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{SECT 0 {PARA 258 "" 0 "" {TEXT -1 77 "                               \+
         Vessiot Tutorial: Recursion Operators" }}{PARA 260 "" 0 "" 
{TEXT 257 7 "Purpose" }}{PARA 257 "" 0 "" {TEXT -1 153 "to create the \+
recursion operators for some well-known integrable  equations and to u
se these operators to generate  higher order generalizer  symmetries.
" }}{PARA 256 "" 0 "" {TEXT -1 7 "Summary" }}{PARA 257 "" 0 "" {TEXT 
-1 119 "Recursion operators for integrable equations typically involve
 the formal  inverse of the total derivative operator.   " }}{PARA 
257 "" 0 "" {TEXT -1 248 "If  g= (x, u, ux, u_xx.. ) is a function on \+
J^k(R,R)  and  f= D_x(g), then the  command inverse_td(f)  produces g,
 (up to an additice constant). This command is easily constructed from
 the horizonal  homotopy operator  in the variational bicomplex." }}
{PARA 256 "" 0 "" {TEXT -1 9 "Reference" }}{PARA 257 "" 0 "" {TEXT -1 
0 "" }{TEXT 260 35 "Olver: Applications of Lie Groups t" }{TEXT -1 1 "
o" }{TEXT 261 23 " Differential Equations" }}{PARA 256 "" 0 "" {TEXT 
-1 5 "Notes" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT 264 8 "See Also" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(Vessiot):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 15 "with(de_appls);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7&%.Lie_reductionG%)Noether1G%)Noether2G%+inverse_tdG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "coord_frame([x], [u]);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3frame~name:~euclidG" }}}{PARA 4 "" 
0 "" {TEXT 259 44 "                                            " }}
{SECT 1 {PARA 256 "" 0 "" {TEXT 262 17 "The  KdV Equation" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 27 "
KdVFlow:= u[3] + u[0]*u[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(KdVF
lowG,&&%\"uG6#\"\"$\"\"\"*&&F'6#\"\"!F*&F'6#F*F*F*" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT 256 35 "Define the  KdV recursion operator:" }}}{EXCHG 
{PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 84 "KdV_Rec:= exp ->  expand(multi
_td(exp,[2]) +2/3*u[0]*exp +1/3*u[1]*inverse_td(exp));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%(KdV_RecGR6#%$expG6\"6$%)operatorG%&arrowGF(-%'e
xpandG6#,(-%)multi_tdG6$9$7#\"\"#\"\"\"*&&%\"uG6#\"\"!F6F3F6#F5\"\"$*&
&F96#F6F6-%+inverse_tdG6#F3F6#F6F=F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 256 81 "Define  the  first  generalized  symmetry Q_0 and genera
te the higher symmetries." }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 
1 0 10 "Q_0:=u[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_0G&%\"uG6#
\"\"\"" }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 18 "Q_1:=KdV_Rec
(Q_0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Q_1G,&&%\"uG6#\"\"$\"\"\"
*&&F'6#\"\"!F*&F'6#F*F*F*" }}}{EXCHG {PARA 0 "euclid > " 0 "" 
{MPLTEXT 1 0 18 "Q_2:=KdV_Rec(Q_1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$Q_2G,*&%\"uG6#\"\"&\"\"\"*&&F'6#\"\"$F*&F'6#\"\"!F*#F)F.*&&F'6#\"
\"#F*&F'6#F*F*#\"#5F.*&)F/F6\"\"\"F7F=#F)\"\"'" }}}{EXCHG {PARA 0 "euc
lid > " 0 "" {MPLTEXT 1 0 18 "Q_3:=KdV_Rec(Q_2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$Q_3G,2*&&%\"uG6#\"\"\"F*&F(6#\"\"%F*\"\"(*&&F(6#\"\"
!F*&F(6#\"\"&F*#F.\"\"$&F(6#F.F**&&F(6#F7F*&F(6#\"\"#F*#\"#NF7*&F;\"\"
\")F0F?FC#FA\"#=*$)F'F7FCFE*(F'FCF0FCF=FC#\"#q\"\"**&)F0F7FCF'FC#FA\"#
a" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 256 66 "Check that the  flow  commute  using the
  generalized Lie bracket." }}}{EXCHG {PARA 0 "euclid > " 0 "" 
{MPLTEXT 1 0 65 "gen_Lie_bracket( KdVFlow &mult vect(u[0]), Q_0 &mult \+
vect(u[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&%$D_xG6#
%!GF%" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 65 "gen_Lie_bra
cket( KdVFlow &mult vect(u[0]), Q_1 &mult vect(u[0]));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&%$D_xG6#%!GF%" }}}{EXCHG {PARA 0 "eu
clid > " 0 "" {MPLTEXT 1 0 65 "gen_Lie_bracket( KdVFlow &mult vect(u[0
]), Q_2 &mult vect(u[0]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G
\"\"\"&%$D_xG6#%!GF%" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 
65 "gen_Lie_bracket( KdVFlow &mult vect(u[0]), Q_3 &mult vect(u[0]));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%#0~G\"\"\"&%$D_xG6#%!GF%" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 256 "" 0 "" {TEXT 263 
25 "The  Sine-Gordon Equation" }}{EXCHG {PARA 0 "" 0 "" {TEXT 256 35 "
Define the  KdV recursion operator:" }}}{EXCHG {PARA 0 "euclid > " 0 "
" {MPLTEXT 1 0 82 "SG_Rec:=exp ->expand(multi_td(exp,[2]) + u[1]^2*exp
 - u[1]*inverse_td(u[2] *exp));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'
SG_RecGR6#%$expG6\"6$%)operatorG%&arrowGF(-%'expandG6#,(-%)multi_tdG6$
9$7#\"\"#\"\"\"*&)&%\"uG6#F6F5\"\"\"F3F6F6*&F9F6-%+inverse_tdG6#*&&F:6
#F5F6F3F<F6!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 81 "Define
  the  first  generalized  symmetry Q_0 and generate the higher symmet
ries." }}}{EXCHG {PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 10 "P_0:=u[1];" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$P_0G&%\"uG6#\"\"\"" }}}{EXCHG 
{PARA 0 "euclid>" 0 "" {MPLTEXT 1 0 17 "P_1:=SG_Rec(P_0);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%$P_1G,&&%\"uG6#\"\"$\"\"\"*$)&F'6#F*F)\"\"\"#
F*\"\"#" }}}{EXCHG {PARA 0 "euclid > " 0 "" {MPLTEXT 1 0 17 "P_2:=SG_R
ec(P_1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$P_2G,*&%\"uG6#\"\"&\"\"
\"*&)&F'6#\"\"#F/\"\"\"&F'6#F*F*#F)F/*&)F1F/F0&F'6#\"\"$F*F3*$)F1F)F0#
F8\"\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 44 "Check \+
that these are generalized symmetries." }}{PARA 263 "" 0 "" {TEXT -1 
0 "" }}{PARA 261 "" 0 "" {TEXT 256 8 "STEP 1. " }}{PARA 262 "" 0 "" 
{TEXT 256 115 "Pullback into the space of two independent variables an
d  construct the  corresponding evolutionary  vector fields." }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "euclid > " 
0 "" {MPLTEXT 1 0 28 "coord_frame([x,t], [u], SG);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%/frame~name:~SGG" }}}{EXCHG {PARA 0 "SG>" 0 "" 
{MPLTEXT 1 0 45 "phi:=transform(SG,euclid,[x=x, u[0]=u[0,0]]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG7$/%\"xGF'/&%\"uG6#\"\"!&F*6$F,
F," }}}{EXCHG {PARA 0 "SG > " 0 "" {MPLTEXT 1 0 49 "phi4:= pr_transfor
m(phi,5,matrix([[1,0],[0,0]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%
phi4G7)/%\"xGF'/&%\"uG6#\"\"!&F*6$F,F,/&F*6#\"\"\"&F*6$F2F,/&F*6#\"\"#
&F*6$F8F,/&F*6#\"\"$&F*6$F>F,/&F*6#\"\"%&F*6$FDF,/&F*6#\"\"&&F*6$FJF,
" }}}{EXCHG {PARA 0 "SG > " 0 "" {MPLTEXT 1 0 44 "Y_0:=evalV( pullback
0(phi4,P_0) * D_u[0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Y_0G*&&
%\"uG6$\"\"\"\"\"!F)&&%$D_uG6#7$F*F*6#%!GF)" }}}{EXCHG {PARA 0 "SG > \+
" 0 "" {MPLTEXT 1 0 44 "Y_1:=evalV( pullback0(phi4,P_1) * D_u[0,0]);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Y_1G*&&,&&%\"uG6$\"\"$\"\"!\"\"\"
*$)&F)6$F-F,F+\"\"\"#F-\"\"#6#%!GF-&&%$D_uG6#7$F,F,6#F6F-" }}}{EXCHG 
{PARA 0 "SG > " 0 "" {MPLTEXT 1 0 44 "Y_2:=evalV( pullback0(phi4,P_2) \+
* D_u[0,0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Y_2G*&&,*&%\"uG6$\"
\"&\"\"!\"\"\"*&)&F)6$\"\"#F,F2\"\"\"&F)6$F-F,F-#F+F2*&)F4F2F3&F)6$\"
\"$F,F-F6*$)F4F+F3#F;\"\")6#%!GF-&&%$D_uG6#7$F,F,6#FAF-" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 256 7 "Step 2:" }}{PARA 0 "" 0 "" {TEXT 256 135 
"Define the  sine-Gordon equation as a Vessiot  differential equation \+
and prolong it to order 5 and  convert to a Vessiot transformation" }}
}{EXCHG {PARA 0 "SG > " 0 "" {MPLTEXT 1 0 28 "Delta0:=u[1,1] -sin(u[0,
0]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'Delta0G,&&%\"uG6$\"\"\"F)F)
-%$sinG6#&F'6$\"\"!F/!\"\"" }}}{EXCHG {PARA 0 "SG > " 0 "" {MPLTEXT 1 
0 49 "Delta:= diffeq([u[1,1] -sin(u[0,0])], [u[1,1]]) ;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&DeltaG7$<#&%\"uG6$\"\"\"F*7#,&F'F*-%$sinG6#&F
(6$\"\"!F2!\"\"" }}}{EXCHG {PARA 0 "_diffeq > " 0 "" {MPLTEXT 1 0 30 "
pr_Delta5:=pr_diffeq(Delta,5):" }}}{EXCHG {PARA 0 "SG > " 0 "" 
{MPLTEXT 1 0 36 "Phi:=diffeq_to_transform(pr_Delta5):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 7 "S
tep 3:" }}{PARA 0 "" 0 "" {TEXT 256 104 "Lie  differentiate the sine-G
ordon equation with respect the  2nd  prolongation of the vector field
 Y_i." }}{PARA 0 "" 0 "" {TEXT 256 52 "Pull the  result  back to the s
ine-Gordon equation. " }}}{EXCHG {PARA 0 "SG > " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "SG > " 0 "" {MPLTEXT 1 0 38 "check0:=Lie0( pr_vec
t(Y_0,2), Delta0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'check0G,&*&&%
\"uG6$\"\"\"\"\"!F*-%$cosG6#&F(6$F+F+F*!\"\"&F(6$\"\"#F*F*" }}}{EXCHG 
{PARA 0 "SG > " 0 "" {MPLTEXT 1 0 21 "pullback(Phi,check0);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "SG > " 0 "" 
{MPLTEXT 1 0 38 "check1:=Lie0( pr_vect(Y_1,2), Delta0);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%'check1G,**&,&&%\"uG6$\"\"$\"\"!\"\"\"*$)&F)6$
F-F,F+\"\"\"#F-\"\"#F--%$cosG6#&F)6$F,F,F-!\"\"&F)6$\"\"%F-F-*&)F0F4F2
&F)6$F4F-F-#F+F4*(&F)6$F-F-F-F0F-&F)6$F4F,F-F+" }}}{EXCHG {PARA 0 "SG \+
> " 0 "" {MPLTEXT 1 0 30 "expand( pullback(Phi,check1));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "SG > " 0 "" {MPLTEXT 
1 0 38 "check2:=Lie0( pr_vect(Y_2,2), Delta0);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%'check2G,6*&,*&%\"uG6$\"\"&\"\"!\"\"\"*&)&F)6$\"\"#F,
F2\"\"\"&F)6$F-F,F-#F+F2*&)F4F2F3&F)6$\"\"$F,F-F6*$)F4F+F3#F;\"\")F--%
$cosG6#&F)6$F,F,F-!\"\"*(F0F-F4F3&F)6$F;F-F-\"#5*&F8F3&F)6$\"\"%F-F-F6
&F)6$\"\"'F-F-*&&F)6$F2F-F-F/F3#\"#:F2*(FRF3F4F3F9F3FI*&FRF3)F4FMF3#FU
F?*(F0F3&F)6$F-F-F-F9F3FI*(F0F3FenF3)F4F;F3FT*(FenF3F4F3&F)6$FMF,F-F+
" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "SG > " 0 "" 
{MPLTEXT 1 0 30 "expand( pullback(Phi,check1));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 265 20 "updated: 01/17/03:IA" 
}}{PARA 256 "" 0 "" {TEXT 258 0 "" }}}{MARK "20 0" 12 }{VIEWOPTS 1 1 
0 3 4 1802 }