第三类自相似性在极端相对论爆炸波中的发现
Self-similarity of the third type in ultra relativistic blastwaves
AI总结：
针对极端相对论爆炸波在幂律密度剖面中传播的问题，发现并求解了一种由非自相似流主导的新型自相似解。
摘要：
在平面平行、极端相对论爆炸波于形如 $ρ\propto z^{-k}$ 的幂律密度剖面中传播的问题中，发现了一种新型自相似性。对于 $k<7/4$，可通过量纲分析求得第一类自相似解。对于 $k>2$ 的更陡密度梯度，通过消除方程中的奇点可获得第二类解。然而，对于中间幂律指数 $7/4<k<2$，流场不遵循任何已知类型的自相似性。相反，这些解属于一个新类别，其自相似动力学由流场中非自相似的部分主导。我们求得了极端相对论流体方程的精确解，发现该非自相似流可描述为向真空的相对论膨胀，由（1）包含大部分能量的加速活塞和（2）与爆炸波内部重合并终止于激波的快速物质前沿组成。活塞自身的动力学是自相似且普适的，不依赖于外部介质。该非自相似流的精确解被用于求解新类别解中的激波问题。
英文摘要：
A new type of self-similarity is found in the problem of a plane-parallel, ultra-relativistic blast wave, propagating in a powerlaw density profile of the form $ρ\propto z^{-k}$. Self-similar solutions of the first kind can be found for $k<7/4$ using dimensional considerations. For steeper density gradients with $k>2$, second type solutions are obtained by eliminating a singularity from the equations. However, for intermediate powerlaw indices $7/4<k<2$ the flow does not obey any of the known types of self-similarity. Instead, the solutions belong to a new class in which the self-similar dynamics are dictated by the non self-similar part of the flow. We obtain an exact solution to the ultra-relativistic fluid equations and find that the non self-similar flow is described by relativistic expansion into vacuum, composed of (1) an accelerating piston that contains most of the energy and (2) a leading edge of fast material that coincides with the interiors of the blastwave and terminates at the shock. The dynamics of the piston itself are self-similar and universal, and do not depend on the external medium. The exact solution of the non self-similar flow is used to solve for the shock in the new class of solutions.
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