(*********************************************************************** 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). 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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[ 1290506, 59372]*) (*NotebookOutlinePosition[ 1291259, 59398]*) (* CellTagsIndexPosition[ 1291215, 59394]*) (*WindowFrame->Normal*) Notebook[{ Cell[CellGroupData[{ Cell["Numerical simulations with a slowly varying potential", "Section"], Cell[CellGroupData[{ Cell["Example with small \[Omega]", "Subsection"], Cell["\<\ For illustration, I will show an example where \[Omega] is such \ that the potential clearly varies on a different length scale than the pulse.\ \ \>", "Text"], Cell["With", "Text"], Cell[BoxData[ \(TraditionalForm \`V(x) = \(-5\)\ \(cos(x\/6)\)\ \((\[Omega] = 1\/6)\)\)], "Input"], Cell[TextData[{ "and ", Cell[BoxData[ \(TraditionalForm\`A\_0 = \ 1\)]], ", ", Cell[BoxData[ \(TraditionalForm\`B\_0 = \ 1\)]], ", the ", ButtonBox["solution envelope ", ButtonData:>{"SVPAnimate.nb", None}, ButtonStyle->"Hyperlink"], " evolves somewhat as predicted." }], "Text"], Cell[BoxData[{ \(\(L\ = \ 12\ \[Pi];\)\), "\[IndentingNewLine]", \(\(n\ = \ 360;\)\), "\[IndentingNewLine]", \(\(grid\ = \ N[Discretize[x, {x, \(-L\), L}, n]];\)\), "\[IndentingNewLine]", \(\(uinit\ = \ \(Soliton[1, 1]\)[grid, \ 0];\)\), "\[IndentingNewLine]", \(\(toutput\ = \ 0.25;\)\), "\[IndentingNewLine]", \(\(dt\ = \ 2. ^\(-6\);\)\)}], "Input", CellOpen->False], Cell[CellGroupData[{ Cell[BoxData[ \(First[ Timing[\(examplebegin\ = \ SplitStepSolve[uinit, \ {\(-L\), L}, {{0, 50, toutput}, dt}, \ Potential \[Rule] \(-5\)\ Cos[x/6], \ Output \[Rule] {Mass, \ Centroid, \ Momentum, \ Value}, \ AbsorbingBoundary \[Rule] True];\)]]\)], "Input", CellOpen->False], Cell[BoxData[ \(43.5`\ Second\)], "Output", CellOpen->False] }, Closed]], Cell[TextData[{ "From this, one would get the idea that the pulse is slowly going to break \ down, but if we continue the simulation, and look at the ", ButtonBox["solution envelope", ButtonData:>{"SVPAnimate.nb", None}, ButtonStyle->"Hyperlink"], " after the solution has evolved for a while, we can see how well it does." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(First[ Timing[\(examplemiddle\ = \ SplitStepSolve[ulast, \ {\(-L\), L}, {{50, 1200, toutput}, dt}, \ Potential \[Rule] \(-5\)\ Cos[x/6], \ Output \[Rule] {Mass, \ Centroid, \ Momentum}, \ AbsorbingBoundary \[Rule] True];\)]]\)], "Input", CellOpen->False], Cell[BoxData[ \(964.2699999999968`\ Second\)], "Output", CellOpen->False] }, Closed]], Cell[CellGroupData[{ Cell[BoxData[ \(First[ Timing[\(exampleend\ = \ SplitStepSolve[ulast, \ {\(-L\), L}, {{1200, 1250, toutput}, dt}, \ Potential \[Rule] \(-5\)\ Cos[x/6], \ Output \[Rule] {Mass, \ Centroid, \ Momentum, \ Value}, \ AbsorbingBoundary \[Rule] True];\)]]\)], "Input", CellOpen->False], Cell[BoxData[ \(43.61000000000058`\ Second\)], "Output", CellOpen->False] }, Closed]], Cell[BoxData[ \(\(example\ = \ Join[examplebegin, \ Drop[examplemiddle, 1], \ Drop[exampleend, 1]]; \)\)], "Input", CellOpen->False], Cell[BoxData[ \(\(example >> "\";\)\)], "Input", CellOpen->False], Cell["\<\ Since it is a bit hard on the eyes to look at the envelopes, a \ better way of tracking the quantities of interest is to look at overall \ values. 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.19615 L .62648 .19628 L .62664 .19638 L .62679 .19645 L .62695 .1965 L .62711 .19652 L .62725 .19652 L .62741 .1965 L .6275 .19647 L .62758 .19644 L .62775 .19636 L .62793 .19624 L .62829 .1959 L .62864 .19545 L .62929 .19429 L .62988 .19289 L .63052 .19097 L .63182 .18591 L .63415 .1731 L .6366 .15505 L .63924 .13173 L .64444 .08042 L .64705 .05598 L .64946 .0362 L .6518 .02082 L .65298 .01474 L .65362 .01199 L .65429 .00949 L Mistroke .65489 .00759 L .65556 .00592 L .65585 .00532 L .65616 .00477 L .65645 .00434 L .65672 .00403 L .65706 .00373 L .65724 .00362 L .65733 .00357 L .65743 .00354 L .6576 .00349 L .65769 .00348 L .65777 .00348 L .65794 .00349 L .6581 .00353 L .65825 .00359 L .65841 .00368 L .6587 .00391 L .65904 .00428 L .65935 .00472 L .65996 .00588 L .66061 .0075 L .66178 .01141 L .66303 .01702 L .6644 .02474 L .6695 .06514 L .6744 .11331 L .6769 .13731 L .67962 .16046 L .68088 .16971 L .68223 .17822 L .68349 .18484 L .68464 .18966 L .68582 .19331 L .68644 .19468 L .68676 .19524 L .68709 .19572 L .68739 .19605 L .68767 .19628 L .6878 .19636 L .68794 .19643 L .68807 .19648 L .6882 .19651 L .68834 .19652 L .68847 .19652 L .68862 .19649 L .68878 .19643 L .68895 .19635 L .6891 .19625 L .6894 .19599 L Mistroke .68973 .19559 L .69004 .19513 L .69061 .194 L .69123 .19243 L .6919 .19032 L .69311 .18544 L .69422 .1798 L .6966 .16432 L .69921 .14293 L .70431 .09352 L .70922 .0475 L .7119 .02742 L .71314 .01999 L .71444 .01352 L .71564 .00894 L .71632 .00694 L .71696 .00547 L .71726 .00492 L .71754 .00448 L .7178 .00415 L .71808 .00387 L .71823 .00374 L .71837 .00365 L .71852 .00357 L .71865 .00352 L .71881 .00349 L .71895 .00348 L .71911 .00349 L .71919 .00351 L .71928 .00353 L .71944 .0036 L .7196 .00368 L .71976 .0038 L .71994 .00396 L .72024 .0043 L .72057 .00477 L .72117 .00592 L .72174 .00732 L .72301 .01158 L .72437 .01782 L .7269 .03351 L .72955 .05491 L .73436 .10102 L .73946 .14955 L .74182 .16801 L .74438 .18344 L .74578 .1895 L .7464 .19158 L .74707 .19346 L .74771 .19483 L Mistroke .74801 .19534 L .74831 .19575 L .74863 .1961 L .74878 .19622 L .74893 .19633 L .74902 .19638 L .7491 .19642 L .74926 .19648 L .74935 .1965 L .74944 .19652 L .74953 .19652 L .74961 .19652 L .74977 .1965 L .74992 .19645 L .75006 .19639 L .7502 .19631 L .75053 .19604 L .75084 .19569 L .7514 .19482 L .75199 .19357 L .75323 .18984 L .75459 .18405 L .75703 .1697 L .75966 .14931 L .76485 .09995 L .76985 .05225 L .77237 .03233 L .77466 .0182 L .77593 .01224 L .77713 .00799 L .77779 .00623 L .7784 .005 L .77872 .00449 L .77907 .00406 L .77924 .00388 L .77943 .00373 L .77961 .00362 L .77977 .00355 L .77992 .0035 L .78006 .00348 L .78021 .00348 L .78038 .00351 L .78054 .00356 L .78062 .00359 L .78071 .00364 L .78102 .00386 L .78131 .00416 L .78162 .00456 L .78217 .00553 L .78288 .00719 L Mistroke .78354 .0092 L .78479 .01409 L .78601 .02024 L .78731 .02814 L .78963 .04543 L .7944 .08959 L .79948 .13914 L .80181 .15907 L .80436 .17681 L .80574 .18428 L .80703 .1897 L .80766 .19181 L .80825 .19344 L .80858 .1942 L .80894 .1949 L .80924 .1954 L .80956 .19583 L .80972 .19601 L .80988 .19615 L .81004 .19627 L .81018 .19636 L .81031 .19643 L .81045 .19648 L .8106 .19651 L .81076 .19652 L .81084 .19652 L .81093 .1965 L .81108 .19646 L .81117 .19643 L .81125 .19639 L .81143 .19628 L .81174 .19602 L .81207 .19563 L .81241 .19512 L .81279 .19443 L .81346 .19283 L .81473 .18866 L .81592 .18337 L .81723 .1761 L .81988 .15733 L .82497 .11066 L .82986 .06278 L .83235 .04138 L .83366 .03159 L .83506 .02256 L .83643 .01529 L .83766 .01014 L .83826 .00816 L .83891 .00641 L .83951 .00513 L Mistroke .84007 .00427 L .84038 .00393 L .84067 .0037 L .84082 .00361 L .84091 .00357 L .84099 .00354 L .84116 .00349 L .8413 .00348 L .84144 .00349 L .84159 .00351 L .84172 .00356 L .84185 .00361 L .842 .0037 L .84216 .00382 L .84245 .00411 L .84279 .00455 L .8431 .00505 L .8438 .00652 L .84445 .0083 L .84506 .01035 L .84629 .01552 L .84758 .02246 L .84991 .03825 L .85469 .08078 L .85978 .13097 L .86213 .152 L .86468 .17133 L .86607 .17983 L .86736 .18625 L .86859 .19099 L .86921 .19285 L .86989 .19446 L .87051 .19552 L .87082 .19591 L .871 .19609 L .87116 .19623 L .8713 .19633 L .87145 .19641 L .87161 .19647 L .87177 .19651 L .87191 .19652 L .87206 .19651 L .87219 .19649 L .87233 .19644 L .87241 .1964 L .8725 .19635 L .87267 .19624 L .87298 .19596 L .87335 .19551 L .87369 .19498 L Mistroke .8743 .19371 L .87496 .19196 L .87625 .18729 L .87766 .1805 L .8802 .16406 L .88537 .11842 L .89035 .06931 L .89285 .04692 L .89514 .02962 L .8976 .01538 L .89825 .01249 L .89895 .00978 L .89961 .00768 L .90023 .00608 L .9008 .00495 L .9011 .00449 L .90142 .00408 L .90175 .00377 L .90193 .00365 L .90202 .0036 L .9021 .00357 L .90226 .00351 L .9024 .00349 L .90256 .00348 L .90265 .00349 L .90273 .0035 L .90288 .00354 L .90304 .00362 L .9032 .00371 L .90334 .00382 L .90369 .00416 L .90401 .0046 L .90467 .00578 L .90538 .00753 L .90668 .01203 L .90786 .01745 L .91051 .03386 L .91557 .07771 L .92043 .12585 L .9229 .14849 L .92561 .16956 L .92686 .17753 L .92819 .18463 L .92945 .18988 L .9306 .19337 L .93121 .19471 L .93154 .19528 L .93185 .19572 L .93212 .19602 L .9324 .19627 L Mistroke .93256 .19637 L .93272 .19644 L .93286 .19649 L .93301 .19652 L .93309 .19652 L .93319 .19652 L .93327 .19651 L .93335 .19649 L .93351 .19644 L .93367 .19637 L .93384 .19626 L .934 .19613 L .93437 .19574 L .93469 .19529 L .93498 .19479 L .93563 .19337 L .93623 .19169 L .93688 .18951 L .93803 .18464 L .94065 .16934 L .9431 .15038 L .9454 .12956 L .95028 .08144 L .95275 .05806 L .95546 .03574 L .95671 .02705 L .95805 .01911 L .9593 .01305 L .96045 .00873 L .96108 .00691 L .96175 .0054 L .96207 .00482 L .96241 .00432 L .96261 .00408 L .96279 .0039 L .96296 .00376 L .96314 .00364 L .96323 .00359 L .96331 .00356 L .96347 .00351 L .96361 .00348 L .96377 .00348 L .96385 .00349 L .96394 .00351 L .96403 .00353 L .96412 .00357 L .96428 .00364 L .96445 .00375 L .96478 .00405 L .96509 .00444 L Mistroke .96568 .00543 L .96626 .00673 L .9669 .00855 L .9682 .01341 L .9694 .01927 L .97052 .02588 L .9754 .06482 L .98059 .11578 L .98318 .14044 L .98559 .16073 L .98792 .17675 L .98921 .1838 L .9904 .18899 L .99101 .19116 L .99169 .19313 L .99237 .19466 L .99274 .19531 L .99309 .19579 L .99325 .19597 L .99342 .19613 L .99357 .19625 L .99373 .19636 L .99388 .19643 L .99404 .19649 L .99418 .19651 L .99432 .19652 L .99448 .1965 L .99456 .19649 L .99465 .19646 L .9948 .19639 L .99495 .19631 L .99528 .19604 L .99563 .19564 L .99619 .19474 L .99671 .19363 L .99788 .19018 L .99891 .18608 L 1 .18068 L Mfstroke 0 g .008 w 0 .1 Mdot .00125 .11246 Mdot .0025 .12472 Mdot .00375 .13654 Mdot .005 .14775 Mdot .00625 .15815 Mdot .0075 .16758 Mdot .00875 .17588 Mdot .01 .18294 Mdot .01125 .18863 Mdot .0125 .19289 Mdot .01375 .19565 Mdot .015 .19688 Mdot .01625 .19656 Mdot .0175 .1947 Mdot .01875 .19132 Mdot .02 .18647 Mdot .02125 .18022 Mdot .0225 .17265 Mdot .02375 .16389 Mdot .025 .15407 Mdot .02625 .14334 Mdot .0275 .13187 Mdot .02875 .11986 Mdot .03 .1075 Mdot .03125 .09502 Mdot .0325 .08262 Mdot .03375 .07053 Mdot .035 .05893 Mdot .03625 .04803 Mdot .0375 .038 Mdot .03875 .02901 Mdot .04 .02118 Mdot .04125 .01465 Mdot .0425 .0095 Mdot .04375 .00582 Mdot .045 .00365 Mdot .04625 .00303 Mdot .0475 .00396 Mdot .04875 .00644 Mdot .05 .01042 Mdot .05125 .01585 Mdot .0525 .02265 Mdot .05375 .03072 Mdot .055 .03994 Mdot .05625 .05016 Mdot .0575 .06121 Mdot .05875 .07292 Mdot .06 .0851 Mdot .06125 .09752 Mdot .0625 .10999 Mdot .06375 .12229 Mdot .065 .13421 Mdot .06625 .14554 Mdot .0675 .15611 Mdot .06875 .16573 Mdot .07 .17425 Mdot .07125 .18155 Mdot .0725 .18753 Mdot .07375 .19208 Mdot .075 .19516 Mdot .07625 .19671 Mdot .0775 .19671 Mdot .07875 .19517 Mdot .08 .19211 Mdot .08125 .18756 Mdot .0825 .18158 Mdot .08375 .17428 Mdot .085 .16575 Mdot .08625 .15612 Mdot .0875 .14555 Mdot .08875 .13422 Mdot .09 .1223 Mdot .09125 .11 Mdot .0925 .09753 Mdot .09375 .0851 Mdot .095 .07293 Mdot .09625 .06122 Mdot .0975 .05017 Mdot .09875 .03996 Mdot .1 .03075 Mdot .10125 .02269 Mdot .1025 .0159 Mdot .10375 .01049 Mdot .105 .00653 Mdot .10625 .00407 Mdot .1075 .00315 Mdot .10875 .00377 Mdot .11 .00594 Mdot .11125 .00962 Mdot .1125 .01475 Mdot .11375 .02127 Mdot .115 .02907 Mdot .11625 .03804 Mdot .1175 .04805 Mdot .11875 .05893 Mdot .12 .0705 Mdot .12125 .08258 Mdot .1225 .09495 Mdot .12375 .10742 Mdot .125 .11975 Mdot .12625 .13174 Mdot .1275 .1432 Mdot .12875 .15392 Mdot .13 .16374 Mdot .13125 .17249 Mdot .1325 .18005 Mdot .13375 .18631 Mdot .135 .19116 Mdot .13625 .19454 Mdot .1375 .1964 Mdot .13875 .19672 Mdot .14 .19549 Mdot .14125 .19272 Mdot .1425 .18846 Mdot .14375 .18276 Mdot .145 .17572 Mdot .14625 .16743 Mdot .1475 .15803 Mdot .14875 .14766 Mdot .15 .13648 Mdot .15125 .12469 Mdot .1525 .11248 Mdot .15375 .10006 Mdot .155 .08764 Mdot .15625 .07543 Mdot .1575 .06364 Mdot .15875 .05247 Mdot .16 .04211 Mdot .16125 .03272 Mdot .1625 .02445 Mdot .16375 .01742 Mdot .165 .01173 Mdot .16625 .00748 Mdot .1675 .00471 Mdot .16875 .00347 Mdot .17 .00377 Mdot .17125 .00561 Mdot .1725 .00897 Mdot .17375 .01379 Mdot .175 .02002 Mdot .17625 .02755 Mdot .1775 .03628 Mdot .17875 .04607 Mdot .18 .05676 Mdot .18125 .06819 Mdot .1825 .08015 Mdot .18375 .09245 Mdot .185 .10488 Mdot .18625 .11722 Mdot .1875 .12927 Mdot .18875 .14082 Mdot .19 .15167 Mdot .19125 .16165 Mdot .1925 .1706 Mdot .19375 .17839 Mdot .195 .18488 Mdot .19625 .18999 Mdot .1975 .19365 Mdot .19875 .19579 Mdot .2 .19641 Mdot .20125 .19547 Mdot .2025 .19301 Mdot .20375 .18906 Mdot .205 .18366 Mdot .20625 .17691 Mdot .2075 .16889 Mdot .20875 .15974 Mdot .21 .14959 Mdot .21125 .1386 Mdot .2125 .12697 Mdot .21375 .11487 Mdot .215 .10252 Mdot .21625 .09014 Mdot .2175 .07792 Mdot .21875 .06609 Mdot .22 .05483 Mdot .22125 .04434 Mdot .2225 .03479 Mdot .22375 .02632 Mdot .225 .01906 Mdot .22625 .01313 Mdot .2275 .0086 Mdot .22875 .00555 Mdot .23 .00402 Mdot .23125 .00403 Mdot .2325 .00557 Mdot .23375 .00864 Mdot .235 .01318 Mdot .23625 .01912 Mdot .2375 .02639 Mdot .23875 .03488 Mdot .24 .04444 Mdot .24125 .05493 Mdot .2425 .06619 Mdot .24375 .07801 Mdot .245 .09021 Mdot .24625 .10257 Mdot .2475 .11488 Mdot .24875 .12694 Mdot .25 .13853 Mdot .25 .13853 Mdot .25125 .14947 Mdot .2525 .15957 Mdot .25375 .16867 Mdot .255 .17663 Mdot .25625 .18332 Mdot .2575 .18866 Mdot .25875 .19256 Mdot .26 .19498 Mdot .26125 .19587 Mdot .2625 .19522 Mdot .26375 .19305 Mdot .265 .18938 Mdot .26625 .18427 Mdot .2675 .17778 Mdot .26875 .17002 Mdot .27 .1611 Mdot .27125 .15117 Mdot .2725 .14037 Mdot .27375 .12888 Mdot .275 .11691 Mdot .27625 .10465 Mdot .2775 .09231 Mdot .27875 .08011 Mdot .28 .06825 Mdot .28125 .05693 Mdot .2825 .04635 Mdot .28375 .03667 Mdot .285 .02804 Mdot .28625 .02061 Mdot .2875 .01448 Mdot .28875 .00974 Mdot .29 .00647 Mdot .29125 .0047 Mdot .2925 .00446 Mdot .29375 .00575 Mdot .295 .00856 Mdot .29625 .01284 Mdot .2975 .01854 Mdot .29875 .02556 Mdot .3 .03381 Mdot .30125 .04316 Mdot .3025 .05346 Mdot .30375 .06455 Mdot .305 .07624 Mdot .30625 .08833 Mdot .3075 .10062 Mdot .30875 .11289 Mdot .31 .12495 Mdot .31125 .13657 Mdot .3125 .14757 Mdot .31375 .15777 Mdot .315 .16699 Mdot .31625 .1751 Mdot .3175 .18197 Mdot .31875 .18751 Mdot .32 .19162 Mdot .32125 .19426 Mdot .3225 .19539 Mdot .32375 .19498 Mdot .325 .19305 Mdot .32625 .18961 Mdot .3275 .18473 Mdot .32875 .17847 Mdot .33 .17091 Mdot .33125 .16219 Mdot .3325 .15242 Mdot .33375 .14178 Mdot .335 .13042 Mdot .33625 .11855 Mdot .3375 .10636 Mdot .33875 .09407 Mdot .34 .08188 Mdot .34125 .07 Mdot .3425 .05863 Mdot .34375 .04797 Mdot .345 .03818 Mdot .34625 .02943 Mdot .3475 .02185 Mdot .34875 .01556 Mdot .35 .01064 Mdot .35125 .00717 Mdot .3525 .00519 Mdot .35375 .00473 Mdot .355 .0058 Mdot .35625 .00839 Mdot .3575 .01245 Mdot .35875 .01793 Mdot .36 .02475 Mdot .36125 .03281 Mdot .3625 .04199 Mdot .36375 .05214 Mdot .365 .06311 Mdot .36625 .0747 Mdot .3675 .08672 Mdot .36875 .09897 Mdot .37 .11124 Mdot .37125 .12331 Mdot .3725 .13499 Mdot .37375 .14607 Mdot .375 .15637 Mdot .37625 .16573 Mdot .3775 .17399 Mdot .37875 .18103 Mdot .38 .18675 Mdot .38125 .19105 Mdot .3825 .19388 Mdot .38375 .19521 Mdot .385 .195 Mdot .38625 .19326 Mdot .3875 .19003 Mdot .38875 .18534 Mdot .39 .17926 Mdot .39125 .17188 Mdot .3925 .16332 Mdot .39375 .15369 Mdot .395 .14317 Mdot .39625 .13191 Mdot .3975 .12011 Mdot .39875 .10796 Mdot .4 .09567 Mdot .40125 .08346 Mdot .4025 .07153 Mdot .40375 .06008 Mdot .405 .04932 Mdot .40625 .0394 Mdot .4075 .0305 Mdot .40875 .02275 Mdot .41 .01627 Mdot .41125 .01116 Mdot .4125 .00748 Mdot .41375 .00529 Mdot .415 .00462 Mdot .41625 .00548 Mdot .4175 .00786 Mdot .41875 .01173 Mdot .42 .01703 Mdot .42125 .02368 Mdot .4225 .03159 Mdot .42375 .04064 Mdot .425 .05068 Mdot .42625 .06156 Mdot .4275 .07309 Mdot .42875 .08508 Mdot .43 .09733 Mdot .43125 .10963 Mdot .4325 .12176 Mdot .43375 .13352 Mdot .435 .14472 Mdot .43625 .15516 Mdot .4375 .16467 Mdot .43875 .1731 Mdot .44 .18033 Mdot .44125 .18624 Mdot .4425 .19075 Mdot .44375 .1938 Mdot .445 .19534 Mdot .44625 .19535 Mdot .4475 .19383 Mdot .44875 .1908 Mdot .45 .1863 Mdot .45125 .18041 Mdot .4525 .1732 Mdot .45375 .16478 Mdot .455 .15528 Mdot .45625 .14485 Mdot .4575 .13365 Mdot .45875 .12189 Mdot .46 .10974 Mdot .46125 .09743 Mdot .4625 .08516 Mdot .46375 .07314 Mdot .465 .06158 Mdot .46625 .05066 Mdot .4675 .04058 Mdot .46875 .03149 Mdot .47 .02354 Mdot .47125 .01684 Mdot .4725 .01149 Mdot .47375 .00758 Mdot .475 .00516 Mdot .47625 .00426 Mdot .4775 .0049 Mdot .47875 .00706 Mdot .48 .01072 Mdot .48125 .01583 Mdot .4825 .02231 Mdot .48375 .03006 Mdot .485 .03898 Mdot .48625 .04891 Mdot .4875 .0597 Mdot .48875 .07118 Mdot .49 .08316 Mdot .49125 .09542 Mdot .4925 .10777 Mdot .49375 .11998 Mdot .495 .13186 Mdot .49625 .14319 Mdot .4975 .1538 Mdot .49875 .1635 Mdot .5 .17214 Mdot .50125 .1796 Mdot .5025 .18575 Mdot .50375 .19051 Mdot .505 .19381 Mdot .50625 .1956 Mdot .5075 .19586 Mdot .50875 .19459 Mdot .51 .19179 Mdot .51125 .18752 Mdot .5125 .18183 Mdot .51375 .1748 Mdot .515 .16654 Mdot .51625 .15716 Mdot .5175 .14683 Mdot .51875 .13571 Mdot .52 .12397 Mdot .52125 .11183 Mdot .5225 .09947 Mdot .52375 .08713 Mdot .525 .075 Mdot .52625 .0633 Mdot .5275 .05222 Mdot .52875 .04195 Mdot .53 .03264 Mdot .53125 .02445 Mdot .5325 .0175 Mdot .53375 .0119 Mdot .535 .00773 Mdot .53625 .00505 Mdot .5375 .00389 Mdot .53875 .00427 Mdot .54 .00618 Mdot .54125 .0096 Mdot .5425 .01448 Mdot .54375 .02075 Mdot .545 .02832 Mdot .54625 .03707 Mdot .5475 .04687 Mdot .54875 .05757 Mdot .55 .06898 Mdot .55125 .08093 Mdot .5525 .09321 Mdot .55375 .1056 Mdot .555 .1179 Mdot .55625 .1299 Mdot .5575 .14138 Mdot .55875 .15217 Mdot .56 .16209 Mdot .56125 .17097 Mdot .5625 .17868 Mdot .56375 .1851 Mdot .565 .19014 Mdot .56625 .19372 Mdot .5675 .1958 Mdot .56875 .19635 Mdot .57 .19535 Mdot .57125 .19281 Mdot .5725 .18879 Mdot .57375 .18332 Mdot .575 .17649 Mdot .57625 .16841 Mdot .5775 .15919 Mdot .57875 .14898 Mdot .58 .13795 Mdot .58125 .12627 Mdot .5825 .11414 Mdot .58375 .10176 Mdot .585 .08935 Mdot .58625 .07713 Mdot .5875 .06529 Mdot .58875 .05404 Mdot .59 .04357 Mdot .59125 .03405 Mdot .5925 .02562 Mdot .59375 .01841 Mdot .595 .01254 Mdot .59625 .00809 Mdot .5975 .00511 Mdot .59875 .00365 Mdot .6 .00374 Mdot .60125 .00536 Mdot .6025 .00851 Mdot .60375 .01312 Mdot .605 .01915 Mdot .60625 .0265 Mdot .6075 .03506 Mdot .60875 .04471 Mdot .61 .05528 Mdot .61125 .06662 Mdot .6125 .07852 Mdot .61375 .09079 Mdot .615 .10322 Mdot .61625 .1156 Mdot .6175 .12771 Mdot .61875 .13935 Mdot .62 .15033 Mdot .62125 .16046 Mdot .6225 .16958 Mdot .62375 .17755 Mdot .625 .18425 Mdot .62625 .18957 Mdot .6275 .19345 Mdot .62875 .19582 Mdot .63 .19666 Mdot .63125 .19594 Mdot .6325 .19369 Mdot .63375 .18993 Mdot .635 .18472 Mdot .63625 .17813 Mdot .6375 .17026 Mdot .63875 .16123 Mdot .64 .15118 Mdot .64125 .14027 Mdot .6425 .12868 Mdot .64375 .11659 Mdot .645 .10422 Mdot .64625 .09177 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While the predicted amplitude is \ very close, it is clear that the predicted period is slightly too small. \ \ \>", "Text"], Cell[TextData[{ "Since the numerical solution is so regular,it is appropriate to find an \ approximate period for it. This is done using a fractional Fourier transform \ method [", ButtonBox["BS91", ButtonData:>{"References.nb", "BS91"}, ButtonStyle->"Hyperlink"], "]" }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(FindPeriod[Centroid\ /. \ example, \ toutput]\)], "Input", CellOpen->False], Cell[BoxData[ \(12.294349122141233`\)], "Output"] }, Open ]], Cell["The relative error from the predicted value is about", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(%\ - \ Re[Period[1/6, \ 5]]\)], "Input", CellOpen->False], Cell[BoxData[ \(0.056779828415187694`\)], "Output"] }, Open ]], Cell["\<\ This error is not surprising considering that the predicted value \ come only from the leading order of the expansion. One possible source of \ error is loss due to radiation, so that the final pulse has a different \ amplitude than the original. 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Mdot .05181 .3517 Mdot .052 .35126 Mdot .05219 .3508 Mdot .05238 .35034 Mdot .05257 .34988 Mdot .05276 .34941 Mdot .05295 .34894 Mdot .05314 .34846 Mdot .05333 .34799 Mdot .05352 .34751 Mdot .05371 .34703 Mdot .0539 .34654 Mdot .0541 .34605 Mdot .05429 .34555 Mdot .05448 .34505 Mdot .05467 .34454 Mdot .05486 .34402 Mdot .05505 .3435 Mdot .05524 .34297 Mdot .05543 .34244 Mdot .05562 .34191 Mdot .05581 .34139 Mdot .056 .34087 Mdot .05619 .34036 Mdot .05638 .33986 Mdot .05657 .33938 Mdot .05676 .33891 Mdot .05695 .33845 Mdot .05714 .33802 Mdot .05733 .3376 Mdot .05752 .3372 Mdot .05771 .33683 Mdot .0579 .33647 Mdot .0581 .33612 Mdot .05829 .3358 Mdot .05848 .33549 Mdot .05867 .33519 Mdot .05886 .33491 Mdot .05905 .33463 Mdot .05924 .33436 Mdot .05943 .3341 Mdot .05962 .33384 Mdot .05981 .33359 Mdot .06 .33334 Mdot .06019 .3331 Mdot .06038 .33286 Mdot .06057 .33263 Mdot .06076 .3324 Mdot .06095 .33219 Mdot .06114 .33199 Mdot .06133 .33179 Mdot .06152 .33162 Mdot .06171 .33145 Mdot .0619 .33131 Mdot .0621 .33117 Mdot .06229 .33105 Mdot .06248 .33094 Mdot .06267 .33085 Mdot .06286 .33077 Mdot .06305 .3307 Mdot .06324 .33064 Mdot .06343 .33058 Mdot .06362 .33053 Mdot .06381 .33049 Mdot .064 .33045 Mdot .06419 .33042 Mdot .06438 .33039 Mdot .06457 .33035 Mdot .06476 .33033 Mdot .06495 .3303 Mdot .06514 .33027 Mdot .06533 .33024 Mdot .06552 .33021 Mdot .06571 .33017 Mdot .0659 .33014 Mdot .0661 .3301 Mdot .06629 .33007 Mdot .06648 .33003 Mdot .06667 .32998 Mdot .06686 .32994 Mdot .06705 .32989 Mdot .06724 .32984 Mdot .06743 .32978 Mdot .06762 .32972 Mdot .06781 .32964 Mdot .068 .32956 Mdot .06819 .32947 Mdot .06838 .32936 Mdot .06857 .32924 Mdot .06876 .32911 Mdot .06895 .32895 Mdot .06914 .32878 Mdot .06933 .3286 Mdot .06952 .3284 Mdot .06971 .32818 Mdot .0699 .32795 Mdot .0701 .32772 Mdot .07029 .32747 Mdot .07048 .32723 Mdot .07067 .32698 Mdot .07086 .32672 Mdot .07105 .32647 Mdot .07124 .32622 Mdot .07143 .32597 Mdot .07162 .32573 Mdot .07181 .32549 Mdot .072 .32525 Mdot .07219 .32501 Mdot .07238 .32478 Mdot .07257 .32454 Mdot .07276 .32431 Mdot .07295 .32407 Mdot .07314 .32383 Mdot .07333 .32359 Mdot .07352 .32334 Mdot .07371 .32309 Mdot .0739 .32284 Mdot .0741 .32259 Mdot .07429 .32233 Mdot .07448 .32208 Mdot .07467 .32183 Mdot .07486 .32158 Mdot .07505 .32133 Mdot .07524 .32109 Mdot .07543 .32086 Mdot .07562 .32063 Mdot .07581 .32041 Mdot .076 .32019 Mdot .07619 .31998 Mdot .07638 .31976 Mdot .07657 .31955 Mdot .07676 .31934 Mdot .07695 .31911 Mdot .07714 .31889 Mdot .07733 .31865 Mdot .07752 .31839 Mdot .07771 .31813 Mdot .0779 .31785 Mdot .0781 .31756 Mdot .07829 .31726 Mdot .07848 .31694 Mdot .07867 .31662 Mdot .07886 .3163 Mdot .07905 .31597 Mdot .07924 .31565 Mdot .07943 .31534 Mdot .07962 .31503 Mdot .07981 .31473 Mdot .08 .31445 Mdot .08019 .31418 Mdot .08038 .31392 Mdot .08057 .31368 Mdot .08076 .31346 Mdot .08095 .31326 Mdot .08114 .31307 Mdot .08133 .3129 Mdot .08152 .31275 Mdot .08171 .31261 Mdot .0819 .31249 Mdot .0821 .31238 Mdot .08229 .31229 Mdot .08248 .3122 Mdot .08267 .31213 Mdot .08286 .31206 Mdot .08305 .312 Mdot .08324 .31194 Mdot .08343 .31189 Mdot .08362 .31185 Mdot .08381 .31181 Mdot .084 .31177 Mdot .08419 .31173 Mdot .08438 .3117 Mdot .08457 .31167 Mdot .08476 .31164 Mdot .08495 .31161 Mdot .08514 .31158 Mdot .08533 .31154 Mdot .08552 .31151 Mdot .08571 .31147 Mdot .0859 .31143 Mdot .0861 .31138 Mdot .08629 .31133 Mdot .08648 .31127 Mdot .08667 .3112 Mdot .08686 .31113 Mdot .08705 .31104 Mdot .08724 .31095 Mdot .08743 .31084 Mdot .08762 .31073 Mdot .08781 .31061 Mdot .088 .31048 Mdot .08819 .31034 Mdot .08838 .3102 Mdot .08857 .31006 Mdot .08876 .30991 Mdot .08895 .30976 Mdot .08914 .30961 Mdot .08933 .30946 Mdot .08952 .30931 Mdot .08971 .30916 Mdot .0899 .30902 Mdot .0901 .30889 Mdot .09029 .30876 Mdot .09048 .30864 Mdot .09067 .30852 Mdot .09086 .30842 Mdot .09105 .30832 Mdot .09124 .30822 Mdot .09143 .30813 Mdot .09162 .30803 Mdot .09181 .30794 Mdot .092 .30785 Mdot .09219 .30775 Mdot .09238 .30764 Mdot .09257 .30753 Mdot .09276 .30742 Mdot .09295 .3073 Mdot .09314 .30717 Mdot .09333 .30704 Mdot .09352 .30691 Mdot .09371 .30678 Mdot .0939 .30665 Mdot .0941 .30653 Mdot .09429 .3064 Mdot .09448 .30628 Mdot .09467 .30616 Mdot .09486 .30604 Mdot .09505 .30593 Mdot .09524 .30581 Mdot .09543 .30569 Mdot .09562 .30557 Mdot .09581 .30544 Mdot .096 .3053 Mdot .09619 .30515 Mdot .09638 .30499 Mdot .09657 .30481 Mdot .09676 .30462 Mdot .09695 .30441 Mdot .09714 .30418 Mdot .09733 .30394 Mdot .09752 .30369 Mdot .09771 .30342 Mdot .0979 .30315 Mdot .0981 .30287 Mdot .09829 .30259 Mdot .09848 .3023 Mdot .09867 .30202 Mdot .09886 .30174 Mdot .09905 .30146 Mdot .09924 .30118 Mdot .09943 .30091 Mdot .09962 .30065 Mdot .09981 .3004 Mdot .1 .30015 Mdot .10019 .29991 Mdot .10038 .29969 Mdot .10057 .29947 Mdot .10076 .29926 Mdot .10095 .29906 Mdot .10114 .29888 Mdot .10133 .2987 Mdot .10152 .29854 Mdot .10171 .29838 Mdot .1019 .29824 Mdot .1021 .29811 Mdot .10229 .298 Mdot .10248 .29789 Mdot .10267 .2978 Mdot .10286 .29773 Mdot .10305 .29766 Mdot .10324 .29761 Mdot .10343 .29756 Mdot .10362 .29752 Mdot .10381 .29748 Mdot .104 .29745 Mdot .10419 .29742 Mdot .10438 .2974 Mdot .10457 .29737 Mdot .10476 .29734 Mdot .10495 .2973 Mdot .10514 .29727 Mdot .10533 .29722 Mdot .10552 .29717 Mdot .10571 .29711 Mdot .1059 .29704 Mdot .1061 .29697 Mdot .10629 .29688 Mdot .10648 .29678 Mdot .10667 .29668 Mdot .10686 .29656 Mdot .10705 .29644 Mdot .10724 .29631 Mdot .10743 .29617 Mdot .10762 .29603 Mdot .10781 .29589 Mdot .108 .29575 Mdot .10819 .29561 Mdot .10838 .29547 Mdot .10857 .29534 Mdot .10876 .29522 Mdot .10895 .2951 Mdot .10914 .295 Mdot .10933 .2949 Mdot .10952 .29482 Mdot .10971 .29475 Mdot .1099 .29468 Mdot .1101 .29463 Mdot .11029 .29458 Mdot .11048 .29453 Mdot .11067 .29448 Mdot .11086 .29444 Mdot .11105 .29439 Mdot .11124 .29433 Mdot .11143 .29427 Mdot .11162 .29421 Mdot .11181 .29413 Mdot .112 .29405 Mdot .11219 .29396 Mdot .11238 .29387 Mdot .11257 .29377 Mdot .11276 .29366 Mdot .11295 .29355 Mdot .11314 .29344 Mdot .11333 .29333 Mdot .11352 .29321 Mdot .11371 .2931 Mdot .1139 .29298 Mdot .1141 .29286 Mdot .11429 .29273 Mdot .11448 .2926 Mdot .11467 .29247 Mdot .11486 .29232 Mdot .11505 .29217 Mdot .11524 .29201 Mdot .11543 .29184 Mdot .11562 .29166 Mdot .11581 .29147 Mdot .116 .29127 Mdot .11619 .29106 Mdot .11638 .29084 Mdot .11657 .29061 Mdot .11676 .29038 Mdot .11695 .29014 Mdot .11714 .2899 Mdot .11733 .28965 Mdot .11752 .2894 Mdot .11771 .28916 Mdot .1179 .28891 Mdot .1181 .28867 Mdot .11829 .28843 Mdot .11848 .2882 Mdot .11867 .28797 Mdot .11886 .28775 Mdot .11905 .28754 Mdot .11924 .28733 Mdot .11943 .28713 Mdot .11962 .28694 Mdot .11981 .28676 Mdot .12 .28659 Mdot .12019 .28642 Mdot .12038 .28627 Mdot .12057 .28612 Mdot .12076 .28598 Mdot .12095 .28585 Mdot .12114 .28573 Mdot .12133 .28561 Mdot .12152 .2855 Mdot .12171 .28539 Mdot .1219 .28528 Mdot .1221 .28517 Mdot .12229 .28506 Mdot .12248 .28495 Mdot .12267 .28485 Mdot .12286 .28473 Mdot .12305 .28462 Mdot .12324 .28451 Mdot .12343 .28439 Mdot .12362 .28427 Mdot .12381 .28414 Mdot .124 .28401 Mdot .12419 .28387 Mdot .12438 .28373 Mdot .12457 .28358 Mdot .12476 .28343 Mdot .12495 .28327 Mdot .12514 .2831 Mdot .12533 .28292 Mdot .12552 .28273 Mdot .12571 .28254 Mdot .1259 .28235 Mdot .1261 .28215 Mdot .12629 .28195 Mdot .12648 .28175 Mdot .12667 .28155 Mdot .12686 .28135 Mdot .12705 .28116 Mdot .12724 .28098 Mdot .12743 .2808 Mdot .12762 .28064 Mdot .12781 .28048 Mdot .128 .28034 Mdot .12819 .2802 Mdot .12838 .28007 Mdot .12857 .27995 Mdot .12876 .27984 Mdot .12895 .27972 Mdot .12914 .2796 Mdot .12933 .27948 Mdot .12952 .27936 Mdot .12971 .27922 Mdot .1299 .27908 Mdot .1301 .27892 Mdot .13029 .27876 Mdot .13048 .27858 Mdot .13067 .2784 Mdot .13086 .27822 Mdot .13105 .27802 Mdot .13124 .27783 Mdot .13143 .27764 Mdot .13162 .27745 Mdot .13181 .27726 Mdot .132 .27708 Mdot .13219 .2769 Mdot .13238 .27674 Mdot .13257 .27658 Mdot .13276 .27643 Mdot .13295 .27629 Mdot .13314 .27616 Mdot .13333 .27603 Mdot .13352 .27591 Mdot .13371 .27579 Mdot .1339 .27568 Mdot .1341 .27556 Mdot .13429 .27545 Mdot .13448 .27533 Mdot .13467 .2752 Mdot .13486 .27508 Mdot .13505 .27495 Mdot .13524 .27481 Mdot .13543 .27466 Mdot .13562 .27452 Mdot .13581 .27436 Mdot .136 .2742 Mdot .13619 .27404 Mdot .13638 .27387 Mdot .13657 .27369 Mdot .13676 .27351 Mdot .13695 .27333 Mdot .13714 .27315 Mdot .13733 .27296 Mdot .13752 .27278 Mdot .13771 .27259 Mdot .1379 .272