论文标题
爱因斯坦圆柱体上的非线性周期波
Non-linear periodic waves on the Einstein cylinder
论文作者
论文摘要
通过研究反对特点时空中的少量幅度的非线性波的研究，尤其是在爱因斯坦方程的时间解决方案中定期的定期存在，我们构建了任意时间 - 周期性解决方案的家族，以构建用于同质的立方波和球体对象的yang yang yang yang-yang yang-bind inbimbinder $ nimbbinbind cybbinbimbbirdiment ungbimmmilbind cybbinbinbimbind cybbinbimbbinbind cybbinbimbbinbbindibn yang imbbind cybbind cybbinbimbbinbbinbbimbbinbimbbinbbirdiment。 \ mathbb {s}^3 $。对于共形立方波方程，我们考虑了球形对称的溶液和综合价值的非球形溶液，并依赖于$ 3 $ -sphere的HOPF纤维。在所有三种情况下，方程式降低到$ 1+1 $ $半线性波方程。我们的证明依赖于Bambusi-paleari的定理，其主要假设是存在种子溶液，这是由与谐振系统相关的非线性操作员的非分类零给定的。对于我们考虑的问题，这种种子解决方案仅由线性化方程的模式解决方案给出。只要可以计算系统的傅立叶系数，则非分类条件就等于求解无限的尺寸线性系统。由于所有三种研究的特征函数均由Jacobi多项式给出，因此我们使用线性化和连接公式以及Jacobi多项式的积分转换得出了不同的傅立叶和谐振系统。在Yang-Mills案例中，Bambusi-Paleari定理的原始版本不适用，因为最小程度的非线性是非谐音的。然后，由于最小程度非线性的反应术语，通过下一个非线性项的下一个非线性项提供了共振项，在这种情况下我们证明了类似的定理。
Motivated by the study of small amplitudes non-linear waves in the Anti-de-Sitter spacetime and in particular the conjectured existence of periodic in time solutions to the Einstein equations, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically-symmetric Yang-Mills equations on the Einstein cylinder $\mathbb{R}\times \mathbb{S}^3$. For the conformal cubic wave equation, we consider both spherically-symmetric solutions and complexed-valued aspherical solutions with an ansatz relying on the Hopf fibration of the $3$-sphere. In all three cases, the equations reduce to $1+1$ semi-linear wave equations. Our proof relies on a theorem of Bambusi-Paleari for which the main assumption is the existence of a seed solution, given by a non-degenerate zero of a non-linear operator associated with the resonant system. For the problems that we consider, such seed solutions are simply given by the mode solutions of the linearized equations. Provided that the Fourier coefficients of the systems can be computed, the non-degeneracy conditions then amount to solving infinite dimensional linear systems. Since the eigenfunctions for all three cases studied are given by Jacobi polynomials, we derive the different Fourier and resonant systems using linearization and connection formulas as well as integral transformation of Jacobi polynomials. In the Yang-Mills case, the original version of the theorem of Bambusi-Paleari is not applicable because the non-linearity of smallest degree is nonresonant. The resonant terms are then provided by the next order non-linear terms with an extra correction due to backreaction terms of the smallest degree non-linearity and we prove an analogous theorem in this setting.
