AI 中文总结
研究固定克尔/史瓦西背景下复标量场的稳态非线性云，发现满足同步条件的有限振幅Q云，四次域可延伸至Ω_H=0，还计算了轴子启发的余弦势的存在域。
AI 中文摘要
我们在固定克尔（Kerr）和史瓦西（Schwarzschild）背景下，研究具有无界标量势的复标量场的稳态非线性云。在克尔背景上，我们发现满足同步条件ω=mΩ_H的有限振幅Q云，它们在(Ω_H,M_K)平面上占据二维区域，且当标量场振幅趋于零时退化为对应的线性云。四次域可延伸至Ω_H=0，此时同步频率和诺特（Noether）荷均消失，解变为固定史瓦西背景上的稳态非线性标量云。我们还计算了受轴子启发的归一化余弦势的存在域。
英文摘要
We study stationary nonlinear clouds of a complex scalar field on fixed Kerr and Schwarzschild backgrounds, with an unbounded scalar potential. On Kerr backgrounds we find finite-amplitude Q-clouds satisfying the synchronisation condition $ω=mΩ_H$. They occupy two-dimensional regions of the ($Ω_H,M_K$) plane and reduce to the corresponding linear clouds as the scalar field amplitude tends to zero. The quartic domain can continue to $Ω_H=0$, where both the synchronised frequency and the Noether charge vanish and the solutions become static nonlinear scalar clouds on a fixed Schwarzschild background. We also compute the existence domain for a normalised, axion-inspired cosine potential.
