Document Zbl 1512.76018

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In this derivation, the restriction of narrow bandwidth constraint is extended, so that this equation will be more appropriate for application to a realistic sea wave spectrum. From this equation, an instability condition is obtained and then instability regions in the perturbed wavenumber space for a uniform wave train are drawn, which are in good agreement with the exact numerical results. As it turns out, the corrections to the stability properties that occur at the fourth-order term arise from an interaction between the mean flow and the frequency-dispersion term. Since the frequency-dispersion term, in the absence of depth-uniform current, for pure capillary waves is of opposite sign for pure gravity waves, so too are the corrections to the instability properties.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7680731">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A76B15" title="MSC2020">76B15</a> </td> <td class="space"> Water waves, gravity waves; dispersion and scattering, nonlinear interaction </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76B45" title="MSC2020">76B45</a> </td> <td class="space"> Capillarity (surface tension) for incompressible inviscid fluids </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76E99" title="MSC2020">76E99</a> </td> <td class="space"> Hydrodynamic stability </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35Q55" title="MSC2020">35Q55</a> </td> <td class="space"> NLS equations (nonlinear Schrödinger equations) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Anonlinear+Schr%C3%B6dinger+equation">nonlinear Schrödinger equation</a>; <a href="/?q=ut%3Amodulational+instability">modulational instability</a>; <a href="/?q=ut%3Afrequency-dispersion+term">frequency-dispersion term</a></div>  <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;">  </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">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1512.76018" data-ciurl="/ci/07680731" data-biburl="/bibtex/07680731.bib" data-amsurl="/amsrefs/07680731.bib" data-xmlurl="/xml/07680731.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07680731.pdf" title="Zbl 1512.76018 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/10.1017/S1446181123000020" aria-label="DOI for “A modification to the Schrödinger equation for broader bandwidth gravity-capillary waves on deep water with depth-uniform current”" title="10.1017/S1446181123000020">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">Benjamin, T. 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Document Zbl 1512.76018 - zbMATH Open