(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 4.0, MathReader 4.0, or any compatible application. 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[ 7340, 239]*) (*NotebookOutlinePosition[ 8087, 265]*) (* CellTagsIndexPosition[ 8043, 261]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Initializations", "Subsubsection"], Cell[CellGroupData[{ Cell[BoxData[ \(SetDirectory["\"]\)], "Input"], Cell[BoxData[ \("C:\\My Documents\\Athens"\)], "Output"] }, Open ]], Cell[BoxData[ \(Needs["\"]\)], "Input"], Cell[BoxData[ \(Needs["\"]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Get["\"];\)\)], "Input"], Cell[BoxData[ \(SetDelayed::"write" \(\(:\)\(\ \)\) "Tag \!\(Variance\) in \!\(Variance[\(ua_, grid_\)]\) is Protected."\)], "Message"] }, Open ]], Cell[BoxData[ \(\(Get["\"];\)\)], "Input"], Cell[BoxData[ \(\(Get["\"];\)\)], "Input"], Cell[BoxData[ \(\(V[x_]\ := \ \(-a\)\ Cos[\[Omega]\ x];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Xmax[\[Omega]_, \ a_]\ = \ xm\ /. \ Last[Solve[V[xm]\ == \ 1\ + \ V[0], xm]]\)], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(Xmax\)\" is similar to \ existing symbol \"\!\(xmax\)\"."\)], "Message"], Cell[BoxData[ \(Solve::"ifun" \(\(:\)\(\ \)\) "Inverse functions are being used by \!\(Solve\), so some solutions may \ not be found."\)], "Message"], Cell[BoxData[ \(ArcCos[\(\(-1\) + a\)\/a]\/\[Omega]\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Period[\[Omega]_, \ a_]\ = \ 2\ \(\[Integral]\_0\%\(Xmax[\[Omega], \ a]\)\(1 \/\@\(1\ + \ V[0]\ - \ V[X]\)\) \[DifferentialD]X\)\)], "Input"], Cell[BoxData[ \(\(4\ EllipticF[1\/2\ ArcCos[\(\(-1\) + a\)\/a], 2\ a]\)\/\[Omega]\)], "Output"] }, Open ]], Cell[BoxData[ \(Xth[\[Omega]_, \ a_, \ tend_]\ := \ Module[{x, \ t}, \ x\ /. \ First[ NDSolve[{\(x''\)[t]\ \[Equal] \ \(-2\)\ a\ \[Omega]\ Sin[\[Omega]\ x[t]], x[0]\ \[Equal] \ 0, \ \(x'\)[0]\ \[Equal] \ 2}, \ x, {t, 0, tend}, \ MaxSteps \[Rule] 100000]]]\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \($Version\)], "Input"], Cell[BoxData[ \("4.0 beta 2 for Microsoft Windows (November 19, 1998)"\)], "Output"] }, Open ]], Cell[BoxData[ \(\(part1\ = \ Get["\"];\)\)], "Input"], Cell[BoxData[ \(\(part2\ = \ Get["\"];\)\)], "Input"], Cell[BoxData[ \(\(part3\ = \ Get["\"];\)\)], "Input"], Cell[BoxData[ \(\(part4\ = \ Get["\"];\)\)], "Input"], Cell[BoxData[ \(\(trapped\ = \ Join[part1, \ part2, \ part3, \ part4];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[{ \(\(rapid\ = \ Get["\"];\)\), "\[IndentingNewLine]", \(\(all\ = \ Join[trapped, rapid];\)\)}], "Input"], Cell[BoxData[ \(General::"spell1" \(\(:\)\(\ \)\) "Possible spelling error: new symbol name \"\!\(all\)\" is similar to \ existing symbol \"\!\(All\)\"."\)], "Message"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["\<\ Numerical simulations of the Cubic Schr\[ODoubleDot]dinger \ equations for Periodic Perturbations.\ \>", "Subtitle", TextAlignment->Center], Cell["\<\ Robert Knapp Wolfram Research, Inc\ \>", "Subsubtitle", TextAlignment->Center], Cell["\<\ In the course of investigating of the effect of a random potential \ on pulses with the cubic nonlinear Schr\[ODoubleDot]dinger equation, \ \>", "Text"], Cell[BoxData[ \(TraditionalForm \`\[ImaginaryI] \[PartialD]u\/\[PartialD]t + \ \[PartialD]\^2 u\/\[PartialD]x\^2 + 2\ \ \(\[LeftBracketingBar]u\[RightBracketingBar]\^2\) u = \(V(x)\)\ u\)], "Input"], Cell["\<\ I became intrigued by what one should define as the effective width \ of a pulse or soliton, particularly in cases which are near-integrable. In an \ attempt to answer that question, I did some numerical simulations with \ periodic potentials and came across an interesting range of behavior \ depending on how the length scales of potential and pulse compare. Some of \ the results are explainable by simple perturbation methods, and some remain \ more mysterious. I will show you the results I have obtained and conclude \ with some observations about the effective width.\ \>", "Text"], Cell[TextData[{ "I will structure this presentation roughly as follows. First, I will give \ a brief description of the ", ButtonBox["numerical method.", ButtonData:>{"SplitStepMethod.nb", None}, ButtonStyle->"Hyperlink"], " Next, I will present a leading order ", ButtonBox["perturbation analysis", ButtonData:>{"SVPTheory.nb", None}, ButtonStyle->"Hyperlink"], " for a slowly varying potential. Then, I will show how well this theory \ bears out in numerical simulations with a ", ButtonBox["slowly varying potential", ButtonData:>{"SVPNumerical.nb", None}, ButtonStyle->"Hyperlink"], ". The results start to get more interesting when the assumption that the \ potential is slowly varying ", ButtonBox["breaks down", ButtonData:>{"Breakdown.nb", None}, ButtonStyle->"Hyperlink"], ". Beyond this, some simulations with a ", ButtonBox["rapidly varying potential", ButtonData:>{"RVPNumeric.nb", None}, ButtonStyle->"Hyperlink"], " show some mildly surprising behavior. Finally, I will try to bridge the \ entire range of length scales with some ", ButtonBox["conclusions and questions", ButtonData:>{"Conclusions.nb", None}, ButtonStyle->"Hyperlink"], "." }], "Text"] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, ScreenStyleEnvironment->"Working", WindowSize->{916, 669}, WindowMargins->{{0, Automatic}, {Automatic, 0}}, Magnification->1, StyleDefinitions -> "ArticleClassic.nb" ] (*********************************************************************** 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. 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