(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.0' Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 24264, 691]*) (*NotebookOutlinePosition[ 25028, 717]*) (* CellTagsIndexPosition[ 24984, 713]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell[" Painlev\[EAcute] Test", "Title"], Cell[BoxData[ \(IMPORTANT\ \(\(MESSAGE\)\(:\)\(\ \)\)\)], "Input"], Cell[BoxData[{ \(It\ can\ happen\ that\ our\ software\ for\ the\ PainleveTestV2\ fails\ \ to\ \ find\ all\ possible\ dominant\ behaviors . \ In\ particular, \ this\ can\ happen\ when\ one\ or\ more\ of\ the\ leading\ order\ \ exponents\ \((alphas)\)\ \ remain\ undetermined . \ \ In\ such\ cases\), "\ \[IndentingNewLine]", \(our\ software\ has\ no\ straightforward\ way\ of\ finding\ \((or\ \ trying)\)\ values\ for\ the\ undetermined\ alphas . \ \ \[IndentingNewLine]\ \[IndentingNewLine]WHEN\ IN\ DOUBT, \ the\ software\ should\ be\ run\ with\ DominantBehaviorMin\ \[Rule] \ \ \(integer1\ \ and/or\ \ DominantBehaviorMax\ \[Rule] \ integer2\), \ where\ \ integer1\ and\ integer2\ can\ be\ specified\ \ by\ the\ user . \ For\ example, \ DominantBehaviorMin\ \[Rule] \ \(\(-3\)\ and/ or\ DominantBehaviorMax\ \[Rule] \ 4\)\ \), "\[IndentingNewLine]", \(were\ choosen, \ the\ the\ code\ will\ consider\ the\ following\ choices\ for\ the\ free\ \ \(alphas : \ \(-3\)\), \(-2\), \(-1\), 0, 1, 2, 3, \ and\ 4. \[IndentingNewLine]\), "\[IndentingNewLine]", \(Default\ \(values : DominantBehaviorMin\ is\ set\ to\ - 3\ and\ \ DominantBehaviorMax\ is\ set\ to\ - 1\ by\ default . \ So\), \ if\ not\ values\ are\ selected, \ the\ code\ will\ run\ the\ test\ for\ the\ default\ values\ for\ the\ \ alphas\ starting\ \ from\ the\ default\ DominantBehaviorMin\ to\ the\ default\ \ DominantBehaviorMax\ value . \ Thus, \ for\ each\ undetermined\ alpha, \ the\ code\ will\ consider\ the\ values\ - 3, \(-2\)\ and\ - 1. \ \[IndentingNewLine]\), "\[IndentingNewLine]", \(Selecting\ appropriate\ values\ for\ DominantBehaviorMax\ is\ important\ \ when\ one\ or\ more\ of\ the\ alphas\ are\ positive\ and\ the\ \), "\ \[IndentingNewLine]", \(other\ alphas\ remain\ undetermined . \ Examples\ are\ given\ at\ the\ bottom\ of\ this\ notebook . \ \[IndentingNewLine]\[IndentingNewLine]Why\ did\ we\ not\ design\ more\ \ sophisticated\ code\ that\ would\ automatically\ compute\ bounds\ for\ the\ \ undetermined\ values?\ \ This\ is\ largely\ due\ to\ the\ lack\ of\ reliable\ \ Mathematica\ functions\ that\ handle\ \(\(inequalities\)\(.\)\)\)}], "Input"], Cell["\<\ To load the package, you must set the directory to where the file is located, \ say c:\\myDirectory, and load the package as follows:\ \>", "Text"], Cell[BoxData[ \(\(\( (*\ \(SetDirectory["\"];\)\ *) \)\(\n\)\( (*\ For\ example, \ SetDirectory["\"]\ *) \)\)\)], "Input"], Cell[BoxData[ \(\(\(\ \)\(SetDirectory["\"]\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(Get["\"]\)], "Input"], Cell[TextData[{ "The companion paper by Baldwin and Hereman, entitled ``Symbolic Software \ for the Painleve Test of Nonlinear Ordinary and Partial\nDifferential \ Equations\" has been published in the Journal of Nonlinear Mathematical \ Physics, vol. 13, number 1, Februrary 2006,\npages 90-110. ", StyleBox[" \n\n", FontSlant->"Italic"], "A short description of how to use the package can be obtained by \ entering:" }], "Text"], Cell[BoxData[ \(?? PainleveTest\)], "Input"], Cell["\<\ As a first example, we consider the equation due to Korteweg and de Vries \ (the KdV equation) :\ \>", "Text"], Cell[BoxData[ \(PainleveTest[{D[u[x, t], \ t] + 6*u[x, t]*D[u[x, t], \ x] + D[u[x, t], \ {x, 3}] \[Equal] 0}, \ \[IndentingNewLine]u[x, t], \ {x, t}\ , \ Verbose\ \[Rule] \ True\ ]\)], "Input", FontSize->10], Cell["\<\ The output of the Painleve Test is in the form {{Dominant Behavior, \ Resonances, {Constants of Integration, Constraints}, ...}. In this example, \ the dominant behavior implies that the singularity manifold given by \ \>", "Text"], Cell[BoxData[ \(\(\(g \((x, t)\) = 0\)\(,\)\)\)], "DisplayFormula"], Cell["\<\ is a double pole and the solutions has a Laurent series expansion of the form\ \ \>", "Text"], Cell[BoxData[ \(\(\(u \((x, t)\)\ = \ \(g\^\(-2\)\) \((x, t)\)\ \ \(\[Sum]\+\(n = 0\)\%\[Infinity]\( u\_n\) \((x, t)\) \(g\^n\) \((x, t)\)\)\)\(,\)\)\)], "DisplayFormula"], Cell[TextData[{ "where ", Cell[BoxData[ \(TraditionalForm\`\(u\_n\)(x, t)\)]], " are analytic functions in the neighborhood of the singularity manifold ", Cell[BoxData[ \(TraditionalForm\`g\ = \ 0. \)]], " The integer resonances ", Cell[BoxData[ \(TraditionalForm\`r\ = \ \(-1\), 4, 6\)]], " indicate that the solution lacks any algebraic branch-points in the \ neighborhood of the singularity manifold. Furthermore, since the \ compatibility conditions were trivially satisfied, the solution lacks \ logarithmic branch-points in the neighborhood of the singularity manifold. \ Indeed, at resonance levels r = 4 and r = 6, there were arbitrary \ coefficients in the series and at all other levels the coefficients (also \ called integration constants) were unambigiously and uniquely determined. \n\n\ Detail can be seen for each of the three main steps of the algorithm by \ setting DominantBehaviorVerbose, ResonancesVerbose, and \ ConstantsOfIntegrationVerbose to a number between 1 and 3 (where larger \ numbers give incrementally more detail). \nWe run the same example to see a \ full trace of the computations. " }], "Text"], Cell[BoxData[ \(PainleveTest[{D[u[x, t], \ t] + 6*u[x, t]*D[u[x, t], \ x] + D[u[x, t], \ {x, 3}] \[Equal] 0}, \ \[IndentingNewLine]u[x, t], \ {x, t}, \ Verbose\ \[Rule] \ True, \ DominantBehaviorVerbose\ \[Rule] \ 3, \ ResonancesVerbose\ \[Rule] \ 3, \ \[IndentingNewLine]ConstantsOfIntegrationVerbose\ \[Rule] \ 3]\)], "Input"], Cell[TextData[{ "The Weiss-Kruskal simplified algorithm is also implemented and can be \ activated with the KruskalSimplification -> x option. \nIf so, the code uses \ the simplified Kruskal ansatz where ", Cell[BoxData[ \(TraditionalForm\`g(x, t)\ \[Congruent] \ x\ - \ h(t)\)]], " for the singularity manifold. The coefficients in the\nLaurent series are \ functions of t only, which drastically simplifies the computations." }], "Text"], Cell[BoxData[ \(\(\(PainleveTest[{D[u[x, t], \ t] + 6*u[x, t]*D[u[x, t], \ x] + D[u[x, t], \ {x, 3}] \[Equal] 0}, \ \[IndentingNewLine]u[x, t], \ {x, t}, \ KruskalSimplification \[Rule] x, \ Verbose\ \[Rule] \ True, \ DominantBehaviorVerbose\ \[Rule] \ 3, \ ResonancesVerbose\ \[Rule] \ 3, \ ConstantsOfIntegrationVerbose\ \[Rule] \ 3]\)\(\ \)\)\)], "Input", FontSize->10], Cell["\<\ Let us now turn to our second example: a system of nonlinearly coupled \ Korteweg-de Vries equations (due to Hirota and Satsuma). This example will \ take a while. So, be patient!\ \>", "Text"], Cell[BoxData[ \(\(\(PainleveTest[{\(-aa\)*D[u[x, t], \ {x, \ 3}] - 6*aa*u[x, t]*D[u[x, t], \ x] + 6*v[x, t]*D[v[x, t], \ x] + D[u[x, t], \ t]\ \[Equal] \ 0, \[IndentingNewLine]D[v[x, t], \ {x, \ 3}] + 3*u[x, t]*D[v[x, t], \ x] + D[v[x, t], \ t]\ \[Equal] \ 0}, \ \[IndentingNewLine]{u[x, t], v[x, t]}, \ \[IndentingNewLine]{x, t}]\)\(\ \)\)\)], "Input", FontSize->10], Cell["\<\ Here, we see that the system has a parameter, aa. Note that (i) there are \ two choices for the dominant behavior of the solution, and (ii) the \ combatibility condition implies that aa = 1/2. Below, the Verbose option is set to True in order for the user to see a \ detailed trace of the computations. A greater level of detail can be see \ for each of the three main steps of the algorithm by setting \ DominantBehaviorVerbose, ResonancesVerbose, or ConstantsOfIntegrationVerbose \ to a number between 1 and 3 (where larger numbers give incrementally more \ detail). \ \>", "Text"], Cell[BoxData[ \(\(\(PainleveTest[{\(-aa\)*D[u[x, t], \ {x, \ 3}] - 6*aa*u[x, t]*D[u[x, t], \ x] + 6*v[x, t]*D[v[x, t], \ x]\ + D[u[x, t], \ t]\ \[Equal] \ 0, \[IndentingNewLine]D[v[x, t], \ {x, \ 3}] + 3*u[x, t]*D[v[x, t], \ x] + D[v[x, t], \ t]\ \[Equal] \ 0}, \ \[IndentingNewLine]{u[x, t], v[x, t]}, \ \[IndentingNewLine]{x, t}, \[IndentingNewLine]Verbose \[Rule] True, \[IndentingNewLine]KruskalSimplification \[Rule] x, \ \ DominantBehaviorVerbose\ \[Rule] \ 3, \ ResonancesVerbose\ \[Rule] \ 3, \ ConstantsOfIntegrationVerbose\ \[Rule] \ 3]\)\(\ \)\)\)], "Input"], Cell[CellGroupData[{ Cell["ODEs", "Section"], Cell[CellGroupData[{ Cell["PI - Painleve Equation I", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[\(w''\)[z] \[Equal] 6*w[z]^2 + z, w[z], z]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["PII - Painleve Equation II", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[\(w''\)[z] \[Equal] 2*w[z]^3 + z*w[z] + \[Alpha], w[z], z]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Chazy Equation: The Chazy equation has only negative resonances.\ \>", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[\[IndentingNewLine]\(y'''\)[x] \[Equal] 2*y[x]*\(y''\)[x] - 3*\(y'\)[x]^2, \[IndentingNewLine]y[x], x, Verbose \[Rule] True]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Example of an ODE with one positive and one negative alpha (Ramani, \ Grammaticos and Bountis).\ \>", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{\[IndentingNewLine]D[x[t], \ t]\ \[Equal] \ x[t]*\((a - x[t] - y[t])\), \[IndentingNewLine]D[y[t], t]\ \[Equal] \ y[t]*\((x[t]\ - 1\ )\)\[IndentingNewLine]}, \ \[IndentingNewLine]{x[t], y[t]}, \ \[IndentingNewLine]{t}, \[IndentingNewLine]Verbose \[Rule] True, \ \ \ DominantBehaviorMin\ \[Rule] \ \(-5\)\ \ , \ \[IndentingNewLine]DominantBehaviorMax\ \[Rule] \ 5\ \[IndentingNewLine]]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell[" F-XVIII (Cosgrove)", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[\[IndentingNewLine]\(y''''\)[ x] \[Equal] \(-5\)*\(y'\)[x]*\(y''\)[x] + 5*y[x]^2*\(y''\)[x] + 5*y[x]*\(y'\)[x]^2 - y[x]^5 + \((\[Lambda]*x + \[Alpha])\)* y[x] + \[Gamma], \[IndentingNewLine]y[x], x]\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Single PDEs", "Section"], Cell[CellGroupData[{ Cell["Fisher Equation", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{D[u[x, t], {x, 2}] + c*D[u[x, t], x] - u[x, t]^2 + u[x, t] \[Equal] 0}, u[x, t], {x, t}, KruskalSimplification \[Rule] x]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Cylindrical KDV Equation", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{a[t]*D[u[x, t], \ x] + D[u[x, t], \ {x, \ 4}] + 6*u[x, t]*D[u[x, t], \ {x, \ 2}] + 6*D[u[x, t], \ x]^2 + D[u[x, t], \ {x, \ 1}, \ {t, \ 1}]}, \ u[x, t], \ {x, t}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Burgers' Equations", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{D[u[x, t], \ t] + u[x, t]*D[u[x, t], \ x] - sigma*D[u[x, t], \ {x, 2}]}, \ u[x, t], \ {x, t}, \ Verbose \[Rule] True]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Modified KdV", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{D[u[x, t], \ t] - 3*\ u[x, t]^2*D[u[x, t], \ x] + 2*sigma^2*D[u[x, t], \ {x, 3}]}, u[x, t], \ {x, t}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Boussinesq Equation", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{D[u[x, t], \ {t, 2}] + 2*u[x, t]*D[u[x, t], \ {x, 2}] + 2*D[u[x, t], \ x]^2 + \((1/3)\)*D[u[x, t], \ {x, 4}]}, \ u[x, t], \ {x, t}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Integrable Nonlinear Klein-Gordon Equations", "Subsection", FontSize->12], Cell[CellGroupData[{ Cell["Case I: Sine-Gorden equation", "Subsubsection"], Cell[BoxData[ \(PainleveTest[{u[x, t]*\((D[u[x, t], \ {t, \ 2}] - D[u[x, t], \ {x, 2}])\) - D[u[x, t], \ t]^2 + D[u[x, t], \ x]^2\ - \ \((1/2)\)*u[x, t]^3 + \((1/2)\)*u[x, t]}, \ u[x, t], \ {x, t}]\)], "Input"], Cell[BoxData[ \(PainleveTest[{\(-2\)*D[u[x, t], \ t]*D[u[x, t], \ x] + 2*u[x, t]*D[u[x, t], \ {x, \ 1}, \ {t, \ 1}] - u[x, t]^3 + u[x, t]}, u[x, t], \ {x, t}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Case II: Liouville equations", "Subsubsection"], Cell[BoxData[ \(PainleveTest[{u[x, t]*\((D[u[x, t], \ {t, \ 2}] - D[u[x, t], \ {x, 2}])\) - D[u[x, t], \ t]^2 + D[u[x, t], \ x]^2\ - \ u[x, t]^3}, \ u[x, t], \ {x, t}]\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Another example, page 180 in Steeb, where alpha[1]\[Rule] -1\ \>", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{2*u[x, t]*D[u[x, t], \ t] + D[u[x, t], \ {x, \ 2}]}, u[x, t], \ {x, t}]\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Systems of PDEs", "Section"], Cell[CellGroupData[{ Cell["One-Dimensional Nonlinear Schrodinger Equation", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{D[u[x, t], \ t] + D[v[x, t], \ {x, 2}] + a*v[x, t]*\((u[x, t]^2 + v[x, t]^2)\), \ D[v[x, t], \ t] - D[u[x, t], \ {x, 2}] - a*u[x, t]*\((u[x, t]^2 + v[x, t]^2)\)}, \ {u[x, t], \ v[x, t]}, \ {x, t}, \ KruskalSimplification \[Rule] x]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Reduced Maxwell Bloch Equations", "Subsection", FontSize->12], Cell[BoxData[ \(\(\(PainleveTest[{D[u[x, t], x] + mu*v[x, t], \ D[v[x, t], \ x] - e[x, t]*w[x, t] - mu*u[x, t], \ D[w[x, t], \ x] + e[x, t]*v[x, t], \ D[e[x, t], \ t] + v[x, t]}, \ {u[x, t], v[x, t], w[x, t], e[x, t]}, {x, t}, KruskalSimplification \[Rule] x]\)\(\ \)\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Lorenz Model of a Dissipative System with lim_{epsilon\[Rule]0}\ \>", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{v[x, t] - D[u[x, t], \ t], \ u[x, t] - u[x, t]*w[x, t] - D[v[x, t], \ t], \ u[x, t]*v[x, t] - D[w[x, t], \ t]}, \ {u[x, t], \ v[x, t], w[x, t]}, \ {x, t}, KruskalSimplification \[Rule] x]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Nonlinearly coupled interacting harmonic oscillators.", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{D[u[x, t], \ {t, \ 2}] + w1^2*u[x, t]\ + \ mu*v[x, t]*w[x, t], \ D[v[x, t], \ {t, \ 2}] + w1^2*v[x, t]\ + \ mu*u[x, t]*w[x, t], \ D[w[x, t], \ {t, \ 2}] + w1^2*w[x, t]\ + \ mu*u[x, t]*v[x, t]}, \ {u[ x, t], \ v[x, t], \ w[x, t]}, \ {x, t}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Two coupled PDEs", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{\(-D[u[x, t], \ t]\) + \((1/ 3)\)*\((\(-D[u[x, t], \ {x, \ 2}]\) - \((7/2)\)* D[u[x, t], \ x]*v[x, t] + \((1/2)\)*u[x, t]* D[v[x, t], \ x] - u[x, t]^2*v[x, t])\), \ \(-D[ v[x, t], {t, \ 2}]\) + D[v[x, t], \ {x, \ 2}] - \((1/2)\)*D[u[x, t], \ x]* v[x, t] + \((3/2)\)*u[x, t]*D[v[x, t], \ x]}, \ {u[x, t], \ v[x, t]}, \ {x, t}]\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["\<\ Equation (2) in \"A note on the Painlev\.b4e property of coupled KdV equations by S.Yu.Sakovich. arXiv:nlin.SI/0402004 v1 2 Feb \ 2004\ \>", "Subsection", FontSize->12], Cell[BoxData[ \(PainleveTest[{D[v[x, t], t]\ + \ 6*v[x, t]*D[v[x, t], \ x]\ + \ D[v[x, t], \ {x, \ 3}]\ \[Equal] \ 0, \[IndentingNewLine]D[w[x, t], \ t]\ + \ 6*a*v[x, t]*D[w[x, t], x\ ]\ + \ 6*\((1 - a)\)*w[x, t]*D[v[x, t], \ x]\ + \ D[w[x, t], \ {x, 3}]\ \[Equal] \ 0} /. \ a\ \[Rule] \ 3/2, \[IndentingNewLine]{v[x, t], \ w[x, t]}, \[IndentingNewLine]{x, t}, \ Verbose \[Rule] True, \ KruskalSimplification \[Rule] x]\)], "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell["Positive DominantBehaviorMax", "Section"], Cell[TextData[{ "Three interesting examples where one of the exponents in the dominant \ behaviors is positive. It is important that at least one of the values of ", Cell[BoxData[ \(TraditionalForm\`\[Alpha]\_i\)]], " is negative. Furthermore, the branch with a positive \[Alpha] is called \ the non-principal (or non-generic) branch." }], "Text"], Cell[BoxData[{ StyleBox[ RowBox[{\(Example\ 1\ is\ a\ coupled\ system\ of\ KdV - mKdV\ equations\ taken\ from\ \ Integrability\ of\ Kersten - Krasilshchik\), " "}], "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}], "\[IndentingNewLine]", StyleBox[\(coupled\ KdV - mKdV\ \(equations : \ singularity\ analysis\ and\ Lax\ pair\ \ by\ \ Kalkani\), \ Sakovich, \ \[IndentingNewLine]and\ Yurdusen . \ J . \ Math . \ Phys . \ 44 \((4)\)\ 2003, \ \(\(pp . \ 1703\)\(-\)\(1708.\)\(\ \)\)\), "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}]}], "Input", PageWidth->Infinity], Cell[BoxData[ StyleBox[\(We\ first\ run\ the\ test\ without\ specifiction\ of\ \ DominantBehaviorMax\ \(\((which\ by\ default\ will\ be\ - 1)\)\(.\)\)\), "Text", FontWeight->"Plain", FontVariations->{"CompatibilityType"->0}]], "Input"], Cell[BoxData[ \(\(\(PainleveTest[{D[u[x, t], \ t] \[Equal] \(-D[u[x, t], \ {x, 3}]\) + 6*u[x, t]*D[u[x, t], \ x] - 3*w[x, t]*D[w[x, t], \ {x, 3}] - 3*D[w[x, t], \ x]*D[w[x, t], \ {x, 2}] + 3*D[u[x, t], \ x]*w[x, t]^2 + 6*u[x, t]*w[x, t]*D[w[x, t], \ x], \ D[w[x, t], \ t] \[Equal] \(-D[w[x, t], \ {x, 3}]\) + 3*w[x, t]^2*D[w[x, t], \ x] + 3*u[x, t]*D[w[x, t], \ x] + 3*w[x, t]*D[u[x, t], \ x]}, \ {u[x, t], \ w[x, t]}, \ {x, t}, KruskalSimplification \[Rule] x]\)\(\[IndentingNewLine]\) \)\)], "Input"], Cell[BoxData[ RowBox[{" ", RowBox[{ StyleBox[\(The\ software\ misses\ two\ non - generic\ \(branches : \ alpha1\)\ = \ \(\(-2\)\ with\ alpha2\ = \ 2\), and\ alpha1 = \(\(-2\)\ with\ alpha2 = \ 3. \)\), "Text"], StyleBox[" ", "Text"], StyleBox["\[IndentingNewLine]", "Text"], StyleBox[" ", "Text"], RowBox[{ StyleBox[\(By\ setting\ DemominantBehaviorMax\ sufficiently\ high\), "Text"], StyleBox[",", "Text"], StyleBox[" ", "Text"], StyleBox[\(in\ this\ case\ 3\ or\ larger\), "Text"], StyleBox[",", "Text"], StyleBox[" ", "Text"], StyleBox[\(these\ branches\ will\ be\ considered . \ \ \ \[IndentingNewLine]Hence\), "Text"], StyleBox[",", "Text"], RowBox[{ StyleBox["run", "Text"], StyleBox[" ", "Text"], StyleBox["the", "Text"], StyleBox[" ", "Text"], StyleBox["test", "Text"], StyleBox[" ", "Text"], StyleBox["again", "Text"], StyleBox[" ", "Text"], StyleBox["with", "Text"], StyleBox[" ", "Text"], StyleBox["the", "Text"], StyleBox[" ", "Text"], StyleBox["following", "Text"], StyleBox[" ", "Text"], RowBox[{ StyleBox["settings", "Text"], StyleBox[":", "Text"], "\[IndentingNewLine]"}]}]}]}]}]], "Input", PageWidth->Infinity], Cell[BoxData[ \(PainleveTest[{D[u[x, t], \ t] \[Equal] \(-D[u[x, t], \ {x, 3}]\) + 6*u[x, t]*D[u[x, t], \ x] - 3*w[x, t]*D[w[x, t], \ {x, 3}] - 3*D[w[x, t], \ x]*D[w[x, t], \ {x, 2}] + 3*D[u[x, t], \ x]*w[x, t]^2 + 6*u[x, t]*w[x, t]*D[w[x, t], \ x], \ D[w[x, t], \ t] \[Equal] \(-D[w[x, t], \ {x, 3}]\) + 3*w[x, t]^2*D[w[x, t], \ x] + 3*u[x, t]*D[w[x, t], \ x] + 3*w[x, t]*D[u[x, t], \ x]}, \ {u[x, t], \ w[x, t]}, \ {x, t}, KruskalSimplification \[Rule] x, \ DominantBehaviorMax \[Rule] 3]\)], "Input"], Cell[BoxData[ \(Example\ 2 : \ A\ coupled\ system\ of\ \(\(ODEs\)\(.\)\)\)], "Input"], Cell[BoxData[ \(PainleveTest[{\[IndentingNewLine]D[x[t], \ t]\ \[Equal] \ x[t]*\((a - x[t] - y[t])\), \[IndentingNewLine]D[y[t], t]\ \[Equal] \ y[t]*\((x[t]\ - 1\ )\)\[IndentingNewLine]}, \ \[IndentingNewLine]{x[t], y[t]}, \ \[IndentingNewLine]{t}, \[IndentingNewLine]Verbose \[Rule] True, \[IndentingNewLine]DominantBehaviorMax\ \[Rule] \ 2\[IndentingNewLine]]\)], "Input"], Cell[BoxData[ \(Example\ 3 : \ \(\(A\)\(\ \)\(coupled\)\(\ \)\(system\)\(\ \)\(of\)\(\ \ \)\(PDEs\)\(\ \)\)\)], "Input"], Cell[BoxData[ \(PainleveTest[\[IndentingNewLine]{D[u[x, t], t]\ \[Equal] \ D[u[x, t], \ {x, \ 2}]\ - \ 2*u[x, t]^2\ + \ 12*v[x, t], \[IndentingNewLine]D[v[x, t], t]\ \[Equal] \ D[v[x, t], \ {x, \ 2}]\ - \ 2*u[x, t]*v[x, t]}, \[IndentingNewLine]{u[x, t], v[x, t]}, \[IndentingNewLine]{x, t}, \[IndentingNewLine]Verbose \[Rule] True, \[IndentingNewLine]DominantBehaviorMax\ \[Rule] \ 3, \[IndentingNewLine]KruskalSimplification \[Rule] x\[IndentingNewLine]]\)], "Input"] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1280}, {0, 941}}, ScreenStyleEnvironment->"Working", WindowSize->{1147, 854}, WindowMargins->{{Automatic, 53}, {Automatic, 16}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, Magnification->1.25 ] (******************************************************************* Cached data follows. If you edit this Notebook file directly, not using Mathematica, you must remove the line containing CacheID at the top of the file. The cache data will then be recreated when you save this file from within Mathematica. *******************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1776, 53, 39, 0, 117, "Title"], Cell[1818, 55, 70, 1, 35, "Input"], Cell[1891, 58, 2290, 37, 455, "Input"], Cell[4184, 97, 157, 3, 38, "Text"], Cell[4344, 102, 181, 3, 56, "Input"], Cell[4528, 107, 93, 1, 35, "Input"], Cell[4624, 110, 60, 1, 35, "Input"], Cell[4687, 113, 440, 9, 122, "Text"], Cell[5130, 124, 48, 1, 35, "Input"], Cell[5181, 127, 120, 3, 38, "Text"], Cell[5304, 132, 236, 4, 54, "Input"], Cell[5543, 138, 242, 4, 59, "Text"], Cell[5788, 144, 71, 1, 30, "DisplayFormula"], Cell[5862, 147, 103, 3, 38, "Text"], Cell[5968, 152, 202, 3, 57, "DisplayFormula"], Cell[6173, 157, 1169, 22, 185, "Text"], Cell[7345, 181, 390, 7, 77, "Input"], Cell[7738, 190, 453, 8, 80, "Text"], Cell[8194, 200, 425, 7, 74, "Input"], Cell[8622, 209, 203, 5, 59, "Text"], Cell[8828, 216, 445, 8, 94, "Input"], Cell[9276, 226, 599, 11, 143, "Text"], Cell[9878, 239, 690, 12, 161, "Input"], Cell[CellGroupData[{ Cell[10593, 255, 23, 0, 92, "Section"], Cell[CellGroupData[{ Cell[10641, 259, 62, 1, 45, "Subsection"], Cell[10706, 262, 88, 1, 35, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[10831, 268, 64, 1, 45, "Subsection"], Cell[10898, 271, 111, 2, 35, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[11046, 278, 111, 3, 45, "Subsection"], Cell[11160, 283, 186, 3, 77, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[11383, 291, 141, 4, 45, "Subsection"], Cell[11527, 297, 515, 9, 203, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[12079, 311, 57, 1, 45, "Subsection"], Cell[12139, 314, 288, 5, 77, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[12476, 325, 30, 0, 92, "Section"], Cell[CellGroupData[{ Cell[12531, 329, 53, 1, 45, "Subsection"], Cell[12587, 332, 188, 3, 56, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[12812, 340, 62, 1, 45, "Subsection"], Cell[12877, 343, 238, 4, 77, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[13152, 352, 56, 1, 45, "Subsection"], Cell[13211, 355, 182, 3, 56, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[13430, 363, 50, 1, 45, "Subsection"], Cell[13483, 366, 159, 2, 56, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[13679, 373, 57, 1, 45, "Subsection"], Cell[13739, 376, 196, 3, 56, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[13972, 384, 81, 1, 45, "Subsection"], Cell[CellGroupData[{ Cell[14078, 389, 54, 0, 34, "Subsubsection"], Cell[14135, 391, 259, 5, 77, "Input"], Cell[14397, 398, 197, 3, 56, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[14631, 406, 53, 0, 34, "Subsubsection"], Cell[14687, 408, 218, 4, 56, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[14954, 418, 106, 3, 45, "Subsection"], Cell[15063, 423, 125, 2, 35, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[15237, 431, 34, 0, 92, "Section"], Cell[CellGroupData[{ Cell[15296, 435, 84, 1, 45, "Subsection"], Cell[15383, 438, 318, 5, 77, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[15738, 448, 69, 1, 45, "Subsection"], Cell[15810, 451, 329, 6, 77, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[16176, 462, 109, 3, 45, "Subsection"], Cell[16288, 467, 261, 4, 56, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[16586, 476, 91, 1, 45, "Subsection"], Cell[16680, 479, 328, 5, 77, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[17045, 489, 54, 1, 45, "Subsection"], Cell[17102, 492, 485, 9, 119, "Input"] }, Open ]], Cell[CellGroupData[{ Cell[17624, 506, 181, 5, 65, "Subsection"], Cell[17808, 513, 542, 9, 119, "Input"] }, Open ]] }, Open ]], Cell[CellGroupData[{ Cell[18399, 528, 47, 0, 92, "Section"], Cell[18449, 530, 361, 7, 59, "Text"], Cell[18813, 539, 733, 17, 77, "Input"], Cell[19549, 558, 258, 5, 35, "Input"], Cell[19810, 565, 604, 10, 140, "Input"], Cell[20417, 577, 1951, 69, 98, "Input"], Cell[22371, 648, 608, 10, 119, "Input"], Cell[22982, 660, 89, 1, 35, "Input"], Cell[23074, 663, 461, 8, 203, "Input"], Cell[23538, 673, 124, 2, 35, "Input"], Cell[23665, 677, 571, 10, 203, "Input"] }, Open ]] }, Open ]] } ] *) (******************************************************************* End of Mathematica Notebook file. *******************************************************************)