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An explicit formula for the discrete power function associated with circle patterns of Schramm type

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$B<9I.Cf!*(B($B$b$&$9$040@.(B)

Explicit solutions to semi-discrete modified KdV equation and motion of discrete plane curve

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$B<9I.Cf!*(B($B$b$&$9$040@.(B)

2011$BG/$K7G:\!&7G:\M=Dj$NO@J8(B

Bilinearization and special solutions to the discrete Schwarzian KdV equation

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Journal of Math-for-Industry3(2011) 53-62.
Discrete Schwarizian KdV (dSKdV)$BJ}Dx<0(B($BJLL>(B cross-ratio equation)$B$H8F$P$l$kN%;62D@QJ,7O$NAP@~7A2=$H&SH!?t$r5DO@$7$^$7$?!%(B $B$3$NJ}Dx<0$O(B4$B$D$NJ#AG?t(B z(n,m), z(n+1,m),z(n,m+1),z(n+1,m+1)$B$KBP$7$FDj$^$kJ#Hf$,(B(n$B$@$1$NH!?t(B)$B$H(B(m$B$@$1$NH!?t(B)$B$N@Q!$$H$$$&7A$r$7$?J}Dx<0$G!$(B $B7A$+$i$7$F0lA0$,$"$j$^$9$,!$(B $B4v2?!$FC$KN%;6HyJ,4v2?$G$O4pK\E*$JJ}Dx<0$H$7$FCN$i$l$F$*$j!$(B $BN%;66J@~$N(BBäcklund$BJQ49$r5-=R$7$?$j!$J#AGH!?t$r]$r5-=R$9$k$b$N$G$O$J$$$?$a$+!$(BKdV $BJ}Dx<0$J$I$HHf$Y$F$"$^$j8&5f$,?J$s$G$$$^$;$s$G$7$?!%(B $B$3$NO@J8$G$O!$$^$:(BdSKdV$BJ}Dx<0$r(BDiscrete Schwarzian KP (dSKP)$BJ}Dx<0$N4JLs$H$7$FDj<02=$7$^$9!%(B $B$=$7$F(BdSKP$BJ}Dx<0$NAP@~7AJ}Dx<0$,(B3$BK\$NN%;6(B2$Bo$N(B Bäcklund$BJQ49$r(B2$B2s7+$jJV$7$?2r$H85$N2r$r$D$J$0J}Dx<0$H$J$C$F$$$k$N$G$9!%(B $B$G$9$N$G!$B>$NN%;62D@QJ,7O$h$j$N(BPainlevé$B7O$G$bF1MM$N$3$H$,4|BT$5$l$^$9!%$=$3$G!$$3$NO@J8$G$O(BPainlevé$B7O$H$7$F(B (A₂+A₁)⁽¹⁾$B7?(B(q-Painlevé III)$B$H(BD₄⁽¹⁾$B7?(B(Painlevé VI)$B$re$2!$(B $B$=$l$i$N&SH!?t$GI=$5$l$k(B dSKdV $BJ}Dx<0$NFC$B&C(B$B$N(BPainlevé VI$B$N&SH!?t$K$h$kL@<(8x<0$r:n$k$3$H$,$G$-!$(B $B$3$l$K$D$$$F$O8=:_O@J8$r=`HwCf$G$9!%N%;6@5B'H!?t$K$D$$$F$O5HED@5>O;a$K$h$kBgJQ$h$$2r@b!J!V?tM}2J3X!W(B2010$BG/(B11$B7n9f!K$,$"$j$^$9!%(B

Motion and Bäcklund transformations of discrete plane curves

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arXiv:1008.2808, to appear in Kyushu Journal of Mathematics (2011)
$B2D@QJ,7O$NM}O@$H8EE5HyJ,4v2?3X!$FC$K6JLL$H6J@~$N4v2?3X$H$N4V$K$OL)@\$J4X78$,$"$k$3$H$,CN$i$l$F$$$^$9!%Nc$($PIi$NDj6JN(6JLL!$(B $BNc$($P5<5e$OE57?E*$J2D@QJ,7O$G$"$k(B sine-Gordon$BJ}Dx<0$GE}@)$5$l$k$3$H$O!$4{$K(B19$B@$5*$KCN$i$l$F$$$^$9!%$^$?!$Cx$B!V6J@~$H%=%j%H%s!W(B$B$Oe$N(B $B6J@~$N1?F0$r07$C$F$*$j!$1?F0$,$3$l$bE57?E*$J2D@QJ,7O$G$"$k(B modified KdV $BJ}Dx<0$GE}@)$5$l$k$3$H$,<($5$l$F$$$^$9!%(B $B$=$l$8$c$"$H$$$&$3$H$G!$N%;62D@QJ,7O$NM}O@$rMQ$$$F2D@QJ,@-$rJ]$C$?$^$^4v2?3XE*%*%V%8%'%/%H$NN%;62=$r9T$($PLLGr$+$m$&$H;W$&$o$1$G$9$,!$(B $B$=$l$OA0$G(B90$BG/Be8eH>$+$i8&5f$r@:NOE*$K?J$a$F$$$F!$(B $BFC$K6JLLO@$NN%;62=$N?JE8$,Cx$7$$$H$3$m$G$9!%$;$C$+$/0f%N8}$5$s$,K\$r=q$$$?$+$i!$(B $B$3$C$A$H$7$F$ON%;66J@~$NN%;61?F0$NM}O@$r:n$C$F$_$?$/$J$k$o$1$G!$$=$&$9$l$P%m%C%I$N@)8f$H$+9)3XE*$J1~MQ$b>-Mh$"$k$+$J$H;W$$$^$7$?!%(B $B$G$b!$0lHL$K$O6JLL$h$j6J@~$NJ}$,>1:$5$s$,$"$kN%;6(Bmodified KdV $BJ}Dx<0(B($BN%;6(Bmodifief KdV$B$H8F$P$l$k:9J,J}Dx<0$OJ#?t$"$k$N$G(B)$B$GE}@)$5$l$k(B $BN%;66J@~$NN%;61?F0$NOHAH$_$r:n$j$^$7$?!%$3$l$OHs>o$K<+A3$J9=@.$K8+$($k$N$G!$$3$l$OK\J*$@!$(B $B0l5$$KM}O@$r40@.$K;}$C$F$$$C$F$7$^$*$&$HCx\:Y$J2r@O$r9T$$!$(B $B%=%j%H%s2r$d%V%j!<%6!<2r$J$I$N87L)2r$dJQ49M}O@$r:n$C$?$j$7$?$N$,$3$NO@J8$G$9!%(B $B!VN%;62D@QJ,7O!&N%;6HyJ,4v2?%A%e!<%H%j%"%k(B2010$B!W(B$B$NB?J,=i$a$F$N6qBNE*[email protected]$G$9!%(B $B$=$7$F!$2D@QJ,7O$N8&5fG/$@$1$G$7$?$,6eBg?tM}$K65

2009$BG/(B-2010$BG/$K7G:\$5$l$?O@J8(B

Projective reduction of the discrete Painlevé system of type (A₂+A₁)⁽¹⁾

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arXiv:0910.4439, Int. Math. Res. Notices doi:10.1093/imrn/rnq089 (2010)
$B1~MQNO3X8&5f=j$NJs9p$K<9I.$7$?O@J8$N@5<0$J%P!<%8%g%s$G$9!%$3$NO@J8$G$O!VBP>N2=!W$N35G0$r0lHL2=$7$F!V(Bprojective reduction$B!W(B($Be$K2+A₁)⁽¹⁾$B7?$N(Bq-$B%Q%s%k%t%'(BIII$BJ}Dx<0$NBP>N2=$r5DO@$7!$(B $B&SH!?t$dD64v2?2r$K8+$i$l$k4qL/$J8=>]$N%a%+%K%:%`2rL@$r$7$?$H$$$&$N$,K\O@J8$NN@-$,9b$$N%;6%Q%s%k%t%'7O$K$OB?$/$NNc$,$"$j!$(B $BNc$($P(BA₃⁽¹⁾$B7?$d(BA₄⁽¹⁾$B7?$NN%;6%Q%s%k%t%'7O$NNc$O6=L#?<$$$G$9!%(B $B:#8e$NO@J8$G>\$7$/5DO@$7$F$$$-$?$$$H;W$C$F$$$^$9!%(B

The discrete potential Boussinesq equation and its multisoliton solutions

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arXiv:0908.1800v1, to appear in Applicable Anaylsis(2010).
Discrete Boussinesq $BJ}Dx<0$K$O$$$m$$$m$J%P!<%8%g%s$,$"$k$N$G$9$,!$0U30$K$-$A$s$H8&5f$5$l$F$$$^$;$s!%(B $B$3$NO@J8$O!$

Bilinearization and Casorati determinant solutions to non-autonomous 1 + 1 dimensional discrete soliton equations

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RIMS Kokyuroku Bessatsu B13(2009) 53-74.
1 + 1 $B

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$BN%;6%Q%s%k%t%'J}Dx<0$NBe?t4v2?3XE*9=B$M}O@$G$"$k:d0fM}O@$K$h$l$P!$N%;6%Q%s%k%t%'J}Dx<0$OJ#AGe$N%/%l%b%JJQ49$,:n$kN%;6NO3X7O$@$HM}2r$5$l$F$$$^$9!%J}Dx<0$O%V%m!<%"%C%W$9$k(B9$BE@$NG[CV$K$h$C$FJ,N`$5$l!$(B $B$b$C$H$b(Bgeneric $B$J>l9g$KBJ1_%Q%s%k%t%'J}Dx<0$,F@$i$l!$B>$NJ}Dx<0$OB`2=$7$?E@G[CV$+$iF@$i$l$^$9!%B`2=?^<0$N$:$C$H2<$NJ}$NJ}Dx<0$r=|$-!$(B $B$[$H$s$I$NJ}Dx<0$O(B1$B3,O"N)J}Dx<0$N7A$r$7$F$$$^$9!%$H$3$m$,N%;6%Q%s%k%t%'J}Dx<0$N8&5f$N=i4|$N:"!$J}Dx<0$OC1FH(B2$B3,$N:9J,J}Dx<0$N7A$G(B $BC5:w$5$l!$$=$l$+$i!VFC0[E@JD$89~$a!W$rMQ$$$F!$2D@QJ,@-$rJ]$C$?$^$^(B1$B3,O"N)J}Dx<0$N7A$K3HD%$5$l$F$-$^$7$?!%(B $B5U$K!$8eN2=!W$H8F$S$^$9!%N2=A08e$G2r$N9=B$$K4qL/$J0c$$$,$"$k$N$G$9!%(B(1) $B8=$l$kD64v2?H!?t$NN2=8e$K9TNs<09=B$$K4qL/$JHsBP>N@-$,8=$l$k!%C1$K%Q%i%a!<%?$NFCe$N0c$$$,$"$k$h$&$J46$8$,$7$^$9!%(B $B;d<+?H$O!$(B(2) $B$N9=B$$,%=%j%H%s7O$J$I$G$O8+$i$l$:!$(B $BN%;6%Q%s%k%t%'7O$N$_$K8+$i$l$k$3$H$rIT;W5D$K;W$$!$$=$l$+$iN%;6%Q%s%k%t%'7O$N8&5f$r;O$a$?$H$$$&7P0^$,$"$j$^$7$F!$$:$C$H5$$K$J$C$F$$$^$7$?!%(B $B$3$NO@J8$G$O$=$N8=>]$NIT;W5D$r2rL@$9$k0l$D$N;n$_$r5DO@$7$^$7$?!%$^$:!$:d0fM}O@$G8=$l$kA4$F$N(Bq-$B%Q%s%k%t%'7O$K$D$$$FBP>N2=$rN2=$7$?J}Dx<0$N$b$C$H$b4JC1$JD64v2?2r$r9=@.$7$^$7$?!%$3$l$O(B(1)$B$r$^$:A4$F$NJ}Dx<0$G8+$F$_$h$&$H$$$&$3$H$G$9!%(B $Bq-$B%Q%s%k%t%'(BIII$BJ}Dx<0$H8F$P$l$kJ}Dx<0$re$2$F!$%o%$%k72BP>N@-$NN)>l$+$i!$>e$N(B(1),(2)$B$NM}O@E*GX7J$r2rL@$7$^$7$?!%(B

Ultradiscretization of a solvable two-dimensional chaotic map assciated with the Hesse cubic curve

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Kyushu Journal of Mathematics 63(No.2)(2009) 315-338.
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2007-2008$BG/$K7G:\$5$l$?O@J8(B

Ultradiscretization of solvable one-dimensional chaotic maps

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J. Phys. A Math. Theor. 41(2008) 395202 (13pp) doi: 10.1088/1751-8113/41/39/395202, arxiv:0803.4062v2
$B2D@QJ,7O$NH>Jb30$K=P$F!$2D@QJ,7O0J30$G0UL#$N$"$kD6N%;62=$K%A%c%l%s%8(B $B$7!$$&$^$/$$$-$^$7$?!%@N!$0lHV4JC1$J2D2r%+%*%97O$G$"$k(Blogistic map (sin²$B$NG\3Q8x<0$+$i:n$i$l$k(B)$B$ND6N%;62=$r;n$_$F(B$B$&$^$/$$(B $B$+$J$$(B$B$H$$$&$3$H$r;(;o!V?tM}2J3X!W$N5-;v$NCf$G=q$$$?$3$H$,$"$j$^(B $B$9(B(1998$BG/(B)$B!%(B$BJ}Dx<0$O$&$^$/D6N%;62=$G$-$k$N$G$9$,!$2r$,6K8B$G$9$C(B $B$H$s$G$7$^$$(B$B!$0UL#$N$"$k2r$,F@$i$l$J$$$N$G$9!%$H$3$m$,:G6a!$D6N%(B $B;62=$K4X$7$F$O%H%m%T%+%k4v2?$H$N4X78$,$"$i$o$K$o$+$C$F$/$k$J$I!$M}O@(B $BE*$JCN8+$,$$$m$$$m$HC_@Q$5$l$F$-$F$$$^$9!%$=$&$$$&6u5$$NCf!$Lnn4$5$s(B $B$,%;%_%J!<$G9V1i$7$KMh$F$/$l$?:]$K!$DEED$5$s$bF~$l$F(B3$B?M$G(Blogistic map $B$NOC$+$i;O$a$F$3$s$J$3$H$G$-$J$$$+$J$!$J$s$F%@%Y$C$F$$$F$$$k$&$A(B $B$K$G$-$?OC$G$9!%(Blogistic map $B$N3HD%$H$7$F!$(Bsn² $B$NG\3Q8x<0(B $B$+$i:n$i$l$k2D2r%+%*%97O$r9M$($^$9!%$3$3$G(B sn $B$OBJ1_H!?t$N0l$D$G(B Jacobi $B$N(B sn $BH!?t$G$9!%$3$NNO3X7O$H2r$O%Q%i%a!<%?(B k (0< k< 1) $B$r4^$_!$(B k=0 $B$N;~$K(Blogistic map $B$H$=$N2r$K5"Ce$7$^$9!%$3$NNO3X7O$r!$(B1871$BG/$K(B $B:G=i$KF3=P$7$?(B Schröder$B$K$A$J$s$G(BSchröder map $B$H8F$V$3$H$K(B $B$7$^$9!%(BSchröder map$B$O$=$N$^$^$G$O<0$KIi9f$r4^$`$N$GD6N%;62=$G(B $B$-$^$;$s$,!$4JC1$J0l$B2r$bD6N%;62=$G$-$F!$(B tent map $B$N2r$rM?$($k$H$$$&$3$H$G$9!%$3$NI{;:J*$H$7$F!$(Btent map $B$KBP(B $B$7$F%H%m%T%+%k4v2?3XE*$JIAA|$rM?$($k$3$H$,$G$-$^$9!%$D$^$j!$(B tent map $B$O$"$k%H%m%T%+%k(B($BBJ1_(B)$B6J@~$N%d%3%SB?MMBN>e$G$NG\3Q$BGX8e$K$OBJ1_(B $B6J@~!&BJ1_H!?t$N@$3&$,$"$k(B$B$H$$$&$3$H$G$9!%$3$N7k2L$O0l

Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation

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J. Phys. Soc. Jpn. 77(2008) 054004, Preprint arxiv:0802.0757v1
$B%?%$%H%kDL$j$NFbMF$G$9$,!$$3$l$OLdBj$H$7$F$O$+$J$j8E$/!$(B1991$BG/$N;d$N(B $BO@J8$K$^$GAL$j$^$9!%(Bdiscrete KP $BJ}Dx<0$dN%;6(B2$Bq-$B:9J,$KOC$r8B$j!$(Breduction $B$r;ve$N(B autonomous$B2=$GF($2$^$7$?!%$H$3$m$,85$,(Bq-$B:9J,$J$N$G!$(B cylindrical Toda lattice $B$N:9J,2=$H2rq-Bessel$BH!?t$H$$$&M3=o@5$7$$FCe$NN%;6%=(B $B%j%H%s7O$O:n$l$J$$$N$+!)9-EDNI8c@h@8$O(B1997$BG/$KN%;6;~4V(B1$B>1:$5$s$,Cf?4%"%U%#%sJ?LL>e(B $B$NN%;66J@~$N1?F0J}Dx<0$H$7$F(B discrete KdV $BJ}Dx<0$N(B non-autonomous $BHG(B $B$rDs=P$7$^$7$?!%$3$N@_Dj$G$O(Bnon-autonomous $B$G$"$k$3$H$,K\

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Hypergeometric solutions to the q-Painlevé equation of type A₄⁽¹⁾

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J. Phys. A: Math. Theor., 40(2007) 12509-12524.

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$BH`$O=$;N(B1$BG/$GBe?t4v2?$r$d$m$&$H$7$F:C@^!$(B $B=$;N(B2$BG/$G$^$C$?$/%<%m$+$i;O$a(B(M2$B$G9)3XIt8~$1$N>oHyJ,J}Dx<0$N=iEy2rK!$NK\$NJY6/$+$i;O$a$?!*(B)$B!$(B 10$B%v7n$G$3$l$r$d$j$-$C$?!%$=$l$b;d$,ESCf$+$i(BSydney$BBg3X$K(B1$BG/9T$C$F$7$^$C$?$+$i(B $B;XF3$G$-$?$N$O

A remark on the Hankel determinant formula for solutions of the Toda equation

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J. Phys. A: Math. Theor., 40(2007) 12661-12675.

$B0J2<$N!V(BPainlevé$B$*$h$S(BToda$BJ}Dx<0$N2r$N(BHankel determinant formula $B$HJd=u@~7ALdBj!W$N(B full paper $BHG$G!$(Bdeterminant formula $B$N(BKP$BM}O@$K$h$kFCD'IU$1$K4X$9$k7k2L$,2CI.$5$l$F$$$^$9!%(B KP$BM}O@$rMQ$$$?8x<0$O!$$-$l$$$K$^$H$^$C$?<0$G$9$,!$$+$J$jIT;W5D$J<0$G$9!%$b$7$+$7$?$i(BToda$B3,AX$=$N$b$N$r(B $BI=8=$9$k<0$J$N$+$bCN$l$J$$$G$9!%$3$&$J$k$H2r$HJ}Dx<0$,^UA30lBN$H$J$C$?@$3&$,!D!%$"$l$@$14JC1$J(B Toda $BJ,;R$N2r$N(B determinant formula$B$NGX8e$K$3$&$$$&$b$N$,95$($F$$$k$H$O!%(B $B$H$j$"$($:4JC1$KFI$a$kO@J8$@$H;W$$$^$9$N$G!$@'Hs$4Mw$/$@$5$$!%(B

Point configurations, Cremona transformations and the elliptic difference Painlevé equation

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Séminaires et Congrès 14(2006) 175-204. Preprint, arXiv:nlin.SI/0411003, MHF2004-5

2005-2006$BG/$K7G:\$5$l$?O@J8(B

Hypergeometric solutions to the q-Painlevé equation of type (A₁+A'₁)⁽¹⁾

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Int. Math. Res. Not. 2006(2006) Article ID 84619, preprint arXiv:nlin.SI/0607065

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Generating function associated with the Hankel determinant formula for the solutions of the Painlevé IV equation

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to appear in Funkcial. Ekvac.(2006), preprint arXiv:nlin.SI/0512041 ($B8m$j$J$I$r=$@5$7$??7$7$$%P!<%8%g%s$K$J$C$F$$$^$9(B)

Astérisque $B$NO@J8$NB3JT!%F1MM$N7W;;$r%Q%s%k%t%'(B IV $BJ}(B $BDx<0$K$D$$$F$d$C$F$_$^$7$?!%7W;;$N>\:Y$O%Q%s%k%t%'(B IV $BFCM-$N(B $B;v>p$K0MB8$9$k$K$b$+$+$o$i$:!$7k2L$O5$;}$A0-$$$/$i$$8+;v$K(B $B%Q%s%k(B $B%t%'(B II $B$N>l9g$H0lCW$7$^$9!%=>$C$F!$$3$3$G4Q;!$5$l$k;vo$K9-$$HO0O$N2D@QJ,7O$K8+$i$l$k$G$7$g$&!%(B

Painlevé$B$*$h$S(BToda$BJ}Dx<0$N2r$N(BHankel determinant formula $B$HJd=u@~7ALdBj(B

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$B6e=#Bg3X1~MQNO3X8&5f=j8&5f=82qJs9p(B $B!VHs@~7AGHF0$*$h$SHs@~7ANO(B $B3X7O$N8=>]$H?tM}!W(B(2006) Article No. 41

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$B$+$i(B

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Generating function associated with the Hankel determinant formula for the solutions of the Painlevé II equation

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Astérisque 274 (2004) 67-78.

2002$BG/$N(B Iwasaki-Kajiwara-Nakamura $B$NB3JT!%:#EY$O(B generic $B$J(B $B2r$KBP$9$k(Bdeterminant formula $B$NMWAG$NJlH!?t$r9M;!$7$F$_$^$7(B $B$?!%JlH!?t$O$d$O$j(B Riccati $BJ}Dx<0$rK~B-$9$k$N$G!$@~7A2=$9$k$H(B $B2?$,=P$k$N$+!$$C$F=P$7$F$_$F$S$C$/$j!$2?$H(B PII $B$NJd=u@~7ALdBj(B (Lax pair)$B$,=P$F$-$?$N$G$7$?!%$J$<$3$s$J$3$H$K$J$k$N$+!)$$$d(B $B$5$C$Q$j$o$+$j$^$;$s$,!$6=L#?<$$$H;W$$$^$9!%Jd=u@~7ALdBj$O#2(B $B3,$N@~7AJ}Dx<0$J$N$GFHN)$J2r$OEvA3Fs$D$"$j!$$=$N0l$D$,>e$NJl(B $BH!?t$H4X78$9$k$o$1$G$9$,!$$G$O;D$j$N0l$D$O0lBN$I$&$J$C$F$$$k(B $B$N!)$H$$$&5?Ld$,M/$-$^$9!%\$7$/$OO@J8$r$I$&(B $B$>!%85!9(BIwasaki-Kajiwara-Nakamura $B$N7k2L$N5/8;$rCN$j$?$/$FD4(B $B$Y;O$a$?$N$G$9$,!$$^$??7$?$JFf$,!%$^$?!$L\>e$O(B2004$BG/$K$J$C$F$7$^$C$?$N$O0lBN!)(B

Soliton solutions for the non-autonomous discrete-time Toda equation

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J. Phys. A: Math. Gen 38(2005) 6363-6370

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Genarating function related to the Okamoto polynomials for the Painlevé IV equation

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Bull. Aust. Math. Soc. 71 (2005) 517-526. (pdf$B%U%!%$%k(B)

Generating Function Associated with the Rational Solutions of the Painlevé II Equation, J. Phys. A: Math. Gen. 35 L207-L211$B$H(BGenerating function associated with the Hankel determinant formula for the solutions of the Painlevé II equation$B$NB3JT!$(B $B$H$$$&$+!$1!@8$@$C$?8eF#7/$H$d$C$?;E;v$r$^$H$a$F6eBg1~NO8&$N(B $B8&5f=82qJs9p$K=P$7$?O@J8$N1Q8lHG!%:#EY$O(B P4$B!%(BJoshi $B$5$s$+$i(B $B!V@'Hs1Q8lHG$,M_$7$$!W$H8@$o$l$?$b$N$G=q$-$^$7$?!%(BOkamoto $BB?(B $B9`<0$N(B Hankel $B9TNs<0I=<($r:n$j!$9TNsMWAG$NJlH!?t$r9M$($k$H!$(B $B$=$l$O(B Riccati $BJ}Dx<0$rK~$?$7$^$9!D$N$G!$@~7A2=$7$F$_$?$/$J$k(B $B$G$7$g!)$I$s$J@~7AJ}Dx<0!)$C$F!$$3$N>l9g$b2A$9$k$3$H$G!$(BOkamoto $BB?9`<0$N(B n>0 $B$N(B $B>l9g$NJlH!?t$O(B Ai $B$N(Blog$BHyJ,!$(Bn<0 $B$N>l9g$O(B Bi $B$N(Blog$BHyJ,$GI=<((B $B$5$l$k$3$H$r<($7$^$7$?!%$^$?(B Airy $B$G$9!%:#2s$O;X?tE*$KH/;6$9(B $B$k2r$H<}B+$9$k2r$,H?BPJ}8~$K8=$l$k$N$C$F$N$O2?$+%*%b%7%m%$$G(B $B$9$M!%$A$J$_$K!$$3$N;(;o!$

Construction of hypergeometric solutions to the q-Painlevé equations

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Int. Math. Res. Not. 2005(24)(2005) 1439-1463.

Hypergeometric solutions for q-Painlevé equations,IMRN 2004:47 (2004) 2497-2521 $B$NB3JT!%A0$NO@J8(B $B$G$OJ?LL(B3$Bq-$BD64v2?H!?t$r3F(Bq-Painlevé$BJ}(B $BDx<0$K$D$$$FF1Dj$7$^$7$?!%$7$+$7$J$,$i!$!VJ88%$K8=$l$k7A!W$N(B $B=>B0JQ?t$KBP$7$F2r$r=q$-2<$9$K$O!J86M}E*$K$O\:Y$r5DO@$7$F$$$^$9!%FC$K(B E₇⁽¹⁾ $B7?$N(B q-Painlevé $BJ}Dx<0$rNc$K$7$F7W;;$r>\$7$/=R$Y$^$7(B $B$?!%(B

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Cubic pencils and Painlevé hamiltonians

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Funkcialaj Ekvacioj 48(1)(2005) 147-160.

q-Painlevé$BJ}Dx<0$ND64v2?2r(B

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q-Painlevé$BJ}Dx<0$ND64v2?2r$K4X$9$k2r@bO@J8!%FC$KJ?LL(B3 $BE₇⁽¹⁾ $B7?$rNc$K$H$C$F2DG=$J8B$j(B $B!J$=$7$F$+$J$j%j%-$rF~$l$F!KCzG+$K=q$$$?$D$b$j$G$9!%@'Hs$*

q-Painlevé II $B$*$h$S(B V $BJ}Dx<0$ND64v2?2r(B

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Hypergeometric solutions for q-Painlevé equations, IMRN 2004:47 (2004) 2497-2521 $B$H(BConstruction of hypergeometric solutions to the q-Painlevé equations, IMRN 2005:24 (2005) 1439-1463$B$NB3JT!%$3$NFs$D(B $B$NO@J8$G$O$b$C$H$b4JC1$JD64v2?2r$r3F(Bq-Painlevé $B$K4X$7$F9=@.$7$?$N$G$9$,!$0lHL$K$O$3$l$K(BBäcklund $BJQ49$r(B $B:nMQ$5$;$k$H(BWeyl $B72:nMQ$N6@1GLL>e$K9TNs<0$GI=<($5$l$kD64v2?2r(B $B$,F@$i$l$k$H9M$($i$l$F$$$^$9!%$3$NJs9p$G$O(BM2$B_@K\7/$H$$$C$7$g(B $B$K(B q-P2 $B$*$h$S(B q-P5 $B$H8F$P$l$k(B q-Painlevé $BJ}Dx<0$K4X$7$F$=$N$h$&$JD64v2?2r$r9=(B $B@.$7$?7k2L$r=q$$$F$$$^$9!%FC$K(Bq-P5 $B$J$s$+$O7k9=LLE]$J(B $B7W;;$,$"$k$N$@$1$l$I!$_@K\7/$OK\Ev$K$h$/4hD%$C$F$/$l$F!$:G8e(B $B$OJ}K!$r40A4$K<+J,$N$b$N$K$7$F$$$^$7$?!%;DG0$J$,$i;f?t$NET9g(B $B$G$[$H$s$I7k2L$@$1$G!$>\:Y$OO@J8$r<9I.Cf!D$$$d!$$=$N$&$A<9I.(B $B$r3+;O$9$k$3$H$G$7$g$&!%Aa$/=q$-;O$a$M$P!%$3$l$G(B D₅⁽¹⁾$B0J2<$N(B q-Painlevé$BJ}Dx<0$K$D$$$F$OD64v2?2r$N9TNs<0I=<((B $B$,$o$+$C$?$3$H$K$J$j$^$9!%(B

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Hypergeometric solutions to the q-Painlevé equations

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$BD9$$4V!$Bg$-$JBP>N@-$r;}$DN%;6%Q%s%k%t%'J}Dx<0$NFCD⁽¹⁾₅$BBP>N@-$r;}$D(Bq-PVI $BJ}Dx<0$ND64v2?2r$O(B Gauss $B$ND64v2?H!?t(B ₂F₁$B$N(B q$B%"%J%m%0$G$"$k(B ₂$B&5(B₁$B$G$"$k$3$H$,$o$+$C$F$*$j!$$b(B $B$A$m$sO"B36K8B$GJ}Dx<0$b2r$b(B PVI $B$K0\9T$7$^$9!%$a$G$?$$!%$H$3(B $B$m$,!$$=$l0J>eBg$-$JBP>N@-$r;}$DN%;6%Q%s%k%t%'J}Dx<0!$:d0fM}(B $BO@$K$h$l$P(B W(E⁽¹⁾₆),W(E⁽¹⁾₇),W(E⁽¹⁾₈) $B$r;}$D(Bq-$B:9J,%Q%s%k%t%'J}Dx<0$,$"$k$3$H$,$o$+$C$F$$$^$9(B $B$,!$$3$l$i$OO"B36K8B$r;}$A$^$;$s!*$G$O$=$l$i$ND64v2?2r$H$7$F(B $B2?$,8=$l$k$N$+6=L#$"$j$^$9$h$M!)$H$3$m$,!$5;=QE*$KHs>o$K:$Fq(B $B$JE@$,$"$j!$D9$$4V$3$l$,$G$-$^$;$s$G$7$?!%$3$NO@J8$G$O!$:rG/(B $B$NO@J8$GDs=P$7$?(BP²$B>e$N#3q-$B%Q%s%k%t%'J}Dx<0$ND64v2?2r$r9=@.$7$^$7$?!%(B($B$?$@$7!$(B $B%Q%s%k%t%'HyJ,J}Dx<0$G$O$o$+$C$F$$$k$h$&$J!$(B $BB>$N(Bq-$B%Q(B $B%s%k%t%'J}Dx<0$,D64v2?2r$r;}$?$J$$$H$$$&>ZL@$d!$%Q%i%a!<%?6u(B $B4V$N$I$3$@$1$KD64v2?2r$,$"$k!$$H$+$$$&OC$O$^$@$"$j$^$;$s!%G_(B $BB
IMRN 2004:47 (2004) 2497-2521 , arXiv:nlin.SI/0403036,

₁₀E₉ solution to the elliptic Painlevé equation

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J. Phys. A: Math. Gen. 36(2003) L263-L272

On a q-Painlevé III Equation. I: Derivations, Symmetry and Riccati Type Solutions

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q-PIII $B$r>\:Y$K5DO@$7$?O@J8$N(B part 1$B!%$3$NO@J8$G$O(B discrete-time relativistic Toda lattice $B$+$i$NF3=P!$BP>N@-!$(B2$B
J. Nonlin. Math. Phys. 10 (2003) 86-102.

On a q-Painlevé III Equation. II: Rational Solutions

$BCx
q-PIII $B$r>\:Y$K5DO@$7$?O@J8$N(B part2. $B$3$NO@J8$G$OM-M}2r$H(B determinant formula $B$K$D$$$F>\$7$/5DO@$7$?!%(B

J. Nonlin. Math. Phys. 10 (2003) 282-303.

q-Painlevé Systems Arising from q-KP Hierarchy

$BCx
q-KP$B3,AX$NDj<02=$rM?$($^$7$?!%$^$?!$$=$N(B similarity reduction $B$+$i(B A⁽¹⁾₁$B!_(B A⁽¹⁾_n-1$B7?%"%U%#%s%o%$%k72BP>N@-$r;}(B $B$DN%;6NO3X7O$rF3$-!$$=$N$b$C$H$b4JC1$J>l9g$,(Bq-PIV$B$H$J(B $B$k$3$H$r<($7$^$7$?!%$^$?!$(Bq-Schur $BH!?t$N(B similarity reduction $B$+$i$=$l$i$NNO3X7O$NM-M}2r$r9=@.$7$^$7$?!%(B

Lett. Math. Phys. 62 (2003) 259-268

2001-2002$BG/$K7G:\$5$l$?O@J8(B

Determinant Formula for the Toda and Discrete Toda Equations

$BCx
Toda $B$*$h$S(B $BN%;6(B Toda $BJ}Dx<0$N0lHL2r$N9TNs<0I=<($rF@!$$=$l$rMQ$$$F(B Painleve $BJ}Dx<0$N(B tau $BH!?t$NB?9`<0@-$r>ZL@$7$?!%(B

Funkcial. Ekvac. 44(2001) 291-307

A Study on the fourth q-Painlevé Equation

$BCx
A$B7?$N%"%U%#%s%o%$%k72$N!V>hK!E*!WI=8=$r9=@.$7!$$=$l$rMQ$$$F(B q-Painlevé IV $BJ}Dx<0$NBP>N7A<0$r:n$C$?!%FC2⁽¹⁾$B!_(BA₁⁽¹⁾ $B7?%k!<%H3J(B $B;R>e$NNO3X7O$H$_$J$9$3$H$,$G$-!$(BSakai $B$K$h$k(B Mul.6 $B7O$HEy2A$G$"$k$3$H$r(B $B<($7$?!%$5$i$K!$D6N%;62=$b2DG=$G$"$k$3$H$r;XE&$7$?!%(B

J. Phys. A: Math. Gen. 34 (2001) 8563-8581

A Determinant Formula for a Class of Rational Solutions of Painlevé V Equation

$BCx
Painlevé V $BJ}Dx<0$N!$%Q%i%a!<%?6u4V$K$*$1$k(B Weyl chamber $B$N=E?4$K(B $B$$$k%/%i%9$NA4$F$NM-M}2r$r(B Universal Character(Schur $BH!?t$N0l
Nagoya Math. J. 168(2002) 1-25.

Discrete dynamical systems with W(A⁽¹⁾_m-1 $B!_(B A⁽¹⁾_n-1) symmetry

$BCx
$B%"%U%#%s%o%$%k72(B W(A⁽¹⁾_m-1 $B!_(B A⁽¹⁾_n-1)$B$NAPM-(B $BM}$Jq-PIV $BJ}Dx<0$r4^$`(B q-$B:9J,%Q%s%k%t%'7?$NNO3X7O$NB2$rF@$k$3$H$b$G$-$k!%$^$?!$(B $B$3$N
Lett. Math. Phys. 60 (2002) 211-219.

Generating Function Associated with the Rational Solutions of the Painlevé II Equation

$BCx
PII $B$NM-M}2r$rFCD'IU$1$kFC]$NK\]$rH/8+$7$?O@J8!%(B

J. Phys. A: Math. Gen. 35(2002) L207-L211

1999-2000$BG/$K7G:\$5$l$?O@J8(B

$B!VO@J8!W$8$c$"$j$^$;$s$,(B
$B!V2D@QJ,7O$N1~MQ?tM}!W>X2ZK<(B(2000)

$BJT>LZJ?(B $B=_B@(B($BN6C+BgM}9)(B)$B!$(B $B@>@.(B $B3hM5(B($BN6C+BgM}9)(B)$B!$3a86(B $B7r;J!$1J0f(B $BFX(B($B:eBg4pAC9)(B)$B!$CfB<(B $B2B@5(B($B:eBg4p(B $BAC9)(B)$B!$EOn4(B $BK'1Q(B($BF1;V

Biliearization of Discrete Soliton Equations through the Singularity Confinement Test

$BCx
Chaos, Solitons & Fractals {\bf 11}(2000) 33-40.

$BFC0[E@JD$89~$aK!$rMQ$$$FM?$($i$l$?N%;6%=%j%H%s7O$N87L)2r$r(B $B=iEyE*$+$DAH?%E*$K9=@.$9$kJ}K!$rO@$8$F$$$^$9!%(B

A Generalization of Determinant Formulas for the Solutions of Painlevé II and XXXIV Equations

$BCx
Darboux $BJQ49$*$h$S(B Hirota $B$ND>@\K!$Nl9g!J=>$C$F2r$,D61[E*$J>l9g!K$KBP$7$F$b(B $B2r$N9TNs<0I=<($rF@$?!#(B

J. Phys. A {\bf 32}(1999) 3763-3778.

On the Umemura Polynomials for the Painlevé III Equation

$BCx
Painlevé III $BJ}Dx<0$NM-M}2r$K8=$l$kFC
Phys. Lett. A{\bf 260}(1999) 462-467.

1997$BG/(B-1998$BG/$K7G:\$5$l$?O@J8(B

Casorati Determinant Solutions for the Discrete Relativistic Toda Lattice Equation

$BCx
$BFC0[E@JD$89~$aK!$rMQ$$$F(B Suris $B$K$h$k(B Discrete Relativistic Toda Lattice Equation $B$rAP@~7A2=$7!"$=$l$+$i(BN-$B%=%j%H%s2r$r6qBNE*$K9=@.$7$?!#$^$?!"AP@~7A7A<0$+$i(B $B$3$NJ}Dx<0$,(B Discrete Time Toda Lattice $BJ}Dx<0$N(B B\"acklund $BJQ49$N7k9g7O$G$"$k$3$H$r(B $BL@$i$+$K$7$?!#(B

Phys. Lett. A {\bf 241}(1998) 335.

Determinant Structure of the Rational Solutions for the Painlevé IV Equation

$BCx
Painlevé IV $BJ}Dx<0$N#3$D$N7ONs$NM-M}2r$KBP$7$F9TNs<0I=<($r9=@.$7$?!#$=$N7k2L!"(B $B#2$D$N7ONs$O(B Hermite $BB?9`<0$rMWAG$H$9$k(B Hankel $B9TNs<0$G!"$^$?#1$D$N7ONs$O(B 3-reduced Schur $BH!?t$NFCl9g$GI=$5$l$k$3$H$r<($7$?!#(B

J. Phys. A {\bf 31}(1998) 2431

Coalescense Cascades and Special Function Solutions for the Continuous and Discrete Painlevé Equations

$BCx
$B$$$/$D$+$N%Q%s%k%Y$*$h$SN%;6%Q%s%k%YJ}Dx<0$NFC
J. Phys. A {\bf 31}(1998) 5799

On an Integrable Discretization of the Rayleigh Quatient Gradient System and the Power Method with the Optimal Shift

$BCx
Rayleigh$B>&8{G[7O$N2D@QJ,$JN%;62=$r9T$J$C$?!#$3$NN%;68{G[7O$O(B $BG$0U$NBg$-$J%9%F%C%W%5%$%:$K(B $BBP$7$FO"B37O$HF1$8J?9UE@$K;X?t4X?tE*$K<}B+$9$k!#(B $B$3$NN%;68{G[7O$OK\hK!$HEy2A$G$"$k$3$H$r<($7!"(B $B87L)2r$rMQ$$$k$3$H$G:GE,$J86E@0\F0NL$r5a$a$k!#(B

to appear in J. Comp. Appl. Math(1998)

Rational Solutions for the Discrete Painlevé II Equation

$BCx
Discrete Painlevé II $BJ}Dx<0$NM-M}2r$rD>@\K!$rMQ$$$F9=@.$7$?!#(B $B2r$O(BLaguerre$BB?9`<0$rMWAG$H(B $B$9$k9TNs<0$GI=<($5$l$k!#$^$?!"$3$N7k2L$OO"B36K8B$G(B Painlevé II $BJ}Dx<0$NM-M}2r$N(BDevisme$BB?9`<0I=8=$K5"Ce$9$k$3$H$r<($7$?!#(B

Phys. Lett. {\bf A232}(1997), 189.

Two-dimensional Soliton Cellular Automaton of Deautonomized Toda-type

A. Nagai, T. Tokihiro, J. Satsuma, R. Willox and K. Kajiwara

2$B
Phys. Lett. {\bf A234}(1997), 301.

Bilinearization of Discrete Soliton Equations and Singularity Confinement

$BCx
$BFC0[E@JD$89~$aK!$OHs@~7A:9J,J}Dx<0$N2D@QJ,@-$rH=Dj$9$k$@$1$G$J$/!"(B $B%=%j%H%s2r$J$I$N9=C[$K(B $B$bM-8z$G$"$k!#=>Mh!"M?$($i$l$?:9J,J}Dx<0$N2r$r9=@.$9$k$3$H$O(B $B:$Fq$H$5$l!"4{CN$NN%;67?%=%j%H%sJ}Dx<0$O!V2r$+$i:n$C$?!W$+!"$b$7$/$O(B $B!VJ}Dx<0$O$"$k$,2r$,(B $B$o$+$i$J$$!W$N$I$A$i$+$G$"$C$?!#$3$NJ}K!$K$h$j!"(B $B>/$J$/$H$bM?$($i$l$?:9J,J}Dx<0$NAP@~7A7A<0$O$[$H$s$I(Broutine work$B$G$o$+$k$h$&$K$J$k!#(B

Physics Letters {\bf A 229} (1997) 173.

Solution and Integrability of a Generalized Derivative Nonlinear Schrodinger Equation

$BCx
$B0lHL2=$5$l$?HyJ,7?Hs@~7A%7%e%l%G%#%s%,!
{\rm i} U_t + {1\over 2} U_{xx} + |U|^2 U+ {\rm i} \alpha|U|^2 U_x
+ {\rm i} \beta U^2 U_x^* = 0,

$B$KBP$7!"G$0U$N%Q%i%a!<%?$KBP$9$k?J9TGH2r$r9=@.$7$?!#$5$i$K!"%Q%s%k%Y%F%9%H$K$h$C$F(B $B2D@QJ,@-$rD4$Y!"?tCM7W;;$K$h$C$F?J9TGH$N>WFM0BDj@-$rD4$Y$?!#(B

Journal of the Physical Society of Japan, 66(1997), 30.

$B;(;oO@J8!J%l%U%'%j!<$,$"$k3X=Q;(;o$K7G:\$5$l$?O@J8!K(B

$B;(;oO@J8!J%l%U%'%j!<$,$"$k3X=Q;(;o$K7G:\$5$l$?O@J8!K(B