(* Content-type: application/mathematica *) (*** Wolfram Notebook File ***) (* http://www.wolfram.com/nb *) (* CreatedBy='Mathematica 6.0' *) (*CacheID: 234*) (* Internal cache information: NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 145, 7] NotebookDataLength[ 22168, 632] NotebookOptionsPosition[ 20127, 559] NotebookOutlinePosition[ 20481, 575] CellTagsIndexPosition[ 20438, 572] WindowFrame->Normal ContainsDynamic->True *) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["The Sine Gordon Equation", "Subtitle"], Cell[BoxData[ RowBox[{"Clear", "[", "\"\\"", "]"}]], "Input", CellChangeTimes->{{3.403977137448413*^9, 3.4039771488290253`*^9}}], Cell[CellGroupData[{ Cell["The Equation", "Subsection"], Cell["The equation is", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ FractionBox[ RowBox[{ SuperscriptBox["\[PartialD]", "2"], "u"}], RowBox[{"\[PartialD]", SuperscriptBox["x", "2"]}]], " ", "-", " ", FractionBox[ RowBox[{ SuperscriptBox["\[PartialD]", "2"], "u"}], RowBox[{"\[PartialD]", SuperscriptBox["t", "2"]}]]}], " ", "=", " ", RowBox[{"sin", " ", "u"}]}]], "DisplayFormula"], Cell[BoxData[ RowBox[{ RowBox[{"sinegordoneq", "[", "u_", "]"}], " ", ":=", " ", RowBox[{ RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", " ", "t"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "2"}], "}"}]}], "]"}], " ", "-", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", " ", "t"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", "2"}], "}"}]}], "]"}]}], " ", "==", " ", RowBox[{"Sin", "[", RowBox[{"u", "[", RowBox[{"x", ",", " ", "t"}], "]"}], "]"}]}], " ", "//", "FullSimplify"}]}]], "Input", CellChangeTimes->{{3.403977771856434*^9, 3.4039777755631123`*^9}}], Cell["\<\ Without the right hand side we would have the wave equation with the wave \ speed, c, set to unity.\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["One Soliton Solutions", "Subsection"], Cell[TextData[{ "We display a soliton solution; first of all for the special case of no time \ dependence. 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Note that 1/", Cell[BoxData[ FormBox[ SqrtBox[ RowBox[{"1", "-", SuperscriptBox["v", "2"]}]], TraditionalForm]]], " is a \"Lorentz contraction\" factor. We verify again that this satisfies \ the equation." }], "Text", CellChangeTimes->{{3.40414804319011*^9, 3.4041480510032587`*^9}}], Cell[BoxData[ RowBox[{ RowBox[{"u", "[", RowBox[{"x_", ",", " ", "t_"}], "]"}], " ", ":=", " ", RowBox[{"4", " ", RowBox[{"ArcTan", "[", RowBox[{"Exp", "[", RowBox[{ RowBox[{"(", RowBox[{"x", " ", "-", " ", RowBox[{"v", " ", "t"}]}], ")"}], "/", SqrtBox[ RowBox[{"1", " ", "-", " ", SuperscriptBox["v", "2"]}]]}], "]"}], "]"}]}]}]], "Input", CellChangeTimes->{{3.40414805710889*^9, 3.404148071603888*^9}}], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"sinegordoneq", "[", "u", "]"}]], "Input", CellChangeTimes->{{3.4039777899988813`*^9, 3.403977791810503*^9}}], Cell[BoxData["True"], "Output", CellChangeTimes->{3.403974392752388*^9, 3.403977028975649*^9, 3.4039771150108624`*^9, 3.403977184299903*^9, 3.4039776582366734`*^9, 3.40397771656817*^9, 3.403977815109735*^9, 3.4041473727735767`*^9, 3.404148104307571*^9}] }, Open ]], Cell["\<\ Now we choose v = 0.8 and produce a simple animation of the soliton. 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