(* 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[ 5840507, 96393] NotebookOptionsPosition[ 5835358, 96231] NotebookOutlinePosition[ 5836030, 96255] CellTagsIndexPosition[ 5835987, 96252] WindowFrame->Normal ContainsDynamic->False*) (* Beginning of Notebook Content *) Notebook[{ Cell[CellGroupData[{ Cell["Solitons in the Korteweg-de Vries Equation (KdV Equation)", "Subtitle"], Cell[CellGroupData[{ Cell["Introduction", "Subsection"], Cell[TextData[{ "The Korteweg-de Vries Equation (KdV equation) describes the theory of water \ waves in shallow channels, such as a canal. It is a non-linear equation which \ exhibits special solutions, known as ", StyleBox["solitons,", FontSlant->"Italic"], " which are stable and do not disperse with time. Furthermore there as \ solutions with more than one soliton which can move towards each other, \ interact and then emerge at the same speed with no change in shape (but with \ a time \"lag\" or \"speed up\"). \n\nThe KdV equation is " }], "Text"], Cell[BoxData[ RowBox[{ FractionBox[ RowBox[{"\[PartialD]", "u"}], RowBox[{"\[PartialD]", "t"}]], "=", " ", RowBox[{ RowBox[{"6", " ", "u", " ", FractionBox[ RowBox[{"\[PartialD]", "u"}], RowBox[{"\[PartialD]", "x"}]]}], "-", " ", FractionBox[ RowBox[{ SuperscriptBox["\[PartialD]", "3"], "u"}], RowBox[{"\[PartialD]", SuperscriptBox["x", "3"]}]]}]}]], "DisplayFormula"], Cell["\<\ Because of the u \[PartialD]u/\[PartialD]x term the equation is non-linear \ (this term increases four times if u is doubled).\ \>", "Text"] }, Open ]], Cell[CellGroupData[{ Cell["One soliton solution", "Subsection"], Cell["The simplest soliton solution is ", "Text"], Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"u", RowBox[{"(", RowBox[{"x", ",", " ", "t"}], ")"}]}], " ", "=", " ", RowBox[{ RowBox[{"-", "2"}], " ", SuperscriptBox["sech", "2"], RowBox[{"(", RowBox[{"x", " ", "-", " ", RowBox[{"4", "t"}]}], ")"}]}]}], ","}]], "DisplayFormula"], Cell["\<\ which is a trough of depth 2 traveling to the right with speed 4 and not \ changing its shape. Let us verify that it does satisfy the equation:\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"uexact", "[", RowBox[{"x_", ",", " ", "t_"}], "]"}], " ", "=", " ", RowBox[{ RowBox[{"-", "2"}], RowBox[{ RowBox[{"Sech", "[", RowBox[{"x", " ", "-", " ", RowBox[{"4", "t"}]}], "]"}], "^", "2"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"-", "2"}], " ", SuperscriptBox[ RowBox[{"Sech", "[", RowBox[{ RowBox[{"4", " ", "t"}], "-", "x"}], "]"}], "2"]}]], "Output", CellChangeTimes->{3.453120097859747*^9, 3.453129845154532*^9, 3.4844844486443768`*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"uexact", "[", RowBox[{"x", ",", " ", "t"}], "]"}], ",", " ", "t"}], "]"}], " ", "\[Equal]", " ", RowBox[{ RowBox[{"6", " ", RowBox[{"uexact", "[", RowBox[{"x", ",", " ", "t"}], "]"}], " ", RowBox[{"D", "[", RowBox[{ RowBox[{"uexact", "[", RowBox[{"x", ",", "t"}], "]"}], ",", "x"}], "]"}]}], " ", "-", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"uexact", "[", RowBox[{"x", ",", " ", "t"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "3"}], "}"}]}], "]"}]}]}], " ", "//", "Simplify"}]], "Input"], Cell[BoxData["True"], "Output", CellChangeTimes->{3.453120098306509*^9, 3.453129845543776*^9, 3.484484448888578*^9}] }, Open ]], Cell[TextData[{ StyleBox["Mathematica", FontSlant->"Italic"], " returns ", StyleBox["True", "Input"], ", indicating that equation is satisfied.\n\n", StyleBox["Mathematica", FontSlant->"Italic"], " function ", StyleBox["NDSolve", "Input"], " can solve partial differential equations in two (but not more than two) \ variables, such as x and t. However, it tends to be very slow and require a \ lot of memory. Nonetheless, if we put in the soliton at the initial time, it \ correctly propagates the soliton in time:" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{ RowBox[{"xmin", " ", "=", " ", RowBox[{"-", "8"}]}], ";", " ", RowBox[{"xmax", " ", "=", " ", "8"}], ";", RowBox[{"sol", " ", "=", RowBox[{"NDSolve", "[", " ", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{ RowBox[{"D", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", " ", "t"}], "]"}], ",", " ", "t"}], "]"}], " ", "\[Equal]", " ", RowBox[{ RowBox[{"6", " ", RowBox[{"u", "[", RowBox[{"x", ",", " ", "t"}], "]"}], " ", RowBox[{"D", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", "t"}], "]"}], ",", "x"}], "]"}]}], " ", "-", " ", RowBox[{"D", "[", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", " ", "t"}], "]"}], ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "3"}], "}"}]}], "]"}]}]}], ",", " ", RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", " ", "0"}], "]"}], " ", "\[Equal]", " ", RowBox[{ RowBox[{"-", "2"}], RowBox[{ RowBox[{"Sech", "[", "x", "]"}], "^", "2"}]}]}], ",", " ", RowBox[{ RowBox[{"u", "[", RowBox[{"xmin", ",", " ", "t"}], "]"}], " ", "\[Equal]", " ", RowBox[{"u", "[", RowBox[{"xmax", ",", " ", "t"}], "]"}]}]}], " ", "}"}], ",", " ", "u", ",", " ", RowBox[{"{", RowBox[{"x", ",", " ", "xmin", ",", " ", "xmax"}], "}"}], ",", " ", RowBox[{"{", RowBox[{"t", ",", " ", RowBox[{"-", "1"}], ",", " ", "1"}], "}"}]}], " ", "]"}]}]}]], "Input"], Cell[BoxData[ RowBox[{ RowBox[{"NDSolve", "::", "\<\"mxsst\"\>"}], RowBox[{ ":", " "}], "\<\"Using maximum number of grid points \\!\\(10000\\) allowed \ by the MaxPoints or MinStepSize options for independent variable \\!\\(x\\). \ \\!\\(\\*ButtonBox[\\\"\[RightSkeleton]\\\", ButtonStyle->\\\"Link\\\", \ ButtonFrame->None, ButtonData:>\\\"paclet:ref/message/NDSolve/mxsst\\\", \ ButtonNote -> \\\"NDSolve::mxsst\\\"]\\)\"\>"}]], "Message", CellChangeTimes->{3.4531200990476627`*^9, 3.453129845909111*^9, 3.484484449133451*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"{", RowBox[{"u", "\[Rule]", TagBox[ RowBox[{"InterpolatingFunction", "[", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"{", RowBox[{ RowBox[{"-", "8.`"}], ",", "8.`"}], "}"}], ",", RowBox[{"{", RowBox[{ RowBox[{"-", "1.`"}], ",", "1.`"}], "}"}]}], "}"}], ",", "\<\"<>\"\>"}], "]"}], False, Editable->False]}], "}"}], "}"}]], "Output", CellChangeTimes->{3.453120139082222*^9, 3.4531298731688957`*^9, 3.484484479555605*^9}] }, Open ]], Cell["\<\ Plotting the solution shows the trough propagating to the right.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ RowBox[{"Plot3D", "[", RowBox[{ RowBox[{ RowBox[{"u", "[", RowBox[{"x", ",", "t"}], "]"}], "/.", "\[InvisibleSpace]", RowBox[{"Flatten", "[", "sol", "]"}]}], ",", RowBox[{"{", RowBox[{"x", ",", RowBox[{"-", "7"}], ",", "7"}], "}"}], ",", RowBox[{"{", RowBox[{"t", ",", RowBox[{"-", "1"}], ",", "1"}], "}"}], ",", RowBox[{"PlotPoints", "\[Rule]", "50"}], ",", RowBox[{"PlotRange", "\[Rule]", "All"}], ",", RowBox[{"AxesLabel", "\[Rule]", RowBox[{"{", 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