Interplay between the Inverse Scattering Method and Fokas's Unified Transform with an Application

Caudrelier Vincent

摘要

Abstract > It is known that the initial‐boundary value problem for certain integrable Partial Differential Equations (PDEs) on the half‐line with integrable boundary conditions can be mapped to a special case of the inverse scattering method (ISM) on the full‐line. This can also be established within the so‐called unified transform (UT) of Fokas for initial‐boundary value problems with linearizable boundary conditions. In this paper, we show a converse to this statement within the Ablowitz‐Kaup‐Newell‐Segur (AKNS) scheme: the ISM on the full‐line can be mapped to an initial‐boundary value problem with linearizable boundary conditions. To achieve this, we need a matrix version of the UT that was introduced by the author to study integrable PDEs on star‐graphs. As an application of the result, we show that the new, nonlocal reduction of the AKNS scheme introduced by Ablowitz and Musslimani to obtain the nonlocal nonlinear Schr?dinger (NLS) equation can be recast as an old, local reduction, thus putting the nonlocal NLS and the NLS equations on equal footing from the point of view of the reduction group theory of Mikhailov. </abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract XMLNS =“http://www.wiley.com/namespaces/wiley”type =“main”xml：lang =“en”> <title type =“main”>抽象</ title> >它是已知具有可集成边界条件的半线上的某些可集偏微分方程（PDE）的初始边界值问题可以映射到全线上的逆散射方法（ISM）的特殊情况。这也可以在Fokas的所谓统一变换（UT）内，用于具有可直链的边界值问题。在本文中，我们在ABLOTZ-KAUP-Newell-Segur（AKNS）方案中显示了对该陈述的反向：全线上的ISM可以映射到具有可直链条化的边界条件的初始边界值问题。为实现这一目标，我们需要由作者引入的矩阵</ i>版本，以研究Star-Graphs上的可集成的PDE。作为结果的应用，我们表明ablowitz和musslimani引入的akns方案的新的，非局部</ i>以获得非局部非线性schr？dinger（nls）等式可以重新增加作为<i >旧的，局部</ i>减少，从而从Mikhailov的减少组理论的角度来看非局部NLS和NLS方程。 </ p> </ abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" >著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-29329/'>《Studies in Applied Mathematics》</a> <b style="margin: 0 2px;">|</b><span>2018年第1期</span><b style="margin: 0 2px;">|</b><span>共24页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Caudrelier Vincent&option=202" target="_blank" rel="nofollow">Caudrelier Vincent;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>University of Leeds;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li > <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span><a href="https://www.zhangqiaokeyan.com/clc/1147.html" title="应用数学">应用数学;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> </p> </li> </ul> </div> </div> <div class="literature cardcommon" id="literaturereference" style="display:none"> <div class="similarity "> <h3 class="all_title" id="enpatent111">引文网络</h3> <div class="referencetab clearfix"> <ul id="referencedaohang"> <li dataid="referenceul">参考文献</li> <li dataid="citationul">引证文献</li> <li dataid="commonreferenceul">共引文献</li> <li dataid="commoncitationul">同被引文献</li> <li dataid="tworeferenceul">二级参考文献</li> <li dataid="twocitationul">二级引证文献</li> </ul> </div> <div class="reference_details" id="referenceList"> <ul id="referenceul"></ul> <ul id="citationul"></ul> <ul id="commonreferenceul"></ul> <ul id="commoncitationul"></ul> <ul id="tworeferenceul"></ul> <ul id="twocitationul"></ul> </div> </div> </div> <div class="literature cardcommon"> <div class="similarity "> <h3 class="all_title" id="enpatent66">相似文献</h3> <div class="similaritytab clearfix"> <ul> <li class="active" >外文文献</li> <li >中文文献</li> <li >专利</li> </ul> </div> <div class="similarity_details"> <ul > <li> <div> <b>1. </b><a class="enjiyixqcontent" href="/journal-foreign-detail/0704024473341.html">Interplay between the Inverse Scattering Method and Fokas's Unified Transform with an 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