Long‐Time Asymptotics for the Focusing Nonlinear Schr?dinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability

Biondini Gino; Mantzavinos Dionyssios

摘要

Abstract > We characterize the long‐time asymptotic behavior of the focusing nonlinear Schr?dinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity by using a variant of the recently developed inverse scattering transform (IST) for such problems and by employing the nonlinear steepest‐descent method of Deift and Zhou for oscillatory Riemann‐Hilbert problems. First, we formulate the IST over a single sheet of the complex plane without introducing the uniformization variable that was used by Biondini and Kova?i? in 2014. The solution of the focusing NLS equation with nonzero boundary conditions is thereby associated with a matrix Riemann‐Hilbert problem whose jumps grow exponentially with time for certain portions of the continuous spectrum. This growth is the signature of the well‐known modulational instability within the context of the IST. We then eliminate this growth by performing suitable deformations of the Riemann‐Hilbert problem in the complex spectral plane. The results demonstrate that the solution of the focusing NLS equation with nonzero boundary conditions remains bounded at all times. Moreover, we show that, asymptotically in time, the xt ‐plane decomposes into two types of regions: a left far‐field region and a right far‐field region, where the solution equals the condition at infinity to leading order up to a phase shift, and a central region in which the asymptotic behavior is described by slowly modulated periodic oscillations. Finally, we show how, in the latter region, the modulus of the leading‐order solution, initially obtained as a ratio of Jacobi theta functions, can be reduced to the well‐known elliptic solutions of the focusing NLS equation. These results provide the first characterization of the long‐time behavior of generic perturbations of a constant </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”> <title type =“main”>抽象</ title> >我们表征了长期渐近行为通过使用最近开发的逆散射变换（IST）的变型，对对称的非氮边界条件的聚焦非线性SCHR？Dinger（NLS）方程在Infinity的逆散射变换（IST）中，采用非线性陡峭的换水和换水周为振荡的riemann-hilbert问题。首先，我们在一张复杂的平面上制定IST，而不引入Biondini和Kova使用的均匀化变量？我？在2014年，具有非零边界条件的聚焦NLS方程的解与矩阵riemann-hilbert问题相关联，其跳跃随着连续频谱的某些部分的时间而呈指数级增长。这种增长是在IST的背景下众所周知的调制不稳定的签名。然后，我们通过在复杂光谱平面中执行黎曼-HILBERT问题的合适变形来消除这种增长。结果表明，聚焦NLS方程与非零边界条件的溶液始终存在界限。此外，我们表明，渐近时刻， xt </ i> -plane分解成两种类型的区域：左远场区域和右远场区域，其中解决方案等于无穷大的条件以通过缓慢调制的周期性振荡描述渐近行为的前导到相移的中心区域。最后，我们示出了如何在后一个区域中，可以减少到最初获得的前导阶层的模量作为Jacobi Theta功能的比率，可以减少到聚焦NLS方程的众所周知的椭圆溶液。这些结果提供了恒定的通用扰动的长时间行为的第一个表征 </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-16958/'>《Communications on Pure and Applied Mathematics》</a> <b style="margin: 0 2px;">|</b><span>2017年第12期</span><b style="margin: 0 2px;">|</b><span>共66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=Biondini Gino&option=202" target="_blank" rel="nofollow">Biondini Gino;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Mantzavinos Dionyssios&option=202" target="_blank" rel="nofollow">Mantzavinos Dionyssios;</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>Department of MathematicsState University of New York at Buffalo BuffaloNY 14260 USA;</p> <p>Department of MathematicsState University of New York at Buffalo BuffaloNY 14260 USA;</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/156.html" title="数学">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> 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