CINXE.COM

Document Zbl 1524.82023 - zbMATH Open

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1524.82023 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">&times;</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools &amp; Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/?q=ra%3Ajiang.xunda+ai%3Aliu.rongsheng"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open &mdash; the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span>&nbsp;<a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="ra:jiang.xunda ai:liu.rongsheng" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span>&nbsp;</span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>&nbsp;&nbsp;</span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>&nbsp;&nbsp;</span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>&nbsp;&nbsp;</span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>&nbsp;&nbsp;</span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>&nbsp;&nbsp;</span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>&nbsp;&nbsp;</span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>&nbsp;&nbsp;</span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>&nbsp;&nbsp;</span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>&nbsp;&nbsp;</span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>&nbsp;&nbsp;</span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>&nbsp;&nbsp;</span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>&nbsp;&nbsp;</span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>&nbsp;&nbsp;</span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>&nbsp;&nbsp;</span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>&nbsp;&nbsp;</span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>&nbsp;&nbsp;</span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>&nbsp;&nbsp;</span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a&nbsp;<span class="item">&</span>&nbsp;b&nbsp;</span><span>&nbsp;</span>logical and (default)</a></li> <li><a href="#"><span class="token">a&nbsp;<span class="item">|</span>&nbsp;b&nbsp;</span><span>&nbsp;</span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab&nbsp;&nbsp;&nbsp;</span><span>&nbsp;</span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>&nbsp;&nbsp;</span><span>&nbsp;</span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab&nbsp;c<span class="item">"</span></span><span>&nbsp;</span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab&nbsp;c<span class="item">)</span></span><span>&nbsp;</span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki &amp; ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &amp;</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b &amp; au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&amp;</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/kong.chao" title="Author Profile">Kong, Chao</a>; <a href="/authors/he.yichen" title="Author Profile">He, Yichen</a>; <a href="/authors/wan.tianzhe" title="Author Profile">Wan, Tianzhe</a>; <a href="/authors/yin.bin" title="Author Profile">Yin, Bin</a>; <a href="/authors/liu.rongsheng" title="Author Profile">Liu, Rongsheng</a>; <a href="/authors/chen.yaqi" title="Author Profile">Chen, Yaqi</a>; <a href="/authors/huang.jianquan" title="Author Profile">Huang, Jianquan</a>; <a href="/authors/lei.dajun" title="Author Profile">Lei, Dajun</a>; <a href="/authors/jiang.chunzhi" title="Author Profile">Jiang, Chunzhi</a>; <a href="/authors/deng.haiming" title="Author Profile">Deng, Haiming</a></div> <h2 class="title"> <strong>Excitation of chaotic atomic population oscillations in a spin-orbit coupled Bose-Einstein condensate with a square-wave driving.</strong> <i>(English)</i> <a class="label nowrap" href="/1524.82023">Zbl 1524.82023</a> </h2> <div class="source"> <a href="/serials/182" title="Journal Profile">Phys. Lett., A</a> <a href="/?q=in%3A491847" title="Articles in this Issue">458, Article ID 128597, 6 p. (2023)</a>. </div> <div class="abstract">Summary: In this paper, we have investigated the excitation of chaotic atomic population oscillations in a spin-orbit (SO) coupled Bose-Einstein condensate (BEC). Unlike previous SOC system with monochrome and bichromatic driving, the present system reveals the chaotic excitation induced by the initial condition in the nondissipative regime and by the driving strength in the dissipative regime under the action of a square-wave driving. The damping-dependent chaotic parameter regions and the modulation of atomic population oscillations by the square-wave driving are studied. The existence of chaos is confirmed by computing the corresponding largest Lyapunov exponents. The results could be significant in the quantum transport of the SO coupled cold-atom system.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A82C10" title="MSC2020">82C10</a> </td> <td class="space"> Quantum dynamics and nonequilibrium statistical mechanics (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A82D03" title="MSC2020">82D03</a> </td> <td class="space"> Statistical mechanics in condensed matter (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81V73" title="MSC2020">81V73</a> </td> <td class="space"> Bosonic systems in quantum theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81Q50" title="MSC2020">81Q50</a> </td> <td class="space"> Quantum chaos </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A81V45" title="MSC2020">81V45</a> </td> <td class="space"> Atomic physics </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ABose-Einstein+condensate">Bose-Einstein condensate</a>; <a href="/?q=ut%3Aspin-orbit+coupling">spin-orbit coupling</a>; <a href="/?q=ut%3Achaotic+oscillations">chaotic oscillations</a>; <a href="/?q=ut%3Aperiodic+driving">periodic driving</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">&times;</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 1524.82023" data-ciurl="/ci/07644901" data-biburl="/bibtex/07644901.bib" data-amsurl="/amsrefs/07644901.bib" data-xmlurl="/xml/07644901.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07644901.pdf" title="Zbl 1524.82023 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/j.physleta.2022.128597" aria-label="DOI for “Excitation of chaotic atomic population oscillations in a spin-orbit coupled Bose-Einstein condensate with a square-wave driving”" title="10.1016/j.physleta.2022.128597">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">Dalfovo, F.; Giorgini, S.; Pitaevskii, L. P.; Stringari, S., Rev. Mod. Phys., 71, 463 (1999); Ketterle, W., Rev. Mod. Phys., 74, 1131 (2002)</td> </tr><tr> <td>[2]</td> <td class="space">Leggett, A. J., Rev. Mod. Phys., 73, 307 (2001)</td> </tr><tr> <td>[3]</td> <td class="space">Smerzi, A.; Fantoni, S.; Giovanzzi, S.; Shenoy, S. R., Phys. Rev. Lett., 79, 4950 (1997); Raghavan, S.; Smerzi, A.; Fantoni, S.; Shenoy, S. R., Phys. Rev. A, 59, 620 (1999)</td> </tr><tr> <td>[4]</td> <td class="space">Juliá-Díaz, B.; Guilleumas, M.; Lewenstein, M.; Polls, A.; Sanpera, A., Phys. Rev. A, 80, Article 023616 pp. (2009)</td> </tr><tr> <td>[5]</td> <td class="space">Sun, B.; Pindzola, M. S., Phys. Rev. A, 80, Article 033616 pp. (2009)</td> </tr><tr> <td>[6]</td> <td class="space">Satija, I. I.; Balakrishnan, R.; Naudus, P.; Heward, J.; Edwards, M.; Clark, C. W., Phys. Rev. A, 79, Article 033616 pp. (2009)</td> </tr><tr> <td>[7]</td> <td class="space">Melé-Messeguer, M.; Juliá-Díaz, B.; Guilleumas, M.; Polls, A.; Sanpera, A., New J. Phys., 13, Article 033012 pp. (2011)</td> </tr><tr> <td>[8]</td> <td class="space">Liberato, S. D.; Foot, C. J., Phys. Rev. A, 73, Article 035602 pp. (2006)</td> </tr><tr> <td>[9]</td> <td class="space">Nigro, M.; Capuzzi, P.; Cataldo, H. M.; Jezek, D. M., Phys. Rev. A, 97, Article 013626 pp. (2018)</td> </tr><tr> <td>[10]</td> <td class="space">Lepers, M.; Zehnlé, V.; Garreau, J. C., Phys. Rev. Lett., 101, Article 144103 pp. (2008)</td> </tr><tr> <td>[11]</td> <td class="space">Martin, J.; Georgeot, B.; Shepelyansky, D. L., Phys. Rev. Lett., 101, Article 074102 pp. (2008)</td> </tr><tr> <td>[12]</td> <td class="space">Kidd, R. A.; Olsen, M. K.; Corney, J. F., Phys. Rev. A, 100, Article 013625 pp. (2019)</td> </tr><tr> <td>[13]</td> <td class="space">Wanzenböck, R.; Donsa, S.; Hofstätter, H.; Koch, O.; Schlagheck, P.; Březinová, I., Phys. Rev. A, 103, Article 023336 pp. (2021)</td> </tr><tr> <td>[14]</td> <td class="space">Li, S. C.; Pezzè, L.; Smerzi, A., Phys. Rev. A, 103, Article 052417 pp. (2021)</td> </tr><tr> <td>[15]</td> <td class="space">Syu, W. C.; Lee, D. S.; Lin, C. Y., Phys. Rev. A, 101, Article 063622 pp. (2020)</td> </tr><tr> <td>[16]</td> <td class="space">Lee, C.; Hai, W.; Shi, L.; Zhu, X.; Gao, K., Phys. Rev. A, 64, Article 053604 pp. (2001)</td> </tr><tr> <td>[17]</td> <td class="space">Bürkle, R.; Vardi, A.; Cohen, D.; Anglin, J. R., Phys. Rev. Lett., 123, Article 114101 pp. (2019)</td> </tr><tr> <td>[18]</td> <td class="space">Dey, A.; Cohen, D.; Vardi, A., Phys. Rev. A, 99, Article 033623 pp. (2019)</td> </tr><tr> <td>[19]</td> <td class="space">Rong, S.; Hai, W.; Xie, Q.; Zhu, Q., Chaos, 19, Article 033129 pp. (2009)</td> </tr><tr> <td>[20]</td> <td class="space">Morales-Molina, L.; Gong, J., Phys. Rev. A, 78 (2008), 041403(R)</td> </tr><tr> <td>[21]</td> <td class="space">Li, D. X.; Shao, X. Q., Sci. Rep., 9, 9023 (2019)</td> </tr><tr> <td>[22]</td> <td class="space">Manson, N. B.; Wei, C.; Martin, J. P.D., Phys. Rev. Lett., 76, 3943 (1996)</td> </tr><tr> <td>[23]</td> <td class="space">Haddadfarshi, F.; Mintert, F., New J. Phys., 18, Article 123007 pp. (2016)</td> </tr><tr> <td>[24]</td> <td class="space">Igarashi, A.; Yamada, H., Phys. Rev. E, 78, Article 026213 pp. (2008)</td> </tr><tr> <td>[25]</td> <td class="space">Lin, L. J.; Jiménez-García, K.; Spielman, I. B., Nature, 471, 83 (2011)</td> </tr><tr> <td>[26]</td> <td class="space">Li, Y.; Pitaevskii, L. P.; Stringari, S., Phys. Rev. Lett., 108, Article 225301 pp. (2012)</td> </tr><tr> <td>[27]</td> <td class="space">Zhou, X.; Zhou, J.; Wu, C., Phys. Rev. A, 84, Article 063624 pp. (2011)</td> </tr><tr> <td>[28]</td> <td class="space">Kartashov, Y. V.; Konotop, V. V.; Zezyulin, D. A.; Torner, L., Phys. Rev. Lett., 117, Article 215301 pp. (2016)</td> </tr><tr> <td>[29]</td> <td class="space">Mæland, K.; Janssønn, A. T.G.; Rygh, J. H.; Sudbø, A., Phys. Rev. A, 102, Article 053318 pp. (2020)</td> </tr><tr> <td>[30]</td> <td class="space">Huang, G.; Luo, G.; Wu, Z.; Xu, Z., Phys. Rev. A, 103, Article 043328 pp. (2021)</td> </tr><tr> <td>[31]</td> <td class="space">Luo, X.; Zeng, Z. Y.; Guo, Y.; Yang, B.; Xiao, J.; Li, L.; Kong, C.; Chen, A. X., Phys. Rev. A, 103, Article 043315 pp. (2021); Luo, X.; Hu, Z.; Zeng, Z. Y.; Luo, Y.; Yang, B.; Xiao, J.; Li, L.; Chen, A. X., Phys. Rev. A, 103, Article 063324 pp. (2021)</td> </tr><tr> <td>[32]</td> <td class="space">Luo, Y.; Wang, X.; Luo, L.; Zhou, Z.; Zeng, Z. Y.; Luo, X., New J. Phys., 22, Article 093041 pp. (2020)</td> </tr><tr> <td>[33]</td> <td class="space">Deng, H.; Wan, T.; He, Y.; Han, T.; Huang, J.; Ding, S.; Li, Y.; Lei, D.; Jiang, C.; Yao, M.; Chen, H.; Kong, C., Results Phys., 31, Article 104926 pp. (2021)</td> </tr><tr> <td>[34]</td> <td class="space">Wang, Q. B.; Hu, J. G.; Su, X. H.; Wen, L. H., Results Phys., 20, Article 103755 pp. (2021); Wang, O. B.; Zhao, W. J.; Wen, L. H., Results Phys., 25, Article 104317 pp. (2021)</td> </tr><tr> <td>[35]</td> <td class="space">Chen, M. N.; Liao, B. J.; Wang, H. C.; Hao Huang, H.; Liu, Y.; Ling, D. X., Results Phys., 18, Article 103304 pp. (2020)</td> </tr><tr> <td>[36]</td> <td class="space">Ye, Z.; Chen, Y.; Zheng, Y.; Chen, X.; Liu, B., Results Phys., 130, Article 109418 pp. (2020) &middot; <a href="/1489.81039" class="nowrap">Zbl 1489.81039</a></td> </tr><tr> <td>[37]</td> <td class="space">Zeng, L.; Milivoj, R. B.; Mihalache, D.; Wang, Q.; Chen, J.; Shi, J.; Cai, Y.; Lu, X.; Li, J., Chaos Solitons Fractals, 152, Article 111406 pp. (2021) &middot; <a href="/1496.35155" class="nowrap">Zbl 1496.35155</a></td> </tr><tr> <td>[38]</td> <td class="space">Fan, Z.; Chen, Z.; Li, Y.; Malomed, B. A., Phys. Rev. A, 101, Article 013607 pp. (2020)</td> </tr><tr> <td>[39]</td> <td class="space">Jiang, X.; Fan, Z.; Chen, Z.; Pang, W.; Li, Y.; Malomed, B. A., Phys. Rev. A, 93, Article 023633 pp. (2016)</td> </tr><tr> <td>[40]</td> <td class="space">Xu, X.; Zhao, F.; Zhou, Y.; Liu, B.; Jiang, X.; Malomed, B. A.; Li, Y., Commun. Nonlinear Sci. Numer. Simul., 117, Article 106930 pp. (2023) &middot; <a href="/1504.35504" class="nowrap">Zbl 1504.35504</a></td> </tr><tr> <td>[41]</td> <td class="space">Chen, Z.; Li, Y.; Malomed, B. A., Phys. Rev. Res., 2, Article 033214 pp. (2020)</td> </tr><tr> <td>[42]</td> <td class="space">Larson, J.; Anderson, B. M.; Altland, A., Phys. Rev. A, 87, Article 013624 pp. (2013)</td> </tr><tr> <td>[43]</td> <td class="space">Kong, C.; Chen, H.; Li, C.; Hai, W., Chaos, 28, Article 023115 pp. (2018) &middot; <a href="/1390.81189" class="nowrap">Zbl 1390.81189</a></td> </tr><tr> <td>[44]</td> <td class="space">Kong, C.; Luo, X.; Chen, H.; Luo, Y.; Hai, W., Chaos, 29, Article 103148 pp. (2019) &middot; <a href="/1429.81011" class="nowrap">Zbl 1429.81011</a></td> </tr><tr> <td>[45]</td> <td class="space">Sakaguchi, H.; Li, B.; Malomed, B. A., Phys. Rev. E, 89, Article 032920 pp. (2014)</td> </tr><tr> <td>[46]</td> <td class="space">Liao, B.; Li, S.; Huang, C.; Luo, Z.; Pang, W.; Tan, H.; Malomed, B. A.; Li, Y., Phys. Rev. A, 96, Article 043613 pp. (2017)</td> </tr><tr> <td>[47]</td> <td class="space">Sakaguchi, H.; Sherman, E. Y.; Malomed, B. A., Phys. Rev. E, 90, Article 032202 pp. (2016)</td> </tr><tr> <td>[48]</td> <td class="space">Sakaguchi, H.; Malomed, B. A., New J. Phys., 18, Article 105005 pp. (2016)</td> </tr><tr> <td>[49]</td> <td class="space">Zhang, Y.; Zhou, Z.; Malomed, B. A.; Pu, H., Phys. Rev. Lett., 115, Article 253902 pp. (2015)</td> </tr><tr> <td>[50]</td> <td class="space">Hu, F. Q.; Wang, J. J.; Yu, Z. F.; Zhang, A. X.; Xue, J. K., Phys. Rev. E, 93, Article 022214 pp. (2016)</td> </tr><tr> <td>[51]</td> <td class="space">Salasnich, L.; Malomed, B. A., Phys. Rev. A, 87, Article 063625 pp. (2013)</td> </tr><tr> <td>[52]</td> <td class="space">Cheng, Y. S.; Tang, G. H.; Adhikari, S. K., Phys. Rev. A, 89, Article 063602 pp. (2014)</td> </tr><tr> <td>[53]</td> <td class="space">Abdullaev, F. K.; Kraenkel, R. A., Phys. Rev. A, 62, Article 023613 pp. (2000)</td> </tr><tr> <td>[54]</td> <td class="space">Hai, W.; Lee, C.; Chong, G.; Shi, L., Phys. Rev. E, 66, Article 026202 pp. (2002); Hai, W.; Zhu, Q.; Rong, S., Phys. Rev. A, 79, Article 023603 pp. (2009); Luo, X.; Hai, W., Chaos, 15, Article 033702 pp. (2005)</td> </tr><tr> <td>[55]</td> <td class="space">Steck, D. A.; Oskay, W. H.; Raizen, M. G., Science, 293, 274 (2001); Steck, D. A., Nature, 461, 736 (2009)</td> </tr><tr> <td>[56]</td> <td class="space">Lin, W. A.; Ballentine, L. F., Phys. Rev. Lett., 65, 2927 (1990); Chaudhury, S.; Smith, A.; Anderson, B. E.; Ghose, S.; Jessen, P. S., Nature, 461, 768 (2009)</td> </tr><tr> <td>[57]</td> <td class="space">Ott, E., Rev. Mod. Phys., 53, 655 (1981) &middot; <a href="/1114.37303" class="nowrap">Zbl 1114.37303</a></td> </tr><tr> <td>[58]</td> <td class="space">Boccaletti, S.; Grebogi, C.; Lai, Y. C.; Mancini, H.; Maza, D., Phys. Rep., 329, 103 (2000)</td> </tr><tr> <td>[59]</td> <td class="space">Tsonis, A. A., Chaos-from Theory to Applications (1992), Plenum Press: Plenum Press New York</td> </tr><tr> <td>[60]</td> <td class="space">Liu, C.; Liu, T.; Liu, L.; Liu, K., Chaos Solitons Fractals, 22, 1031 (2004) &middot; <a href="/1060.37027" class="nowrap">Zbl 1060.37027</a></td> </tr><tr> <td>[61]</td> <td class="space">Kengne, J.; Njitacke, Z. T.; Fotsin, H. B., Nonlinear Dyn., 83, 751 (2016)</td> </tr><tr> <td>[62]</td> <td class="space">Chong, G.; Hai, W.; Xie, Q., Chaos, 14, 217 (2004); Chong, G.; Hai, W.; Xie, Q., Phys. Rev. E, 71, Article 016202 pp. (2005); Hai, W.; Rong, S.; Zhu, Q., Phys. Rev. E, 78, Article 066214 pp. (2008); Jun, X.; Luo, X., Chin. Phys. Lett., 26, Article 040305 pp. (2009)</td> </tr><tr> <td>[63]</td> <td class="space">Melnikov, V. K., Trans. Mosc. Math. Soc., 12, 1 (1963) &middot; <a href="/0135.31001" class="nowrap">Zbl 0135.31001</a></td> </tr><tr> <td>[64]</td> <td class="space">Sharma, A.; Patidar, V.; Purohit, G.; Sud, K. K., Commun. Nonlinear Sci. Numer. Simul., 17, 2254 (2021)</td> </tr><tr> <td>[65]</td> <td class="space">Kadau, H.; Schmitt, M.; Wenzel, M.; Wink, C.; Maier, T.; Ferrier-Barbut, I.; Pfau, T., Nature, 530, 194 (2016)</td> </tr><tr> <td>[66]</td> <td class="space">Ferrier-Barbut, I.; Kadau, H.; Schmitt, M.; Wenzel, M.; Pfau, T., Phys. Rev. Lett., 116, Article 215301 pp. (2016)</td> </tr><tr> <td>[67]</td> <td class="space">Malomed, B. A., Front. Phys., 16, Article 22504 pp. (2021)</td> </tr><tr> <td>[68]</td> <td class="space">Luo, Z.; Pang, W.; Liu, B.; Li, Y.; Malomed, B. A., Front. Phys., 16, Article 32201 pp. (2021)</td> </tr><tr> <td>[69]</td> <td class="space">Zhang, X.; Xu, X.; Zheng, Y.; Chen, Z.; Liu, B.; Huang, C.; Malomed, B. A.; Li, Y., Phys. Rev. Lett., 123, Article 133901 pp. (2019)</td> </tr><tr> <td>[70]</td> <td class="space">Li, Y.; Luo, Z.; Liu, Y.; Chen, Z.; Huang, C.; Fu, S.; Tan, H.; Malomed, B. A., New J. Phys., 19, Article 113043 pp. (2017) &middot; <a href="/1516.74059" class="nowrap">Zbl 1516.74059</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH&nbsp;Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> &copy; 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms &amp; Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none">&nbsp;<span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

Pages: 1 2 3 4 5 6 7 8 9 10