DBI模型中的焦散形成：平面畴壁上的波传播
Caustic formation in DBI models: Wave propagation on planar domain walls
AI总结：
研究DBI模型中平面畴壁上的波传播，证明在2D平坦时空双曲情形下无奇点，但在扩展宇宙等更一般情形下特征曲线不再平行，但DBI仍无焦散，仅因双曲性丧失才形成尖点。
摘要：
我们研究了由标量 Dirac-Born-Infeld 模型（DBI）有效描述的薄平面畴壁上的一般波的传播。我们特别关注焦散形成的可能性——这一过程可能导致畴壁发射大量粒子。我们证明，在二维平坦时空的双曲情形下，如果从光滑初始条件出发，DBI 中不会出现奇点。从技术上讲，这是因为相关偏微分方程的同一族特征线在所有时刻保持平行，尽管它们通常不是直线。关键在于，在二维平坦时空的简化 DBI 设置之外，特征曲线不再平行。特别地，这在 $D>2$ 的球面波、膨胀宇宙以及为避免宇宙学畴壁问题而对 DBI 进行最小变形的情况下被证明成立。然而，我们证明 DBI 在所有这些物理相关情形下在双曲情形中仍然无焦散。这强烈表明，平面畴壁上的焦散只能由于双曲性丧失而形成，并且具有尖点轮廓。我们展示了本工作中揭示的二维平坦时空设置之外的 DBI 特征线的非平凡结构如何显著影响尖点形成。
英文摘要：
We investigate propagation of generic waves on thin planar domain walls effectively described by the scalar Dirac-Born-Infeld model (DBI). We pay a particular attention to the possibility of caustic formation - the process, which may lead to intensive particle emission by domain walls. It is demonstrated that no singularities arise in DBI in 2D flat spacetime in the hyperbolic case, if one starts from smooth initial conditions. Technically, this happens because the same family characteristics of the relevant partial differential equation remain parallel at all the times, albeit not being straight lines generically. Crucially, characteristic curves cease to be parallel beyond the simplified setup of DBI in 2D flat spacetime. In particular, this is shown to be the case in $D>2$ for spherical waves, in an expanding Universe, and in the case of a minimal deformation of DBI necessary for avoiding the domain wall problem in cosmology. However, we prove that DBI remains caustic free in the hyperbolic case in all these physically relevant situations. This strongly suggests that caustics can form on planar domain walls only due to the loss of hyperbolicity, and they have a cusp profile. We demonstrate, how the non-trivial structure of DBI characteristics beyond the 2D flat spacetime setup uncovered in this work can significantly affect cusp formation.
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