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We also split the deformation space into trivial and non -trivial directions." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 223 "The def ormation operator is \\hbar (alpha15\\partial_\{t_\{\\infty^\{(1)\},5 \} +alpha14\\partial_\{t_\{\\infty^\{(1)\},4\}+alpha13*\\partial_\{t_ \{\\infty^\{(1)\},3\} +alpha12*\\partial_\{t_\{\\infty^\{(1)\},2\}+alp ha11*\\partial_\{t_\{\\infty^\{(1)\},1\} )) " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }{MPLTEXT 1 0 11 "rinfty:=3:\n" } {MPLTEXT 1 0 21 "with(LinearAlgebra):\n" }{MPLTEXT 1 0 21 "tinfty25:=- tinfty15;\n" }{MPLTEXT 1 0 21 "tinfty23:=-tinfty13;\n" }{MPLTEXT 1 0 21 "tinfty21:=-tinfty11;\n" }{MPLTEXT 1 0 21 "tinfty20:=-tinfty10;\n" }{MPLTEXT 1 0 20 "tinfty24:=tinfty14;\n" }{MPLTEXT 1 0 20 "tinfty22:=t infty12;\n" }{MPLTEXT 1 0 13 "tinfty10:=0;\n" }{MPLTEXT 1 0 34 "Pinfty 11:=-(tinfty14+tinfty24)/2;\n" }{MPLTEXT 1 0 34 "Pinfty01:=-(tinfty12+ tinfty22)/2;\n" }{MPLTEXT 1 0 31 "Pinfty32 := -(1/4)*tinfty15^2;\n" } {MPLTEXT 1 0 55 "Pinfty22 := -(1/2)*tinfty15*tinfty13+(1/4)*tinfty14^2 ;\n" }{MPLTEXT 1 0 79 "Pinfty12 := -(1/2)*tinfty15*tinfty11+(1/2)*tinf ty14*tinfty12-(1/4)*tinfty13^2;\n" }{MPLTEXT 1 0 29 "P1:=x-> Pinfty01+ Pinfty11*x;\n" }{MPLTEXT 1 0 54 "P2:=x-> Pinfty02+Pinfty12*x+Pinfty22* x^2+Pinfty32*x^3;" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 137 "c1Asymp := (6*alpha15*tinfty12*tinfty15-6*alpha15*tinfty13*t infty14+10*alpha13*tinfty14*tinfty15-15*alpha12*tinfty15^2)/(30*tinfty 15^2):\n" }{MPLTEXT 1 0 65 "c2Asymp :=(4*alpha15*tinfty14-5*alpha14*ti nfty15)/(20*tinfty15):\n" }{MPLTEXT 1 0 134 "muAsymp := -(2*(3*alpha15 *tinfty11*tinfty15-3*alpha15*tinfty13^2+5*alpha13*tinfty13*tinfty15-15 *alpha11*tinfty15^2))/(15*tinfty15^3):\n" }{MPLTEXT 1 0 9 "nuAsymp:=" }{MPLTEXT 1 0 62 " -(2*(3*alpha15*tinfty13-5*alpha13*tinfty15))/(15*ti nfty15^2):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "dP1dlambda:=unapply(diff(P1(lambda),lambda),lambda):\n" }{MPLTEXT 1 0 53 "dP2dlambda:=unapply(diff(P2(lambda),lambda),lambda):\n" } {MPLTEXT 1 0 18 "L:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 11 "L[1,1]:=0:\n" }{MPLTEXT 1 0 11 "L[1,2]:=1:\n" }{MPLTEXT 1 0 50 "L[2,1]:=-P2(lambda)+ Pinfty02 +C - p*h/(lambda-q):\n" }{MPLTEXT 1 0 37 "L[2,2]:= P1(lambda) +h/(lambda-q) :\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "A:=Matrix( 2,2,0):\n" }{MPLTEXT 1 0 10 "A[1,1]:=c2" }{MPLTEXT 1 0 38 "*lambda^2+c 1*lambda+c0+ rho/(lambda-q)" }{MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 9 "A[ 1,2]:=2" }{MPLTEXT 1 0 26 "*alpha15/5/tinfty15*lambda" }{MPLTEXT 1 0 20 "+nu+ mu/(lambda-q):\n" }{MPLTEXT 1 0 22 "A[2,1]:=AA21(lambda):\n" }{MPLTEXT 1 0 22 "A[2,2]:=AA22(lambda):\n" }{MPLTEXT 1 0 26 "dAdlambda :=Matrix(2,2,0):\n" }{MPLTEXT 1 0 88 "for i from 1 to 2 do for j from \+ 1 to 2 do dAdlambda[i,j]:=diff(A[i,j],lambda): od: od:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 3 "L;\n" }{MPLTEXT 1 0 3 "A;\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 18 "J:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 11 "J[1, 1]:=1:\n" }{MPLTEXT 1 0 11 "J[1,2]:=0:\n" }{MPLTEXT 1 0 23 "J[2,1]:=-p /(lambda-q):\n" }{MPLTEXT 1 0 22 "J[2,2]:=1/(lambda-q):\n" }{MPLTEXT 1 0 26 "dJdlambda:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 87 "for i from 1 to 2 do for j from 1 to 2 do dJdlambda[i,j]:=diff(J[i,j],lambda): od: od :\n" }{MPLTEXT 1 0 3 "J:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 24 "Cur lyLJ:=Matrix(2,2,0):\n" }{MPLTEXT 1 0 17 "CurlyLJ[1,1]:=0:\n" } {MPLTEXT 1 0 17 "CurlyLJ[1,2]:=0:\n" }{MPLTEXT 1 0 78 "CurlyLJ[2,2]:=d iff(J[2,2],q)*CurlyLq+diff(J[2,2],p)*CurlyLp+h*diff(J[2,2],t):\n" } {MPLTEXT 1 0 78 "CurlyLJ[2,1]:=diff(J[2,1],q)*CurlyLq+diff(J[2,1],p)*C urlyLp+h*diff(J[2,1],t):\n" }{MPLTEXT 1 0 9 "CurlyLJ:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "L new:=simplify(Multiply(Multiply(J,L),J^(-1))+h*Multiply(dJdlambda,J^(- 1))):\n" }{MPLTEXT 1 0 73 "Anew:=simplify(Multiply(Multiply(J,A),J^(-1 ))+Multiply(CurlyLJ,J^(-1))):\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 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GQ&1.0exF'/%.columnspacingGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlines GF[z/%&frameGF[z/%-framespacingGQ,0.4em~0.5exF'/%*equalrowsGFgn/%-equa lcolumnsGFgn/%-displaystyleGFgn/%%sideGQ&rightF'/%0minlabelspacingGFhy F;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(-F,6)-F,6%-I#miGF$6&Q#c2F'/%'italicGQ %trueF'/%+foregroundGQ([0,0,0]F'/%,mathvariantGQ'italicF'-I#moGF$6-Q1& InvisibleTimes;F'/FFQ'normalF'/%&fenceGQ&falseF'/%*separatorGFP/%)stre tchyGFP/%*symmetricGFP/%(largeopGFP/%.movablelimitsGFP/%'accentGFP/%'l spaceGQ&0.0emF'/%'rspaceGFin-I%msupGF$6%-F<6&Q)λF'/F@FPFBFL-I#m nGF$6%Q\"2F'FBFL/%1superscriptshiftGQ\"0F'-FI6.Q\"+F'FBFLFNFQFSFUFWFYF en/FhnQ,0.2222222emF'/F[oF^p-F,6%-F<6&Q#c1F'F?FBFEFHF_oFjo-F<6&Q#c0F'F ?FBFEFjo-I&mfracGF$6)-F,6#-F<6&Q&ρF'FboFBFL-F,6#-F,6%F_o-FI6.Q(&mi nus;F'FBFLFNFQFSFUFWFYFenF]pF_p-F<6&Q\"qF'F?FBFE/%.linethicknessGQ\"1F '/%+denomalignGQ'centerF'/%)numalignGF_r/%)bevelledGFPFB/%)rowalignGQ! F'/%,columnalignGFfr/%+groupalignGFfr/%(rowspanGF\\r/%+columnspanGF\\r -F56(-F,6'-F,6%-Fip6)Fco-Fdo6%Q\"5F'FBFLFjqF]rF`rFbrFBFH-Fip6)-F,6%-F< 6&Q*α15F'F?FBFEFHF_o-F,6#-F<6&Q)tinfty15F'F?FBFEFjqF]rF`rFbrFBFj o-F<6&Q%νF'FboFBFLFjo-Fip6)-F,6#-F<6&Q%μF'FboFBFLF`qFjqF]rF`rFbr FBFdrFgrFirF[sF]sFdrFgrFir-F26'-F56(-F,6%-F<6&Q%AA21F'F?FBFE-FI6-Q0&Ap plyFunction;F'FLFNFQFSFUFWFYFenFgnFjn-F#6%-F,6#F_oFBFLFdrFgrFirF[sF]s- F56(-F,6%-F<6&Q%AA22F'F?FBFEFiuF\\vFdrFgrFirF[sF]sFdrFgrFir/%&alignGQ% axisF'/FerQ)baselineF'/FhrF_r/FjrQ'|frleft|hrF'/%/alignmentscopeGFA/%, columnwidthGQ%autoF'/%&widthGFcw/%+rowspacingGQ&1.0exF'/%.columnspacin gGQ&0.8emF'/%)rowlinesGQ%noneF'/%,columnlinesGF^x/%&frameGF^x/%-frames pacingGQ,0.4em~0.5exF'/%*equalrowsGFP/%-equalcolumnsGFP/%-displaystyle GFP/%%sideGQ&rightF'/%0minlabelspacingGF[xFBFL/%%openGQ\"[F'/%&closeGQ \"]F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 82 "Using the compatibility equation \\mathcal\{L\}[L]=h\\partial_x A+[A,L] we can obtain " } {TEXT 226 58 "A[2,1] et A[2,2] from the trivial first line of L(\\lamb da)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "CurlyL:=h*dAdlambda+ (Multiply(A,L)-Multiply(L,A)):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 22 "Entry11:=CurlyL[1,1]:\n" }{MPLTEXT 1 0 22 "Entry12:=CurlyL[1,2]:\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 53 "AA21:=unapply(solve(Entry11=0,AA 21(lambda)),lambda):\n" }{MPLTEXT 1 0 41 "AA21bis:=h*dAdlambda[1,1]+A[ 1,2]*L[2,1]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simplify(AA21(l ambda)-AA21bis);\n" }{MPLTEXT 1 0 53 "AA22:=unapply(solve(Entry12=0,AA 22(lambda)),lambda):\n" }{MPLTEXT 1 0 48 "AA22bis:=h*dAdlambda[1,2]+A[ 1,1]+A[1,2]*L[2,2]:\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 32 "simplify (AA22(lambda)-AA22bis);\n" }{MPLTEXT 1 0 19 "simplify(Entry11);\n" } {MPLTEXT 1 0 19 "simplify(Entry12);\n" }{MPLTEXT 1 0 50 "CurlyL:=h*dAd lambda+(Multiply(A,L)-Multiply(L,A)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 120 "The next step is to use the compatibili ty equation on L[2,2] et L[2,1] to get the evolutions of the Darboux c oordinates:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Entry22:=sim plify(CurlyL[2,2]):\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaMinusqCube:= factor(residue(Entry22*(lambda-q)^2,lambda=q));\n" }{MPLTEXT 1 0 76 "E ntry22TermLambdaMinusqCarre:=factor(residue(Entry22*(lambda-q),lambda= q));\n" }{MPLTEXT 1 0 60 "Entry22TermLambdaMinusq:=factor(residue(Entr y22,lambda=q));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 77 "Entry22TermL ambdaInfty4:=factor(-residue(Entry22/lambda^5,lambda=infinity));\n" } {MPLTEXT 1 0 77 "Entry22TermLambdaInfty3:=factor(-residue(Entry22/lamb da^4,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty2:= factor(-residue(Entry22/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry22TermLambdaInfty1:=factor(-residue(Entry22/lambda^2,lambda=i nfinity));\n" }{MPLTEXT 1 0 75 "Entry22TermLambdaInfty0:=factor(-resid ue(Entry22/lambda,lambda=infinity));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 96 "simplify( Entry22-(Entry22TermLambdaMinusqCarre/(lambda-q)^2+ Entry22TermLambdaMinusq/(lambda-q)\n" }{MPLTEXT 1 0 159 "+Entry22TermL ambdaInfty0+Entry22TermLambdaInfty1*lambda+Entry22TermLambdaInfty2*lam bda^2+Entry22TermLambdaInfty3*lambda^3+Entry22TermLambdaInfty4*lambda^ 4) );\n" }{MPLTEXT 1 0 8 "L[2,2];\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">II=Entry22TermLambdaMinusq CarreG6\",$*(I\"hGF$\"\"\",,**I#muGF$F(I\"qGF$F(I)tinfty14GF$F(I)tinft y15GF$F(!\"&*(I(alpha15GF$F(F'F(F,F(\"\"#*(F'F(I#nuGF$F(F.F(\"\"&*(F+F (I)tinfty12GF$F(F.F(F/*&I$rhoGF$F(F.F(\"#5F(F.!\"\"#F;F5" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I8Entry22TermLambdaMinusqG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty1G6\",$*(I\"hGF$\"\"\",&*&I(a lpha15GF$F(I)tinfty14GF$F(F(*&I#c2GF$F(I)tinfty15GF$F(!\"&F(F/!\"\"#! \"%\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry22TermLambdaInfty0G6 \",$*(I\"hGF$\"\"\",(*(I#nuGF$F(I)tinfty14GF$F(I)tinfty15GF$F(\"\"&*&I (alpha15GF$F(I)tinfty12GF$F(\"\"#*&I#c1GF$F(F-F(!#5F(F-!\"\"#F6F." }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ", (*&I)tinfty14G6\"\"\"\"I'lambdaGF%F&!\"\"I)tinfty12GF%F(*&I\"hGF%F&,&F 'F&I\"qGF%F(F(F&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 115 "Since the d eformation operator is \\hbar (alpha15\\partial_\{t_\{\\infty^\{(1)\} ,5\} +alpha14\\partial_\{t_\{\\infty^\{(1)\},4\}+" }{TEXT 226 7 "alpha 13" }{TEXT 226 31 "*\\partial_\{t_\{\\infty^\{(1)\},3\} +" }{TEXT 226 7 "alpha12" }{TEXT 226 30 "*\\partial_\{t_\{\\infty^\{(1)\},2\}+" } {TEXT 226 7 "alpha11" }{TEXT 226 122 "*\\partial_\{t_\{\\infty^\{(1)\} ,1\} ) we get the evolutions of q and so consistency equations for th e singular part at infinity" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "L22OrderLambda2:=-residue(L[2,2]/lambda^3,lambda=infinity);\n" } {MPLTEXT 1 0 60 "L22OrderLambda1:=-residue(L[2,2]/lambda^2,lambda=infi nity);\n" }{MPLTEXT 1 0 60 "L22OrderLambda0:=-residue(L[2,2]/lambda^1, lambda=infinity);\n" }{MPLTEXT 1 0 147 "Equation1:=factor(simplify(h*( alpha15*diff(L22OrderLambda1,tinfty15)+alpha14*diff(L22OrderLambda1,ti nfty14)+alpha13*diff(L22OrderLambda1,tinfty13)+" }{MPLTEXT 1 0 7 "alph a12" }{MPLTEXT 1 0 32 "*diff(L22OrderLambda1,tinfty12)+" }{MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 61 "*diff(L22OrderLambda1,tinfty11))- Entr y22TermLambdaInfty1));\n" }{MPLTEXT 1 0 108 "Equation2:=factor(simplif y(h*(alpha15*diff(L22OrderLambda0,tinfty15)+alpha14*diff(L22OrderLambd a0,tinfty14)+" }{MPLTEXT 1 0 7 "alpha13" }{MPLTEXT 1 0 32 "*diff(L22Or derLambda0,tinfty13)+" }{MPLTEXT 1 0 7 "alpha12" }{MPLTEXT 1 0 32 "*di ff(L22OrderLambda0,tinfty12)+" }{MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 61 "*diff(L22OrderLambda0,tinfty11))- Entry22TermLambdaInfty0));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda1G6\",$I)tinfty14GF$!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I0L22OrderLambda0G6\",$I)tinfty12GF$!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation1G6\",$*(I\"hGF$\"\"\", (*&I(alpha14GF$F(I)tinfty15GF$F(\"\"&*&I(alpha15GF$F(I)tinfty14GF$F(! \"%*&I#c2GF$F(F,F(\"#?F(F,!\"\"#F5F-" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I*Equation2G6\",$*(I\"hGF$\"\"\",**(I#nuGF$F(I)tinfty14GF$F(I)tinfty1 5GF$F(!\"&*&I(alpha12GF$F(F-F(\"\"&*&I(alpha15GF$F(I)tinfty12GF$F(!\"# *&I#c1GF$F(F-F(\"#5F(F-!\"\"#F9F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "CurlyLq:=factor(Entry22TermLambdaMinusqCarre/h);\n" } {MPLTEXT 1 0 59 "CurlyLqbis:=-mu*P1(q)-2*rho-h*nu-2/5/tinfty15*alpha15 *h*q;\n" }{MPLTEXT 1 0 50 "factor(simplify(series(CurlyLq-CurlyLqbis,n u=0)));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I(CurlyLqG6\",$*&,,**I#muGF$ \"\"\"I\"qGF$F*I)tinfty14GF$F*I)tinfty15GF$F*!\"&*(I(alpha15GF$F*I\"hG F$F*F+F*\"\"#*(F1F*I#nuGF$F*F-F*\"\"&*(F)F*I)tinfty12GF$F*F-F*F.*&I$rh oGF$F*F-F*\"#5F*F-!\"\"#F;F5" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+Curly LqbisG6\",**&I#muGF$\"\"\",&*&I\"qGF$F(I)tinfty14GF$F(!\"\"I)tinfty12G F$F-F(F-I$rhoGF$!\"#*&I\"hGF$F(I#nuGF$F(F-**I)tinfty15GF$F-I(alpha15GF $F(F2F(F+F(#F0\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 70 "Let us now look at the compatibility equ ation for \\mathcal\{L\}[L_\{2,1\}]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Entry21:=simplify(CurlyL[2,1]):\n" }{MPLTEXT 1 0 77 " Entry21TermLambdaMinusqCube:=factor(residue(Entry21*(lambda-q)^2,lambd a=q));\n" }{MPLTEXT 1 0 76 "Entry21TermLambdaMinusqCarre:=factor(resid ue(Entry21*(lambda-q),lambda=q));\n" }{MPLTEXT 1 0 60 "Entry21TermLamb daMinusq:=factor(residue(Entry21,lambda=q));\n" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 77 "Entry21TermLambdaInfty6:=factor(-residue(Entry21/lamb da^7,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty5:= factor(-residue(Entry21/lambda^6,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty4:=factor(-residue(Entry21/lambda^5,lambda=i nfinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty3:=factor(-resid ue(Entry21/lambda^4,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21Ter mLambdaInfty2:=factor(-residue(Entry21/lambda^3,lambda=infinity));\n" }{MPLTEXT 1 0 77 "Entry21TermLambdaInfty1:=factor(-residue(Entry21/lam bda^2,lambda=infinity));\n" }{MPLTEXT 1 0 75 "Entry21TermLambdaInfty0: =factor(-residue(Entry21/lambda,lambda=infinity));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 137 "simplify( Entry21-(Entry21TermLambdaMinusqCube /(lambda-q)^3+Entry21TermLambdaMinusqCarre/(lambda-q)^2+Entry21TermLam bdaMinusq/(lambda-q)\n" }{MPLTEXT 1 0 122 "+Entry21TermLambdaInfty0+En try21TermLambdaInfty1*lambda+Entry21TermLambdaInfty2*lambda^2+Entry21T ermLambdaInfty3*lambda^3\n" }{MPLTEXT 1 0 100 "+Entry21TermLambdaInfty 4*lambda^4+Entry21TermLambdaInfty5*lambda^5+Entry21TermLambdaInfty6*la mbda^6\n" }{MPLTEXT 1 0 5 ") );\n" }{MPLTEXT 1 0 8 "L[2,1];\n" } {MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">II=Entry21TermLambdaMinusqCar reG6\",$*(I\"hGF$\"\"\",8*(I#muGF$F(I\"qGF$\"\"$I)tinfty15GF$F-\"\"&** F+F(F,\"\"#I)tinfty13GF$F(F.F1\"#5**F+F(F,F1I)tinfty14GF$F1F.F(!\"&**F +F(F,F(I)tinfty11GF$F(F.F1F3*,F+F(F,F(I)tinfty12GF$F(F5F(F.F(!#5**F+F( F,F(F2F1F.F(F/**I(alpha15GF$F(F'F(I\"pGF$F(F,F(!\"%**F'F(I#nuGF$F(F?F( F.F(F;**F,F(I$rhoGF$F(F5F(F.F(F3*(I\"CGF$F(F+F(F.F(\"#?*(FDF(F:F(F.F(F 3F(F.!\"\"#FIF3" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaM inusqG6\",$*(I\"hGF$\"\"\",6*(I#muGF$F(I\"qGF$\"\"#I)tinfty15GF$\"\"$ \"#:**F+F(F,F(I)tinfty13GF$F(F.F-\"#?**F+F(F,F(I)tinfty14GF$F-F.F(!#5* *I#c2GF$F(F'F(F,F(F.F(\"#S*(F+F(I)tinfty11GF$F(F.F-\"#5**F+F(I)tinfty1 2GF$F(F5F(F.F(F6*(F+F(F2F-F.F(\"\"&*(I(alpha15GF$F(F'F(I\"pGF$F(\"\")* (I#c1GF$F(F'F(F.F(F3*(I$rhoGF$F(F5F(F.F(F3F(F.!\"\"#FIF3" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty6G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty5G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty4G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty3G6\",$*(I\"hGF$\"\"\"I(alph a15GF$F(I)tinfty15GF$F(#F(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8En try21TermLambdaInfty2G6\",$*(I\"hGF$\"\"\",**&I#nuGF$F(I)tinfty15GF$\" \"$\"#:*(I(alpha15GF$F(I)tinfty13GF$F(F,F(\"#;*&F0F(I)tinfty14GF$\"\"# !\")*(I#c2GF$F(F4F(F,F(\"#SF(F,!\"\"#F(\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I8Entry21TermLambdaInfty1G6\",$*(I\"hGF$\"\"\",2*&I#muGF $F(I)tinfty15GF$\"\"$\"\"&*(I#nuGF$F(I)tinfty13GF$F(F,\"\"#\"#?*(F0F(I )tinfty14GF$F2F,F(!#5*(I(alpha15GF$F(I)tinfty11GF$F(F,F(\"#7*(F8F(I)ti nfty12GF$F(F5F(!#7*&F8F(F1F2\"\"'*(I#c1GF$F(F5F(F,F(F3*(I#c2GF$F(FI8Entry21TermL ambdaInfty0G6\",$*(I\"hGF$\"\"\",.*(I#muGF$F(I\"qGF$F(I)tinfty15GF$\" \"$!\"&*(I#nuGF$F(I)tinfty11GF$F(F-\"\"#\"#5**F1F(I)tinfty12GF$F(I)tin fty14GF$F(F-F(!#5*(F1F(I)tinfty13GF$F3F-F(\"\"&*(I#c1GF$F(F6F(F-F(\"#? *&I\"CGF$F(I(alpha15GF$F(\"#;F(F-!\"\"#F(F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",,*&,(*&I)tinfty15 G6\"\"\"\"I)tinfty11GF'F(#!\"\"\"\"#*&I)tinfty14GF'F(I)tinfty12GF'F(#F (F,*$I)tinfty13GF'F,#F+\"\"%F(I'lambdaGF'F(F+*&,&*&F&F(F2F(F**$F.F,#F( F4F(F5F,F+*&F&F,F5\"\"$F:I\"CGF'F(*(I\"pGF'F(I\"hGF'F(,&F5F(I\"qGF'F+F +F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 53 "We now solve the compatib ility equation for each term" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "rho:=factor(solve(Entry21TermLambdaMinusqCube,rho));\n" } {MPLTEXT 1 0 23 "simplify(rho-(-p*mu));\n" }{MPLTEXT 1 0 39 "simplify( Entry21TermLambdaMinusqCube);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$rh oG6\",$*&I\"pGF$\"\"\"I#muGF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "L21OrderLambda4:=-residue(L[2,1]/lambda^5,lambda= infinity);\n" }{MPLTEXT 1 0 60 "L21OrderLambda3:=-residue(L[2,1]/lambd a^4,lambda=infinity);\n" }{MPLTEXT 1 0 60 "L21OrderLambda2:=-residue(L [2,1]/lambda^3,lambda=infinity);\n" }{MPLTEXT 1 0 60 "L21OrderLambda1: =-residue(L[2,1]/lambda^2,lambda=infinity);\n" }{MPLTEXT 1 0 60 "L21Or derLambda0:=-residue(L[2,1]/lambda^1,lambda=infinity);\n" }{MPLTEXT 1 0 108 "Equation3:=factor(simplify(h*(alpha15*diff(L21OrderLambda3,tinf ty15)+alpha14*diff(L21OrderLambda3,tinfty14)+" }{MPLTEXT 1 0 7 "alpha1 3" }{MPLTEXT 1 0 32 "*diff(L21OrderLambda3,tinfty13)+" }{MPLTEXT 1 0 7 "alpha12" }{MPLTEXT 1 0 32 "*diff(L21OrderLambda3,tinfty12)+" } {MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 61 "*diff(L21OrderLambda3,tinft y11))- Entry21TermLambdaInfty3));\n" }{MPLTEXT 1 0 108 "Equation4:=fac tor(simplify(h*(alpha15*diff(L21OrderLambda2,tinfty15)+alpha14*diff(L2 1OrderLambda2,tinfty14)+" }{MPLTEXT 1 0 7 "alpha13" }{MPLTEXT 1 0 32 " *diff(L21OrderLambda2,tinfty13)+" }{MPLTEXT 1 0 7 "alpha12" }{MPLTEXT 1 0 32 "*diff(L21OrderLambda2,tinfty12)+" }{MPLTEXT 1 0 7 "alpha11" } {MPLTEXT 1 0 61 "*diff(L21OrderLambda2,tinfty11))- Entry21TermLambdaIn fty2));\n" }{MPLTEXT 1 0 108 "Equation5:=factor(simplify(h*(alpha15*di ff(L21OrderLambda1,tinfty15)+alpha14*diff(L21OrderLambda1,tinfty14)+" }{MPLTEXT 1 0 7 "alpha13" }{MPLTEXT 1 0 32 "*diff(L21OrderLambda1,tinf ty13)+" }{MPLTEXT 1 0 7 "alpha12" }{MPLTEXT 1 0 32 "*diff(L21OrderLamb da1,tinfty12)+" }{MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 61 "*diff(L21O rderLambda1,tinfty11))- Entry21TermLambdaInfty1));\n" }{MPLTEXT 1 0 40 "Equation1:=factor(simplify(Equation1));\n" }{MPLTEXT 1 0 40 "Equat ion2:=factor(simplify(Equation2));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda4G6\"\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda3G6\",$*$I)tinfty15 GF$\"\"##\"\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambd a2G6\",&*&I)tinfty15GF$\"\"\"I)tinfty13GF$F(#F(\"\"#*$I)tinfty14GF$F+# !\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda1G6\",(*& I)tinfty15GF$\"\"\"I)tinfty11GF$F(#F(\"\"#*&I)tinfty14GF$F(I)tinfty12G F$F(#!\"\"F+*$I)tinfty13GF$F+#F(\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I0L21OrderLambda0G6\"I\"CGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equ ation3G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation4G6\",$*(I \"hGF$\"\"\",.*&I#nuGF$F(I)tinfty15GF$\"\"$!#:*&I(alpha13GF$F(F,\"\"# \"#5*(I(alpha14GF$F(I)tinfty14GF$F(F,F(!#5*(I(alpha15GF$F(I)tinfty13GF $F(F,F(!\"'*&F8F(F5F1\"\")*(I#c2GF$F(F5F(F,F(!#SF(F,!\"\"#F(\"#?" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation5G6\",$*(I\"hGF$\"\"\",:*&I#m uGF$F(I)tinfty15GF$\"\"$!\"&*(I#nuGF$F(I)tinfty13GF$F(F,\"\"#!#?*(F0F( I)tinfty14GF$F2F,F(\"#5*&I(alpha11GF$F(F,F2F6*(I(alpha12GF$F(F5F(F,F(! #5*(I(alpha13GF$F(F1F(F,F(F6*(I(alpha14GF$F(I)tinfty12GF$F(F,F(F;*(I(a lpha15GF$F(I)tinfty11GF$F(F,F(!\"#*(FBF(F@F(F5F(\"#7*&FBF(F1F2!\"'*(I# c1GF$F(F5F(F,F(F3*(I#c2GF$F(F@F(F,F(!#SF(F,!\"\"#F(\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation1G6\",$*(I\"hGF$\"\"\",(*&I(alpha14GF$F( I)tinfty15GF$F(\"\"&*&I(alpha15GF$F(I)tinfty14GF$F(!\"%*&I#c2GF$F(F,F( \"#?F(F,!\"\"#F5F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Equation2G6\",$ *(I\"hGF$\"\"\",**(I#nuGF$F(I)tinfty14GF$F(I)tinfty15GF$F(!\"&*&I(alph a12GF$F(F-F(\"\"&*&I(alpha15GF$F(I)tinfty12GF$F(!\"#*&I#c1GF$F(F-F(\"# 5F(F-!\"\"#F9F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "solve(\{ Equation1,Equation2,Equation4,Equation5\},\{c2,c1,mu,nu\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "<&/I#c1G6\",$*&,**&I(alpha12GF%\"\"\"I)tinfty1 5GF%\"\"#\"#:*(I(alpha13GF%F+I)tinfty14GF%F+F,F+!#5*(I(alpha15GF%F+I)t infty12GF%F+F,F+!\"'*(F4F+I)tinfty13GF%F+F1F+\"\"'F+F,!\"##!\"\"\"#I/I #c2GF%,$*&,&*&I(alpha14GF%F+F,F+\"\"&*&F4F+F1F+!\"%F+F,F<#F<\"#?/I#muG F%,$*&,**&I(alpha11GF%F+F,F-F.*(F0F+F8F+F,F+!\"&*(F4F+I)tinfty11GF%F+F ,F+!\"$*&F4F+F8F-\"\"$F+F,FU#F-F./I#nuGF%,$*&,&*&F0F+F,F+FE*&F4F+F8F+F UF+F,F:FX" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "c1bis := (6*al pha15*tinfty12*tinfty15-6*alpha15*tinfty13*tinfty14+10*alpha13*tinfty1 4*tinfty15-15*" }{MPLTEXT 1 0 7 "alpha12" }{MPLTEXT 1 0 30 "*tinfty15^ 2)/(30*tinfty15^2);\n" }{MPLTEXT 1 0 63 "c2bis:= (4*alpha15*tinfty14-5 *alpha14*tinfty15)/(20*tinfty15);\n" }{MPLTEXT 1 0 63 "mubis:=(2*(-3*a lpha15*tinfty11*tinfty15+3*alpha15*tinfty13^2-5*" }{MPLTEXT 1 0 7 "alp ha13" }{MPLTEXT 1 0 22 "*tinfty13*tinfty15+15*" }{MPLTEXT 1 0 7 "alpha 11" }{MPLTEXT 1 0 31 "*tinfty15^2))/(15*tinfty15^3);\n" }{MPLTEXT 1 0 33 "nubis:=-(2*(3*alpha15*tinfty13-5*" }{MPLTEXT 1 0 7 "alpha13" } {MPLTEXT 1 0 29 "*tinfty15))/(15*tinfty15^2);\n" }{MPLTEXT 1 0 25 "sim plify(c1bis-c1Asymp);\n" }{MPLTEXT 1 0 25 "simplify(c2bis-c2Asymp);\n" }{MPLTEXT 1 0 25 "simplify(mubis-muAsymp);\n" }{MPLTEXT 1 0 24 "simpl ify(nubis-nuAsymp);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&c1bisG6\",$*&, **&I(alpha12GF$\"\"\"I)tinfty15GF$\"\"#!#:*(I(alpha13GF$F*I)tinfty14GF $F*F+F*\"#5*(I(alpha15GF$F*I)tinfty12GF$F*F+F*\"\"'*(F3F*I)tinfty13GF$ F*F0F*!\"'F*F+!\"##F*\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&c2bisG6 \",$*&,&*&I(alpha14GF$\"\"\"I)tinfty15GF$F*!\"&*&I(alpha15GF$F*I)tinft y14GF$F*\"\"%F*F+!\"\"#F*\"#?" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&mubi sG6\",$*&,**&I(alpha11GF$\"\"\"I)tinfty15GF$\"\"#\"#:*(I(alpha13GF$F*I )tinfty13GF$F*F+F*!\"&*(I(alpha15GF$F*I)tinfty11GF$F*F+F*!\"$*&F3F*F0F ,\"\"$F*F+F5#F,F-" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&nubisG6\",$*&,&* &I(alpha13GF$\"\"\"I)tinfty15GF$F*!\"&*&I(alpha15GF$F*I)tinfty13GF$F* \"\"$F*F+!\"##F1\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "CurlyLpfunction:=unapply(-Entry21TermLambdaMinusq/h,C ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Equation7:=simplify(E ntry21TermLambdaMinusqCarre-(-p*h*CurlyLq)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "Csol:=solve(Equation7,C):\n" }{MPLTEXT 1 0 38 "Cs olbis:=p^2- P1(q)*p+P2(q)-Pinfty02:\n" }{MPLTEXT 1 0 33 "factor(series (Csol-Csolbis,p=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "CurlyLp:=factor(simplify(Cur lyLpfunction(Csol))):\n" }{MPLTEXT 1 0 48 "CurlyLpbis:=mu*(p*diff(P1(q ),q) -diff(P2(q),q))\n" }{MPLTEXT 1 0 26 "+h*2/5/tinfty15*alpha15*p\n" }{MPLTEXT 1 0 15 "+2*h*c2*q+h*c1:" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 39 "factor(series(CurlyLp-CurlyLpbis,q=0));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 226 153 "Conclusio n: We have verified that the formulas for c1,c2,mu, are compatible wit h the theory. Moreover, we get the evolutions of the Darboux coordinat es:\n" }{TEXT 226 5 "L[q]=" }{TEXT 226 6 "2*mu*p" }{TEXT 226 14 "-mu*P 1(q)-h*nu" }{TEXT 226 25 "-2/5/tinfty15*alpha15*h*q" }{TEXT 226 1 ";" }}{PARA 0 "" 0 "" {TEXT 226 7 "L[p] = " }{TEXT 226 35 "mu*(p*diff(P1(q ),q) -diff(P2(q),q))" }{TEXT 226 1 "+" }{TEXT 226 24 "h*2/5/tinfty15*a lpha15*p" }{TEXT 226 14 "+2*h*c2*q+h*c1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "CurlyLqbis:=" }{MPLTEXT 1 0 6 "2*p*mu" }{MPLTEXT 1 0 14 "-mu*P1(q)-h*nu" }{MPLTEXT 1 0 25 "-2/5/tinfty15*alpha15*h*q" } {MPLTEXT 1 0 2 ":\n" }{MPLTEXT 1 0 48 "factor(simplify(series(CurlyLq- CurlyLqbis,q=0)))" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Hamiltonian:= mu*( p^2+P2(q)-" }{MPLTEXT 1 0 16 "p*P1(q))-h*nu*p-" }{MPLTEXT 1 0 45 "2/5/ tinfty15*h*alpha15*p*q -h*c2*q^2-h*c1*q:\n" }{MPLTEXT 1 0 42 "simplify (CurlyLp-(-diff(Hamiltonian,q)));\n" }{MPLTEXT 1 0 40 "simplify(CurlyL q-(diff(Hamiltonian,p)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "C:=Csolbis:\n" }{MPLTEXT 1 0 11 "c1:=c1bis:\n" } {MPLTEXT 1 0 11 "c2:=c2bis:\n" }{MPLTEXT 1 0 11 "mu:=mubis:\n" } {MPLTEXT 1 0 10 "nu:=nubis:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 224 74 "Decomposition of the tangent space into trivial and \+ non-trivial directions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "T 1:=tinfty13/(2*rinfty-3)*(1/2*tinfty15)^(-(2*rinfty-3)/(2*rinfty-1)); \n" }{MPLTEXT 1 0 37 "T2:=(1/2*tinfty15)^(2/(2*rinfty-1));\n" } {MPLTEXT 1 0 66 "tau:=(1/2*tinfty15)^((2*rinfty-3-2*1)/(2*rinfty-1))*1 /2*tinfty11;\n" }{MPLTEXT 1 0 17 "checkq:=T2*q+T1;\n" }{MPLTEXT 1 0 31 "checkp:=T2^(-1)*(p-1/2*P1(q));\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "tau1theo:" }{MPLTEXT 1 0 170 "=unapply( (1/2*tinfty15)^(-(2*rin fty-5)/(2*rinfty-1))*1/2*tinfty11- (2*rinfty-5)/(2*rinfty-3)/2*(1/2*ti nfty13)^2*(1/2*tinfty15)^(-2*(2*rinfty-3)/(2*rinfty-1)), tinfty11);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 89 "tinfty11theo:=unapply( 2*T2^(1/2 )*(tau1+T1^2*(2*rinfty-3)*(2*rinfty-5)/2^2/(2!)), tau1);\n" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 12 "tinfty11theo" }{MPLTEXT 1 0 1 "(" }{MPLTEXT 1 0 8 "tau1theo" }{MPLTEXT 1 0 6 "(t)));" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#T1G6\",$*(I)tinfty13GF$\"\"\"\"\"##\"\"$\"\"&I)tinfty 15GF$#!\"$F,#F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I#T2G6\",$*&\"\"## \"\"$\"\"&I)tinfty15GF$#F'F*#\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I$tauG6\",$*(\"\"##\"\"%\"\"&I)tinfty15GF$#\"\"\"F*I)tinfty11GF$F-#F -F)" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'checkqG6\",&*(\"\"##\"\"$\"\"& I)tinfty15GF$#F'F*I\"qGF$\"\"\"#F.F'*(I)tinfty13GF$F.F'F(F+#!\"$F*#F.F )" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I'checkpG6\"*(\"\"##F&\"\"&I)tinft y15GF$#!\"#F(,(I\"pGF$\"\"\"*&I)tinfty14GF$F.I\"qGF$F.#F.F&I)tinfty12G F$F2F." }}{PARA 11 "" 1 "" {XPPMATH 20 ">I)tau1theoG6\"f*6#I)tinfty11G F$F$6$I)operatorGF$I&arrowGF$F$,&*(\"\"##\"\"\"\"\"&I)tinfty15GF$#!\" \"F09$F/#F/F-*(I)tinfty13GF$F-F-F.F1#!\"'F0#F3\"#7F$F$F$" }}{PARA 11 " " 1 "" {XPPMATH 20 ">I-tinfty11theoG6\"f*6#I%tau1GF$F$6$I)operatorGF$I &arrowGF$F$*(\"\"##\"\"\"F,*&F,#\"\"$\"\"&I)tinfty15GF$#F,F2F-,&9$F.*( I)tinfty13GF$F,F,#F.F2F3#!\"'F2#F.\"#7F.F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "I\"tG6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Lu Minus1T1:=h/2*(5*" }{MPLTEXT 1 0 9 "tinfty15*" }{MPLTEXT 1 0 18 "diff( T1,tinfty15)+" }{MPLTEXT 1 0 2 "4*" }{MPLTEXT 1 0 9 "tinfty14*" } {MPLTEXT 1 0 20 "diff(T1,tinfty14)+3*" }{MPLTEXT 1 0 9 "tinfty13*" } {MPLTEXT 1 0 18 "diff(T1,tinfty13)+" }{MPLTEXT 1 0 2 "2*" }{MPLTEXT 1 0 9 "tinfty12*" }{MPLTEXT 1 0 18 "diff(T1,tinfty12)+" }{MPLTEXT 1 0 2 "1*" }{MPLTEXT 1 0 9 "tinfty11*" }{MPLTEXT 1 0 20 "diff(T1,tinfty11)); \n" }{MPLTEXT 1 0 12 "LuMinus1T2:=" }{MPLTEXT 1 0 7 "h/2*(5*" } {MPLTEXT 1 0 125 "tinfty15*diff(T2,tinfty15)+4*tinfty14*diff(T2,tinfty 14)+3*tinfty13*diff(T2,tinfty13)+2*tinfty12*diff(T2,tinfty12)+1*tinfty 11*" }{MPLTEXT 1 0 19 "diff(T2,tinfty11));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 17 "LuMinus1T2-h*T2);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 21 "LuMinus1P1:=unapply( " }{MPLTEXT 1 0 7 "h/2*(5*" }{MPLTEXT 1 0 51 "tinfty15*diff(P1(lambda),tinfty15)+ 4*tinfty14*diff(" }{MPLTEXT 1 0 10 "P1(lambda)" }{MPLTEXT 1 0 27 ",tin fty14)+3*tinfty13*diff(" }{MPLTEXT 1 0 10 "P1(lambda)" }{MPLTEXT 1 0 27 ",tinfty13)+2*tinfty12*diff(" }{MPLTEXT 1 0 10 "P1(lambda)" } {MPLTEXT 1 0 22 ",tinfty12)+1*tinfty11*" }{MPLTEXT 1 0 5 "diff(" } {MPLTEXT 1 0 10 "P1(lambda)" }{MPLTEXT 1 0 11 ",tinfty11))" }{MPLTEXT 1 0 10 ",lambda);\n" }{MPLTEXT 1 0 25 "CurlyLqfunction:=unapply(" } {MPLTEXT 1 0 27 "CurlyLqbis,alpha15,alpha14," }{MPLTEXT 1 0 7 "alpha13 " }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 7 "alpha12" }{MPLTEXT 1 0 1 "," } {MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 3 "):\n" }{MPLTEXT 1 0 25 "Curl yLpfunction:=unapply(" }{MPLTEXT 1 0 27 "CurlyLpbis,alpha15,alpha14," }{MPLTEXT 1 0 23 "alpha13,alpha12,alpha11" }{MPLTEXT 1 0 3 "):\n" } {MPLTEXT 1 0 11 "LuMinus1q:=" }{MPLTEXT 1 0 17 "CurlyLqfunction(5" } {MPLTEXT 1 0 11 "*tinfty15/2" }{MPLTEXT 1 0 2 ",4" }{MPLTEXT 1 0 11 "* tinfty14/2" }{MPLTEXT 1 0 2 ",3" }{MPLTEXT 1 0 11 "*tinfty13/2" } {MPLTEXT 1 0 2 ",2" }{MPLTEXT 1 0 11 "*tinfty12/2" }{MPLTEXT 1 0 2 ",1 " }{MPLTEXT 1 0 11 "*tinfty11/2" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 11 "LuMinus1p:=" }{MPLTEXT 1 0 17 "CurlyLpfunction(5" }{MPLTEXT 1 0 64 "*tinfty15/2,4*tinfty14/2,3*tinfty13/2,2*tinfty12/2,1*tinfty11/2)" }{MPLTEXT 1 0 2 ";\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "LuMinus1c heckq:=" }{MPLTEXT 1 0 9 "simplify(" }{MPLTEXT 1 0 223 "h/2*(5*tinfty1 5*diff(checkq,tinfty15)+4*tinfty14*diff(checkq,tinfty14)+3*tinfty13*di ff(checkq,tinfty13)+2*tinfty12*diff(checkq,tinfty12)+1*tinfty11*diff(c heckq,tinfty11)) +diff(checkq,q)*LuMinus1q+diff(checkq,p)*LuMinus1p);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 248 "LuMinus1checkp:=simplify(h/2* (5*tinfty15*diff(checkp,tinfty15)+4*tinfty14*diff(checkp,tinfty14)+3*t infty13*diff(checkp,tinfty13)+2*tinfty12*diff(checkp,tinfty12)+1*tinft y11*diff(checkp,tinfty11)) +diff(checkp,q)*LuMinus1q+diff(checkp,p)*Lu Minus1p);" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+LuMinus1T1G6\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I+LuMinus1T2G6\",$*(I\"hGF$\"\"\"\"\"## \"\"$\"\"&I)tinfty15GF$#F)F,#F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+LuMinus1P1G6\"f*6#I'lambdaGF$F $6$I)operatorGF$I&arrowGF$F$,$*&I\"hGF$\"\"\",&*&9$F.I)tinfty14GF$F.! \"%I)tinfty12GF$!\"#F.#F.\"\"#F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 " >I*LuMinus1qG6\",$*&I\"hGF$\"\"\"I\"qGF$F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*LuMinus1pG6\"*&I\"pGF$\"\"\"I\"hGF$F'" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/LuMinus1checkqG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I/LuMinus1checkpG6\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "Lu0T1:=h/2*(3*tinfty15*diff(T1,tinfty13)+2*tinfty14* diff(T1,tinfty12)+1*tinfty13*diff(T1,tinfty11));\n" }{MPLTEXT 1 0 22 " simplify(Lu0T1-h*T2);\n" }{MPLTEXT 1 0 7 "Lu0T2:=" }{MPLTEXT 1 0 94 "h /2*(3*tinfty15*diff(T2,tinfty13)+2*tinfty14*diff(T2,tinfty12)+1*tinfty 13*diff(T2,tinfty11));\n" }{MPLTEXT 1 0 16 "simplify(Lu0T2);" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 16 "Lu0P1:=unapply( " }{MPLTEXT 1 0 58 "h/2*(3*tinfty15*diff(P1(lambda),tinfty13)+2*tinfty14*diff(" } {MPLTEXT 1 0 10 "P1(lambda)" }{MPLTEXT 1 0 27 ",tinfty12)+1*tinfty13*d iff(" }{MPLTEXT 1 0 10 "P1(lambda)" }{MPLTEXT 1 0 20 ",tinfty11)),lamb da);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 67 "Lu0q :=CurlyLqfunction(0,0,3*tinfty15/2,2*tinfty14/2,1*tinfty13/2);\n" } {MPLTEXT 1 0 67 "Lu0p:=CurlyLpfunction(0,0,3*tinfty15/2,2*tinfty14/2,1 *tinfty13/2);\n" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 20 "Lu0checkq:=sim plify(" }{MPLTEXT 1 0 21 "h/2*(3*tinfty15*diff(" }{MPLTEXT 1 0 6 "chec kq" }{MPLTEXT 1 0 27 ",tinfty13)+2*tinfty14*diff(" }{MPLTEXT 1 0 6 "ch eckq" }{MPLTEXT 1 0 27 ",tinfty12)+1*tinfty13*diff(" }{MPLTEXT 1 0 6 " checkq" }{MPLTEXT 1 0 11 ",tinfty11))" }{MPLTEXT 1 0 44 " +diff(checkq ,q)*Lu0q+diff(checkq,p)*Lu0p);\n" }{MPLTEXT 1 0 20 "Lu0checkp:=simplif y(" }{MPLTEXT 1 0 21 "h/2*(3*tinfty15*diff(" }{MPLTEXT 1 0 6 "checkp" }{MPLTEXT 1 0 27 ",tinfty13)+2*tinfty14*diff(" }{MPLTEXT 1 0 6 "checkp " }{MPLTEXT 1 0 27 ",tinfty12)+1*tinfty13*diff(" }{MPLTEXT 1 0 6 "chec kp" }{MPLTEXT 1 0 11 ",tinfty11))" }{MPLTEXT 1 0 43 " +diff(checkp,q)* Lu0q+diff(checkp,p)*Lu0p);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&Lu0T1G6\",$*(I\"hGF$\"\"\"\"\"##\"\"$\"\"&I)tinfty15GF $#F)F,#F(F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&Lu0T2G6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 ">I&Lu0P1G6\"f*6#I'lambdaGF$F$6$I)ope ratorGF$I&arrowGF$F$,$*&I\"hGF$\"\"\"I)tinfty14GF$F.!\"\"F$F$F$" }} {PARA 11 "" 1 "" {XPPMATH 20 ">I%Lu0qG6\",$I\"hGF$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I%Lu0pG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "> I*Lu0checkqG6\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I*Lu0checkpG6\" \"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {TEXT 216 24 "We define tdp:=p-P1(q )/2" }{MPLTEXT 1 225 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "p:=tdp+P1(q)/2:\n" }{MPLTEXT 1 0 53 "CurlyLtdp:=simplify( CurlyLp-dP1 dlambda(q)/2*CurlyLq\n" }{MPLTEXT 1 0 6 "- 1/2*" }{MPLTEXT 1 0 45 "h*( alpha15*diff(P1(q),tinfty15)+alpha14*diff(" }{MPLTEXT 1 0 5 "P1(q)" } {MPLTEXT 1 0 24 ",tinfty14)+alpha13*diff(" }{MPLTEXT 1 0 5 "P1(q)" } {MPLTEXT 1 0 24 ",tinfty13)+alpha12*diff(" }{MPLTEXT 1 0 5 "P1(q)" } {MPLTEXT 1 0 24 ",tinfty12)+alpha11*diff(" }{MPLTEXT 1 0 5 "P1(q)" } {MPLTEXT 1 0 15 ",tinfty11)) ): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "CurlyLtdpbis:=mu*(diff(P1(q)^2/4-P2(q),q))+" }{MPLTEXT 1 0 26 "h*2/5/tinfty15*alpha15*tdp" }{MPLTEXT 1 0 1 ":" }{MPLTEXT 1 0 1 " \n" }{MPLTEXT 1 0 43 "factor(series(CurlyLtdp-CurlyLtdpbis,q=0));" } {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "simplify(CurlyLq- ( 2*mu*tdp-h*nu-2 /5/tinfty15*alpha15*h*q));" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 43 "We get tha t \\mathcal\{L\}[q]=2*mu*tdp-h*nu-h*" }{TEXT 216 23 "2/5/tinfty15*alph a15*q;" }{TEXT 216 1 "\n" }{TEXT 216 1 "\n" }{TEXT 216 10 "mathcal\{L \}" }{TEXT 216 8 "[tdp] = " }{TEXT 216 29 "mu*(diff(P1(q)^2/4-P2(q),q) )+" }{TEXT 216 27 "h*2/5/tinfty15*alpha15*tdp\n" }{TEXT 216 1 "\n" } {TEXT 216 22 "Hamiltonian= mu*tdp^2 " }{TEXT 216 30 "-h*2/5/tinfty15*a lpha15*q*tdp " }{TEXT 216 31 "-mu*(P1(q)^2/4-P2(q)) -h*nu*tdp" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "mufunction:=unapply(mubis,al pha15,alpha14," }{MPLTEXT 1 0 7 "alpha13" }{MPLTEXT 1 0 1 "," } {MPLTEXT 1 0 7 "alpha12" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 7 "alpha11" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 23 "nufunction:=unapply(nu," } {MPLTEXT 1 0 15 "alpha15,alpha14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 23 "alpha13,alpha12,alpha11" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 38 " c2function:=unapply(c2,alpha15,alpha14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 23 "alpha13,alpha12,alpha11" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 38 "c1function:=unapply(c1,alpha15,alpha14" }{MPLTEXT 1 0 1 "," } {MPLTEXT 1 0 23 "alpha13,alpha12,alpha11" }{MPLTEXT 1 0 3 ");\n" } {MPLTEXT 1 0 33 "CurlyLqfunction:=unapply(CurlyLq," }{MPLTEXT 1 0 15 " alpha15,alpha14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 23 "alpha13,alpha12 ,alpha11" }{MPLTEXT 1 0 9 ",q,tdp):\n" }{MPLTEXT 1 0 52 "CurlyLtdpfunc tion:=unapply(CurlyLtdp,alpha15,alpha14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 23 "alpha13,alpha12,alpha11" }{MPLTEXT 1 0 10 ",q,tdp):\n\n" } {MPLTEXT 1 0 393 "CurlyLcheckqfunction:=unapply( h*(diff(T2,tinfty15)* alpha15+diff(T2,tinfty14)*alpha14+diff(T2,tinfty13)*alpha13+diff(T2,ti nfty12)*alpha12+diff(T2,tinfty11)*alpha11)*qfunction+ T2*CurlyLqfuncti on(alpha15,alpha14,alpha13,alpha12,alpha11,qfunction,tdpfunction)+h*(d iff(T1,tinfty15)*alpha15+diff(T1,tinfty14)*alpha14+diff(T1,tinfty13)*a lpha13+diff(T1,tinfty12)*alpha12+diff(T1,tinfty11)*alpha11),\n" } {MPLTEXT 1 0 42 "alpha15,alpha14,alpha13,alpha12,alpha11):\n" } {MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 272 "CurlyLcheckpfunction:=unapply(- h/T2^2*(diff(T2,tinfty15)*alpha15+diff(T2,tinfty14)*alpha14+diff(T2,ti nfty13)*alpha13+diff(T2,tinfty12)*alpha12+diff(T2,tinfty11)*alpha11)*t dpfunction+T2^(-1)*CurlyLtdpfunction(alpha15,alpha14,alpha13,alpha12,a lpha11,qfunction,tdpfunction),\n" }{MPLTEXT 1 0 41 "alpha15,alpha14,al pha13,alpha12,alpha11):" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+mufunction G6\"f*6'I(alpha15GF$I(alpha14GF$I(alpha13GF$I(alpha12GF$I(alpha11GF$F$ 6$I)operatorGF$I&arrowGF$F$,$*&,**&9(\"\"\"I)tinfty15GF$\"\"#\"#:*(9&F 4I)tinfty13GF$F4F5F4!\"&*(9$F4I)tinfty11GF$F4F5F4!\"$*&F=F4F:F6\"\"$F4 F5F?#F6F7F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+nufunctionG6\"f*6' I(alpha15GF$I(alpha14GF$I(alpha13GF$I(alpha12GF$I(alpha11GF$F$6$I)oper atorGF$I&arrowGF$F$,$*&,&*&9&\"\"\"I)tinfty15GF$F4!\"&*&9$F4I)tinfty13 GF$F4\"\"$F4F5!\"##F;\"#:F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+c2 functionG6\"f*6'I(alpha15GF$I(alpha14GF$I(alpha13GF$I(alpha12GF$I(alph a11GF$F$6$I)operatorGF$I&arrowGF$F$,$*&,&*&9%\"\"\"I)tinfty15GF$F4!\"& *&9$F4I)tinfty14GF$F4\"\"%F4F5!\"\"#F4\"#?F$F$F$" }}{PARA 11 "" 1 "" {XPPMATH 20 ">I+c1functionG6\"f*6'I(alpha15GF$I(alpha14GF$I(alpha13GF$ I(alpha12GF$I(alpha11GF$F$6$I)operatorGF$I&arrowGF$F$,$*&,**&9'\"\"\"I )tinfty15GF$\"\"#!#:*(9&F4I)tinfty14GF$F4F5F4\"#5*(9$F4I)tinfty12GF$F4 F5F4\"\"'*(F=F4I)tinfty13GF$F4F:F4!\"'F4F5!\"##F4\"#IF$F$F$" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 216 25 "Evolution in direction w2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mufunction(0,1,0,0,0);\n" } {MPLTEXT 1 0 22 "nufunction(0,1,0,0,0);" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 23 "c2function(0,1,0,0,0);\n" }{MPLTEXT 1 0 23 "c1functio n(0,1,0,0,0);\n" }{MPLTEXT 1 0 31 "CurlyLqfunction(0,1,0,0,0,q,tdp" } {MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 33 "CurlyLtdpfunction(0,1,0,0,0,q, tdp" }{MPLTEXT 1 0 2 ");" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 9 "simpli fy(" }{MPLTEXT 1 0 21 "CurlyLcheckqfunction(" }{MPLTEXT 1 0 9 "0,1,0,0 ,0" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 30 "simplify(CurlyLcheckpfun ction(" }{MPLTEXT 1 0 9 "0,1,0,0,0" }{MPLTEXT 1 0 3 "));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "# !\"\"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 216 25 "Evolution in direction w1" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "mufunction(0,0,0,1,0);\n" } {MPLTEXT 1 0 23 "nufunction(0,0,0,1,0);\n" }{MPLTEXT 1 0 23 "c2functio n(0,0,0,1,0);\n" }{MPLTEXT 1 0 23 "c1function(0,0,0,1,0);\n" }{MPLTEXT 1 0 34 "CurlyLqfunction(0,0,0,1,0,q,tdp);\n" }{MPLTEXT 1 0 35 "CurlyL tdpfunction(0,0,0,1,0,q,tdp);" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 " simplify(CurlyLcheckqfunction(" }{MPLTEXT 1 0 9 "0,0,0,1,0" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 30 "simplify(CurlyLcheckpfunction(" } {MPLTEXT 1 0 9 "0,0,0,1,0" }{MPLTEXT 1 0 3 "));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "#!\"\"\"\" #" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 216 25 "Evolution \+ in direction u0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "mufuncti on(" }{MPLTEXT 1 0 4 "0,0," }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 11 "3 *tinfty15," }{MPLTEXT 1 0 14 "1/2*2*tinfty14" }{MPLTEXT 1 0 1 "," } {MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 8 "tinfty13" }{MPLTEXT 1 0 3 ");\n " }{MPLTEXT 1 0 11 "nufunction(" }{MPLTEXT 1 0 4 "0,0," }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 11 "3*tinfty15," }{MPLTEXT 1 0 14 "1/2*2*tinfty 14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 8 "tinfty 13" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 11 "c2function(" }{MPLTEXT 1 0 4 "0,0," }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 11 "3*tinfty15," } {MPLTEXT 1 0 14 "1/2*2*tinfty14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 4 " 1/2*" }{MPLTEXT 1 0 8 "tinfty13" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 11 "c1function(" }{MPLTEXT 1 0 4 "0,0," }{MPLTEXT 1 0 4 "1/2*" } {MPLTEXT 1 0 11 "3*tinfty15," }{MPLTEXT 1 0 14 "1/2*2*tinfty14" } {MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 8 "tinfty13" } {MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 16 "CurlyLqfunction(" }{MPLTEXT 1 0 4 "0,0," }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 11 "3*tinfty15," } {MPLTEXT 1 0 14 "1/2*2*tinfty14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 4 " 1/2*" }{MPLTEXT 1 0 8 "tinfty13" }{MPLTEXT 1 0 18 ",q,tdp);\nsimplify( " }{MPLTEXT 1 0 18 "CurlyLtdpfunction(" }{MPLTEXT 1 0 4 "0,0," } {MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 11 "3*tinfty15," }{MPLTEXT 1 0 14 "1/2*2*tinfty14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 8 "tinfty13" }{MPLTEXT 1 0 9 ",q,tdp));" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 30 "simplify(CurlyLcheckqfunction(" }{MPLTEXT 1 0 46 "0,0 ,1/2*3*tinfty15,1/2*2*tinfty14,1/2*tinfty13" }{MPLTEXT 1 0 4 "));\n" } {MPLTEXT 1 0 30 "simplify(CurlyLcheckpfunction(" }{MPLTEXT 1 0 46 "0,0 ,1/2*3*tinfty15,1/2*2*tinfty14,1/2*tinfty13" }{MPLTEXT 1 0 3 "));" }} {PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$I\"hG6\"!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 " \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 216 29 "Evolution in direction u_\{-1\}" }{TEXT 216 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "mufunction(" }{MPLTEXT 1 0 72 "1/2*5*tinfty15,1/2*4*tinfty14,1/2*3*tinfty13,1/2*2*tinfty12,1/2*ti nfty11" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 11 "nufunction(" } {MPLTEXT 1 0 72 "1/2*5*tinfty15,1/2*4*tinfty14,1/2*3*tinfty13,1/2*2*ti nfty12,1/2*tinfty11" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 11 "c2functi on(" }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 11 "5*tinfty15," }{MPLTEXT 1 0 14 "1/2*4*tinfty14" }{MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 4 "1/2*" } {MPLTEXT 1 0 11 "3*tinfty13," }{MPLTEXT 1 0 14 "1/2*2*tinfty12" } {MPLTEXT 1 0 1 "," }{MPLTEXT 1 0 4 "1/2*" }{MPLTEXT 1 0 8 "tinfty11" } {MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 11 "c1function(" }{MPLTEXT 1 0 72 "1/2*5*tinfty15,1/2*4*tinfty14,1/2*3*tinfty13,1/2*2*tinfty12,1/2*tinft y11" }{MPLTEXT 1 0 3 ");\n" }{MPLTEXT 1 0 16 "CurlyLqfunction(" } {MPLTEXT 1 0 72 "1/2*5*tinfty15,1/2*4*tinfty14,1/2*3*tinfty13,1/2*2*ti nfty12,1/2*tinfty11" }{MPLTEXT 1 0 18 ",q,tdp);\nsimplify(" }{MPLTEXT 1 0 18 "CurlyLtdpfunction(" }{MPLTEXT 1 0 72 "1/2*5*tinfty15,1/2*4*tin fty14,1/2*3*tinfty13,1/2*2*tinfty12,1/2*tinfty11" }{MPLTEXT 1 0 9 ",q, tdp));" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 30 "simplify(CurlyLcheckqfu nction(" }{MPLTEXT 1 0 72 "1/2*5*tinfty15,1/2*4*tinfty14,1/2*3*tinfty1 3,1/2*2*tinfty12,1/2*tinfty11" }{MPLTEXT 1 0 4 "));\n" }{MPLTEXT 1 0 30 "simplify(CurlyLcheckpfunction(" }{MPLTEXT 1 0 72 "1/2*5*tinfty15,1 /2*4*tinfty14,1/2*3*tinfty13,1/2*2*tinfty12,1/2*tinfty11" }{MPLTEXT 1 0 3 "));" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 ",$*&I\"hG6 \"\"\"\"I\"qGF%F&!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "*&I\"hG6\"\"\" \"I$tdpGF$F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{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 }