标题： Klein-Gordon方程中平面波的局部化摄动轨道稳定性 显示英文标题

标题： Orbital Stability of Plane Waves in the Klein-Gordon Equation against Localized Perturbations

摘要： 我们研究了复 Klein-Gordon 方程中受局域扰动影响的空间周期性平面波的稳定性及其长期行为。 此类扰动使波既非局域化也非周期化，将其稳定性分析置于 Grillakis、Shatah 和 Strauss 所发展的经典哈密顿系统轨道稳定性理论范围之外。 受到 Z hidkov 在非线性 Schrödinger 方程中关于时间周期、空间均匀态稳定性工作的启发，我们发展了一种替代方法，该方法基于振幅-相位分解，并利用与扰动方程匹配的守恒量。 我们建立了平面波的一种局部一致空间轨道稳定性结果，能够涵盖$L^2$-局域化扰动以及非局域化相位调制。 在某些情况下，我们的方法甚至允许无界调制。 从某种意义上说，我们的结果是尖锐的，因为它在谱稳定性边界范围内成立。 显示英文摘要

摘要： We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its stability analysis outside the scope of the classical orbital stability theory for Hamiltonian systems developed by Grillakis, Shatah, and Strauss. Inspired by Zhidkov's work on the stability of time-periodic, spatially homogeneous states in the nonlinear Schr\"odinger equation, we develop an alternative method that relies on an amplitude-phase decomposition and leverages conserved quantities tailored to the perturbation equation. We establish an orbital stability result of plane waves that is locally uniform in space, accommodating $L^2$-localized perturbations as well as nonlocalized phase modulations. In certain regimes, our method even allows for unbounded modulations. Our result is sharp in the sense that it holds up to the spectral stability boundary.