(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of 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). 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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[ 225240, 6512]*) (*NotebookOutlinePosition[ 244703, 7060]*) (* CellTagsIndexPosition[ 244343, 7045]*) (*WindowFrame->Normal*) Notebook[{ Cell["\[Copyright] Copyright Notice", "Section", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, AspectRatioFixed->True, CellTags->"Copyright"], Cell[TextData[{ "Copyright \[Copyright] by \.86nal G\.9aktas and Willy Hereman (Department \ of Mathematical and Computer Sciences, Colorado School of Mines, Golden, \ Colorado, USA). No part of the ", StyleBox["Integrability ", FontSlant->"Italic"], "package can be sold or reproduced without written consent of the authors. \ ", ButtonBox["Wolfram Research", ButtonData:>{ URL[ "http://www.wolfram.com"], None}, ButtonStyle->"Hyperlink"], ", Inc. (Champaign, Illinois, USA) is the holder of the copyright to the ", StyleBox["Mathematica", FontSlant->"Italic"], " software system." }], "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True, CellTags->"Copyright"], Cell[TextData[{ "About the Package: ", StyleBox["Integrability", FontFamily->"Helvetica", FontSlant->"Italic"] }], "Section", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, AspectRatioFixed->True, CellTags->"Introduction"], Cell[TextData[{ "Nonlinear partial differential equations (PDEs) and differential \ difference equations (DDEs) possess a number of remarkable properties \ reflecting their rich mathematical structure. Such properties include the \ Painlev\.8e property, nontrivial prolongation and bi-Hamiltonian structures, \ soliton solutions, Lax pairs, B\.8acklund transformations, symmetries and \ invariants. They reveal the adequacy of PDEs and DDEs as models for physical \ relevant phenomena. The search for these intrinsic properties becomes more \ attractive due to the availability of CAS such as ", StyleBox["Mathematica.", FontSlant->"Italic"], " Indeed, ", StyleBox["Mathematica", FontSlant->"Italic"], " is well suited and often necessary tool to perform the computations \ inherent in the investigation of integrability." }], "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, AspectRatioFixed->True, CellTags->"Introduction"], Cell[TextData[{ StyleBox["Integrability ", FontSlant->"Italic"], "is a collection of ", StyleBox["Mathematica", FontSlant->"Italic"], " functions addressing computations of invariants and symmetries in \ analyzing nonlinear PDEs and DDEs. The main topics covered in the ", StyleBox["Integrability", FontSlant->"Italic"], " package are as follows:\n\[FilledVerySmallSquare] Computations of \ invariants for systems of nonlinear PDEs and DDEs,\n\[FilledVerySmallSquare] \ Computations of symmetries for systems of nonlinear PDEs and DDEs." }], "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->0, AspectRatioFixed->True, CellTags->"Introduction"], Cell[TextData[{ "A complete set of mathematical definitions and the concepts used in the \ package and this manual are beyond the scope of this document. For the formal \ definitions of the concepts and the theory behind the functions in the ", StyleBox["Integrability", FontSlant->"Italic"], " package, consult the references given in the References section. 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So, in addition to the execution time of a function, some \ delay will occur when a package is first loaded." }], "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True, CellTags->"Loading"], Cell[TextData[{ "If the directory in which the package is located is not on the ", StyleBox["Mathematica", FontSlant->"Italic"], " ", StyleBox["$Path", FontWeight->"Bold"], ", a warning message will be generated. To solve this problem, the \ directory containing the package should be included on the path ", StyleBox["$Path", FontWeight->"Bold"], ". 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One space variable and one time \ variable are allowed." }], "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True, CellTags->"Scope"], Cell["\<\ Theoretically, there is no limit on the number of equations. In \ practice, for large systems, the computations may take a long time or require \ a lot of memory. Computational speed depends primarily on the amount of \ memory.\ \>", "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True, CellTags->"Scope"], Cell[TextData[{ "Only polynomial invariants and symmetries are computed. By design, ", StyleBox["Integrability", FontSlant->"Italic"], " constructs only invariants and symmetries that are uniform in rank. The \ uniform rank assumption allows one to compute independent invariants and \ symmetries piece by piece, without having to split linear combinations. Due \ to the superposition principle, a linear combination of invariants of unequal \ rank is still an invariant. This situation arises frequently when parameters \ with weights are introduced. " }], "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True, CellTags->"Scope"], Cell["\<\ The systems may have parameters, which are assumed to be nonzero. \ If a system has parameters, the program will compute the conditions for these \ parameters so that invariants or symmetries of a given rank might exist. The \ assumption that parameters are nonzero is necessary. As a result of setting \ parameters to zero in an equation, the weights and therefore the ranks might \ change.\ \>", "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True, CellTags->"Scope"], Cell["\<\ The assumption that the systems are uniform in rank is not very \ restrictive. If the uniform rank condition is violated, users can introduce \ parameters with weights. This also allows some flexibility in the form of the \ invariants and symmetries such that invariants and symmetries may not be \ uniform in rank. In cases where it is not clear whether or not parameters \ with weights should be introduced, one should start without introducing any. \ If this causes incompatibilities in the assignment of weights, program tries \ to cure the incompatibility by introducing parameters with weights. In cases \ of persistent incompatibility, corresponding diagnostic messages are \ printed.\ \>", "Text", CellMargins->{{10.75, 24.125}, {Inherited, Inherited}}, Evaluatable->False, CellLabelMargins->{{11.5625, Inherited}, {Inherited, Inherited}}, TextAlignment->Left, TextJustification->1, AspectRatioFixed->True, CellTags->"Scope"], Cell["\<\ Negative weights on the dependent variables are not permitted. Zero \ weights are allowed, but at least one of the dependent variables must have \ positive weight. Fractional weights and ranks are permitted. 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