同调双曲坐标下的半线性波动方程与尾迹衰减
Semilinear wave equations in homothetic hyperboloidal coordinates and tail decay
浏览论文内容
中文总结 AI 辅助
本文采用适配尾迹标度结构的同调双曲坐标，解决半线性波动方程晚期波尾数值演化的过渡难题，经3+1维伪谱模拟重现Rinne猜想的通用衰减速率，还发现类光无穷远处主导尾迹系数非通用抵消的数值证据。
中文摘要 AI 辅助
晚期波尾在未来类光无穷远和有限半径类时世界线上以不同速率衰减。紧致化数值演化必须同时表征类光无穷远的较慢衰减与内部更快的衰减，从而在两种衰减 regime 之间产生日益尖锐的过渡。我们针对闵氏时空中的半线性波动方程，采用适配尾迹标度结构的同调双曲坐标解决这一难题。在该坐标下，尾迹趋近于光滑径向剖面，且在每个紧致化半径处具有相同的衰减速率。该公式因此避免了定态双曲演化中出现的径向剖面陡化问题，且其达到晚期的步数仅随推迟时间对数增长。我们通过3+1维伪谱模拟验证了该方法，重现了Rinne猜想的通用衰减速率，还提供了数值证据，与类光无穷远处主导尾迹系数的非通用余维1抵消一致，该抵消导致更快的衰减速率。
英文摘要
Late-time wave tails decay at different rates along future null infinity and along timelike worldlines at finite radius. A compactified numerical evolution must represent both the slower decay at null infinity and the faster interior decay, producing an increasingly sharp transition between the two regimes. We address this difficulty for semilinear wave equations in Minkowski spacetime using homothetic hyperboloidal coordinates adapted to the scaling structure of the tail. In these coordinates, the tail approaches a smooth radial profile with the same decay rate at every compactified radius. The formulation therefore avoids the steepening of the radial profile seen in stationary hyperboloidal evolutions, and it reaches late times in a number of steps that grows only logarithmically with retarded time. We demonstrate this approach using pseudospectral simulations in 3+1 dimensions and reproduce the generic decay rates conjectured by Rinne. We also provide numerical evidence consistent with a nongeneric codimension-one cancellation of the leading tail coefficient at null infinity, resulting in a faster decay rate.
