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R.</a>; <a href="/authors/segre.enrico" title="Author Profile">Segre, E.</a></div> <h2 class="title"> <strong>Numerical solutions of the Korteweg-de Vries equation using the periodic scattering transform \(\mu\)-representation.</strong> <i>(English)</i> <a class="label nowrap" href="/0707.65089">Zbl 0707.65089</a> </h2> <div class="source"> <a href="/serials/1030" title="Journal Profile">Physica D</a> <a href="/?q=in%3A174530" title="Articles in this Issue">44, No. 3, 575-604 (1990)</a>. </div> <div class="abstract">A numerical procedure is developed, based on the periodic scattering transform, for determining the spatial and temporal evolution of solutions of the Korteweg-de Vries equation with periodic boundary conditions. In this approach solutions of KdV are found in terms of the “\(\mu\)-representation”; the wave amplitude is given by a trace formula consisting of a linear superposition of hypoelliptic functions, governed by a system of nonlinear, ordinary differential equations. These equations are integrated numerically, using a second order Taylor series expansion with a variable step size.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A2469">Michael Sever (Jerusalem)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A4162209">Cited in <strong>9</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65Z05" title="MSC2020">65Z05</a> </td> <td class="space"> Applications to the sciences </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65M20" title="MSC2020">65M20</a> </td> <td class="space"> Method of lines for initial value and initial-boundary value problems involving PDEs </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65L05" title="MSC2020">65L05</a> </td> <td class="space"> Numerical methods for initial value problems involving ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q53" title="MSC2020">35Q53</a> </td> <td class="space"> KdV equations (Korteweg-de Vries equations) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amu-representation">mu-representation</a>; <a href="/?q=ut%3Aperiodic+scattering+transform">periodic scattering transform</a>; <a href="/?q=ut%3AKorteweg-de+Vries+equation">Korteweg-de Vries equation</a>; <a href="/?q=ut%3Aperiodic+boundary+conditions">periodic boundary conditions</a>; <a href="/?q=ut%3Awave+amplitude">wave amplitude</a>; <a href="/?q=ut%3Atrace+formula">trace formula</a>; <a href="/?q=ut%3Ahypoelliptic+functions">hypoelliptic functions</a>; <a href="/?q=ut%3Asecond+order+Taylor+series+expansion">second order Taylor series expansion</a>; <a href="/?q=ut%3Avariable+step+size">variable step size</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;"> <!-- alert for connection errors etc --> </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0707.65089" data-ciurl="/ci/04162209" data-biburl="/bibtex/04162209.bib" data-amsurl="/amsrefs/04162209.bib" data-xmlurl="/xml/04162209.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/04162209.pdf" title="Zbl 0707.65089 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/0167-2789(90)90163-J" aria-label="DOI for “Numerical solutions of the Korteweg-de Vries equation using the periodic scattering transform \(\mu\)-representation”" title="10.1016/0167-2789(90)90163-J">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ablowitz, M. 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