(*********************************************************************** 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[ 18495, 601]*) (*NotebookOutlinePosition[ 19566, 634]*) (* CellTagsIndexPosition[ 19522, 630]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Slowly varying potential", "Section"], Cell[TextData[{ "Following Wang and Benney [", ButtonBox["WB96", ButtonData:>{"References.nb", "YB96"}, ButtonStyle->"Hyperlink"], "], we develop an approximate theory for the case of a slowly varying \ potential. A more rigorous development can be found therein." }], "Text"], Cell[BoxData[ \(TraditionalForm \`\(\(Suppose\ that\ the\ potential\ is\ slowly\ varying, \ i . e . \)\(\ \)\)\)], "Text"], Cell[BoxData[ \(TraditionalForm\`V = \(V(X) = V(\[Epsilon]\ x)\)\)], "Input"], Cell["and make the ansatz that ", "Text"], Cell[BoxData[ \(TraditionalForm \`u(x, t) = A\ \(sech( A(x\ - x\_0 - 2 B\ t))\)\ \[ExponentialE]\^\(\(i(B\ \((x\ - \ 2\ B\ t)\)\ \ + \ \((A\^2 - B\^2\ + \ V(X))\)\ t\ + \ \[Theta]\_0)\)\(\ \)\)\)], "Input"], Cell[TextData[{ "such that for t = 0, we have a soliton solution of the equation with ", Cell[BoxData[ \(TraditionalForm\`V(\ x) \[Congruent] 0\)], FontWeight->"Bold"], ". Further, suppose that the pulse remains intact as single entity." }], "Text"], Cell["Conservation of \"Mass\" ,", "Text"], Cell[BoxData[ \(TraditionalForm \`M\ = \ \(\[Integral]\[LeftBracketingBar]u(x, t)\[RightBracketingBar]\^2\ \ \[DifferentialD]x = \ \[Integral]\(A\^2\) \(\(sech\^2\)(A(x\ - x\_0 - 2 B\ t))\)\ \[DifferentialD]x\)\)], "Input"], Cell[TextData[{ "implies that to leading order, ", Cell[BoxData[ \(TraditionalForm\`A\)], FontWeight->"Bold"], " must be a constant." }], "Text"], Cell["Conservation of \"Energy\", implies to leading order", "Text"], Cell[BoxData[ \(TraditionalForm \`H\ = \ \[Integral]\[LeftBracketingBar]\(u\_x\)(x, t)\[RightBracketingBar]\^2\ - \[LeftBracketingBar]u(x, t)\[RightBracketingBar]\^4 + \ \(V(X)\)\ \[LeftBracketingBar]u(x, t)\[RightBracketingBar]\^2\ \ \ \[DifferentialD]x \[TildeEqual] \ 2 \( A(B\^2 + V(X)\ - \ \(1\/3\) A\^2)\)\)], "Input"], Cell["which gives the result that", "Text"], Cell[BoxData[ \(TraditionalForm\`B\^2 + V(X)\ = \ constant\)], "Input"], Cell["\<\ This condition can be used to derive some qualitative results.\ \>", "Text"], Cell["\<\ First, we observe that a slowly varying potential will not allow \ transmission if \ \>", "Text"], Cell[BoxData[ \(TraditionalForm\`max(V(X))\ > \ B\_0\^2 + V(X\_0)\)], "Input"], Cell[TextData[{ "In fact, for a periodic potential with period long compared to the initial \ soliton pulse, the pulse can oscillate between reflective boundaries. To \ model this, consider X(T) to be the position of the center of the pulse at \ time T. Take ", Cell[BoxData[ \(TraditionalForm\`\(\(\(X\_0\)\(\ \)\(=\)\)\(\ \)\)\)]], "X(0) = 0, at the center of the well, and " }], "Text"], Cell[BoxData[ \(TraditionalForm \`X(T)\ = 2\ \(\[Integral]\_0\%T B \[DifferentialD]t\)\)], "Input"], Cell["\<\ So we can use the condition as an energy integral to find the \ period\ \>", "Text"], Cell[BoxData[ \(TraditionalForm \`T\ = \ \ 2 \(\[Integral]\_0\%X\_max\( 1\/\@\(B\_0\^2 + V(0)\ - \ V(X)\)\) \[DifferentialD]X\)\)], "Input"], Cell[TextData[{ "Where ", Cell[BoxData[ \(TraditionalForm\`\(\(X\_max\ is\ the\ solution\ of\)\(\ \)\)\)]] }], "Text"], Cell[BoxData[ \(TraditionalForm\`V(X\_max) = B\_0\^2 + V(0)\)], "Input"], Cell[TextData[{ "For a periodic potential V(x) = -a cos(\[Omega] x), and ", Cell[BoxData[ \(TraditionalForm\`B\_0\ = \ 1, \ \ we\ obtain\)]] }], "Text"], Cell[BoxData[ \(\(V[x_]\ := \ \(-a\)\ Cos[\[Omega]\ x];\)\)], "Input", CellOpen->False], Cell[CellGroupData[{ Cell[BoxData[ \(X\_max\ = \ xm\ /. \ Last[Solve[V[xm]\ == \ 1\ + \ V[0], xm]]\)], "Input", CellOpen->False], Cell[BoxData[ \(ArcCos[\(\(-1\) + a\)\/a]\/\[Omega]\)], "Output", CellOpen->False] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(Period[\[Omega]_, \ a_]\ = \ 2 \(\[Integral]\_0\%X\_max\( 1\/\@\(1\ + \ V[0]\ - \ V[X]\)\) \[DifferentialD]X\)\)], "Input", CellOpen->False], Cell[BoxData[ \(\(4\ EllipticF[1\/2\ ArcCos[\(\(-1\) + a\)\/a], 2\ a]\)\/\[Omega]\)], "Output", CellOpen->False] }, Closed]], Cell[BoxData[ \(TraditionalForm \`\(4 \( F(1\/2\ \(\(cos\^\(-1\)\)(\(a - 1\)\/a)\) \[VerticalSeparator] 2\ \ a)\)\)\/\[Omega]\)], "Input"], Cell["For a = 5, ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(\(Plot[Period[\[Omega], 5], {\[Omega], \ 2, 1/8}];\)\)], "Input", CellOpen->False], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def %% Graphics %%IncludeResource: font Courier %%IncludeFont: Courier /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.47619 0.0147151 0.0360736 [ [.2619 .00222 -9 -9 ] [.2619 .00222 9 0 ] [.5 .00222 -3 -9 ] [.5 .00222 3 0 ] [.7381 .00222 -9 -9 ] [.7381 .00222 9 0 ] [.97619 .00222 -3 -9 ] [.97619 .00222 3 0 ] [.01131 .1049 -18 -4.5 ] [.01131 .1049 0 4.5 ] [.01131 .19508 -6 -4.5 ] [.01131 .19508 0 4.5 ] [.01131 .28527 -18 -4.5 ] [.01131 .28527 0 4.5 ] [.01131 .37545 -12 -4.5 ] [.01131 .37545 0 4.5 ] [.01131 .46564 -24 -4.5 ] [.01131 .46564 0 4.5 ] [.01131 .55582 -12 -4.5 ] [.01131 .55582 0 4.5 ] [ 0 0 0 0 ] [ 1 .61803 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .25 Mabswid [ ] 0 setdash .2619 .01472 m .2619 .02097 L s [(0.5)] .2619 .00222 0 1 Mshowa .5 .01472 m .5 .02097 L s [(1)] .5 .00222 0 1 Mshowa .7381 .01472 m .7381 .02097 L s [(1.5)] .7381 .00222 0 1 Mshowa .97619 .01472 m .97619 .02097 L s [(2)] .97619 .00222 0 1 Mshowa .125 Mabswid .07143 .01472 m .07143 .01847 L s .11905 .01472 m .11905 .01847 L s .16667 .01472 m .16667 .01847 L s .21429 .01472 m .21429 .01847 L s .30952 .01472 m .30952 .01847 L s .35714 .01472 m .35714 .01847 L s .40476 .01472 m .40476 .01847 L s .45238 .01472 m .45238 .01847 L s .54762 .01472 m .54762 .01847 L s .59524 .01472 m .59524 .01847 L s .64286 .01472 m .64286 .01847 L s .69048 .01472 m .69048 .01847 L s .78571 .01472 m .78571 .01847 L s .83333 .01472 m .83333 .01847 L s .88095 .01472 m .88095 .01847 L s .92857 .01472 m .92857 .01847 L s .25 Mabswid 0 .01472 m 1 .01472 L s .02381 .1049 m .03006 .1049 L s [(2.5)] .01131 .1049 1 0 Mshowa .02381 .19508 m .03006 .19508 L s [(5)] .01131 .19508 1 0 Mshowa .02381 .28527 m .03006 .28527 L s [(7.5)] .01131 .28527 1 0 Mshowa .02381 .37545 m .03006 .37545 L s [(10)] .01131 .37545 1 0 Mshowa .02381 .46564 m .03006 .46564 L s [(12.5)] .01131 .46564 1 0 Mshowa .02381 .55582 m .03006 .55582 L s [(15)] .01131 .55582 1 0 Mshowa .125 Mabswid .02381 .03275 m .02756 .03275 L s .02381 .05079 m .02756 .05079 L s .02381 .06883 m .02756 .06883 L s .02381 .08686 m .02756 .08686 L s .02381 .12294 m .02756 .12294 L s .02381 .14097 m .02756 .14097 L s .02381 .15901 m .02756 .15901 L s .02381 .17705 m .02756 .17705 L s .02381 .21312 m .02756 .21312 L s .02381 .23116 m .02756 .23116 L s .02381 .24919 m .02756 .24919 L s .02381 .26723 m .02756 .26723 L s .02381 .3033 m .02756 .3033 L s .02381 .32134 m .02756 .32134 L s .02381 .33938 m .02756 .33938 L s .02381 .35741 m .02756 .35741 L s .02381 .39349 m .02756 .39349 L s .02381 .41152 m .02756 .41152 L s .02381 .42956 m .02756 .42956 L s .02381 .4476 m .02756 .4476 L s .02381 .48367 m .02756 .48367 L s .02381 .50171 m .02756 .50171 L s .02381 .51975 m .02756 .51975 L s .02381 .53778 m .02756 .53778 L s .02381 .57386 m .02756 .57386 L s .02381 .59189 m .02756 .59189 L s .02381 .60993 m .02756 .60993 L s .25 Mabswid .02381 0 m .02381 .61803 L s 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath .5 Mabswid .08333 .60332 m .09175 .53041 L .10097 .46881 L .11955 .38065 L .12943 .34642 L .14 .31624 L .15906 .27377 L .17903 .24043 L .19791 .21595 L .23392 .18146 L .25252 .1679 L .27224 .15575 L .30771 .13813 L .34548 .12364 L .38182 .11258 L .42047 .10304 L .4577 .09546 L .49351 .08931 L .53161 .08371 L .56829 .07906 L .60728 .07476 L .64484 .07113 L .68098 .06803 L .71942 .06508 L .75644 .06254 L .79204 .06032 L .82994 .05818 L .86641 .0563 L .90519 .05447 L .94255 .05285 L .97619 .0515 L s % End of Graphics MathPictureEnd \ \>"], "Graphics", ImageSize->{288, 177.938}, ImageMargins->{{43, 0}, {0, 0}}, ImageRegion->{{0, 1}, {0, 1}}, ImageCache->GraphicsData["Bitmap", "\<\ CF5dJ6E]HGAYHf4PAg9QL6QYHg_oo00<00?ooool0aOoo000Kool00`00ooooo`0h ool20038ool001_oo`8003Ooo`800<[oo`006ooo00<00?ooool0=Ooo00<00?ooool0b_oo000Kool0 0`00ooooo`0dool00`00ooooo`3;ool001?oo`<000Goo`03003ooooo03?oo`03003ooooo03oo`002_oo 00<00?ooool01_oo00<00?ooool01Ooo00<00?ooool07ooo00<00?ooool0h?oo0007ool01000oooo o`002?oo00<00?ooool01Ooo00<00?ooool07_oo00<00?ooool0hOoo0007ool40008ool40004ool0 0`00ooooo`0Nool00`00ooooo`3Qool001_oo`8001koo`03003ooooo0>;oo`006ooo00<00?ooool0 7Ooo00<00?ooool0h_oo000Kool00`00ooooo`0Lool00`00ooooo`3Sool001_oo`03003ooooo01co o`03003ooooo0>?oo`006ooo00<00?ooool07?oo00<00?ooool0hooo000Kool2000Lool00`00oooo o`3Tool001_oo`03003ooooo01_oo`03003ooooo0>Coo`006ooo00<00?ooool06ooo00<00?ooool0 i?oo000Kool00`00ooooo`0Jool00`00ooooo`3Uool001_oo`8001_oo`03003ooooo0>Goo`006ooo 00<00?ooool06Ooo00<00?ooool0i_oo000Kool00`00ooooo`0Iool00`00ooooo`3Vool001_oo`03 003ooooo01Woo`03003ooooo0>Koo`006ooo00<00?ooool06?oo00<00?ooool0iooo000Kool2000I ool00`00ooooo`3Wool001_oo`03003ooooo01Soo`03003ooooo0>Ooo`006ooo00<00?ooool05ooo 00<00?ooool0j?oo000=ool40003ool20005ool00`00ooooo`0Gool00`00ooooo`3Xool000ooo`05 003oooooool00002ool00`00ooooo`02ool00`00ooooo`0Gool00`00ooooo`3Xool000ooo`05003o ooooool00002ool00`00ooooo`02ool2000Gool00`00ooooo`3Yool000ooo`05003oooooool00002 ool00`00ooooo`02ool00`00ooooo`0Fool00`00ooooo`3Yool000ooo`05003oooooool00002ool0 0`00ooooo`02ool00`00ooooo`0Fool00`00ooooo`3Yool000goo`<000?oo`04003ooooo0004ool0 0`00ooooo`0Fool00`00ooooo`3Yool000ooo`03003ooooo00;oo`8000Goo`03003ooooo01Goo`03 003ooooo0>[oo`006ooo0P005_oo00<00?ooool0j_oo000Kool00`00ooooo`0Eool00`00ooooo`3Z ool001_oo`03003ooooo01Goo`03003ooooo0>[oo`006ooo00<00?ooool05Ooo00<00?ooool0j_oo 000Kool00`00ooooo`0Dool00`00ooooo`3[ool001_oo`8001Goo`03003ooooo0>_oo`006ooo00<0 0?ooool05?oo00<00?ooool0jooo000Kool00`00ooooo`0Dool00`00ooooo`3[ool001_oo`03003o oooo01Coo`03003ooooo0>_oo`006ooo0P005?oo00<00?ooool0k?oo000Kool00`00ooooo`0Cool0 0`00ooooo`3/ool001_oo`03003ooooo01?oo`03003ooooo0>coo`006ooo00<00?ooool04ooo00<0 0?ooool0k?oo000Kool00`00ooooo`0Bool00`00ooooo`3]ool001_oo`8001?oo`03003ooooo0>go o`006ooo00<00?ooool04_oo00<00?ooool0kOoo000Kool00`00ooooo`0Bool00`00ooooo`3]ool0 0003ool00000008000;oo`@000?oo`03003ooooo00;oo`<000Goo`03003ooooo01;oo`03003ooooo 0>goo`000ooo00D00?ooooooo`0000koo`03003ooooo00;oo`03003ooooo017oo`03003ooooo0>ko o`000ooo00<00?ooool00_oo00<00?ooool02ooo00<00?ooool00_oo0P004_oo00<00?ooool0k_oo 0003ool00`00ooooo`03ool00`00ooooo`07ool30005ool00`00ooooo`0Aool00`00ooooo`3^ool0 00?oo`03003ooooo00Coo`03003ooooo00Koo`03003ooooo00Goo`03003ooooo017oo`03003ooooo 0>koo`0000Coo`00000000?oo`04003ooooo0008ool00`00ooooo`05ool00`00ooooo`0Aool00`00 ooooo`3^ool000?oo`03003ooooo00;oo`8000Woo`@000Coo`03003ooooo017oo`03003ooooo0>ko o`006ooo0P004_oo00<00?ooool0k_oo000Kool00`00ooooo`0@ool00`00ooooo`3_ool001_oo`03 003ooooo013oo`03003ooooo0>ooo`006ooo00<00?ooool04?oo00<00?ooool0kooo000Kool2000A ool00`00ooooo`3_ool001_oo`03003ooooo013oo`03003ooooo0>ooo`006ooo00<00?ooool04?oo 00<00?ooool0kooo000Kool00`00ooooo`0@ool00`00ooooo`3_ool001_oo`03003ooooo013oo`03 003ooooo0>ooo`006ooo0P004?oo00<00?ooool0l?oo000Kool00`00ooooo`0?ool00`00ooooo`3` ool001_oo`03003ooooo00ooo`03003ooooo0?3oo`006ooo00<00?ooool03ooo00<00?ooool0l?oo 000Kool00`00ooooo`0?ool00`00ooooo`3`ool001_oo`80013oo`03003ooooo0?3oo`006ooo00<0 0?ooool03ooo00<00?ooool0l?oo000Kool00`00ooooo`0?ool00`00ooooo`3`ool000goo`@000;o o`<000Goo`03003ooooo00ooo`03003ooooo0?3oo`003ooo00<00?ooool01?oo00<00?ooool00_oo 00<00?ooool03_oo00<00?ooool0lOoo000?ool00`00ooooo`04ool00`00ooooo`02ool2000?ool0 0`00ooooo`3aool000ooo`04003oooooool30005ool00`00ooooo`0>ool00`00ooooo`3aool000oo o`05003oooooool00007ool00`00ooooo`0>ool00`00ooooo`3aool000goo`<000?oo`03003ooooo 00Goo`03003ooooo00koo`03003ooooo0?7oo`003ooo00@00?ooooooo`@000Coo`03003ooooo00ko o`03003ooooo0?7oo`006ooo0P003ooo00<00?ooool0lOoo000Kool00`00ooooo`0>ool00`00oooo o`3aool001_oo`03003ooooo00koo`03003ooooo0?7oo`006ooo00<00?ooool03_oo00<00?ooool0 lOoo000Kool2000>ool00`00ooooo`3bool001_oo`03003ooooo00goo`03003ooooo0?;oo`006ooo 00<00?ooool03Ooo00<00?ooool0l_oo000Kool00`00ooooo`0=ool00`00ooooo`3bool001_oo`03 003ooooo00goo`03003ooooo0?;oo`006ooo0P00oooo1?oo000Kool00`00ooooo`3oool3ool001_o o`03003ooooo0?ooo`?oo`00oooo8Ooo003ooolQool00001\ \>"], ImageRangeCache->{{{0, 287}, {176.938, 0}} -> {-0.215729, -1.4994, 0.00789457, 0.104212}}] }, Open ]] }, Open ]] }, FrontEndVersion->"4.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 695}}, ScreenStyleEnvironment->"Presentation", WindowSize->{520, 600}, WindowMargins->{{63, Automatic}, {47, Automatic}}, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PaperSize"->{612, 792}, "PaperOrientation"->"Portrait", "PostScriptOutputFile":>FrontEnd`FileName[{"home", "rknapp", "Research", "NLS", "Periodic", "Athens"}, "SlowlyVaryingPotential.nb.ps", CharacterEncoding -> "WindowsANSI"], "Magnification"->1}, 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. The cache data will then be recreated when you save this file from within Mathematica. ***********************************************************************) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[CellGroupData[{ Cell[1739, 51, 43, 0, 69, "Section"], Cell[1785, 53, 291, 7, 95, "Text"], Cell[2079, 62, 135, 3, 41, "Text"], Cell[2217, 67, 81, 1, 39, "Input"], Cell[2301, 70, 41, 0, 45, "Text"], Cell[2345, 72, 252, 6, 68, "Input"], Cell[2600, 80, 265, 6, 70, "Text"], Cell[2868, 88, 42, 0, 45, "Text"], Cell[2913, 90, 266, 7, 56, "Input"], Cell[3182, 99, 161, 6, 45, "Text"], Cell[3346, 107, 68, 0, 45, "Text"], Cell[3417, 109, 373, 7, 107, "Input"], Cell[3793, 118, 43, 0, 45, "Text"], Cell[3839, 120, 76, 1, 40, "Input"], Cell[3918, 123, 89, 3, 45, "Text"], Cell[4010, 128, 107, 3, 70, "Text"], Cell[4120, 133, 83, 1, 40, "Input"], Cell[4206, 136, 402, 8, 120, "Text"], Cell[4611, 146, 108, 2, 57, "Input"], Cell[4722, 150, 94, 3, 45, "Text"], Cell[4819, 155, 172, 4, 83, "Input"], Cell[4994, 161, 128, 4, 45, "Text"], Cell[5125, 167, 76, 1, 40, "Input"], Cell[5204, 170, 163, 4, 45, "Text"], Cell[5370, 176, 94, 2, 19, "Input", CellOpen->False], Cell[CellGroupData[{ Cell[5489, 182, 120, 3, 19, "Input", CellOpen->False], Cell[5612, 187, 88, 2, 50, "Output", CellOpen->False] }, Closed]], Cell[CellGroupData[{ Cell[5737, 194, 185, 4, 19, "Input", CellOpen->False], Cell[5925, 200, 121, 3, 50, "Output", CellOpen->False] }, Closed]], Cell[6061, 206, 146, 3, 72, "Input"], Cell[6210, 211, 27, 0, 45, "Text"], Cell[CellGroupData[{ Cell[6262, 215, 104, 2, 19, "Input", CellOpen->False], Cell[6369, 219, 12098, 378, 186, 3788, 271, "GraphicsData", "PostScript", "Graphics"] }, Open ]] }, Open ]] } ] *) (*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)