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1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }{TEXT 258 68 "Tutorial on Differential Elimination Completion for PDE with r ifsimp" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT 259 1 " " }{URLLINK 17 "Greg Reid " 4 "htt p://www.orcca.on.ca/~reid" "" }{TEXT 260 41 " (University of Western \+ Ontario) and " }{URLLINK 17 "Allan Wittkopf" 4 "http://www.cecm.sfu .ca/~wittkopf" "" }{TEXT -1 24 " (Maplesoft and CECM)" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 54 "Special Thanks a lso to Edgardo Cheb-Terrab (Maplesoft)" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 17 "October 26, 2006" }}{PARA 256 "" 0 "" {TEXT -1 43 "Workshop on Software for Algebraic Geometry" }} {PARA 256 "" 0 "" {TEXT -1 70 "Institute for Mathematics and its Appli cations, Minneapolis, Minnesota" }}{PARA 256 "" 0 "" {TEXT -1 16 "Spec ial Year on " }{TEXT 263 34 "Applications of Algebraic Geometry" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 62 "This do cument will be updated before the tutorial on Thursday." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 798 "The main purpose of this self-paced tutorial is to give an easy-paced hands-on informal i ntroduction to differential elimination completion using the rifsimp p ackage. Technical/theoretical details occur elsewhere. While softw are for polynomial systems has become common-place the same can not be said for PDE. We especially point the participants to the talk on \+ Tuesday by Evelyne Hubert (with associated software diffalg); and the \+ tutorial by Thomas Wolf (Reduce, for large and difficult overdetermin ed PDE systems on Thursday evening). For newer developments based on \+ Numerical Algebraic Geometry, we point to our poster on HybridRif (Rei d, Verschelde, Wittkopf and Wu) and also the poster by Robin Scott (Ap proximate Grobner Bases - a backwards approach). And finally Greg Rei d's talk " }{TEXT 264 51 "Application of Numerical Algebraic Geometry \+ to PDE " }{TEXT -1 9 "(Friday)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 82 "We are very interested in your feedback a nd features that you want to see added. " }}{PARA 0 "" 0 "" {TEXT -1 87 "Please email us (reid@uwo.ca and wittkopf@shaw.ca) with your quest ions and suggestions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 0 "" }{TEXT 266 32 "Part I: \+ Linear Systems of PDE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 60 " Click on this tab if you are not familia r at all with Maple" }}{PARA 0 "" 0 "" {TEXT -1 65 "You actually need \+ to know very little Maple to do this tutorial. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 219 "The main thing you need \+ to know is how to enter differential equations using the diff notation , end Maple commands with a ; and execute them by striking enter. A lso you open the section tabs by left clicking on them." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 213 "Some help can also \+ be obtained from the new user's tour (go to the Help Tab. Top middle l eft above, and select new user's tour, and differential equations, but most of it is not directly relevant to the tutorial)." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 55 " If you forget everything else, then just remember ... " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 261 0 "" } {TEXT 262 9 "? rifsimp" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "This giv es extensive documentation in Maple on the package (approx. 100 printe d pages)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "Or click on the Help button in the Horizontal ToolBar to gain acce ss to Maple's Menu Driven Help" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 51 " Getting started - E ntering systems of PDE in Maple" }}{PARA 0 "" 0 "" {TEXT -1 61 "To ent er the following PDE is done in standard Maple notation" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "pde := diff(u(x,t),x)-t*u(x,t) = 0;" "6#>%$pdeG/,& -%%diffG6$-%\"uG6$%\"xG%\"tGF-\"\"\"*&F.F/-F+6$F-F.F/!\"\"\"\"!" }} {PARA 0 "" 0 "" {TEXT -1 76 "by typing the followin (then enter to see the output, the =0's are optional)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "pde := diff(u(x,t),x) - t*u (x,t) = 0;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Now try entering th e following system" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "sys1 := [diff(u(x ,t),x)-t*u(x,t) = 0, diff(u(x,t),t)-u(x,t) = 0];" "6#>%%sys1G7$/,&-%%d iffG6$-%\"uG6$%\"xG%\"tGF.\"\"\"*&F/F0-F,6$F.F/F0!\"\"\"\"!/,&-F)6$-F, 6$F.F/F/F0-F,6$F.F/F4F5" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 " Answer" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 70 "asys1 := [diff(u(x,t),x) - t*u(x,t) = 0, diff( u(x,t),t) - u(x,t) = 0];" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 48 " A pplying rifsimp - simple illustrative examples" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 110 "rifsimp is part of the DEtools package in Maple (it c omes with distributed Maple since Maple 7). To load it:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "with(DEtools): infolevel[rifsimp]: = 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "rpde1 := rifsimp([p de]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rpde1GK%&TABLEG6#7#/%'Solv edG7#/-%%diffG6$-%\"uG6$%\"xG%\"tGF3*&F4\"\"\"F0F6Q)pprint176\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Note that all rifsimp did was rew rite the pde in solved form with respect to its highest derivative. F or a single PDE rifsimp is often not very helpful." }}{PARA 0 "" 0 "" {TEXT -1 93 "However for the system of 2 PDE for 1 dependent variable \+ (i.e. an overdetermined PDE system):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "rasys1:= rifsimp(asys1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "So one of rifsimp's main purposes is simplification, and it is most useful for overdetermined PDE systems." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Now consider the following system" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "sys2:= [diff(u(x,y),y,y) - diff(u(x ,y),x), diff(u(x,y),x,y) - u(x,y)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "rsys2:= rifsimp(sys2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "Try to recognize the above system and its rifsimp form a s isomorphic to a certain polynomial basis in a certain ring." }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 7 " Answer" }}{PARA 0 "" 0 "" {TEXT -1 48 "Under the map between the differential ring Q [ " }{XPPEDIT 18 0 "D[x],D[y];" "6$&%\"DG6#%\"xG&F$6#%\"yG" }{TEXT -1 23 " ] and Q[X, \+ Y] where " }{XPPEDIT 18 0 "D[x];" "6#&%\"DG6#%\"xG" }{TEXT -1 8 "-> X , " }{XPPEDIT 18 0 "D[y];" "6#&%\"DG6#%\"yG" }{TEXT -1 98 "-> Y this has given the same result as a total degree Groebner Basis of the pol ynomial system \{ " }{XPPEDIT 18 0 "Y^2-X,X*Y-1;" "6$,&*$%\"YG\"\"# \"\"\"%\"XG!\"\",&*&F(F'F%F'F'F'F)" }{TEXT -1 182 " \} . For consta nt coefficient linear homogeneous PDE in 1 dependent variable rifsimp \+ computes a Groebner basis in the differential ring, isomorphic to a po lynomial Groebner Basis." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "One ca n convert back and forth using the following Maple commands" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "ps ys2:= PolynomialTools[PDEToPolynomial](sys2, [x,y], [u]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "gb_psys2:= Groebner[Basis](psys2, t deg(x,y));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "PolynomialToo ls[PolynomialToPDE](gb_psys2, [x,y], [u]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "which was the same answer as we got before using rifsimp. " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 62 " Computation of Initial Data and Formal Power Series Solutions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "de3:= [diff(u(x),x,x) - \+ a*u(x) = 0];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$de3G7#/,&-%%diffG6$ -%\"uG6#%\"xG-%\"$G6$F.\"\"#\"\"\"*&%\"aGF3F+F3!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "rde3 := rifsimp(de3);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%rde3GK%&TABLEG6#7#/%'SolvedG7#/-%%diffG6$-%\" uG6#%\"xG-%\"$G6$F3\"\"#*&%\"aG\"\"\"F0F:Q)pprint186\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "id3 := initialdata(rde3, [u]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$id3GK%&TABLEG6#7$/%)InfiniteG7\"/%' FiniteG7$/-%\"uG6#&%\"xG6#\"\"!%$_C1G/--%\"DG6#F1F2%$_C2GQ)pprint196\" " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "Notice that if this was a pol ynomial system " }{XPPEDIT 18 0 "X^2-a = 0;" "6#/,&*$%\"XG\"\"#\"\"\" %\"aG!\"\"\"\"!" }{TEXT -1 86 " by the bijection above then the numbe r of complex roots including multiplicity is 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 164 "For the differential equ ation, the corresponding quantity is the number of initial conditions \+ needed to specify a formal power series solution at an initial point \+ " }{XPPEDIT 18 0 "x = x[0];" "6#/%\"xG&F$6#\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "Now we compute the formal power series solution (up to degree 4) with the above initial data:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "fps_deg4 := rtaylor(rde3[ Solved], id3, order = 4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)fps_de g4G7$/-%\"uG6#%\"xG,,%$_C1G\"\"\"*&%$_C2GF-,&F*F-&F*6#\"\"!!\"\"F-F-*& #F-\"\"#F-*(%\"aGF-F,F-)F0F7F-F-F-*&#F-\"\"'F-*(F9F-F/F-)F0\"\"$F-F-F- *&#F-\"#CF-*()F9F7F-F,F-)F0\"\"%F-F-F-/F9F9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 77 " Symmetr y Analysis a success story for Differential Grobner Bases (examples) " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 372 "Most importantly seeking the \+ linearized forms of the point symmetries leaving a PDE invariant leads to a system of generally overdetermined linear PDE for the linearized symmetries. Many programs have been developed for both producing the linear overdetermined system and reducing it to a Differential Grobne r Basis (or closely related objects, such as involutive Bases)." }} {PARA 0 "" 0 "" {TEXT -1 110 "For background see Olver's, Applications of Lie Groups to Differential Equations, or for a few brief comments: " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 99 " More remarks about symmetri es of PDE (skip this if you don't care where the PDE systems came from )" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "The point symmetries of a PD E F with independent variables x, t and dependent variable u are the diffeomorphisms " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 3 ": \+ " }{XPPEDIT 18 0 "C^3;" "6#*$%\"CG\"\"$" }{TEXT -1 5 " -> " } {XPPEDIT 18 0 "C^3;" "6#*$%\"CG\"\"$" }{TEXT -1 16 " of form " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 63 " (x, t , u) = (x*, t*, u*) = (X(x,t,u), T(x,t,u),U(x,t,u)) " }}{PARA 0 " " 0 "" {TEXT -1 57 "which leave the family of solutions of the PDE inv ariant." }}{PARA 0 "" 0 "" {TEXT -1 139 "Imposing this invariance usua lly leads to highly nonlinear overdetermined PDE for the unknown funct ions X, T, U determining the symmetries." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Lie characterized such pseudo-grou ps by their linearization about the identity symmetry:" }}{PARA 0 "" 0 "" {TEXT -1 18 " x* = X(x,t,u) = " }{XPPEDIT 18 0 "x+xi(x,t,u)*epsi lon+O(epsilon^2);" "6#,(%\"xG\"\"\"*&-%#xiG6%F$%\"tG%\"uGF%%(epsilonGF %F%-%\"OG6#*$F,\"\"#F%" }{TEXT -1 4 " , " }}{PARA 0 "" 0 "" {TEXT -1 19 " t* = T(x,t,u) = " }{XPPEDIT 18 0 "t+tau(x,t,u)*epsilon+O(epsilo n^2);" "6#,(%\"tG\"\"\"*&-%$tauG6%%\"xGF$%\"uGF%%(epsilonGF%F%-%\"OG6# *$F,\"\"#F%" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 18 " u* = U( x,t,u) = " }{XPPEDIT 18 0 "u+eta(x,t,u)*epsilon+O(epsilon^2);" "6#,(% \"uG\"\"\"*&-%$etaG6%%\"xG%\"tGF$F%%(epsilonGF%F%-%\"OG6#*$F,\"\"#F%" }{TEXT -1 7 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 35 "The corresponding Lie vector field " }{XPPEDIT 18 0 "xi *D[x]+tau*D[t]+eta*D[u];" "6#,(*&%#xiG\"\"\"&%\"DG6#%\"xGF&F&*&%$tauGF &&F(6#%\"tGF&F&*&%$etaGF&&F(6#%\"uGF&F&" }{TEXT -1 388 " generates Li e symmetries leaving the PDE invariant. These vector fields span a ve ctor space endowed with a commutator whose structure constants determi ne the Lie algebra of the group. Most importantly the awkward problem of dealing the nonlinear overdetermined PDE for the X, T, U was repla ced with a much easier problem of dealing with linear overdetermined \+ PDE for the components " }{XPPEDIT 18 0 "xi(x,t,u);" "6#-%#xiG6%%\"xG %\"tG%\"uG" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "tau(x,t,u);" "6#-%$tau G6%%\"xG%\"tG%\"uG" }{TEXT -1 6 " , " }{XPPEDIT 18 0 "eta(x,t,u);" "6#-%$etaG6%%\"xG%\"tG%\"uG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 200 "" 0 "" {TEXT -1 63 " Ex (small linear): Symmetries for Scalar Telegraph Equation " }{XPPEDIT 19 1 "u[t, t] \+ = -2*u[x]^2/u^3+u[x, x]/u^2-u[x]/u^2" "6#/&%\"uG6$%\"tGF',(*(\"\"#\"\" \"*$&F%6#%\"xGF*F+*$F%\"\"$!\"\"F2*&&F%6$F/F/F+*$F%F*F2F+*&&F%6#F/F+*$ F%F*F2F2" }{TEXT 214 1 " " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "Thes e are automatically generated by many available programs (which maps t he solutions of the scalar telegraph equation to itself, and have many important applications):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 567 "telsymsys:= \n[diff(xi(x,t,u),t)=0,diff(tau(x,t,u),x) = 0, diff(t au(x,t,u),u) = 0, diff(xi(x,t,u),u) = 0,\ndiff(diff(eta(x,t,u),u),u) = 0, diff(diff(tau(x,t,u),t),t)-2*diff(diff(eta(x,t,u),t),u) = 0, eta(x ,t,u)-u*diff(tau(x,t,u),t)+diff(xi(x,t,u),x)*u = 0, \n-diff(diff(eta(x ,t,u),x),x)+diff(eta(x,t,u),x)+diff(diff(eta(x,t,u),t),t)*u^2 = 0, \nd iff(eta(x,t,u),u)*u-2*diff(xi(x,t,u),x)*u-3*eta(x,t,u)+2*u*diff(tau(x, t,u),t) = 0, \n-diff(diff(xi(x,t,u),x),x)*u+diff(xi(x,t,u),x)*u-4*diff (eta(x,t,u),x)+2*diff(diff(eta(x,t,u),u),x)*u+2*eta(x,t,u)-2*u*diff(ta u(x,t,u),t) = 0];" }}{PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Here the unknowns " }{XPPEDIT 18 0 "xi(x,t,u);" "6#-%#xiG6%%\"xG%\"tG%\"uG" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "tau(x,t,u);" "6#-%$tauG6%%\"xG%\"tG%\"uG" } {TEXT -1 6 " , " }{XPPEDIT 18 0 "eta(x,t,u);" "6#-%$etaG6%%\"xG%\"t G%\"uG" }{TEXT -1 41 " are coefficients of a Lie vector field " } {XPPEDIT 18 0 "xi*D[x]+tau*D[t]+eta*D[u];" "6#,(*&%#xiG\"\"\"&%\"DG6#% \"xGF&F&*&%$tauGF&&F(6#%\"tGF&F&*&%$etaGF&&F(6#%\"uGF&F&" }{TEXT -1 57 " that generate Lie symmetries leaving the PDE invariant." }} {PARA 0 "" 0 "" {TEXT -1 143 "Note that the fully explicit notation be comes cumbersome, so we use, as is conventional abbreviated subscript \+ notation for partial derivatives." }}{PARA 0 "" 0 "" {TEXT -1 44 "To d o this we use the Maple command declare:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(PDEtools):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "declare(telsymsys);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Now telsymsys displays much more p leasantly as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "telsymsys; " }}}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 32 "rtels ymsys:= rifsimp(telsymsys);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 82 "The ranking of derivatives used to reduce the system (up to different ial order 2):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 70 "checkran k([xi(x,t,u),tau(x,t,u),eta(x,t,u)], indep=[x,t,u], degree=2);" }}} {EXCHG {PARA 202 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 128 "The automated existence and uniqueness theorem giving i nitial data at a point determining a unique formal power series soluti on:" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 35 "idtelsys:= initial data(rtelsymsys);" }}}{EXCHG {PARA 201 "" 0 "" {TEXT 200 128 "Thus the re is a 4 dimensional group of symmetries, since there are 4 arbitrary constants _C1, _C2, _C3, _C4 in the initial data." }}{PARA 0 "" 0 "" {TEXT -1 99 "Below we calculate the formal power series solution with \+ the given initial data, to various orders:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "fps_soln1:= rtaylor(rtelsymsys[Solved], idtelsys, o rder = 1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "fps_soln2:= r taylor(rtelsymsys[Solved], idtelsys, order = 2);" }}}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 200 "" 0 "" {TEXT -1 55 " \+ Ex (large linear): Symmetries for the CNLS Equation " }{XPPEDIT 19 1 "Delta*Psi+i*Diff(Psi, t)+(abs(Psi)^2+abs(Phi)^2)*Psi = 0, Delta*Ph i+i*Diff(Phi, t)+(abs(Psi)^2+abs(Phi)^2)*Phi = 0" "6$/,(*&%&DeltaG\"\" \"%$PsiGF'F'*&%\"iGF'-%%DiffG6$F(%\"tGF'F'*&,&*$-%$absG6#F(\"\"#F'*$-F 36#%$PhiGF5F'F'F(F'F'\"\"!/,(*&F&F'F9F'F'*&F*F'-F,6$F9F.F'F'*&,&*$-F36 #F(F5F'*$-F36#F9F5F'F'F9F'F'F:" }{TEXT 215 1 " " }}{PARA 0 "" 0 "" {TEXT -1 133 "The cubic nonclassical Schrodinginger system occurs in m any applications. Here we look for its classical Lie symmetries gener ated by" }}{PARA 0 "" 0 "" {TEXT -1 24 "vector fields of form " } {XPPEDIT 18 0 "tau*D[t]+eta*D[y]+xi*D[x]+theta*D[z]+phi*D[p]+psi*D[q]+ alpha*D[r]+beta*D[s];" "6#,2*&%$tauG\"\"\"&%\"DG6#%\"tGF&F&*&%$etaGF&& F(6#%\"yGF&F&*&%#xiGF&&F(6#%\"xGF&F&*&%&thetaGF&&F(6#%\"zGF&F&*&%$phiG F&&F(6#%\"pGF&F&*&%$psiGF&&F(6#%\"qGF&F&*&%&alphaGF&&F(6#%\"rGF&F&*&%% betaGF&&F(6#%\"sGF&F&" }{TEXT -1 24 " which act on solutions." }} {PARA 0 "" 0 "" {TEXT -1 104 "Our purpose is not to describe the backg round (see Mansfield, Reid and Clarkson (2001) for the details)." }} {PARA 0 "" 0 "" {TEXT -1 102 "But instead note that the system for the se classical symmetries is very large and very overdetermined." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "read \"TutSystems.txt\":" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%hoThis~file~contains~several~systems: ~~Wolf's~challenge~system~WolfConsSys~fromG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%bpconservation~laws~-~a~system~of~148~PDE~in~2~depende nt~variables~qne~10~indep~variablesG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%\\pThis~file~also~contains~the~classical~symmetry~defining~system~o f~1888~PDE~-~csysG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Yfor~symmetries ~of~the~Cubic~Nonlinear~Schrodinger~SystemG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 60 "Again we turn on the abbreviated partial derivative not ation" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 21 "declare(csys, qu iet):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 11 "nops(csys);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"%))=" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "It has 1888 PDE! To see some of these:" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 29 "seq(csys[i], i = 600 .. 700);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6aq,$*&\"\"#\"\"\"-%%diffG6%%#xiG%\"qG%\"sGF&! \"\",$*&F%F&-F(6%%%betaGF+F,F&F&,$*&F%F&-F(6%%&alphaGF+%\"rGF&F&,$*&F% F&-F(6%%$tauGF+F,F&F&F#,$*&F%F&-F(6%F7%\"pGF+F&F&,$*&F%F&-F(6%F2FBF+F& F&,$*&F%F&-F(6%F*FBF+F&F-F#,$*&F%F&-F(6$F7F+F&F&,$*&F%F&-F(6%F=F8F,F&F &,$*&F%F&-F(6%F*F8F,F&F-,$*&F%F&-F(6%F7FBF8F&F&,$*&F%F&-F(6%F*FBF8F&F- -F(6$F2F8,6**\"\"%F&-F(6$F*F+F&F8F&F+F&F-*(F%F&-F(6$F*F8F&)F8F%F&F-*(F %F&)F,F%F&FaoF&F-*(\"\"'F&FaoF&)F+F%F&F-**F%F&-F(6%F*F+F8F&)FBF%F&F+F& F-*(F%F&F\\pF&FaoF&F-**F%F&FjoF&F+F&FeoF&F-*&F%F&-F(6%%$psiGF+F8F&F-*( F%F&FjoF&)F+\"\"$F&F-**F%F&FjoF&F+F&FcoF&F-,$*&F%F&FjoF&F-FSFgp,6**F%F &F'F&F\\pF&F+F&F-*(F%F&F'F&FdpF&F-**F%F&F'F&F+F&FeoF&F-*(F%F&F\\pF&-F( 6$F*F,F&F-*&F%F&-F(6%FbpF+F,F&F-**F%F&F'F&F+F&FcoF&F-*(FgoF&FhoF&F^qF& F-*(F%F&FcoF&F^qF&F-*(F%F&FeoF&F^qF&F-**F]oF&F,F&F+F&F^oF&F-,0**F]oF&F ,F&F+F&FaoF&F-*(F%F&FUF&FdpF&F-**F%F&FUF&F\\pF&F+F&F-*&F%F&-F(6%FbpF8F ,F&F-**F%F&FUF&F+F&FcoF&F-**F%F&FUF&F+F&FeoF&F-**F]oF&F8F&F+F&F^qF&F-, $*&F%F&-F(6%F2FBF8F&F&FSFen,$*&F%F&-F(6%F7F8F,F&F&,$FaoF-,$*&F%F&-F(6% F2F8F,F&F&,2*&F%F&-F(6%F7F8%\"xGF&F&**F%F&FgnF&F\\pF&F+F&F-**F]oF&FaoF &FBF&F+F&F-**F%F&FgnF&F+F&FeoF&F-*(F%F&FgnF&FdpF&F-**F]oF&-F(6$F*FBF&F 8F&F+F&F-**F%F&FgnF&F+F&FcoF&F-*&F%F&-F(6%FbpFBF8F&F-,$*&F%F&-F(6%F=FB F8F&F&Fen,$*&F%F&-F(6%F*FBF,F&F-,$*&F%F&-F(6%F=FBF,F&F&Fct-F(6$F7F8Fct ,2*&F%F&-F(6%FbpFBF,F&F-**F]oF&F^qF&FBF&F+F&F-**F]oF&FisF&F,F&F+F&F-** F%F&FetF&F\\pF&F+F&F-*(F%F&FetF&FdpF&F-*&F%F&-F(6%F7F,FcsF&F&**F%F&Fet F&F+F&FcoF&F-**F%F&FetF&F+F&FeoF&F-,$*&F%F&-F(6%F2FBF,F&F&,$*&F%F&-F(6 %F7FBF,F&F&FenFctFenFGFen,&-F(6$F2FBF&*&F%F&-F(6%F*F+%\"zGF&F-FG,$*&F% F&-F(6$F*FhvF&F-FGF_tF>,$*&F%F&-F(6$F*FcsF&F-,$*&F%F&-F(6%F7F+F,F&F&Fi vF.F.Faw-F(6$F=-%\"$G6$F,F%F#,2**F]oF&F^qF&FBF&F+F&F-*&F%F&-F(6%F2F,Fh vF&F&**F%F&FetF&F+F&FcoF&F-*&F%F&F_uF&F-**F]oF&FisF&F,F&F+F&F-*(F%F&Fe tF&FdpF&F-**F%F&FetF&F+F&FeoF&F-**F%F&FetF&F\\pF&F+F&F-FSF#F]w,8*(F%F& FeoF&FisF&F-*&F%F&-F(6%F7F+FcsF&F&*(F%F&F\\pF&FisF&F-*&F%F&-F(6%FbpFBF +F&F-**F%F&FIF&F\\pF&F+F&F-*(F%F&FisF&FcoF&F-*(F%F&FIF&FdpF&F-**F%F&FI F&F+F&FcoF&F-*(FgoF&FhoF&FisF&F-**F]oF&F^oF&FBF&F+F&F-**F%F&FIF&F+F&Fe oF&F-F#F9FGFGF#,$*&F%F&-F(6%F2F+F8F&F&FGFipF3Fgp,$*&F%F&-F(6$F*%\"yGF& F-,2**F]oF&FaoF&FBF&F+F&F-**F]oF&FisF&F8F&F+F&F-**F%F&FgnF&F+F&FeoF&F- **F%F&FgnF&F\\pF&F+F&F-*&F%F&-F(6%F=F8F]zF&F&*&F%F&F]tF&F-**F%F&FgnF&F +F&FcoF&F-*(F%F&FgnF&FdpF&F-Fgp,$*&F%F&-F(6%F=F+F8F&F&Fiy,$*&F%F&-F(6% F=FBF+F&F&FCFgpFiyFeyFgpF3F[o,8*(F%F&FIF&FdpF&F-*&F%F&-F(6%F2F+FhvF&F& *(F%F&F\\pF&FisF&F-**F%F&FIF&F+F&FeoF&F-**F]oF&F^oF&FBF&F+F&F-*&F%F&F \\yF&F-*(FgoF&FhoF&FisF&F-*(F%F&FisF&FcoF&F-*(F%F&FeoF&FisF&F-**F%F&FI F&F+F&FcoF&F-**F%F&FIF&F\\pF&F+F&F-Fgp,$*&F%F&-F(6%F2F+FcsF&F&FipFgpFi zF[oFhq,2**F]oF&FaoF&FBF&F+F&F-**F]oF&FisF&F8F&F+F&F-**F%F&FgnF&F+F&Fc oF&F-*&F%F&-F(6%F2F8FhvF&F&*(F%F&FgnF&FdpF&F-**F%F&FgnF&F\\pF&F+F&F-** F%F&FgnF&F+F&FeoF&F-*&F%F&F]tF&F-Fiv,$*&F%F&-F(6%F=F+FcsF&F&FizF]wFeyF gpFG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 149 "Regarded as a purely algebraic system it has 280 indeterminants, 1132 PDE with order 2 partial derivatives and the rem aining 756 PDE are first order." }}{PARA 0 "" 0 "" {TEXT -1 177 "Conse quently if this was constant coefficient (and its not); then the analo gous system of polynomials would have 1132 degree 2 polynomials and a \+ total (Bezout) degree of 2^1132!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Increasing levels of information of the operation of rifsimp can \+ be requested for setting infolevel (the default is 0)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 288 "Apply rifsimp to reduce the system. How many parameters are there in the solution? Compute a formal series s olution to degree 0 and 1. Note that the series has new terms at orde r 1. Given that there are no new terms at order 2, what can you say a bout the solution to order 1, and why?" }}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 7 " Answer" }}{PARA 0 "" 0 "" {TEXT -1 107 "To see increasing information about the internal processing of rifsimp we set infolevel (the default is 0):" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 23 "in folevel[rifsimp]:= 2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Apply ri fsimp:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "rcsys:= rifsimp(c sys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KThe~system~has~been~identif ied~as~follows:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RThe~system~has~t he~following~dependent~variables:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6* %&alphaG%%betaG%$etaG%$phiG%$psiG%$tauG%&thetaG%#xiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TThe~system~has~the~following~independent~variables :G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%\"tG%\"yG%\"xG%\"zG%\"qG%\"rG% \"pG%\"sG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TThe~following~are~to~be ~treated~as~solve~variables:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%&alp haG%%betaG%$etaG%$phiG%$psiG%$tauG%&thetaG%#xiG" }}{PARA 6 "" 1 "" {TEXT -1 30 " Initial State of System:" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 0 #NL: 0 #UnC: 268 #NonP: 0 #Piv: \+ 0" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #1 Time:1.121" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 67 #Sol: 4 #NL: 0 #UnC : 64 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End it eration #2 Time:.190" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 67 \+ #Sol: 9 #NL: 0 #UnC: 41 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #3 Time:.451" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 68 #Sol: 15 #NL: 0 #UnC: 34 #NonP: 0 \+ #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #4 Time: .651" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 71 #Sol: 22 #NL: 0 #UnC: 25 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 33 " * *End iteration #5 Time:.971" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT : 79 #Sol: 22 #NL: 0 #UnC: 13 #NonP: 0 #Piv: 0" }}{PARA 6 " " 1 "" {TEXT -1 33 " **End iteration #6 Time:.701" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 83 #Sol: 21 #NL: 0 #UnC: 7 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #7 Tim e:.481" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 83 #Sol: 25 #NL: 0 #UnC: 30 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 33 " \+ **End iteration #8 Time:.741" }}{PARA 6 "" 1 "" {TEXT -1 65 " # OT: 83 #Sol: 29 #NL: 0 #UnC: 36 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #9 Time:1.142" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 88 #Sol: 33 #NL: 0 #UnC: 35 #NonP : 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 35 " **End iteration #10 \+ Time:1.071" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 92 #Sol: 34 \+ #NL: 0 #UnC: 25 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 35 " **End iteration #11 Time:1.232" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 98 #Sol: 35 #NL: 0 #UnC: 20 #NonP: 0 #Piv: 0 " }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #12 Time:.561" } }{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 98 #Sol: 36 #NL: 0 #UnC : 20 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End it eration #13 Time:.581" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 98 \+ #Sol: 36 #NL: 0 #UnC: 13 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 35 " **End iteration #14 Time:1.512" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 98 #Sol: 39 #NL: 0 #UnC: 11 #NonP: 0 \+ #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 35 " **End iteration #15 Time :2.303" }}{PARA 6 "" 1 "" {TEXT -1 65 " #OT: 113 #Sol: 40 #NL : 0 #UnC: 3 #NonP: 0 #Piv: 0" }}{PARA 6 "" 1 "" {TEXT -1 35 " \+ **End iteration #16 Time:1.362" }}{PARA 6 "" 1 "" {TEXT -1 65 " \+ #OT: 108 #Sol: 34 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&rcsysGK%&TABLEG6#7#/%'SolvedG7`o/-%%diffG 6$%&thetaG-%\"$G6$%\"yG\"\"#\"\"!/-F.6%F0%\"xGF4F6/-F.6%F0F4%\"zGF6/-F .6$%$tauG-F26$F:F5F6/-F.6$F0FCF6/-F.6%FBF:F>F6/-F.6%F0F:F>F6/-F.6$%&al phaG-F26$F>F5F6/-F.6$FBFRF6/-F.6$F0FRF6/-F.6$%$psiG-F26$%\"sGF5F6/-F.6 $FQ%\"tG,$*(F5\"\"\"-F.6$F0F:Fao%\"rG!\"\"Fao/-F.6$%%betaGF^o,$*(F5Fao -F.6$F0F>FaoFdoFeoFao/-F.6$%$etaGF^oF6/-F.6$%$phiGF^oF6/-F.6$FgnF^oF6/ -F.6$FBF^o,$*(F5Fao-F.6$F0F4FaoFdoFeoFao/-F.6$F0F^oF6/-F.6$%#xiGF^o*&, **(F5FaoFepFaoFdoFaoFeo*(F5FaoFgnFao%\"pGFaoFeo*(F5FaoF0FaoFjnFaoFeo*( F5FaoFapFao%\"qGFaoFeoFao,**$)FdoF5FaoFao*$)F[rF5FaoFao*$)FjnF5FaoFao* $)F^rF5FaoFaoFeo/-F.6$FQF4,$-F.6$FBF:Feo/-F.6$FioF4,$-F.6$FBF>Feo/-F.6 $FapF4*(F^qFaoF[rFaoFdoFeo/-F.6$FepF4,$*(F^qFaoFdoFeoFjnFaoFeo/-F.6$Fg nF4,$*(F^qFaoF^rFaoFdoFeoFeo/-F.6$FBF4*&,**&FepFaoFdoFaoFeo*&FgnFaoF[r FaoFeo*&F0FaoFjnFaoFeo*&FapFaoF^rFaoFeoFaoF_rFeo/-F.6$FfqF4F6/-F.6$FQF :Fet/-F.6$FioF:,$-F.6$FQF>Feo/-F.6$FapF:*(FboFaoF[rFaoFdoFeo/-F.6$FepF :,$*(FboFaoFjnFaoFdoFeoFeo/-F.6$FgnF:,$*(FboFaoF^rFaoFdoFeoFeo/-F.6$Ff qF:F6/-F.6$FioF>Fet/-F.6$FapF>*(F\\pFaoF[rFaoFdoFeo/-F.6$FepF>,$*(F\\p FaoFjnFaoFdoFeoFeo/-F.6$FgnF>,$*(F\\pFaoF^rFaoFdoFeoFeo/-F.6$FfqF>F6/- F.6$FQF^rF6/-F.6$FioF^rF6/-F.6$FapF^r*&,*FgtFaoFhtFaoFitFaoFjtFaoFaoF_ rFeo/-F.6$FepF^r-F.6$FgnFjn/-F.6$FgnF^r*(,H*(FarFaoF^rFaoFepFaoFao**Fd oFaoF^rFaoFgnFaoF[rFaoFao**FdoFaoFjnFaoF^rFaoF0FaoFao*&)Fdo\"\"%FaoFjx FaoFeo*(FcrFaoFjxFaoFarFaoFeo**F5FaoFarFaoFjxFaoFerFaoFeo*(FgrFaoFjxFa oFarFaoFeo*(FgnFaoFjnFaoFarFaoFao*&FgnFao)Fjn\"\"$FaoFao*(FgnFaoFjnFao FgrFaoFao**FepFaoFdoFaoF[rFaoFjnFaoFeo*(F0FaoF[rFaoFerFaoFeo**F[rFaoF^ rFaoFapFaoFjnFaoFeo*&)FdoF]zFaoFapFaoFeo*(FdoFaoFapFaoFcrFaoFeo*(FdoFa oFapFaoFerFaoFeo*(FcrFaoFerFaoFjxFaoFeo*&)FjnFfyFaoFjxFaoFeo*(FgrFaoFj xFaoFerFaoFeoFaoF_rFeo,&*&FdoFaoF[rFaoFao*&FjnFaoF^rFaoFaoFeo/-F.6$FBF ^rF6/-F.6$F0F^r*(,B*(FepFaoFdoFaoFcrFaoFao*(F0FaoFjnFaoFcrFaoFao*(FepF aoFdoFaoFgrFaoFao*(F0FaoFjnFaoFgrFaoFao*(F^rFaoFjxFaoFczFaoFeo**F^rFao FjxFaoFdoFaoFcrFaoFeo**F^rFaoFjxFaoFdoFaoFerFaoFeo*()F^rF]zFaoFjxFaoFd oFaoFeo*(F^rFaoFapFaoFarFaoFeo*(F^rFaoFapFaoFerFaoFeo**FjnFaoF[rFaoFjx FaoFarFaoFao*(FjnFao)F[rF]zFaoFjxFaoFao*(F\\zFaoF[rFaoFjxFaoFao**FjnFa oF[rFaoFjxFaoFgrFaoFao*(FgnFaoF[rFaoFarFaoFeo*(FgnFaoF[rFaoFerFaoFeoFa oF_rFeoFjzFeo/-F.6$FfqF^rF6/-F.6$FQFdoF6/-F.6$FioFdoF6/-F.6$FapFdo,$Fj xFeo/-F.6$FepFdoFex/-F.6$FgnFdo*(,BFe[lFeoFf[lFeoFg[lFeoFh[lFeoFi[lFao Fj[lFaoF[\\lFaoF\\\\lFaoF^\\lFaoF_\\lFaoF`\\lFeoFa\\lFeoFc\\lFeoFd\\lF eoFe\\lFaoFf\\lFaoFaoF_rFeoFjzFeo/-F.6$FBFdoF6/-F.6$F0Fdo*(,HFizFao*(F [rFaoFgrFaoF0FaoFaoFiyFao**F5FaoFgrFaoFjxFaoFcrFaoFaoFazFeo*(FdoFaoFgr FaoFapFaoFaoFbyFaoFcyFao*(FepFaoFerFaoF^rFaoFeo*(FepFaoF^rFaoFcrFaoFeo F_zFeo*(FarFaoF[rFaoF0FaoFaoFgyFao*(FcrFaoFgnFaoFjnFaoFeo*&FepFaoF]\\l FaoFeo*&)F^rFfyFaoFjxFaoFao*&Fb\\lFaoF0FaoFaoFfzFao*&)F[rFfyFaoFjxFaoF aoFaoF_rFeoFjzFeo/-F.6$FfqFdoF6/-F.6$FQF[rF6/-F.6$FioF[rF6/-F.6$FapF[r *(,HFayFeoFbyFeoFcyFeoFdyFaoFgyFao**F5FaoFarFaoFjxFaoFerFaoFaoFiyFaoFj yFeoF[zFeoF^zFeoF_zFaoF`zFaoFazFaoFbzFaoFdzFaoFezFaoFfzFaoFgzFaoFizFao FaoF_rFeoFjzFeo/-F.6$FepF[rFc[l/-F.6$FgnF[rFex/-F.6$FBF[rF6/-F.6$F0F[r Fc]l/-F.6$FfqF[rF6/-F.6$FQFjnF6/-F.6$FioFjnF6/-F.6$FapFjnFj]l/-F.6$Fep Fjn*(,HFizFeoFd^lFeoFiyFeo**F5FaoFgrFaoFjxFaoFcrFaoFeoFazFaoFf^lFeoFby FeoFcyFeoFg^lFaoFh^lFaoF_zFaoFi^lFeoFgyFeoFj^lFaoF[_lFaoF\\_lFeoF^_lFe oFfzFeoF__lFeoFaoF_rFeoFjzFeo/-F.6$FBFjnF6/-F.6$F0FjnFex/-F.6$FfqFjnF6 Q)pprint206\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "idcsys:= i nitialdata(rcsys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KThe~system~has ~been~identified~as~follows:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RThe ~system~has~the~following~dependent~variables:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%&alphaG%%betaG%$etaG%$phiG%$psiG%$tauG%&thetaG%#xiG" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%TThe~system~has~the~following~indepe ndent~variables:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%\"tG%\"yG%\"xG% \"zG%\"qG%\"rG%\"pG%\"sG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TThe~foll owing~are~to~be~treated~as~solve~variables:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*%&alphaG%%betaG%$etaG%$phiG%$psiG%$tauG%&thetaG%#xiG" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'idcsysGK%&TABLEG6#7$/%)InfiniteG7 \"/%'FiniteG71/-%&alphaG6*&%\"tG6#\"\"!&%\"yGF5&%\"xGF5&%\"zGF5&%\"qGF 5&%\"rGF5&%\"pGF5&%\"sGF5%$_C1G/--&%\"DG6#\"\"%6#F1F2%$_C2G/-%%betaGF2 %$_C3G/-%$etaGF2%$_C4G/-%$phiGF2%$_C5G/-%$psiGF2%$_C6G/--&FJ6#\"\")6#F gnF2%$_C7G/-%$tauGF2%$_C8G/--&FJ6#\"\"$6#FcoF2%$_C9G/--FIF[pF2%%_C10G/ -%&thetaGF2%%_C11G/--&FJ6#\"\"#6#FcpF2%%_C12G/--FhoF[qF2%%_C13G/--FIF[ qF2%%_C14G/-%#xiGF2%%_C15GQ)pprint216\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "Thus there are 15 parameters in the formal series soluti on so the cubic nonlinear Schrodinger system has a 15 parameter Lie gr oup of symmetries." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Series to d egree 0 is:" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 54 "fps_soln1: = rtaylor(rcsys[Solved], idcsys, order = 0);" }}}{EXCHG {PARA 202 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "fps_so ln2:= rtaylor(rcsys[Solved], idcsys, order = 1);" }}}{PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "You can che ck that the expansion to degree 2 is the same, thus the exact solution is given above, since the point about which the expansion was made is arbitrary." }}}}{EXCHG {PARA 202 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 71 " Use of rifsimp in PDE (and ODE) solving in Maple (Edgardo Cheb-Terra b)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[rifsimp]:= 2 :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 39 "Maple's pdsolve can be calle d simply by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pdsolve(csys );" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 84 "Nice we get to the solution without using rifsimp (and besides in a nicer form!)...." }}{PARA 0 " " 0 "" {TEXT -1 9 "BUT ....." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[rifsimp]:= 2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "pdsolve(csys);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "As you ca n see pdsolve makes repeated calls to rifsimp in its solving routines! " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Also calls are made to rifsim p in ode solving." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 53 " Try to solve the Wolf Conservation System Challenge!" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "read \"TutSystems.txt\":" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%hoThis~file~contains~several~systems: ~~Wolf's~challenge~system~WolfConsSys~fromG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%bpconservation~laws~-~a~system~of~148~PDE~in~2~depende nt~variables~qne~10~indep~variablesG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%\\pThis~file~also~contains~the~classical~symmetry~defining~system~o f~1888~PDE~-~csysG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Yfor~symmetries ~of~the~Cubic~Nonlinear~Schrodinger~SystemG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "nops(WolfConsSys);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$[\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "declare(Wol fConsSys, quiet);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infole vel[rifsimp]:= 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "rWolfC onSys:= rifsimp(WolfConsSys, casesplit):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "caseplo t(rWolfConSys, [Q1, Q2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,Case~1: ~3-dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,Case~2:~5-dG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%,Case~3:~4-dG" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%,Case~4:~1-dG" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6:-%'CURVESG6$7$7$$!3+++++++]()!#=$!\"\"\"\"!7$$!3+++++++D;!#<$!\"# F--%'COLOURG6&%$RGBGF-F-F--F$6$7$F.7$$!3+++++++D@F1$!\"$F-F4-F$6$7$F.7 $$!3+++++++D6F1F>F4-F$6$7$F'7$$!3+++++++]7F*F2F4-F$6$7$7$$F-F-FP7$$\"3 +++++++]()F*F+F4-F$6$7$FOF'F4-%%TEXTG6%7$$!++++]9!\"*$!++++]8FgnQ#<>6 \"F4-FX6%7$$!+)z1')4#Fgn$!+,m>)Q#FgnFjnF4-FX6%7$$!++++D@Fgn$!#KF,Q\"1F [oF4-FX6%7$Ffo$!$X$F3Q$3-dF[oF4-FX6%7$$!+-KR^6FgnFaoQ\"=F[oF4-FX6%7$$! ++++D6FgnFhoQ\"2F[oF4-FX6%7$FjpF^pQ$5-dF[oF4-FX6%7$$!+++++I!#5FhnFfpF4 -FX6%7$$!++++]7Ffq$!#AF,Q\"3F[oF4-FX6%7$Fjq$!$X#F3Q$4-dF[oF4-FX6%7$$\" +Q " 0 "" {MPLTEXT 1 0 22 "print(rWolfConSys[1]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7%/%%CaseG7%7$0%\"bG\"\"!&% #Q1G6$%#m2G%$m22G7$0,&F,\"\"\"\"\"#!\"\"F-&%#Q2G6#%#u1G7$0,&\"\"$F8F,F 6F-&F:6#%\"uG/%'SolvedG76/&F/6$F2F2F-/&F/6#%\"tGF-/&F:FLF-/&F/6#%\"xGF -/&F:FRF-/&F/6#%\"mG**,6*.F7F6F,F6FYF6&F/6#F2F6F2F6%#u2GF6F8*()F1F7F6F gnF6FinF6F6**F,F6FYF6F/F6FinF6F6**)F,F7F6)FYF7F6FgnF6FinF6F8**F^oF6F_o F6F1F6FgnF6F6*(F,F6FYF6F:F6F8*,F@F6F[oF6F,F6FgnF6FinF6F6*,F^oF6FYF6Fgn F6F2F6FinF6F8**F^oF6FYF6F/F6FinF6F8*,F7F6FgnF6F[oF6F^oF6FinF6F6F6FinF8 F,!\"#FYFfo/&F:FX,$*(FgnF6F1F6,&F,F6F6F6F6F8/&F/FBF-/FAF-/&F/6#F1,$*,F gnF6F1F6,&F6F6*&F7F6F,F6F6F6F,F8FYF8F8/&F:Fap*(FgnF6F,F6FYF6/&F/F;F-/F 9F-/&F/6#FinF-/&F:F^q*&,&**F1F6F,F6FYF6FgnF6F8F:F6F6FinF8/&F:FhnF-/&F/ 6#%$u11GF-/&F:FhqF-/&F/6#%$u12GF-/&F:F^rF-/%'PivotsG7%F+F4F>Q)pprint14 6\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "initialdata(rWolfCon Sys[1]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7%/%'FiniteG7&/ -%#Q1G6,&%\"tG6#\"\"!&%\"xGF0&%\"mGF0&%\"uGF0&%#m2GF0&%#u1GF0&%#u2GF0& %$m22GF0&%$u11GF0&%$u12GF0%$_C1G/--&%\"DG6#\"\")6#F,F-%$_C2G/-%#Q2GF-% $_C3G/%\"bG%$_C4G/%'PivotsG&&%,rWolfConSysG6#\"\"\"6#FV/%)InfiniteG7\" Q)pprint156\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "pdsolve(rW olfConSys[1][Solved]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<$/%#Q2G,$*( ,**2\"\"#\"\"\",&%\"bGF+F+F+F+)%#m2GF*F+%$_C1GF+,&#F+F*F+F-F+F+%\"mGF+ %#u2GF+)F3*&,&F+!\"\"*&\"\"$F+F-F+F8F+F-F8F+F+*,F4F+F3F+)F-F*F+F0F+)F3 *&,&F+F8F-F8F+F-F8F+F8*(F0F+,(*&,&*&,&*&F*F+F.F+F8*$)F3F*F+F+F+F4F+F+* &FHF+F/F+F8F+F " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "print(rWolfConSys[2]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7$/%%CaseG7%7$0%\"bG\"\"!&% #Q1G6$%#m2G%$m22G7$0,&F,\"\"\"\"\"#!\"\"F-&%#Q2G6#%#u1G7$/,&\"\"$F8F,F 6F-&F:6#%\"uG/%'SolvedG78/&F:6$%\"xGFJ*&,0**\"\"%F6&F:6#%#u2GF6FQF6)FC F7F6F8*,\"#7F6F1F6%\"mGF6&F/6#F2F6FRF6F8*(FNF6F:F6FRF6F6*,FTF6)FQF7F6F 1F6FUF6FVF6F6*(FNF6)FQF@F6FOF6F6*(FNF6FZF6F:F6F8**F7F6FQF6&F:6#FJF6FCF 6F8F6FC!\"#/&F:6$FQFJ*&,(**\"\"'F6F1F6FUF6FVF6F6*(F7F6FOF6FQF6F6*&F7F6 F:F6F8F6FCF8/&F:6$FQFQF-/&F/6$F2F2F-/&F/6#%\"tGF-/&F:F]pF-/&F/Fjn*&,(F :F6*&FOF6FQF6F8**F@F6F1F6FUF6FVF6F8F6FCF8/&F/6#FU,$*&#F6\"\"*F6*&,,** \"#:F6FUF6FVF6F2F6F8*(\"#GF6)F1F7F6FVF6F6*(FboF6FUF6F/F6F8*(F]qF6)FUF7 F6FVF6F8*(F@F6FUF6FOF6F8F6FUF[oF6F6/&F:Fip,$*(FNF6FVF6F1F6F8/&F/FBF-/F AFcp/&F/6#F1,$*&#\"\"(F@F6*(FVF6F1F6FUF8F6F8/&F:Fbr,$*(F@F6FVF6FUF6F6/ &F/F;F-/F9F-/&F/FPF-/&F:FWF-/&F/6#%$u11GF-/&F:FesF-/&F/6#%$u12GF-/&F:F [tF-/F,F@Q)pprint166\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "i nitialdata(rWolfConSys[2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABL EG6#7$/%'FiniteG7'/-%#Q1G6,&%\"tG6#\"\"!&%\"xGF0&%\"mGF0&%\"uGF0&%#m2G F0&%#u1GF0&%#u2GF0&%$m22GF0&%$u11GF0&%$u12GF0%$_C1G/--&%\"DG6#\"\")6#F ,F-%$_C2G/-%#Q2GF-%$_C3G/--&FI6#\"\"#6#FPF-%$_C4G/--&FI6#\"\"(FXF-%$_C 5G/%)InfiniteG7\"Q)pprint176\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel[rifsimp]:= 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "pdsolve(subs(b=3, rWolfConSys[2][Solved]));" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#<$/%#Q1G,0*(%$_C1G\"\"\"%\"mG#!\"(\"\"$%$m22GF)F)*,\" \"(F)\"\"'!\"\"F(F)F*#!#5F-%#m2G\"\"#F2**F-F)F6F2%$_C5GF)F*#!\"#F-F2** F-F)F6F2F(F)F*#!\"%F-F)%$_C2GF)*&%$_C3GF)-%$expG6#,$*&F6F)%\"xGF)F2F)F )*&%$_C4GF)-FB6#,$*&F6F)FFF)F)F)F)/%#Q2G*&,**,F6F),&%\"uGF)%#u2GF)F)F@ F))F*#\"#;F-F)FAF)F2*,F6F)FHF),&FTF2FSF)F)FUF)FIF)F)**F6F)FTF)FUF)F>F) F2**F-F)F(F)F5F))F*\"\"%F)F)F)F*#!#;F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "print(rWolfConSys[3]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7$/%%CaseG7$7$0%\"bG\"\"!&% #Q1G6$%#m2G%$m22G7$/,&F,\"\"\"\"\"#!\"\"F-&%#Q2G6#%#u1G/%'SolvedG77/&F :6$%#u2GFCF-/&F/6$F2F2F-/&F/6#%\"tGF-/&F:FIF-/&F/6#%\"xGF-/&F:FOF-/&F/ 6#%\"mG,$*&#F6\"\"%F6*(,2*,FZF6)FVF7F6%\"uGF6F1F6&F/6#F2F6F8*,F7F6FVF6 FinF6&F:6#FCF6FCF6F8**F7F6FVF6FinF6F:F6F6*,\"\")F6FVF6FjnF6F2F6F " 0 "" {MPLTEXT 1 0 28 "initialdata (rWolfConSys[3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7$/%'F initeG7&/-%#Q1G6,&%\"tG6#\"\"!&%\"xGF0&%\"mGF0&%\"uGF0&%#m2GF0&%#u1GF0 &%#u2GF0&%$m22GF0&%$u11GF0&%$u12GF0%$_C1G/--&%\"DG6#\"\")6#F,F-%$_C2G/ -%#Q2GF-%$_C3G/--&FI6#\"\"(6#FPF-%$_C4G/%)InfiniteG7\"Q)pprint196\"" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "pdsolve(subs(b=2, rWolfCon Sys[3][Solved]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/%#Q2G,$*(\"\"& !\"\",&*&,&*(F(\"\"\"%#u2GF.%$_C4GF.F)*(F(F.%#u1GF.%$_C1GF.F.F.)%\"mG# \"#6\"\"#F.F.**\"\")F.%#m2GF.%$_C2GF.)F5\"\"%F.F)F.F5#!#6F8F./%#Q1G*&F 5#!\"(F8,,*&,&*&F3F.%\"uGF.F.F0F.F.)F5#\"\"(F8F.F.*(F8F.)F5\"\"$F.%$_C 3GF.F)**F>F.F(F)F5F8FF.F(F)F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 22 "print(rWolfConSys[4]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7$/%%CaseG7#7$/%\"bG\"\"!&%#Q1G6$%#m2G%$m22G /%'SolvedG7./&%#Q2G6#%\"tGF-/&F86#%\"xGF-/&F86#%\"mGF-/&F86#%\"uGF-/&F 86#F1F-/&F86#%#u1GF-/&F86#%#u2G*&F8\"\"\"FQ!\"\"/&F86#F2F-/&F86#%$u11G F-/&F86#%$u12GF-/F/FRF+Q)pprint206\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "initialdata(rWolfConSys[4]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7$/%'FiniteG7#/-%#Q2G6,&%\"tG6#\"\"!&%\"xGF0 &%\"mGF0&%\"uGF0&%#m2GF0&%#u1GF0&%#u2GF0&%$m22GF0&%$u11GF0&%$u12GF0%$_ C1G/%)InfiniteG7\"Q)pprint216\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "pdsolve(subs(b=0, rWolfConSys[4][Solved]));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#<$/%#Q1G%$_C1G/%#Q2G*&F&\"\"\"%#u2GF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 39 " Solve W olf's 15 PDE Challenge Problem" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "with(DEtools): with(PDEtools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2613 "WolfSys15 := [diff(G(phi),phi) = 2*a*G(phi), X1(u,v ,f,phi,chi)-diff(X1(u,v,f,phi,chi\n),f)*f = 0, diff(X2(u,v,f,phi,chi), f)-u*diff(X4(u,v,f,phi,chi),f)+v*diff(X5(u,\nv,f,phi,chi),f)+diff(X1(u ,v,f,phi,chi),chi) = 0, 4*diff(X3(u,v,f,phi,chi),f)*f\n^2+diff(X1(u,v, f,phi,chi),phi) = 0, u*diff(X2(u,v,f,phi,chi),f)-diff(X4(u,v,f,\nphi,c hi),f)*u^2+2*diff(X4(u,v,f,phi,chi),f)/G(phi)*f+u*v*diff(X5(u,v,f,phi, chi\n),f)-diff(X1(u,v,f,phi,chi),v) = 0, v*diff(X2(u,v,f,phi,chi),f)-u *v*diff(X4(u,\nv,f,phi,chi),f)+diff(X5(u,v,f,phi,chi),f)*v^2-2*diff(X5 (u,v,f,phi,chi),f)*G(\nphi)*f+diff(X1(u,v,f,phi,chi),u) = 0, f*diff(X2 (u,v,f,phi,chi),chi)-f*u*diff(\nX4(u,v,f,phi,chi),chi)+f*v*diff(X5(u,v ,f,phi,chi),chi)-X1(u,v,f,phi,chi) = 0,\n4*diff(X3(u,v,f,phi,chi),chi) *f^2+diff(X2(u,v,f,phi,chi),phi)-u*diff(X4(u,v,f,\nphi,chi),phi)+v*dif f(X5(u,v,f,phi,chi),phi) = 0, f*u*diff(X2(u,v,f,phi,chi),\nchi)-f*diff (X4(u,v,f,phi,chi),chi)*u^2+2*f^2*diff(X4(u,v,f,phi,chi),chi)/G(phi\n) +f*u*v*diff(X5(u,v,f,phi,chi),chi)-2*X1(u,v,f,phi,chi)*u+X5(u,v,f,phi, chi)*f-\nf*diff(X2(u,v,f,phi,chi),v)+f*u*diff(X4(u,v,f,phi,chi),v)-f*v *diff(X5(u,v,f,\nphi,chi),v) = 0, f*v*diff(X2(u,v,f,phi,chi),chi)-f*u* v*diff(X4(u,v,f,phi,chi),\nchi)+f*diff(X5(u,v,f,phi,chi),chi)*v^2-2*f^ 2*diff(X5(u,v,f,phi,chi),chi)*G(phi\n)-2*X1(u,v,f,phi,chi)*v+X4(u,v,f, phi,chi)*f+f*diff(X2(u,v,f,phi,chi),u)-f*u*\ndiff(X4(u,v,f,phi,chi),u) +f*v*diff(X5(u,v,f,phi,chi),u) = 0, diff(X3(u,v,f,phi\n,chi),phi) = 0, u*diff(X2(u,v,f,phi,chi),phi)-diff(X4(u,v,f,phi,chi),phi)*u^2+\n2*dif f(X4(u,v,f,phi,chi),phi)/G(phi)*f+u*v*diff(X5(u,v,f,phi,chi),phi)-4*di ff(\nX3(u,v,f,phi,chi),v)*f^2 = 0, v*diff(X2(u,v,f,phi,chi),phi)-u*v*d iff(X4(u,v,f,\nphi,chi),phi)+diff(X5(u,v,f,phi,chi),phi)*v^2-2*diff(X5 (u,v,f,phi,chi),phi)*G(\nphi)*f+4*diff(X3(u,v,f,phi,chi),u)*f^2 = 0, - f*u*diff(X2(u,v,f,phi,chi),v)+f*\ndiff(X4(u,v,f,phi,chi),v)*u^2-2*f^2* diff(X4(u,v,f,phi,chi),v)/G(phi)+f*u*X5(u,\nv,f,phi,chi)-f*u*v*diff(X5 (u,v,f,phi,chi),v)+X1(u,v,f,phi,chi)*f/G(phi)-X1(u,v\n,f,phi,chi)*u^2+ 2*X3(u,v,f,phi,chi)/G(phi)*a*f^2 = 0, f*v*diff(X2(u,v,f,phi,\nchi),v)- f*u*X4(u,v,f,phi,chi)-f*u*v*diff(X4(u,v,f,phi,chi),v)+f*diff(X5(u,v,f, \nphi,chi),v)*v^2-f*v*X5(u,v,f,phi,chi)-2*f^2*G(phi)*diff(X5(u,v,f,phi ,chi),v)+2\n*X1(u,v,f,phi,chi)*u*v-f*u*diff(X2(u,v,f,phi,chi),u)+f*dif f(X4(u,v,f,phi,chi),\nu)*u^2-2*f^2/G(phi)*diff(X4(u,v,f,phi,chi),u)-f* u*v*diff(X5(u,v,f,phi,chi),u)\n= 0, -f*v*diff(X2(u,v,f,phi,chi),u)-f*v *X4(u,v,f,phi,chi)+f*u*v*diff(X4(u,v,f,\nphi,chi),u)-f*diff(X5(u,v,f,p hi,chi),u)*v^2+2*f^2*diff(X5(u,v,f,phi,chi),u)*G(\nphi)-X1(u,v,f,phi,c hi)*G(phi)*f+X1(u,v,f,phi,chi)*v^2+2*X3(u,v,f,phi,chi)*a*f^\n2*G(phi) \+ = 0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "declare(WolfSys15) : WolfSys15;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"GG6#%$phiG\"\" \"%9will~now~be~displayed~asGF(F%F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*(-%#X1G6'%\"uG%\"vG%\"fG%$phiG%$chiG\"\"\"%9will~now~be~displayed~as GF,F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#X2G6'%\"uG%\"vG%\"fG%$ phiG%$chiG\"\"\"%9will~now~be~displayed~asGF,F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#X4G6'%\"uG%\"vG%\"fG%$phiG%$chiG\"\"\"%9will~now~b e~displayed~asGF,F%F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#X5G6'%\" uG%\"vG%\"fG%$phiG%$chiG\"\"\"%9will~now~be~displayed~asGF,F%F," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#X3G6'%\"uG%\"vG%\"fG%$phiG%$chiG \"\"\"%9will~now~be~displayed~asGF,F%F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#72/&%\"GG6#%$phiG,$*(\"\"#\"\"\"%\"aGF,F&F,F,/,&%#X1GF,*&&F06#% \"fGF,F4F,!\"\"\"\"!/,*&%#X2GF3F,*&%\"uGF,&%#X4GF3F,F5*&%\"vGF,&%#X5GF 3F,F,&F06#%$chiGF,F6/,&*(\"\"%F,&%#X3GF3F,)F4F+F,F,&F0F'F,F6/,,*&FFDF,F5*(F4F,F@F,&FBFDF,F,F0F5F6/,** (FIF,&FKFDF,FLF,F,&F:F'F,*&FF'F,F5*&F@F,&FBF'F,F,F6/,4*(F4F,FFVF,F,*(F4F,F@F,&F BFVF,F5F6/,4*(F4F,F@F,F]oF,F,**F4F,FF,F4F,F,*&F4F,&F:FinF,F,*(F4F,F FinF,F5*(F4F,F@F,&FBFinF,F,F6/&FKF'F6/,,*&FF,F5**F4F,FF,F5**F4F,F " 0 "" {MPLTEXT 1 0 43 "rWolfSys15:= rifsimp(WolfSys15, casesplit):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "caseplot(rWolfSys15, [X1,X2,X3,X4,X5]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,Case~1:~5-dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,Case~2:~8-dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,Cas e~3:~9-dG" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "64-%' CURVESG6$7$7$$\"\"!F)F(7$$!3++++++++v!#=$!\"\"F)-%'COLOURG6&%$RGBGF)F) F)-F$6$7$F'7$$\"3++++++++vF-F.F0-F$6$7$F77$$\"3++++++++DF-$!\"#F)F0-F$ 6$7$F77$$\"3+++++++]7!#6\"F0 -FJ6%7$$!+++++vFO$!#7F/Q\"1FSF0-FJ6%7$FW$!$X\"FAQ$5-dFSF0-FJ6%7$$\"+++ +]dFOFPQ\"=FSF0-FJ6%7$$\"+@?$Rw#FO$!+,m>)Q\"!\"*FRF0-FJ6%7$$\"+++++DFO $!#AF/Q\"2FSF0-FJ6%7$F]p$!$X#FAQ$8-dFSF0-FJ6%7$$\"+)z1OA\"FioFgoFaoF0- FJ6%7$$\"++++]7FioF_pQ\"3FSF0-FJ6%7$F`qFepQ$9-dFSF0-%&TITLEG6#%.Rif~Ca se~TreeG-%+AXESLABELSG6$Q!FSF^r-%*AXESSTYLEG6#%%NONEG-%%VIEWG6$%(DEFAU LTGFfr" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "print(rWolfSys15[1]);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7%/%%CaseG7#7$0*&%\"aG\"\" \",&\"\"$!\"\"*$)F-\"\"#F.F.F.\"\"!&%#X4G6#%$chiG/%'SolvedG7 " 0 "" {MPLTEXT 1 0 27 "initialdata(rWolfSys15[1]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7%/%'FiniteG7)/-%\"GG6#&%$phiG6#\"\"!%$_C1G/ -%#X1G6'&%\"uGF0&%\"vGF0&%\"fGF0F.&%$chiGF0%$_C2G/-%#X2GF6%$_C3G/-%#X3 GF6%$_C4G/-%#X4GF6%$_C5G/-%#X5GF6%$_C6G/%\"aG%$_C7G/%'PivotsG&&%+rWolf Sys15G6#\"\"\"6#FT/%)InfiniteG7\"Q(pprint16\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "pdsolve(rWolfSys15[1][Solved]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(/%#X4G,&*(\"\"#!\"\",&%$_C2G\"\"\"*(F(F,%$_C1GF, %\"aGF,F,F,%\"vGF,F,%$_C3GF,/%#X3GF./%#X5G,&*(F(F),&F+F,*(F(F,F.F,F/F, F)F,%\"uGF,F,%$_C4GF,/%\"GG*&%$_C6GF,-%$expG6#,$*(F(F,F/F,%$phiGF,F,F, /%#X1G*&F+F,%\"fGF,/%#X2G,**&F;F,F0F,F,*&F+F,%$chiGF,F,*&F1F,F:F,F)%$_ C5GF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "print(rWolfSys15[2 ]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7&/%%CaseG7$7$/*&%\" aG\"\"\",&\"\"$!\"\"*$)F-\"\"#F.F.F.\"\"!&%#X4G6#%$chiG7$0F-F5&%#X5G6# %\"vG/%'SolvedG7,&*&FNF.FQF.F1*(F4F.FgnF.FSF.F./&FVF>,$*&#F.F4F.*(,0*(FSF.F?F .FQF.F1*(FEF.FNF.F]oF.F.*(F[oF.F\\oF.F6F.F1*.F4F.FSF.F?F.FNF.FgnF.F]oF .F1**F4F.F=F.FSF.F]oF.F1*.F4F.FNF.F]oF.F_oF.F-F.FSF.F1*,F4F.)FNF4F.F]o F.FSF.F6F.F.F.FSF1F]oF1F.F1/&F_oF>*&FgnF.F-F./&F7F>,$*&#F.F4F.*&,(FEF1 **F4F.F_oF.F-F.FSF.F1**F4F.FSF.FNF.F6F.F.F.FSF1F.F1/F<,$*&#F.F4F.*&,&* *F4F.FNF.FgnF.F]oF.F.FQF.F.F]oF1F.F1/&FE6#FS*&FEF.FSF1/&FVF_s,$*&,(*(F NF.)F]oF4F.FgnF.F1F\\pF1F[pF.F.F]oF1F1/&F_oF_sF5/&F7F_s*&FgnF.F]oF./&F =F_s,$*&F6F.F]oF1F1/&F]o6#%$phiG,$*(F4F.F-F.F]oF.F./&FEFctF5/&FVFct,$* ,F4F.F-F.FSF.,&FfsF1F\\pF.F.F]oF1F1/&F_oFctF5/&F7Fct,$*,F4F.FgnF.F-F.F ]oF.FSF.F./&F=Fct,$*,F4F.F-F.FSF.F6F.F]oF1F./&FVF8,$*&,(*(FSF.FNF.F6F. F1*(FSF.F?F.FgnF.F.FEF1F.FSF1F1/&F_oF8F5/%+ConstraintG7#/F/F5/%'Pivots G7$0F]oF5F;Q(pprint26\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " initialdata(rWolfSys15[2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABL EG6#7&/%'FiniteG7,/-%\"GG6#&%$phiG6#\"\"!%$_C1G/-%#X1G6'&%\"uGF0&%\"vG F0&%\"fGF0F.&%$chiGF0%$_C2G/--&%\"DG6#\"\"&6#F5F6%$_C3G/-%#X2GF6%$_C4G /-%#X3GF6%$_C5G/-%#X4GF6%$_C6G/--FC6#FSF6%$_C7G/-%#X5GF6%$_C8G/--FC6#F fnF6%$_C9G/%\"aG%%_C10G/%+ConstraintG&&%+rWolfSys15G6#\"\"#6#Fao/%'Piv otsG&Fco6#Fio/%)InfiniteG7\"Q(pprint36\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "pdsolve(rWolfSys15[2][Solved]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(/%#X1G*&%$_C1G\"\"\"%\"fGF(/%\"GG*&%$_C3GF(-%$expG6#, $*(\"\"#F(%\"aGF(%$phiGF(F(F(/%#X3G%$_C2G/%#X4G,&*(F3!\"\",&F'F(*(F3F( F8F(F4F(F(F(%\"vGF(F(%$_C4GF(/%#X2G,**&%$_C5GF(F@F(F(*&F'F(%$chiGF(F(* &FAF(%\"uGF(F=%$_C6GF(/%#X5G,&*(F3F=,&F'F(*(F3F(F8F(F4F(F=F(FJF(F(FFF( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "print(rWolfSys15[3]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7%/%%CaseG7$7$/*&%\"aG\" \"\",&\"\"$!\"\"*$)F-\"\"#F.F.F.\"\"!&%#X4G6#%$chiG7$/F-F5&%#X5G6#%\"v G/%'SolvedG7>/&F=6$F?F?,$*(F0F.F6F.%\"GG!\"#F./&F=6$F9F?F5/&%#X1G6$F9F 9F5/&F7FPF5/&F=FPF5/&FO6#%\"uG,&*&F?F.&FOF8F.F.*(F4F.F6F.%\"fGF.F1/&%# X2GFW,$*&#F.F4F.*&,0*&FOF.F?F.F.**\"\"%F.)FgnF4F.&F=F8F.FHF.F.**F4F.Fg nF.FdoF.)F?F4F.F1*(F4F.F7F.FgnF.F1*,F4F.FgnF.FXF.)FHF4F.F,&*&FXF.FenF.F1*(F4F.FdoF.Fg nF.F./&FjnF>,$*&F]oF.*(,,*(FOF.FXF.FHF.F1*(FboF.FcoF.F6F.F.*,F4F.FgnF. F?F.FF5/&F7F>,$*&F]oF.*&,&FOF.**F4F.FgnF.FXF.F6F.F1F.FgnF1F.F ./&FO6#Fgn*&FOF.FgnF1/&FjnFhr*&,(FepF.FdpF.*&FHF.FenF.F1F.FHF1/&F_pFhr F5/&F7Fhr*&FdoF.FHF./&F=Fhr,$*&F6F.FHF1F1/&FH6#%$phiGF5/&FOFjsF5/&FjnF jsF5/&F_pFjsF5/&F7FjsF5/&F=FjsF5/&FjnF8*&,(*(FgnF.FXF.F6F.F.*(FgnF.F?F .FdoF.F1FOF.F.FgnF1/&F_pF8F5F;/%'PivotsG7#0FHF5Q(pprint46\"" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "initialdata(rWolfSys15[3]); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#K%&TABLEG6#7%/%'FiniteG7,/-%\"GG6# &%$phiG6#\"\"!%$_C1G/-%#X1G6'&%\"uGF0&%\"vGF0&%\"fGF0F.&%$chiGF0%$_C2G /--&%\"DG6#\"\"&6#F5F6%$_C3G/-%#X2GF6%$_C4G/-%#X3GF6%$_C5G/-%#X4GF6%$_ C6G/--FC6#FSF6%$_C7G/-%#X5GF6%$_C8G/--&FD6#\"\"#6#FfnF6%$_C9G/--FCF^oF 6%%_C10G/%'PivotsG&&%+rWolfSys15G6#\"\"$6#Feo/%)InfiniteG7\"Q(pprint56 \"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pdsolve(subs(a = 0, r WolfSys15[3][Solved]));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6'<(/%#X5G,&* (\"\"#!\"\"%$_C2G\"\"\"%\"uGF+F+%$_C3GF+/%#X3G%$_C1G/%#X4G,&*(F(F)F*F+ %\"vGF+F+%$_C4GF+/%\"GG%$_C6G/%#X1G*&F*F+%\"fGF+/%#X2G,**&F-F+F5F+F+*& F*F+%$chiGF+F+*&F6F+F,F+F)%$_C5GF+<(F.F:/F%,(*(F(F)F*F+F,F+F+FAF+F6F+/ F8,$*&F-F),$*&F-F+FEF+F)#F+F(F)/F?,**&F6F+F5F+F+FBF+*&F9F+F,F+F)%$_C7G F+/F2,(*(F(F)F*F+F5F+F+*&FEF+F,F+F+F9F+<(F.F:FGFPFU/F8FL<(F./F;*&,&*&F *F+F5F+F+F-F+F+F=F+/F2,,*(\"\"%F)F*F+F5F(F+*(F(F)F-F+F5F+F+*&FTF+F=F+F +*,\"\"$F+F,F+FTF(,&*&F*F+F,F+F+*(F]oF+FaoF)F6F+F+F+F*!\"#F)F9F+/F?,0* (F,F+FTF+F=F+F+*&,&*(F(F)F*F+F5F+F+F-F+F+FCF+F+**F]oF)F*F+F,F+F5F(F)*& FEF+F5F+F+FSF)*(F*F)FTF(F,FaoF)%$_C8GF+/F8,$*(F(F+FTF+F*F)F+/F%,**(F(F ),&FCF+*(F(F+F,F+F5F+F+F+F*F+F+FEF+*(F(F)F-F+F,F+F+FRF+<(F./F8,$*&F+F+ *&F(F+F_pF+F)F)/F?,$*(\"#KF),.**\"#;F+)F5FaoF+,&FAF+*&F(F+F*F+F)F+)F_p F]oF+F)*,FaqF+F=F+F5F+,&F*F)FAF+F+)F_pFaoF+F)*&,.*&,&*(\"\")F+FEF+F,F+ F)*(FarF+)F,F(F+F-F+F)F+)F5F(F+F+*&,(*(FdqF+FCF+FEF+F+*&FaqF+FTF+F+*(F arF+FcrF+F*F+F+F+F5F+F+*&,&*(FdqF+F*F+FCF+F+*&FaqF+%$_C9GF+F)F+F,F+F+* &,&*&FdqF+)F=F(F+F)*&FdqF+)FCF(F+F+F+F-F+F+*&FaqF+%%_C10GF+F+*(FaqF+F6 F+FCF+F+F+)F_pF(F+F+*,FarF+F,F+F=F+,&FEF+*&F-F+F,F+F+F+F_pF+F)*(F(F+FE F+)F,FaoF+F)*&)F,F]oF+F-F+F)F+F_pFeoF+/F%,2*(FhsF+F-F+FeqF+F+**FaoF+Fh sF+F*F+FdrF+F)*(F]oF),**(F]oF+FEF+F,F+F+*&FcrF+F-F+F+*&F]oF+F9F+F+**F] oF+F_pF+F=F+F-F+F+F+F5F+F+*(F]oF)F,F(F*F+F+*(F]oF),&*&F(F+F6F+F+*(F(F+ F-F+FCF+F+F+F,F+F+*(F(F)FCF+FEF+F+FTF+*(F_pF+F=F+F*F+F)/F;*&,*FcoF+F]p F+*&F-F+FCF+F+F6F+F+F=F+/F2,$*(FdqF),(*&,,**F]oF+F5F+,&FAF+*&F]oF+F*F+ F)F+F,F+F)*(F]oF+FdrF+FEF+F+*&,&*(FarF+F-F+FCF+F+*&FarF+F6F+F+F+F5F+F+ *(FarF+F*F+FCF+F)*&FdqF+F^sF+F+F+FhsF+F+**F]oF+FjsF+F=F+F_pF+F)*&F,F+, (FgtF+*(FaoF+FEF+F,F+F+*&F]oF+F9F+F+F+F)F+F_pFeoF+" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 " Summary for Part I" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " " }{TEXT 267 87 "1. Constant Coeff linear homogeneous P DE in 1 dependent variables rifsimp is just a GB" }}{PARA 0 "" 0 "" {TEXT -1 316 "We have illustrated by example that for constant coeffic ient systems in 1 dependent variable rifsimp is simply a Groebner Basi s in a ring of differential operators (and more generally for a module , if there are more than 1 dependent variable). There are many other \+ (and better) packages for doing such computations." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " " }{TEXT 268 77 "2. Non-constant coeff linear homogeneous PDE is a Differential Grobner Basis" }}{PARA 0 "" 0 "" {TEXT -1 144 "For nonconstant coefficient linear PDE, rifsimp produces a Differential Groebner Basis. So in analogy with Groebner Bases the applications are:" }}{PARA 0 "" 0 "" {TEXT -1 40 " * Finding normal form of such systems" }}{PARA 0 "" 0 "" {TEXT -1 58 " * Testing the ir consistency and properties of solutions" }}{PARA 0 "" 0 "" {TEXT -1 98 " * Rendering them in a form suitable for subsequent applicati on of exact or numerical techniques" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 96 "Software for producing such Differential Bases for linear systems is becoming reasonably common." }}{PARA 0 " " 0 "" {TEXT -1 99 "The spoly's in such approaches correspond to integ rability conditions (or differential s-poly's). " }}{PARA 0 "" 0 "" {TEXT -1 102 "The Hilbert Functions which for polynomial ideals gives \+ the number of roots in the differential case, " }}{PARA 0 "" 0 "" {TEXT -1 64 "gives the number of parameters in formal power series sol utions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT 269 108 " 3. Showed the basic application to computing initial data and corresponding formal power series solutions" }{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 270 75 " 4. Illustrated on some ove rdetermined PDE systems from Symmetry Analysis" }{TEXT -1 104 ". This can be regarded as a success story in that despite their size they ar e amenable to these methods" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " \+ " }{TEXT 271 81 "5. Open problem: why are overdetermined systems for symmetries often tractable?" }}{PARA 0 "" 0 "" {TEXT -1 122 "An aethe stic reason is that these are the most beautiful overdetermined system s. But can a complexity argument be given?" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 2 " " }{TEXT 272 77 "6. Illustrated the tight integratio n of rifsimp into the PDE and ODE solvers" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 38 "Part II: Nonlinear Sy stems of PDE " }}}{EXCHG {PARA 204 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 200 "" 0 "" {TEXT -1 30 " Diverse History & Motivations" }} {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 207 11 "Anal ytic: " }{TEXT 206 145 "analytic pde and solns, analytic Existence an d Uniqueness (EU) theorems,\n(Cauchy-Kovalevskya (Kov 1875)), Tresse 1 894, Riquier 1910, Janet 1920)." }{TEXT 207 13 "\n\nAlgebraic: " } {TEXT 206 231 "polynomially nonlinear pde, Differential Rings, Fields \+ etc (Janet 1920, Ritt 50, Kolchin 50, Seidenberg 56, Rosenfeld 59, Buc hberger 65, Carra-Ferro 87, Ollivier 91, Mansfield 91, Wu 91, Boulier \+ et al 95, Hubert 97, Yufu-Gao 2003)." }{TEXT 207 14 "\n\nGeometric: \+ " }{TEXT 206 140 "Jet manifolds, etc (Lie 1890's, Cartan 1946, Ehresma n 50, Kuranishi 58, Spencer 60, Goldschmidt 69, Pommaret 78, Bryant et al 91, Seiler 94)" }}{PARA 204 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 200 "" 0 "" {TEXT -1 13 " Applications" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 206 43 "* Numerical Solution \+ of Constrained System" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 " " 0 "" {TEXT 206 12 "* Geometric" }}{PARA 201 "" 0 "" {TEXT -1 0 "" } }{PARA 201 "" 0 "" {TEXT 206 45 "* Commutative Algebra and Algebraic \+ Geometry" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 47 "* Simplification of Feynman Diagram s (Tarasov)" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 200 " " 0 "" {TEXT -1 50 " Rankings, Leading Linear & Leading Non-linear PDE " }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 208 11 "Definition:" }{TEXT 209 2 " \+ " }{TEXT 210 91 "a ranking < is a total order of the set of derivativ es satisfying for all derivatives u, v:" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 210 29 "* u < v => D u < D v" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 210 13 "* \+ u < D u" }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 208 8 "Example:" }{TEXT 210 52 " For 1 independent var there is only one ranking: " }{XPPEDIT 19 1 "u" "6#%\"uG" }{TEXT 211 1 " " } {TEXT 210 3 " < " }{XPPEDIT 19 1 "u[x]" "6#&%\"uG6#%\"xG" }{TEXT 211 1 " " }{TEXT 210 3 " < " }{XPPEDIT 19 1 "u[x,x]" "6#&%\"uG6$%\"xGF&" } {TEXT 211 1 " " }{TEXT 210 2 "< " }{XPPEDIT 19 1 "u[x,x,x]" "6#&%\"uG6 %%\"xGF&F&" }{TEXT 211 1 " " }{TEXT 210 6 " < ..." }}{PARA 201 "" 0 " " {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 210 73 "Classification of al l rankings given in Rust (1998); Rust & Reid (1997)." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 208 11 "Definition:" } {TEXT 212 2 " " }{TEXT 210 66 "HD(F) := highest derivative in F with \+ respect to the given ranking" }}{PARA 201 "" 0 "" {TEXT 210 90 " \+ A DE is leading linear (nonlinear) if it is linear (nonlin ear) in HD(F)." }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{PARA 201 "" 0 "" {TEXT 210 8 "The DE " }{XPPEDIT 19 1 "F = u+u[x]^3" "6#/%\"FG,&%\"uG \"\"\"*$&F&6#%\"xG\"\"$F'" }{TEXT 213 1 " " }{TEXT 210 5 " has " } {XPPEDIT 19 1 "HD(F) = u[x]" "6#/-%#HDG6#%\"FG&%\"uG6#%\"xG" }{TEXT 213 1 " " }{TEXT 210 32 " and is leading nonlinear while " }{XPPEDIT 19 1 "G = x*u[x,x] + u[x]^2 + u^3;" "6#/%\"GG,(*&%\"xG\"\"\"&%\"uG6$F' F'F(F(*$&F*6#F'\"\"#F(*$F*\"\"$F(" }{TEXT 213 1 " " }{TEXT 210 19 " is leading linear." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 64 " Simplest E xample: a Clairaut Equation (Evelyne discussed this)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "ceqn:= [ diff(u(x),x)^2 + x*diff(u(x),x) \+ - u(x) = 0 ]: declare(ceqn,quiet): ceqn;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 14 "rifsimp(ceqn);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "But Evelyne obtained another case, to obtain all cases" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rifsimp(ceqn, casesplit);" }}}} {SECT 1 {PARA 206 "" 0 "" {TEXT -1 67 " Maximal Soln Structures - Pot ential Nonlinear Telegraph Equation " }{XPPEDIT 19 1 "v[x] = u[t], v[t ] = B(u)*u[x]+C(u)" "6$/&%\"vG6#%\"xG&%\"uG6#%\"tG/&F%6#F+,&*&-%\"BG6# F)\"\"\"&F)6#F'F4F4-%\"CG6#F)F4" }{TEXT 217 1 " " }}{PARA 201 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 202 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "The symmetry defining system is automatically generate d and is:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 975 "telpotsymsys: = \n[B(u)*diff(tau(x,t,u,v),v)-diff(xi(x,t,u,v),u) = 0, -diff(xi(x,t,u ,v),v)+diff(tau(x,t,u,v),u) = 0, diff(xi(x,t,u,v),v)-diff(tau(x,t,u,v) ,u) = 0, -diff(eta(x,t,u,v),v)*C(u)-diff(tau(x,t,u,v),x)*C(u)-diff(eta (x,t,u,v),t)+diff(phi(x,t,u,v),x) = 0, diff(tau(x,t,u,v),t)-diff(xi(x, t,u,v),x)-diff(eta(x,t,u,v),u)+diff(phi(x,t,u,v),v) = 0, -diff(eta(x,t ,u,v),v)*B(u)+diff(xi(x,t,u,v),t)-diff(tau(x,t,u,v),x)*B(u)-diff(tau(x ,t,u,v),u)*C(u)+diff(phi(x,t,u,v),u)+diff(xi(x,t,u,v),v)*C(u) = 0, -di ff(xi(x,t,u,v),v)*C(u)-diff(eta(x,t,u,v),v)*B(u)-diff(tau(x,t,u,v),u)* C(u)-diff(xi(x,t,u,v),t)+diff(phi(x,t,u,v),u)+diff(tau(x,t,u,v),x)*B(u ) = 0, -diff(tau(x,t,u,v),t)*B(u)+diff(phi(x,t,u,v),v)*B(u)-B(u)*diff( eta(x,t,u,v),u)-eta(x,t,u,v)*diff(B(u),u)-2*diff(tau(x,t,u,v),v)*B(u)* C(u)+B(u)*diff(xi(x,t,u,v),x) = 0, diff(phi(x,t,u,v),t)-diff(tau(x,t,u ,v),t)*C(u)-diff(tau(x,t,u,v),v)*C(u)^2-eta(x,t,u,v)*diff(C(u),u)+diff (phi(x,t,u,v),v)*C(u)-B(u)*diff(eta(x,t,u,v),x) = 0]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 29 "declare(telpotsymsys, quiet):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 13 "telpotsymsys;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7+/,&*&%\"BG\"\"\"&%$tauG6#%\"vGF(F(&%#xiG6#%\"uG!\"\"\"\"!/,&&F .F+F1&F*F/F(F2/,&F5F(F6F1F2/,**&&%$etaGF+F(%\"CGF(F1*&&F*6#%\"xGF(F>F( F1&F=6#%\"tGF1&%$phiGFAF(F2/,*&F*FDF(&F.FAF1&F=F/F1&FGF+F(F2/,.*&FF(F1&FGF/F(*&F5F(F>F(F(F2/,.FUF1FPF1FS F1FQF1FTF(FRF(F2/,.*&FJF(F'F(F1*&FMF(F'F(F(*&F'F(FLF(F1*&F=F(&F'F/F(F1 **\"\"#F(F)F(F'F(F>F(F1*&F'F(FKF(F(F2/,.&FGFDF(*&FJF(F>F(F1*&F)F()F>Fj nF(F1*&F=F(&F>F/F(F1*&FMF(F>F(F(*&F'F(&F=FAF(F1F2" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 76 "checkrank([xi(x,t,u,v),tau(x,t,u,v),eta (x,t,u,v),phi(x,t,u,v)], degree = 1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#76&%$etaG6#%\"xG&%$phiGF&&%$tauGF&&%#xiGF&&F%6#%\"tG&F)F/&F+F/&F-F/ &F%6#%\"uG&F)F5&F+F5&F-F5&F%6#%\"vG&F)F;&F+F;&F-F;F%F)F+F-" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 118 "rsys := maxdimsystems([op(telpot symsys),diff(B(u),u)<>0,diff(C(u),u)<>0], [xi,tau,eta,phi], mindim = 3 , output = rif):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%KThe~system~has~b een~identified~as~follows:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%RThe~s ystem~has~the~following~dependent~variables:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%#xiG%$tauG%$etaG%$phiG%\"BG%\"CG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%TThe~system~has~the~following~independent~variables:G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%\"xG%\"tG%\"uG%\"vG" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%TThe~following~are~to~be~treated~as~solve~varia bles:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6(%#xiG%$tauG%$etaG%$phiG%\"BG %\"CG" }}{PARA 6 "" 1 "" {TEXT -1 30 " Initial State of System:" }}{PARA 6 "" 1 "" {TEXT -1 62 " #OT: 0 #Sol: 0 #NL: 0 #UnC: 9 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 35 " **End iter ation #1 Time:.30e-1" }}{PARA 6 "" 1 "" {TEXT -1 62 " #OT: 0 # Sol: 3 #NL: 0 #UnC: 6 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #2 Time:.180" }}{PARA 6 "" 1 "" {TEXT -1 62 " #OT: 0 #Sol: 7 #NL: 0 #UnC: 4 #NonP: 0 #P iv: 4" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #3 Time:.13 0" }}{PARA 6 "" 1 "" {TEXT -1 62 " #OT: 0 #Sol: 9 #NL: 1 #U nC: 4 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End i teration #4 Time:.241" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 0 \+ #Sol: 11 #NL: 2 #UnC: 5 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #5 Time:.250" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 0 #Sol: 12 #NL: 3 #UnC: 6 #NonP: 0 # Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #6 Time:.6 81" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 0 #Sol: 14 #NL: 6 \+ #UnC: 6 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #7 Time:.120" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 \+ #Sol: 14 #NL: 12 #UnC: 0 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 " " {TEXT -1 35 "Split on case #1 at depth 1: [`<>`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 14 #NL: 10 #UnC: 2 #NonP: 0 \+ #Piv: 5" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #8 Time:. 660" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 14 #NL: 12 \+ #UnC: 5 #NonP: 0 #Piv: 5" }}{PARA 6 "" 1 "" {TEXT -1 33 " **E nd iteration #9 Time:.691" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: \+ 0 #Sol: 15 #NL: 16 #UnC: 3 #NonP: 0 #Piv: 5" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #10 Time:.160" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 15 #NL: 19 #UnC: 0 #NonP: 0 \+ #Piv: 5" }}{PARA 6 "" 1 "" {TEXT -1 41 "Split on case #1 at depth 2: [ `<>`, `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 15 \+ #NL: 18 #UnC: 1 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #11 Time:.411" }}{PARA 6 "" 1 "" {TEXT -1 64 " \+ #OT: 0 #Sol: 15 #NL: 19 #UnC: 2 #NonP: 0 #Piv: 6" }} {PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #12 Time:.130" }} {PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 15 #NL: 21 #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 47 "Split on case \+ #1 at depth 3: [`<>`, `<>`, `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " \+ #OT: 0 #Sol: 15 #NL: 19 #UnC: 2 #NonP: 0 #Piv: 7" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #13 Time:.220" }}{PARA 6 " " 1 "" {TEXT -1 64 " #OT: 0 #Sol: 16 #NL: 16 #UnC: 2 #Non P: 0 #Piv: 7" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #14 Time:.321" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 6 \+ #NL: 8 #UnC: 0 #NonP: 0 #Piv: 7" }}{PARA 6 "" 1 "" {TEXT -1 53 " Split on case #1 at depth 4: [`<>`, `<>`, `<>`, `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 6 #NL: 7 #UnC: 1 #NonP: 0 \+ #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #15 Tim e:.401" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 7 #NL: \+ 5 #UnC: 2 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 34 " * *End iteration #16 Time:.170" }}{PARA 6 "" 1 "" {TEXT -1 63 " #O T: 11 #Sol: 7 #NL: 7 #UnC: 0 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 59 "Split on case #1 at depth 5: [`<>`, `<>`, `<>`, `<>` , `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 11 #Sol: 7 #NL: 5 #UnC: 2 #NonP: 0 #Piv: 10" }}{PARA 6 "" 1 "" {TEXT -1 36 " \+ **End iteration #17 Time:.90e-1" }}{PARA 6 "" 1 "" {TEXT -1 64 " \+ #OT: 11 #Sol: 7 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 10" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #18 Time:.40e-1" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 13 #Sol: 0 #NL: 0 #UnC: 0 #N onP: 0 #Piv: 10" }}{PARA 6 "" 1 "" {TEXT -1 58 "Split on case #2 at \+ depth 5: [`<>`, `<>`, `<>`, `<>`, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 7 #NL: 5 #UnC: 3 #NonP: 0 #Piv: 8" }} {PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #17 Time:.130" }} {PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 8 #NL: 5 #UnC: \+ 0 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 64 "Split on case # 2 at depth 6: [`<>`, `<>`, `<>`, `<>`, `=`, `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 8 #NL: 4 #UnC: 1 #NonP: 0 # Piv: 9" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #18 Time:. 320" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 9 #NL: 1 \+ #UnC: 1 #NonP: 0 #Piv: 9" }}{PARA 6 "" 1 "" {TEXT -1 34 " **En d iteration #19 Time:.100" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: \+ 12 #Sol: 8 #NL: 2 #UnC: 0 #NonP: 0 #Piv: 9" }}{PARA 6 "" 1 " " {TEXT -1 70 "Split on case #2 at depth 7: [`<>`, `<>`, `<>`, `<>`, ` =`, `<>`, `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 12 #Sol: \+ 8 #NL: 0 #UnC: 2 #NonP: 0 #Piv: 10" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #20 Time:.61e-1" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 12 #Sol: 8 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 1 0" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #21 Time:.110" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 13 #Sol: 1 #NL: 0 #UnC : 0 #NonP: 0 #Piv: 10" }}{PARA 6 "" 1 "" {TEXT -1 69 "Split on cas e #3 at depth 7: [`<>`, `<>`, `<>`, `<>`, `=`, `<>`, `=`]" }}{PARA 6 " " 1 "" {TEXT -1 63 " #OT: 12 #Sol: 8 #NL: 0 #UnC: 3 #NonP : 0 #Piv: 9" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #20 \+ Time:.261" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 12 #Sol: 9 # NL: 0 #UnC: 0 #NonP: 0 #Piv: 9" }}{PARA 6 "" 1 "" {TEXT -1 63 "S plit on case #4 at depth 6: [`<>`, `<>`, `<>`, `<>`, `=`, `=`]" }} {PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 8 #NL: 4 #UnC: \+ 2 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End itera tion #18 Time:.240" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #S ol: 8 #NL: 0 #UnC: 2 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #19 Time:.90e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 12 #Sol: 6 #NL: 0 #UnC: 2 #NonP: 0 # Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #20 Time:. 41e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 5 #NL: 2 #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 69 "Split \+ on case #4 at depth 7: [`<>`, `<>`, `<>`, `<>`, `=`, `=`, `<>`]" }} {PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 5 #NL: 0 #UnC: \+ 2 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End itera tion #21 Time:.30e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 \+ #Sol: 5 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #22 Time:.30e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 1 #NL: 0 #UnC: 0 #NonP: 0 # Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 68 "Split on case #5 at depth 7: [` <>`, `<>`, `<>`, `<>`, `=`, `=`, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 5 #NL: 0 #UnC: 3 #NonP: 0 #Piv: 6" }} {PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #21 Time:.30e-1" }} {PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 6 #NL: 0 #UnC: \+ 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 52 "Split on case # 6 at depth 4: [`<>`, `<>`, `<>`, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 6 #NL: 7 #UnC: 2 #NonP: 0 #Piv: 7" }} {PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #15 Time:.291" }} {PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 7 #NL: 5 #UnC: \+ 3 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End itera tion #16 Time:.200" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #S ol: 8 #NL: 6 #UnC: 3 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #17 Time:.180" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 9 #NL: 6 #UnC: 4 #NonP: 0 # Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #18 Time:. 100" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 13 #Sol: 7 #NL: 10 \+ #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 58 "Split o n case #6 at depth 5: [`<>`, `<>`, `<>`, `=`, `<>`]" }}{PARA 6 "" 1 " " {TEXT -1 63 " #OT: 13 #Sol: 7 #NL: 4 #UnC: 6 #NonP: 0 \+ #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #19 Time :.110" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 6 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 36 " ** End iteration #20 Time:.30e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " # OT: 13 #Sol: 1 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 8" }}{PARA 6 " " 1 "" {TEXT -1 57 "Split on case #7 at depth 5: [`<>`, `<>`, `<>`, `= `, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 7 #NL: 4 #UnC: 7 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 36 " \+ **End iteration #19 Time:.61e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " \+ #OT: 13 #Sol: 8 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #20 Time:.40e-1" }}{PARA 6 " " 1 "" {TEXT -1 63 " #OT: 14 #Sol: 7 #NL: 0 #UnC: 0 #NonP : 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 46 "Split on case #8 at dept h 3: [`<>`, `<>`, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 \+ #Sol: 15 #NL: 19 #UnC: 3 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #13 Time:.390" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 16 #NL: 4 #UnC: 15 #NonP: 0 \+ #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #14 Time: .621" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 9 #NL: 1 \+ #UnC: 2 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 36 " **E nd iteration #15 Time:.71e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #O T: 13 #Sol: 7 #NL: 3 #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 52 "Split on case #8 at depth 4: [`<>`, `<>`, `=`, `<>`] " }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 7 #NL: 0 #U nC: 3 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End i teration #16 Time:.40e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 1 3 #Sol: 7 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 " " {TEXT -1 36 " **End iteration #17 Time:.30e-1" }}{PARA 6 "" 1 " " {TEXT -1 63 " #OT: 13 #Sol: 1 #NL: 0 #UnC: 0 #NonP: 0 \+ #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 51 "Split on case #9 at depth 4: \+ [`<>`, `<>`, `=`, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 \+ #Sol: 7 #NL: 0 #UnC: 4 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #16 Time:.50e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 13 #Sol: 8 #NL: 0 #UnC: 0 #NonP: 0 # Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 41 "Split on case #10 at depth 2: [ `<>`, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 15 # NL: 18 #UnC: 2 #NonP: 0 #Piv: 5" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #11 Time:.480" }}{PARA 6 "" 1 "" {TEXT -1 64 " \+ #OT: 0 #Sol: 16 #NL: 18 #UnC: 0 #NonP: 0 #Piv: 5" }} {PARA 6 "" 1 "" {TEXT -1 47 "Split on case #10 at depth 3: [`<>`, `=`, `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 16 #NL: \+ 7 #UnC: 11 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " \+ **End iteration #12 Time:.651" }}{PARA 6 "" 1 "" {TEXT -1 64 " # OT: 0 #Sol: 17 #NL: 6 #UnC: 17 #NonP: 0 #Piv: 6" }}{PARA 6 " " 1 "" {TEXT -1 34 " **End iteration #13 Time:.891" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 9 #NL: 3 #UnC: 7 #NonP: \+ 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #14 T ime:.160" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 11 #Sol: 9 #NL : 10 #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 53 "Sp lit on case #10 at depth 4: [`<>`, `=`, `<>`, `<>`]" }}{PARA 6 "" 1 " " {TEXT -1 63 " #OT: 11 #Sol: 9 #NL: 9 #UnC: 1 #NonP: 0 \+ #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #15 Time :.110" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 8 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 8" }}{PARA 6 "" 1 "" {TEXT -1 36 " ** End iteration #16 Time:.30e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " # OT: 13 #Sol: 1 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 8" }}{PARA 6 " " 1 "" {TEXT -1 52 "Split on case #11 at depth 4: [`<>`, `=`, `<>`, `= `]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 11 #Sol: 9 #NL: 9 \+ #UnC: 2 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #15 Time:.110" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 1 1 #Sol: 10 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 " " {TEXT -1 36 " **End iteration #16 Time:.41e-1" }}{PARA 6 "" 1 " " {TEXT -1 63 " #OT: 14 #Sol: 7 #NL: 0 #UnC: 0 #NonP: 0 \+ #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 46 "Split on case #12 at depth 3: [`<>`, `=`, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 16 #NL: 7 #UnC: 12 #NonP: 0 #Piv: 5" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #12 Time:.240" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 0 #Sol: 17 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 5" } }{PARA 6 "" 1 "" {TEXT -1 35 "Split on case #13 at depth 1: [`=`]" }} {PARA 6 "" 1 "" {TEXT -1 64 " #OT: 0 #Sol: 14 #NL: 10 #UnC: 3 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iter ation #8 Time:.321" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 0 #So l: 15 #NL: 0 #UnC: 4 #NonP: 0 #Piv: 3" }}{PARA 6 "" 1 "" {TEXT -1 33 " **End iteration #9 Time:.210" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 4 #Sol: 14 #NL: 1 #UnC: 7 #NonP: 0 # Piv: 3" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #10 Time:. 110" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 3 #Sol: 14 #NL: 3 \+ #UnC: 6 #NonP: 0 #Piv: 3" }}{PARA 6 "" 1 "" {TEXT -1 34 " **En d iteration #11 Time:.151" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: \+ 3 #Sol: 14 #NL: 8 #UnC: 4 #NonP: 0 #Piv: 3" }}{PARA 6 "" 1 " " {TEXT -1 34 " **End iteration #12 Time:.110" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 3 #Sol: 14 #NL: 12 #UnC: 0 #NonP: 0 \+ #Piv: 3" }}{PARA 6 "" 1 "" {TEXT -1 41 "Split on case #13 at depth 2: \+ [`=`, `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 3 #Sol: 14 \+ #NL: 3 #UnC: 9 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #13 Time:.100" }}{PARA 6 "" 1 "" {TEXT -1 63 " \+ #OT: 3 #Sol: 15 #NL: 3 #UnC: 8 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 34 " **End iteration #14 Time:.271" }}{PARA 6 " " 1 "" {TEXT -1 63 " #OT: 13 #Sol: 8 #NL: 1 #UnC: 2 #NonP : 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #15 \+ Time:.60e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 14 #Sol: 7 \+ #NL: 3 #UnC: 0 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 47 "Split on case #13 at depth 3: [`=`, `<>`, `<>`]" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 14 #Sol: 7 #NL: 0 #UnC: 3 #NonP: 0 # Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #16 Time:. 40e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 14 #Sol: 7 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 "" 1 "" {TEXT -1 36 " ** End iteration #17 Time:.30e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " # OT: 13 #Sol: 1 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 6" }}{PARA 6 " " 1 "" {TEXT -1 46 "Split on case #14 at depth 3: [`=`, `<>`, `=`]" }} {PARA 6 "" 1 "" {TEXT -1 63 " #OT: 14 #Sol: 7 #NL: 0 #UnC: \+ 4 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End itera tion #16 Time:.50e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 14 \+ #Sol: 8 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 4" }}{PARA 6 "" 1 "" {TEXT -1 40 "Split on case #15 at depth 2: [`=`, `=`]" }}{PARA 6 "" 1 "" {TEXT -1 64 " #OT: 3 #Sol: 14 #NL: 3 #UnC: 10 #NonP: 0 #Piv: 3" }}{PARA 6 "" 1 "" {TEXT -1 36 " **End iteration #13 Ti me:.80e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " #OT: 3 #Sol: 15 #N L: 0 #UnC: 0 #NonP: 0 #Piv: 3" }}{PARA 6 "" 1 "" {TEXT -1 36 " \+ **End iteration #14 Time:.30e-1" }}{PARA 6 "" 1 "" {TEXT -1 63 " \+ #OT: 3 #Sol: 14 #NL: 0 #UnC: 0 #NonP: 0 #Piv: 3" }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 0 33 "caseplot(rsys, [xi,tau,eta ,phi]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,Case~1:~3-dG" }}{PARA 11 "" 1 "" 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