摇摆扭结的后续故事
The Continuing Story of the Wobbling Kink
摘要：
摆动扭结是具有激发内模的$ϕ^4$模型的孤子。我们概述了这种粒子样的解决方案，考虑到几个空间和时间尺度的共存的渐近建设。在大距离的渐近展开的故障是防止通过引入远程变量“untied”从短程振荡。我们制定了一个定量的理论，由于二次谐波辐射的扭结的摆动的衰落，解释摆动模式的寿命，并讨论如何补偿辐射损失。通过在各种谐振频率下对扭结进行空间均匀驱动（外部或参数驱动）来实现补偿。对于给定的驱动强度值，扭结振荡的最大振幅由{ \it参数\/}泵浦维持-在其自然摆动频率。这种类型的强迫也产生了最宽的阿诺德舌在“驱动强度与驱动频率”参数平面。对于具有相同频率的{ \it外部\/}驱动器，它带来了一个有趣的齿条和齿轮机构，将外部振荡的能量转换为扭结的平移运动。
英文摘要：
The wobbling kink is the soliton of the $ϕ^4$ model with an excited internal mode. We outline an asymptotic construction of this particle-like solution that takes into account the coexistence of several space and time scales. The breakdown of the asymptotic expansion at large distances is prevented by introducing the long-range variables "untied" from the short-range oscillations. We formulate a quantitative theory for the fading of the kink's wobbling due to the second-harmonic radiation, explain the wobbling mode's longevity and discuss ways to compensate the radiation losses. The compensation is achieved by the spatially uniform driving of the kink, external or parametric, at a variety of resonant frequencies. For the given value of the driving strength, the largest amplitude of the kink's oscillations is sustained by the {\it parametric\/} pumping --- at its natural wobbling frequency. This type of forcing also produces the widest Arnold tongue in the "driving strength versus driving frequency" parameter plane. As for the {\it external\/} driver with the same frequency, it brings about an interesting rack and pinion mechanism that converts the energy of external oscillation to the translational motion of the kink.
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