Scholarlink

<p>The <i>short-pulse (SP) equation</i> serves as a fundamental model for describing the propagation of ultra-short optical pulses in silica fibers. In this paper, we rigorously establish the <i>finite-time blow-up phenomenon</i> for a class of large initial data in the Sobolev space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^{s}(\mathbb {R})\cap \dot{H}^{-1}(\mathbb {R}),~s&gt;\frac{3}{2}.\)</EquationSource> </InlineEquation> In addition, we prove a <i>sharp ill-posedness result</i> for the SP equation with initial data in the critical space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(H^{\frac{3}{2}}(\mathbb {R}).\)</EquationSource> </InlineEquation> Combined with the local well-posedness for initial data in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H^{s}(\mathbb {R})\)</EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(s&gt;\frac{3}{2},\)</EquationSource> </InlineEquation> these results provide a complete characterization of the <i>critical Sobolev regularity</i> for the SP equation, yielding a comprehensive understanding of its well-posedness and ill-posedness dynamics.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Blow up phenomenon and sharp ill-posedness for the short-pulse equation

  • 期刊论文

  • 发表日期：

    2025年12月29日

查看全文
  • Yingying Guo,
  • Min Li,
  • Yue Liu,
  • Weikui Ye,
  • Zhaoyang Yin

摘要

The short-pulse (SP) equation serves as a fundamental model for describing the propagation of ultra-short optical pulses in silica fibers. In this paper, we rigorously establish the finite-time blow-up phenomenon for a class of large initial data in the Sobolev space \(H^{s}(\mathbb {R})\cap \dot{H}^{-1}(\mathbb {R}),~s>\frac{3}{2}.\) In addition, we prove a sharp ill-posedness result for the SP equation with initial data in the critical space \(H^{\frac{3}{2}}(\mathbb {R}).\) Combined with the local well-posedness for initial data in \(H^{s}(\mathbb {R})\) with \(s>\frac{3}{2},\) these results provide a complete characterization of the critical Sobolev regularity for the SP equation, yielding a comprehensive understanding of its well-posedness and ill-posedness dynamics.