II型强爆炸的离散自相似性
Discrete Self-Similarity in Type-II Strong Explosions
摘要：
我们提出了新的解决方案，在非幂律密度分布的强爆炸问题。Waxman和Shvarts发现的未扰动自相似解描述了强牛顿激波传播到密度分布为$r^{-ω}$的冷气体中，其中$ω>3$（II型解）。我们考虑的扰动是球对称的和对数周期的半径。虽然未扰动解是连续自相似的，但密度扰动的对数周期性导致扰动的离散自相似性，即解在离散时间间隔上重复自身。我们讨论这些解决方案，并验证他们对数值积分的时间相关的流体动力学方程。最后，我们表明，这种方法可以推广到处理任何小的，球对称密度扰动采用傅立叶分解。
英文摘要：
We present new solutions to the strong explosion problem in a non-power law density profile. The unperturbed self-similar solutions discovered by Waxman & Shvarts describe strong Newtonian shocks propagating into a cold gas with a density profile falling off as $r^{-ω}$, where $ω>3$ (Type-II solutions). The perturbations we consider are spherically symmetric and log-periodic with respect to the radius. While the unperturbed solutions are continuously self-similar, the log-periodicity of the density perturbations leads to a discrete self-similarity of the perturbations, i.e. the solution repeats itself up to a scaling at discrete time intervals. We discuss these solutions and verify them against numerical integrations of the time dependent hydrodynamic equations. Finally we show that this method can be generalized to treat any small, spherically symmetric density perturbation by employing Fourier decomposition.
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