爱因斯坦约束共形表述的数值分岔分析
Numerical Bifurcation Analysis of Conformal Formulations of the Einstein Constraints
AI总结：
本文利用分岔理论和数值同伦方法，系统研究了爱因斯坦约束方程共形表述中出现的二次折叠分岔现象，并通过AUTO软件包验证了折叠的存在。
摘要：
爱因斯坦约束方程已经研究了五十多年。20世纪70年代引入的共形方法作为爱因斯坦方程初始数据的参数化，促使人们对约束的完整解理论产生更大兴趣，其中常平均曲率（CMC）空间切片和闭流形的理论在1995年完全发展。第一个一般的非CMC存在性结果由Holst等人于2008年建立，2009年扩展到粗糙数据，并由Maxwell于2009年扩展到真空时空。非CMC理论仍然大部分未解决；此外，Maxwell最近关于特定对称模型的工作揭示了共形方法在非CMC设置中作为参数化的基本非唯一性问题。与这些数学发展并行，计算物理学家在爱因斯坦约束的扩展共形薄三明治公式的数值解中发现了令人惊讶的行为。特别是，数值证据表明存在具有二次折叠的多重解，最近对简化模型的分析支持了这一结论。在本文中，我们使用分岔理论和数值同伦方法的现代技术，系统地研究了这种明显的分岔现象。我们首先回顾时间对称情况下哈密顿约束中存在分岔的证据。我们简要介绍了分析分岔现象的数学框架，然后发展了构建数值同伦（或路径跟踪）方法以分析分岔现象的主要思想。接着，我们将延续性软件包AUTO应用于此问题，并通过基于同伦的数值方法验证了折叠的存在。
英文摘要：
The Einstein constraint equations have been the subject of study for more than fifty years. The introduction of the conformal method in the 1970's as a parameterization of initial data for the Einstein equations led to increased interest in the development of a complete solution theory for the constraints, with the theory for constant mean curvature (CMC) spatial slices and closed manifolds completely developed by 1995. The first general non-CMC existence result was establish by Holst et al. in 2008, with extensions to rough data by Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC theory remains mostly open; moreover, recent work of Maxwell on specific symmetry models sheds light on fundamental non-uniqueness problems with the conformal method as a parameterization in non-CMC settings. In parallel with these mathematical developments, computational physicists have uncovered surprising behavior in numerical solutions to the extended conformal thin sandwich formulation of the Einstein constraints. In particular, numerical evidence suggests the existence of multiple solutions with a quadratic fold, and a recent analysis of a simplified model supports this conclusion. In this article, we examine this apparent bifurcation phenomena in a methodical way, using modern techniques in bifurcation theory and in numerical homotopy methods. We first review the evidence for the presence of bifurcation in the Hamiltonian constraint in the time-symmetric case. We give a brief introduction to the mathematical framework for analyzing bifurcation phenomena, and then develop the main ideas behind the construction of numerical homotopy, or path-following, methods in the analysis of bifurcation phenomena. We then apply the continuation software package AUTO to this problem, and verify the presence of the fold with homotopy-based numerical methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
