%0 Journal Article
%T 具有4 &#215; 4 Lax对的NLS方程的黎曼&#8211;希尔伯特问题<br>The Riemann-Hilbert Problem for the NLS Equation with a 4 &#215; 4 Lax Pair
%A 曹一苇
%J Pure Mathematics
%P 108-116
%@ 2160-7605
%D 2025
%I Hans Publishing
%R 10.12677/pm.2025.155159
%X 本文首先阐述了具有4 &#215;4 Lax对的NLS方程的来历、表示形式&#65292;并将其表示为矩阵的分块形式&#65292;通过对Lax对、特征函数以及对称性的分析&#65292;得出全局关系&#65292;并表示出<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mstyle mathvariant='bold' mathsize='normal'><mi>s</mi></mstyle><mrow><mo>(</mo>
   <mstyle mathvariant='bold' mathsize='normal'><mi>k</mi></mstyle>
  <mo>)</mo></mrow></mrow>
</math>

和<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mstyle mathvariant='bold' mathsize='normal'><mi>S</mi></mstyle><mrow><mo>(</mo>
   <mstyle mathvariant='bold' mathsize='normal'><mi>k</mi></mstyle>
  <mo>)</mo></mrow></mrow>
</math>

的形式&#65292;进而构造黎曼&#8211;希尔伯特问题&#65292;并通过<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mstyle mathvariant='bold' mathsize='normal'><mi>M</mi></mstyle>
   <mstyle mathvariant='bold' mathsize='normal'><mi>n</mi></mstyle>
  </msub>
  </mrow>
</math>

之间的关系&#65292;计算出其跳跃矩阵。<br>In this paper, the origin and representation of the square matrix NLS equation with a 4 &#215; 4 Lax pair are first expounded, and it is expressed as the block form of the matrix. Through the analysis of lax pair, eigenfunction and symmetry, we establish the global relation and derive the explicit forms of <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mstyle mathvariant='bold' mathsize='normal'><mi>s</mi></mstyle><mrow><mo>(</mo>
   <mstyle mathvariant='bold' mathsize='normal'><mi>k</mi></mstyle>
  <mo>)</mo></mrow></mrow>
</math>

 and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mstyle mathvariant='bold' mathsize='normal'><mi>S</mi></mstyle><mrow><mo>(</mo>
   <mstyle mathvariant='bold' mathsize='normal'><mi>k</mi></mstyle>
  <mo>)</mo></mrow></mrow>
</math>

. And then the Riemann-Hilbert problem is constructed, and its jump matrix is calculated through the relationship between <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msub>
   <mstyle mathvariant='bold' mathsize='normal'><mi>M</mi></mstyle>
   <mstyle mathvariant='bold' mathsize='normal'><mi>n</mi></mstyle>
  </msub>
  </mrow>
</math>

.
%K 具有4 &#215
%K 4 Lax对的NLS方程&#65292
%K Lax对&#65292
%K 黎曼&#8211
%K 希尔伯特问题<br>The NLS Equation with a 4 &#215
%K 4 Lax Pair
%K Lax Pair
%K Riemann-Hilbert Problem
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=114884