论文标题
调制区域中TODA冲击波的长期渐近学
Long-time asymptotics for Toda shock waves in the modulation region
论文作者
论文摘要
我们表明，TODA冲击波渐近地接近分隔孤子和椭圆波区域的区域中的调制有限间隙溶液。我们先前在所有主要区域中这些冲击波的渐近扩展的领先术语得出了公式，并推测在调制区域中，下一个项为$ o（t^{ - 1}）$。在本文中，我们证明了这一事实并研究了共鸣和特征值如何影响领先的渐近行为。我们的主要贡献是解决局部参数Riemann-Hilbert问题的解决方案和分析的严格理由。特别是，这涉及构建适当的单数矩阵模型解决方案。
We show that Toda shock waves are asymptotically close to a modulated finite gap solution in the region separating the soliton and the elliptic wave regions. We previously derived formulas for the leading terms of the asymptotic expansion of these shock waves in all principal regions and conjectured that in the modulation region the next term is of order $O(t^{-1})$. In the present paper we prove this fact and investigate how resonances and eigenvalues influence the leading asymptotic behaviour. Our main contribution is the solution of the local parametrix Riemann-Hilbert problems and a rigorous justification of the analysis. In particular, this involves the construction of a proper singular matrix model solution.
