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{SECT 0 {EXCHG {PARA 264 "" 0 "" {TEXT -1 40 "Maple code accompanying \+
the paper titled" }}{PARA 266 "" 0 "" {TEXT 260 39 "'Global classifica
tion of two-component" }}{PARA 269 "" 0 "" {TEXT 259 45 "approximately
 integrable evolution equations'" }}}{EXCHG {PARA 263 "" 0 "" {TEXT 
257 134 "\nPeter H. van der Kamp\n\nDepartment of Mathematics,\nLa Tro
be University, Victoria, 3086, Australia\n\nEmail: P.vanderkamp@latrob
e.edu.au " }}}{EXCHG {PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 268 "" 0 
"" {TEXT -1 202 "The Maple-code provided allows one to calculate the q
uadratic parts of\napproximate symmetries of an evolution equation, us
ing the symbolic method.\nAlso an implementation of the Lie-bracket is
 provided.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 774 "g:=proc(k,
n,i,j,a,b,x,y) if k=2 then return(g(1,n,j,i,b,a,y,x)) fi:\na*(add(x[q]
^n,q=1..i+1)-(add(x[q],q=1..i+1)+add(y[q],q=1..j))^n)\n+b*add(y[q]^n,q
=1..j) end:\n\nTRANS:=proc(P,d) local R,e,i,Q: R:=0: Q:=expand(P): if \+
type(Q,`+`)\nthen Q:=convert(Q,list) else Q:=[Q] fi: for e in Q do for
 i to d\ndo e:=e*u[degree(e,x[i])]/x[i]^degree(e,x[i]) od: for i to 2-
d do\ne:=e*v[degree(e,y[i])]/y[i]^degree(e,y[i]) od: R:=R+e od: R end:
\n\nProd:=proc(A,B) [seq(A[i]*B[i],i=1..6)] end:\n\ndiv:=proc(a,b) if \+
factor(a)=0 and factor(b)=0 then return(nnli)\nfi: factor(a/b) end:\n
\nDiv:=proc(A,B) [seq(div(A[i],B[i]),i=1..6)] end:\n\nG:=proc(a,b,n) [
seq(g(1,n,1-k,k,a,b,x,y),k=0..2),\nseq(g(2,n,k,1-k,a,b,x,y),k=0..2)] e
nd:\n\nTRA:=proc(K) [add(TRANS(K[i],3-i),i=1..3),\nadd(TRANS(K[i],i-4)
,i=4..6)]: end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 553 "VAR:=pr
oc(P) local e,R: R:=NULL: for e in indets(P) do\nif evalb(op(0,e) in \+
\{u,v\}) then R:=R,e fi od: \{R\} end:\n\nDD:=proc(P,n) local R,i,e,Q:
 R:=P: for i to n do Q:=0: for e in VAR(R)\ndo Q:=Q+op(0,e)[op(1,e)+1]
*diff(R,e) od: R:=diff(R,x)+Q od: R end:\n\nFR:=proc(x,A,B) local e,R:
 R:=0: for e in VAR(A) do\nif op(0,e)=x then R:=R+diff(A,e)*DD(B,op(1,
e)) fi od: R end:\n\nMFR:=proc(A,B) local R,i,j,Q,U: U:=[u,v]: R:=[]: \+
for i to 2 do\nQ:=0: for j to 2 do Q:=Q+FR(U[j],A[i],B[j]) od: R:=[op(
R),Q] od:\nR end:\n\nLIE:=proc(K,S) RETURN(expand(MFR(S,K)-MFR(K,S))) \+
end:" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 118 "To calculate the fi
rst approximate symmetry of equation 0.3, we proceed as follows.\nThe \+
linear part of equation 0.3 is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 25 "n:=0:\neq0:=[a*u[n],v[n]]:" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 25 "The constant tuple is set" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 17 "K:=[e,f,g,h,i,j]:" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 59 "which is translated into the quadratic part of the e
quation" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "eq1:=TRA(K):" }}
}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 47 "The first approximate symmet
ry appears at order" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:=1:
" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 22 "It has has linear part" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sy0:=[c*u[m],d*v[m]]:" }}
}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 18 "and quadratic part" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "sy1:=TRA(Div(Prod(K,G(c,d,m)
),G(a,1,n)));\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sy1G7$,***%\"fG
\"\"\"%\"cGF)&%\"uG6#\"\"!F)&%\"vG6#F)F)!\"\"**F(F)%\"dGF)F+F)F/F)F)*.
\"\"#F)%\"gGF),&%\"aGF)F6F2F2F*F)F/F)&F0F-F)F)*.F6F)F7F)F8F2F4F)F/F)F:
F)F2,**,%\"iGF)F9F2F*F)&F,F1F)F:F)F)*,F>F)F9F2F4F)F?F)F:F)F2*.F6F)%\"j
GF),&F)F2*&F6F)F9F)F)F2F4F)F?F)F+F)F2*.F6F)FBF)FCF2F*F)F?F)F+F)F)" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 60 "To check that this is an appr
oximate symmetry we verify that" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "factor(LIE(eq0,sy1) + LIE(eq1,sy0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#7$\"\"!F$" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 366 
"One can do the same calculation, but with a=0, i=0 from the start. Th
e program will tell\nyou that the linear part is not nonlinear injecti
ve by introducing a constant nnli in the\napproximate symmetry. The ab
ove procedure can be used to calculate any approximate symmetry\nof an
y equation in our list, which is useful to further classify with respe
ct to integrability." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
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