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-1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption Reference" -1 235 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small208209" -1 236 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 237 "Tim es" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 237 336 "In this Maple file, we \+ compare our Lax pair and Hamiltonian evolutions with the one given by \+ H. Chiba in \"Multi-poisson approach to the Painlev\351 equations: fro m the isospectral deformation to the isomonodromic deformation\".\nNot e that we place ourselves directly after the symplectic reduction. In \+ particular \\check\{q\}=q and \\check\{p\}=p." }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 238 45 "Loading the Lax matrices obtained by H. Chiba" } {TEXT 238 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 23 "Sigma3:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 16 "Sigma3[1, 1]:=1:\n" }{MPLTEXT 1 0 17 "Sigma3[2,2]:=-1:\n" }{MPLTEXT 1 0 8 "Sigma 3;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 23 "LChiba:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 5 "Chiba" }{MPLTEXT 1 0 58 "[1,1]:= lambda^3+lambda*(2*t2-1/2*V2)+t1-1/2*V3-1/2*V2*W1:\n" }{MPLTEXT 1 0 1 "L" }{MPLTEXT 1 0 5 "Chiba" }{MPLTEXT 1 0 66 "[1,2]:=V2*lambda^2+V3*la mbda+alpha4+t2*V2-1/4*V2^2-V2*tdW2-V3*W1:\n" }{MPLTEXT 1 0 1 "L" } {MPLTEXT 1 0 5 "Chiba" }{MPLTEXT 1 0 35 "[2,1]:=lambda^2+lambda*W1+tdW 2+t2:\n" }{MPLTEXT 1 0 67 "LChiba[2,2]:=-lambda^3-(2*t2-1/2*V2)*lambda -(t1-1/2*V3-1/2*V2*W1):\n" }{MPLTEXT 1 0 8 "LChiba;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "A1Chiba:=Matrix(2,2,0): \n" }{MPLTEXT 1 0 25 "A1Chiba[1,1]:=lambda-W1:\n" }{MPLTEXT 1 0 18 "A1 Chiba[1,2]:=V2:\n" }{MPLTEXT 1 0 17 "A1Chiba[2,1]:=1:\n" }{MPLTEXT 1 0 26 "A1Chiba[2,2]:=-lambda+W1:\n" }{MPLTEXT 1 0 9 "A1Chiba;\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "A2Chiba:=Matrix(2,2,0):\n" } {MPLTEXT 1 0 39 "A2Chiba[1,1]:=lambda^2+t2-1/2*V2-tdW2:\n" }{MPLTEXT 1 0 28 "A2Chiba[1,2]:=lambda*V2+V3:\n" }{MPLTEXT 1 0 25 "A2Chiba[2,1]: =lambda+W1:\n" }{MPLTEXT 1 0 42 "A2Chiba[2,2]:=-lambda^2-(t2-1/2*V2-td W2):\n" }{MPLTEXT 1 0 8 "A2Chiba;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I( mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I' mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#mnGF$6%Q\"1F'/%+foregroundGQ([0,0, 0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+group alignGFC/%(rowspanGF:/%+columnspanGF:-F56(-F86%Q\"0F'F;F>FAFDFFFHFJFAF DFF-F26'FL-F56(-F,6$-I#moGF$6.Q*&uminus0;F'F;F>/%&fenceGQ&falseF'/%*se paratorGFgn/%)stretchyGFgn/%*symmetricGFgn/%(largeopGFgn/%.movablelimi tsGFgn/%'accentGFgn/%'lspaceGQ,0.2222222emF'/%'rspaceGFfoF7FAFDFFFHFJF 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+q2^3+p1-p2)*lambda/(q1-q2)+(q1^3*q2+(-q2^3+p2)*q1-p1*q2)/(q1-q2):\n" }{MPLTEXT 1 0 42 "tdL[1,2]:=lambda^2+(-q1-q2)*lambda+q1*q2:\n" } {MPLTEXT 1 0 10 "tdL[2,1]:=" }{MPLTEXT 1 0 275 " (tau2-(2*(p1-p2))/(q1 -q2)+2*q1^2+2*q1*q2+2*q2^2)*lambda^2+((q1+q2)*tau2+tau1-(2*(p1*q1-p2*q 2))/(q1-q2)+(2*(q1+q2))*(q1^2+q2^2))*lambda+(q1+q2)*tau1+(1/4)*tau2^2+ (q1^2+q1*q2+q2^2)*tau2-(p1-p2)^2/(q1-q2)^2+(2*(p1*q2-p2*q1))*(q1+q2)/( q1-q2)+q1^4+q2^4-q1^2*q2^2-2*h+2*tinfty10:\n" }{MPLTEXT 1 0 21 "tdL[2, 2]:=-tdL[1,1]:\n" }{MPLTEXT 1 0 5 "tdL;\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 21 "tdA1:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 43 "tdA1[1,1]:= (1/2)*lambda+(1/2)*q1+(1/2)*q2:\n" }{MPLTEXT 1 0 16 "tdA1[1,2]:=1/2:\n " }{MPLTEXT 1 0 63 "tdA1[2,1]:=((q1-q2)*tau2+2*q1^3-2*q2^3-2*p1+2*p2)/ (2*q1-2*q2):\n" }{MPLTEXT 1 0 44 "tdA1[2,2]:=-(1/2)*lambda-(1/2)*q1-(1 /2)*q2:\n" }{MPLTEXT 1 0 6 "tdA1;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 21 "tdA2:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 88 "tdA2[1,1]:=(q1-q2)*lam bda^2/(4*q1-4*q2)+(-q1^3-q1^2*q2+q1*q2^2+q2^3+p1-p2)/(4*q1-4*q2):\n" } {MPLTEXT 1 0 43 "tdA2[1,2]:=(1/4)*lambda-(1/4)*q1-(1/4)*q2:\n" } {MPLTEXT 1 0 153 "tdA2[2,1]:=(1/2)*((p2-p1)/(q1-q2)+q1^2+q1*q2+q2^2+(1 /2)*tau2)*lambda+(1/4)*(q1+q2)*tau2+(1/4)*tau1+(-p1*q1+p2*q2)/(2*(q1-q 2))+(1/2)*(q1+q2)*(q1^2+q2^2):\n" }{MPLTEXT 1 0 23 "tdA2[2,2]:=-tdA2[1 ,1]:\n" }{MPLTEXT 1 0 5 "tdA2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfe ncedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mta bleGF$66-I$mtrGF$6'-I$mtdGF$6(-F,6'-F,6#-I%msupGF$6%-I#miGF$6&Q)&lambd a;F'/%'italicGQ&falseF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'norma lF'-I#mnGF$6%Q\"3F'FEFH/%1superscriptshiftGQ\"0F'-I#moGF$6.Q\"+F'FEFH/ %&fenceGFD/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%. movablelimitsGFD/%'accentGFD/%'lspaceGQ,0.2222222emF'/%'rspaceGF`o-I&m fracGF$6)-F,6%-F#6%-F,6*-FS6.Q*&uminus0;F'FEFHFVFXFZFfnFhnFjnF\\oF^oFa o-F,6#-F<6%-F?6&Q#q1F'/FCQ%trueF'FE/FIQ'italicF'FKFOFR-F,6#-F<6%-F?6&Q 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This implies also to renormalize the auxilia ry matrices accordingly." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "tau1:=2*t1:\n" }{MPLTEXT 1 0 12 "tau2:=4*t2:\n" }{MPLTEXT 1 0 18 "tdA 1norm:=2*tdA1:\n" }{MPLTEXT 1 0 17 "tdA2norm:=4*tdA2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 212 122 "We shall prove that we may identigy Chi ba's coordinates with ours so that Chiba's Lax matrices are the transp osed of ours." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "V2:=-2*(p1 -p2)/(q1-q2)+2*q1^2+2*q1*q2+2*q2^2+4*t2;\n" }{MPLTEXT 1 0 86 "V3:=-2*( p1*q1-p2*q2)/(q1-q2)+2*q1^3+2*q1^2*q2+2*q1*q2^2+2*q2^3+4*q1*t2+4*q2*t2 +2*t1; \n" }{MPLTEXT 1 0 14 "W1:=-(q1+q2);\n" }{MPLTEXT 1 0 16 "tdW2:= q1*q2-t2;\n" }{MPLTEXT 1 0 23 "alpha4:=2*tinfty10-2*h;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 35 "simplify((tdL-Transpose(LChiba)));\n" } {MPLTEXT 1 0 41 "simplify((tdA1norm-Transpose(A1Chiba)));\n" }{MPLTEXT 1 0 40 "simplify((tdA2norm-Transpose(A2Chiba)));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#V2G6\",,*&,&I#p1GF$\"\"\"I#p2GF$!\"\"F),&I#q1GF$F)I#q2 GF$F+F+!\"#*$F-\"\"#F1*&F-F)F.F)F1*$F.F1F1I#t2GF$\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#V3G6\",2*&,&*&I#p1GF$\"\"\"I#q1GF$F*F**&I#p2GF$F* I#q2GF$F*!\"\"F*,&F+F*F.F/F/!\"#*$F+\"\"$\"\"#*&F+F4F.F*F4*&F+F*F.F4F4 *$F.F3F4*&F+F*I#t2GF$F*\"\"%*&F.F*F9F*F:I#t1GF$F4" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#W1G6\",&I#q1GF$!\"\"I#q2GF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%tdW2G6\",&*&I#q1GF$\"\"\"I#q2GF$F(F(I#t2GF$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I'alpha4G6\",&I)tinfty10GF$\"\"#I\"hGF$ !\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I, TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdG F$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'normalF' /%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/%+c olumnspanGFJF4FAFDFFF1/%&alignGQ%axisF'/FBQ)baselineF'/FEQ&rightF'/FGQ '|frleft|hrF'/%/alignmentscopeGQ%trueF'/%,columnwidthGQ%autoF'/%&width GFen/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%non eF'/%,columnlinesGF`o/%&frameGF`o/%-framespacingGQ,0.4em~0.5exF'/%*equ alrowsGQ&falseF'/%-equalcolumnsGFjo/%-displaystyleGFjo/%%sideGFS/%0min labelspacingGF]oF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 '-I%mrowGF$6#-I'mtableGF$66-I$mtrGF$6'-I$mtdGF$6(-I#mnGF$6%Q\"0F'/%+fo regroundGQ([0,0,0]F'/%,mathvariantGQ'normalF'/%)rowalignGQ!F'/%,column alignGFC/%+groupalignGFC/%(rowspanGQ\"1F'/%+columnspanGFJF4FAFDFFF1/%& alignGQ%axisF'/FBQ)baselineF'/FEQ&rightF'/FGQ'|frleft|hrF'/%/alignment scopeGQ%trueF'/%,columnwidthGQ%autoF'/%&widthGFen/%+rowspacingGQ&1.0ex F'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF`o/%&f rameGF`o/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalc olumnsGFjo/%-displaystyleGFjo/%%sideGFS/%0minlabelspacingGF]oF;F>/%%op enGQ\"[F'/%&closeGQ\"]F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "-I(mfencedG6 #/I+modulenameG6\"I,TypesettingGI(_syslibGF'6'-I%mrowGF$6#-I'mtableGF$ 66-I$mtrGF$6'-I$mtdGF$6(-I#mnGF$6%Q\"0F'/%+foregroundGQ([0,0,0]F'/%,ma thvariantGQ'normalF'/%)rowalignGQ!F'/%,columnalignGFC/%+groupalignGFC/ %(rowspanGQ\"1F'/%+columnspanGFJF4FAFDFFF1/%&alignGQ%axisF'/FBQ)baseli neF'/FEQ&rightF'/FGQ'|frleft|hrF'/%/alignmentscopeGQ%trueF'/%,columnwi dthGQ%autoF'/%&widthGFen/%+rowspacingGQ&1.0exF'/%.columnspacingGQ&0.8e mF'/%)rowlinesGQ%noneF'/%,columnlinesGF`o/%&frameGF`o/%-framespacingGQ ,0.4em~0.5exF'/%*equalrowsGQ&falseF'/%-equalcolumnsGFjo/%-displaystyle GFjo/%%sideGFS/%0minlabelspacingGF]oF;F>/%%openGQ\"[F'/%&closeGQ\"]F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }