半直线上的短波方程的统一变换方法
The Short Wave equation on the half-line by the Unified Transform Method
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中文总结 AI 辅助
本文用统一变换方法研究半直线短波方程的初边值问题，根据边界迹的符号动态区分两种谱公式，并构造Riemann-Hilbert问题以重构解。
中文摘要 AI 辅助
我们研究了半直线$x\ge 0$上短波方程的初边值问题。该问题的一个显著特征是边界$x=0$不能先验地刻画为流入或流出边界：其性质由未知迹$u(0,t)$的符号动态决定。这导致相关特征函数具有不同的解析性质，从而在$u(0,t)\le0$和$u(0,t)\ge0$两种情形下产生不同的谱公式。利用统一变换方法（又称Fokas方法），我们将解表示为矩阵Riemann-Hilbert问题。我们构造了相关的谱函数，推导了全局关系，并展示了如何从相应的Riemann-Hilbert问题重构解。在$u(0,t)\le0$的情形下，解仅由初始数据决定（假设当$x\to \infty$时有适当的衰减），而在$u(0,t)\ge0$的情形下，构造还需要相容的边界数据。
英文摘要
We study the initial-boundary value problem for the Short Wave equation on the half-line $x\ge 0$. A distinctive feature of this problem is that the boundary $x=0$ cannot be characterized \emph{a priori} as an inflow or outflow boundary: its character is determined dynamically by the sign of the unknown trace $u(0,t)$. This leads to different analyticity properties of the associated eigenfunctions and, consequently, to different spectral formulations in the regimes $u(0,t)\le0$ and $u(0,t)\ge0$. Using the Unified Transform Method (aka the Fokas method), we formulate the solution in terms of matrix Riemann--Hilbert problems. We construct the associated spectral functions, derive the global relations, and show how the solution is reconstructed from the corresponding Riemann--Hilbert problem. In the case $u(0,t)\le0$, the solution is determined by the initial data alone (assuming an appropriate decay as $x\to \infty$), whereas for $u(0,t)\ge0$, compatible boundary data are also required for the construction.
发表机构
- University of Vienna（维也纳大学）
- B. Verkin Institute for Low Temperature Physics and Engineering（B.维尔金低温物理与工程研究所）
- V.N.Karazin Kharkiv National University（V.N.卡拉津哈尔科夫国立大学）
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