论文标题
按时定期解决爱因斯坦圆柱体上保形的立方波方程的解决方案
On time periodic solutions to the conformal cubic wave equation on the Einstein cylinder
论文作者
论文摘要
我们考虑了爱因斯坦圆柱体上的共形波方程，并具有偏置的立方非线性性。由Rostworowski-Maliborski开发的一种方法，该方法是针对球形对称的爱因斯坦 - 克莱因 - 戈登系统的时间周期性解决方案的，我们将围绕零溶液的扰动作为正式系列扩展研究，并假定扰动从$ 1- $ $ $ $ $ $ $模式中。在这项工作的中心，这是一个严格的证据，说明人们如何选择初始数据以取消谐振系统中的所有世俗项。有趣的是，我们的分析表明，存在时间周期性解决方案的唯一可能选择是扩展中的错误项与主要的$ 1- $模式成正比。最后，我们使用来自普通微分方程的技术，并建立与线性化运算符的第一种模式成正比的时间周期解决方案的存在。
We consider the conformal wave equation on the Einstein cylinder with a defocusing cubic non-linearity. Motivated by a method developed by Rostworowski-Maliborski on the existence of time periodic solutions to the spherically symmetric Einstein-Klein-Gordon system, we study perturbations around the zero solution as a formal series expansion and assume that the perturbations bifurcate from $1-$mode. In the center of this work stands a rigorous proof on how one can choose the initial data to cancel out all secular terms in the resonant system. Interestingly, our analysis reveals that the only possible choice for the existence of time periodic solutions is when the error terms in the expansion are all proportional to the dominant $1-$mode. Finally, we use techniques from ordinary differential equations and establish the existence of time periodic solutions for initial data proportional to the first mode of the linearized operator.
