1}(Y- 1,\zeta)}-\frac{f_{4}(\gamma- 1;\zeta)}{f_{1}(Y- 1;\zeta)}\}F(\alpha, \beta, Y- 1;\zeta)=0$’, (5.11) $(5.11)^{\uparrow}$ $f_{j}(7;\zeta),$ $j=1,2,3,4$.. $(5.11)^{1}$ $F(\alpha, \beta, 7\pm 1;\xi)$ 7 2.6.[16][17] .,. ,.6. 1, .(6.1) $\frac{dy}{dx}=\frac{y}{x}$ ,(6.1) $x=0$ 1 , $y=\frac{y0}{x0}x$. , .2 ,(62) $( x-\xi)^{2}\frac{d^{2}y}{dx^{2}}+(x-\xi)P_{1}(x)\frac{dy}{dx}+P_{2}(x)y=0$ ,$P_{i}(x)$ $x=\xi$ . x$=\xi$ , $\xi$(62) 7- $\grave\sqrt{}$ .(6.2) $P_{1},$ $P_{2}$(6.3) $\Phi(z)=z^{p}\sum_{n\geq 0}^{00}C_{n}$ $C_{0}=1,$ $z=x-\xi,$ $P_{j}(x)=\sum_{i\geq 0}^{00}p^{(i)_{j}}z^{i},$ $j=1,2$(62) , $z^{n}$ $0$58. $n=0$ ( ),(6.4) $p(p- 1)+pp^{(1)_{0}}+p^{(2)_{0}}=0$ .2 $\rho_{1},$ $\rho_{2}$ $f_{H}^{\epsilon}$ .1. $\rho_{1},$ $\rho_{2}$ 2 .2. ..3. $\rho_{1},$ $\rho_{2}$ , $n$,. ., ,(6.5) $\frac{dy}{dx}=y$ $k\neq 0$ ., $C$ ,(6.6) $y=\varphi(x;k)=k^{-1/k}(C- x)^{-1/k}$ .$C$ $x=C$ , $k=1$, ., , ., .$x$ , $y0,$ $y1,$ $y2,\ldots,$ $yn$ $F(y0, y1, y2,\ldots, y_{n})$, .(6.7) $F(x, y, \frac{dy}{dx},\ldots, \frac{d^{n}y}{dx^{n}})=0$ ,, 2 , 1), 2) , 3) , 4) 6( ) ..(6.8) $PI$ : $\frac{d^{2_{W}}}{dt^{2}}=6w^{2}+t$ ,59$($6.9$)$ PII: $\frac{d^{2_{W}}}{dt^{2}}=2w^{3}+$ tw $+a$ ,$($6.10$)$ PIII: $\frac{d^{2_{W}}}{dt^{2}}=\perp w(\frac{dw}{dt})^{2}-\perp t(\frac{dw}{dt})+\perp t(\alpha w^{2}+\beta)+Yw^{2}+\frac{6}{w}$ ,$($6.11$)$ PIV: $\frac{d^{2_{W}}}{dt^{2}}=\frac{1}{2w}(\frac{dw}{dt})^{2}+2_{w^{2}+}34tw^{2}+2(t^{2}-\alpha)w+\frac{\beta}{w}$,$($6.12$)$ $PV:\frac{d^{2_{W}}}{dt^{2}}=\{\frac{1}{2w}-\frac{1}{w- 1}\}(\frac{dw}{dt})^{2}-\perp t(\frac{dw}{dt})+\frac{(w- 1)^{2}}{t^{2}}(\alpha w-\frac{\beta}{w})+\frac{Y^{w}}{t}+\frac{6w(w+1)}{w- 1}$,(6.13) $Pw$ : $\frac{d^{2_{W}}}{dt^{2}}=^{1}2\{w_{\overline{w}- 1}^{-}\perp\perp+\frac{1}{w- t}\}(\frac{dw}{dt})^{2}-\{1t^{+\frac{1}{t- 1}}+\frac{1}{w- t}\}(\frac{dw}{dt})$$+ \frac{w(w- 1)(w- t)}{t^{2}(t- 1)^{2}}\{\alpha+\frac{\beta t}{w}+\frac{Y(t- 1)}{(w- 1)^{2}}+\frac{6t(t- 1)}{(w- t)^{2}}\}$ .. (6.8)-(6.13) $0,1$ ,$t,$ $\infty$ , $w$(6.14) $\frac{d^{2}u}{d_{7}^{2}}+P_{1}(w, t)\frac{du}{dz}+P_{2}(w, t)u=0$. ,(6.15) $x=w$, $y={\rm Res} P(x, t),$ $H=-{\rm Res} P(x, t)$ ,(6.16) $\frac{dx}{d\tau}=p\frac{\partial H(x,y,t)}{\partial y},-\frac{dy}{d\tau}=p\frac{\partial H(x,y,t)}{\partial x},$ $\frac{dt}{d\tau}=p,$ $\frac{dH}{d\tau}=p\frac{\partial H}{\partial t}$ , $p=1$$($6.17$)$ HI: $\frac{1}{2}y^{2}- 2x^{3_{-}}$ tx,(6.18) HII: $\frac{1}{2}y^{2_{+}}(x211$(6.19) Hiii: $\frac{1}{t}\{2x^{2}y^{2}-[2\eta 00ty^{2}+(2\Theta_{0}+1)x- 2\eta 0t]y+\eta 0o(\Theta_{0}+\Theta_{00})t\lambda\}$ ,(6.20) HIV: $2xy^{2_{-}}\{x^{2}+2tx+2\kappa_{0}|y+\Theta$oox;(6.21) $HV$: $\frac{1}{t}\{x(x- 1)^{2}y^{2_{-}}[\kappa_{0}(x- 1)^{2}+\Theta x(x- 1)-\eta k]y+\kappa(x- 1)\}$ ,(6.22) $HW$: $\frac{1}{t(t- 1)}\{x(x- 1)(x- t)y^{2}-[\kappa o(x- 1)(x- t)+\kappa_{1}x(x- t)+(\Theta- 1)x(x- 1)]_{y}+\kappa(x- t)\}$ .1 , $F(x, y, \frac{dy}{dx})=0$ ,, . 1)$(dy/dx=a(x)y^{2}+b(x)y+c(x))$ , 2) $((dy/dx)^{2}- 4y^{3}+g_{2}y+g_{3}=0$ ,$g_{2},g_{3}$ ), 3) .QRT ,60, HVI( 6 ), QRT., ,, QRT ,.6. 2 ,,, 2. , ,( ), ,, .6 ,(622) HVI: $\frac{1}{t(t- 1)}\{x(x- 1)(x- t)y^{2_{-}[K_{0}}$ $1)(x- t)+\kappa_{1}x(x- t)+(a1)x(x- 1)]_{y}+K(X- t)\}$$\frac{dx}{dt}=\frac{\partial H(x,y,t)}{\partial y},-\frac{dy}{dt}=\frac{\partial H(x,y,t)}{\partial x}$ ,$\frac{dx}{dt}=\frac{1}{t(t- 1)}(2y x($$1)(x- t)\{\kappa o(x- 1)(x- t)+\kappa_{1}x(x- t)+(\Theta- 1)x(x- 1)])$ ,(6,$\cdot$23)$- \frac{dy}{dt}=\frac{1}{t(t- 1)}(\frac{d}{dx}[x(x- 1)$ $t)y-[*]]y+\kappa)$ ,. $\kappa=0$ ,$y(t)=0$ ,(6.24) $= \frac{t(t- 1)}{\kappa_{0}+\kappa_{1}+\Theta- 1}\frac{d}{dt}.\log((t- 1)^{K_{0}}u(t))$ ,. $u(t)$ $\alpha=1-\kappa_{1},$ $\beta=\Theta+1,$ $Y=\kappa_{0}+\Theta- 1$,$(5.22)^{1}$ $t(1- t)\frac{d^{2}u}{dt^{2}}+[Y^{-}(\alpha+\beta+1)t]\frac{du}{dt}-\alpha\beta u$61. ,, 6.6 ,(6.13) $PW$ : $\frac{d^{2_{W}}}{dt^{2}}=^{1}2\{w_{\overline{w}- 1}^{-}\perp\llcorner+_{w- t}\perp\}(\frac{dw}{dt})^{2}- 1_{t}^{1}+_{t- 1}\perp+_{w- t}\perp\}(\frac{dw}{dt})$$+ \frac{w(w- 1)(w- t)}{t^{2}(t- 1)^{2}}\{\alpha+\frac{\beta t}{w}+\frac{Y(t- 1)}{(w- 1)^{2}}+\frac{6t(t- 1)}{(w- t)^{2}}\}$ .$t=t_{0},$ $w(t_{0})=0$ .(625) $\frac{d^{2_{W}}}{dt^{2}}=\frac{1}{2w}(\frac{dw}{dt})_{w}^{2_{+\frac{\beta}{(t- 1)^{2}}\perp}}+S(t, w, \frac{dw}{dt})$ ,,(6.26) $\hat{w}(t)=(t-\ddagger 0)^{m}\sum_{j--0}^{00}$ $aj$ $(t- to)^{j},$ $m>0,$ $a0\neq 0$ ,(6.25) ( ) ,, $y_{0}$ .[18] .$\frac{dw}{dt}=\frac{\alpha}{t- 1}+wy0$ ,(6.27)$\frac{dyo}{dt}=Ao(w, t)y^{2}0+Bo(w, t)yo+Co(w, t)$ ,. $w=\xi=1,$ $t,$ $\infty$,$\frac{dw}{dt}=c_{\xi}(w, t)+b_{\xi}(w, t)y_{\xi}$ ,(628)$\frac{dy\xi}{dt}=A_{\xi}(w, t)y_{\xi}^{2}+B_{\xi}(w, t)y_{\xi}+C_{\xi}(w, t)$. $A_{\xi},$ $B_{\xi},$ $C_{\xi},$ $b{}_{\xi}C_{\xi}$ $\xi$ ,$b{}_{\xi}C_{\xi}$ $w$ 2 ., 4 $($0, 1, $t,$ $\infty)$, $y_{\xi}=g_{\xi\eta}(w, t)y_{\eta}+f_{\xi\eta}(w, t)$..627. WTC ,KP WTC ,7. 1, ,. ,, . ,, .7. 2 (WTC [14][15])– , $z_{1},$ $z_{2},$ $\ldots,$ $z_{n}$ $f(z_{1}, z_{2}, \ldots, z_{n})$ ,(7.1) $\phi(Z1, Z2, \ldots, z_{n})=0$ ,. (7.1) .(7.2) $q\iota=K\lceil q]$ , $q=q(x, t)$ ,$q(x, t)$ , $\phi(x, t)=0$ , ,(7.2) . $q(x, t)$(7.3) qO $= \frac{1}{\phi^{a}(x,t)}\sum^{00}j_{--0}$ q (x$\phi^{j}(x, t)$. $Q$ , $q_{j}$. ,, .7. 3 WTC ,,1 $)$ wrc ,,. , .2 $)$ (QRT) .3 $)$ 12 ,.63. 1, 2QRT .4 $)$ ,. .. WTC, ,,,. WTC .$\Phi(Z1,$$\ldots$ , Zn$)=0,$ $u=u(Zl,\ldots$ ,Zn $)$ ,.(7.4) $u=\Phi^{a}\sum_{j\overline{-}0}^{00}$ ujd, $uo\neq 0$$u_{j},$$\Phi^{j}$ $\Phi=$0 $\alpha$ ..1:Kadomtsev-Petviashvili (KP)(7.5) uyy $+\partial/\partial_{X}(u_{t}+uu_{X}+e)u_{XXX})=0$Leading order $\alpha=- 2$ ,$u=\Phi^{- 2_{j}}\sum_{--0}^{00}uj\Phi^{j}$ , $j=- 1,4,5,6$ .$j=0$ , $uo$ $=- 120\Phi_{x}^{2}$ , $j=1,$ $u1=120\Phi_{xx}$ ,$j=2,$ $\Phi_{t}\Phi_{x}+40\Phi_{x}\Phi_{xxx}- 30\Phi_{xx}^{2}+\Phi_{y}^{2}+u2\Phi_{X}^{2}=0$ ,$j=3,$ $\Phi_{xt}+0\Phi_{xxxx}+\Phi_{yy+u2}\Phi_{xx3}- u\Phi_{x}^{2}=0$ ,(7.6) $j=4$ , (Resonance) $u4(Eq. j=2)_{xx}=0$ ,$j=5$ , (Resonance) $u5(Eq. j=3)_{xx}=0$ ,$j=6$ , Conpatibility condiotion between $j=2$ and $j=3$$u_{4},$ $u_{5},$ $u_{6}$ . $u_{4},$ $u_{5},$ $u_{6}=0$ . $u_{j}$$j$ , KP. $u_{4},$ $u_{5},$ $u_{6}=0$ $u_{3}=0$ ., $\Phi$ . $\Phi$. $u_{2}$ $j=2,$ $j=3$. uo$=- 120\Phi_{x}^{2}$ , $u1=120\Phi_{XX}$ ,64(7.7)$u=u0\Phi u\Phi u=120\partial/\partial_{X2}(\ln\Phi)+u2- 2_{+1}- 1_{+2}$ ,$u2+\Phi_{t}/\Phi_{x}+40\Phi_{xxx}/\Phi_{X}- 30\Phi_{xx}^{2}/\Phi_{X}^{2}+\Phi_{y}^{2}/\Phi_{x}^{2}=0$ ,$\partial/\partial y(\Phi_{t}/\Phi_{x})+\partial/\partial x(\Phi_{t}/\Phi_{x}+\{\Phi:x\}+1/2\Phi_{y}^{2}/\Phi_{x}^{2})=0$ ,$\{\Phi:x\}=\partial/\partial x(\Phi_{xx}/\Phi_{x})- 1/2(\Phi_{xx}/\Phi_{x})^{2}$ ,$\Phi$ $(\phi=(a\phi+b)/(c\phi+d), ad- bc\neq 0)$.2 : $D\Delta KdV$ ( Kdv )(78) $\frac{du_{n}}{dt}=\frac{- 2p^{2}}{(u_{n+1}- u_{n- 1})^{2}}(\frac{1}{u_{n+2}- u_{n}}+\frac{1}{u_{n}- u_{n- 2}})+\frac{1}{u_{n+1}- u_{n- 1}}$ ,(7.9) $u_{n}(t)\equiv(ot+\overline{u}_{n},$ $x_{n}\equiv\frac{1}{\overline{u}_{n+1}-\overline{u}_{n- 1}}$ ,(7.10) 2 $p^{2}x_{n}^{2}(x_{n+1}+x_{n-1})+\omega-$ Xn $=0$ . $arrow QRT$ ( ),. , QRT.7 . 4 (SCM), WTC, ,. .7. 5 2 SIM SCM,.2 ( $n$ ) .(7.11) $f(xi+1, xi, x\iota- 1, A(i))=0$.$A(i)$ $x$ $i$ .$xi+$ ixn $- A_{1}(i)h_{1}$ (xi)(xi$+$ l $+$xn)$+$h2( (i) xi $+$ (i)) $=0$ .$f(Xi+1, xi, xi- l, A(i))=f_{i+1},$ $f(Xi, xn, f_{i- 2}, A(i- 1))=$ fi .65$f(xi+1, Xi, xn, A(i))=0$ $x$ , .1 $)$ , $x_{i+1},$ $x_{i},$ $x_{i- 1}$ 3 .$x_{i}$ $x_{i+1},$ $x_{i- 1}$ . 3 $\overline{7}$.2 $)$ $(x_{i- 1}, x_{i})arrow x_{i+1},$ $(x_{i- 1}, x_{i})arrow(x_{i}, x_{i+1})$ .3 $)$ $x_{i}arrow x_{j}$ $f_{i}=f_{j}=0$ ( ) .2 $)$ , 3) , $f_{i}$ $(x_{i- 1}, x_{i})arrow(x_{i}, x_{i+1})$ ,SCM $f_{i}arrow f_{i+1}$ . .$Xi+1$ , xi, xn, $A(i))=0$ . $f_{i}$ .$($7.12$)$ dfi $\frac{\partial f_{i}}{-\partial xi+1}$ dxi $+ i+\frac{\partial f_{i}}{\partial xi}$ dxi $+ \frac{\partial f_{i}}{\partial xi- 1}dxi- i+\frac{\partial f_{i}}{\partial A(i)}dA(i)=0$ , $i=1,\ldots$ ,$dx_{I+1}$(7.13) $dxi+i=-(\frac{\overline\lrcorner}{\partial xi+1}\overline{)}(\frac{\partial f_{i}}{\partial xi}dxi+\frac{\partial f_{i}}{\partial xi- 1}d_{Xi- 1}+\frac{\partial f_{i}}{\partial A(i)}dA(i))$ , $i=1,\ldots$ ,SCM . .$dx_{i- 1},$ $dA(i)$ , $dx_{i}$ $dx_{i+1}arrow\infty$. $x_{i}$ $dx_{i}$.(7.14) $fi$$(0O, x00, xi- 1, A(i))=0$ , $xi$ $arrow$ $xo$ $0$$x_{\infty}$ $f_{i}$ . $x_{i+1}arrow\infty$ .$1$66(7.13) $dx_{i+2}$ ,(7.15) $d_{Xi+2}=-\frac{}{(\frac{\partial f_{i+1}1}{\partial xi+2})}(\frac{\partial f_{i+1}}{\partial xi+1}d_{Xi+1}+\frac{\partial f_{i+1}}{\partial xi}d_{Xi}+\frac{\partial f_{i+1}}{\partial A(i)}dA(i))$ ,(7.13) $dx_{i+1}$ (7.15) , $dx_{i+2}$.(7.16)$-$$\partial$x$\partial$$=0 arrow\frac{}{\frac{\partial f(xxx)\partial f(x,xx\partial xi+1}{\partial xi}}-\frac\frac\frac{\partial f(x,x,x)}{\partial f(xx,x)\partial xi+1,\partial xi}\frac$$f_{i}$ (Xi$+$ l’ $xJ$$f_{i+1}$ $(x_{i+1}, x_{i})$. $f_{i}$ $f_{i+1}$., ,,$f_{j}$ .8..1 $)$. , .2 $)$ , QRT.3 $)$67QRT ,.4 $)$,.5 $)$ ... ,.. 1) WTC( ) , () ., . 2)( ) , , . 3), ,( ) . ,. 4) , 5), ,. ,, 6 ,,, ,,, ,, ,, $\tau$.. ,,,,, .68,, .6 ..[1] Agaiwal R. P., Diffeoence $Equ\theta tioiis$ and IiiequaIities (1992), Marcel Dekker.[2] Ahlfors L. A., Complex Apdysis. 3rd ed. (1979), McGraw-Hill.[3] Gibbon J. D. and Tabor M. , J. $Mat\Lambda$ . $P\Lambda ys.,$ $26,8$ (1985) 1956-1960.[4] Grammaticos B., Ramani A. and Papageorgiou V. , PBys. Rev. Lett., 67, 14 (1991),1825-1828.[5] Itoh T. and Cai D., , 3, 4 (1993), 337-351.[6] Kajiwara K., Ohta Y. and Satsuma $J.$ , (1993)[7] Mickens R. E., Diffetence $Equ\theta tioiis$ 2nd ed. (1987), Van Nostrand Reinhold, New York.[8] Papageorgiou V. G., Nijhoff F. W., Grammaticos B. and Ramani A., PBys. Lett. $A,$ $164$(1992) 57-64.[9] Ouispel G. R. W., Roberts J. G. A. and Thompson C. J., $P\Lambda ys$. Lett. $A,$ $\cdot 126,7$ (1988),419-421.[10] Ouispel G. R. W., Roberts J. A. G. and Tbompson C. J., PBysica $D,$ $34$ (1989), 183-192.[11] Ramaiu A., Grammaticos B. and Hietarinta J., PBys. Rev. Lett., 67, 14 (1991),1829-1831.[12] Ramani A. , Grammaticos B. and Satuma J. , PAysics Iaeaets $A,$ $169$ (1992), 323-328.[13] $=$ , (1985), .[14] Weiss J., Tabor M., and Camevale G., J. $M_{\theta}tB$ . PBys. 24, 3 (1983), 522-526.[15] Weiss J., J. $M_{\theta}tB$ . PBys., 26, 9 (1985) 2174-2181.[16] , , 19.[17] K. Okamoto, $P1zysi_{C\theta}2D$ (1981), 525-535.[18] K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, Ftom Gauss to $PamIeve$ , Vieweg1991.[19] Clarkson. P. A., Plzys. Lett. $A$, to be published.$\overline{T}657$ 2- 1. Email: titoh\copyright icluna.kobe-u.ac.jp$\mp 305$ 1- 1- 1. Email: cai\copyright iris.is.tsukuba.ac.jp69

1ジ ブンスウガタ テンヘンカン リサン リッカチ ホウテイシキ ト リサン パンルヴェ ホウテイシキ ヒセンケイ カセキブンケイ ニヨル オウヨウ カイセキ

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1ジブンスウガタテンヘンカンリサンリッカチホウテイシキトリサンパンルヴェホウテイシキヒセンケイカセキブンケイニヨルオウヨウカイセキ

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1ジ ブンスウガタ テンヘンカン リサン リッカチ ホウテイシキ ト リサン パンルヴェ ホウテイシキ ヒセンケイ カセキブンケイ ニヨル オウヨウ カイセキ

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1ジブンスウガタテンヘンカンリサンリッカチホウテイシキトリサンパンルヴェホウテイシキヒセンケイカセキブンケイニヨルオウヨウカイセキ