In this paper, we consider the 1 + 1-dimensional vector-valued principal chiral field model (PCF) obtained as a simplification of the vacuum Einstein field equations under the Belinski–Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step toward the study of its nonlinear stability.

在本文中，我们考虑 1 + 1 维向量值主手性场模型 (PCF)，该模型是 Belinski-Zakharov 对称性下真空爱因斯坦场方程的简化。PCF 是一个可积模型，但对其演变的严格描述还远未完成。在这里，我们提供了在适当选择的能量空间中局部解的存在性，以及在一定非简并条件下的小全局平滑解的存在性。我们还构建了维里泛函，它提供了光锥内平滑全局解的衰减的清晰描述。最后，介绍了 PCF 孤子的一些应用，这是研究其非线性稳定性的第一步。