Oxtoby O F, Barashenkov I V
CSIR Computational Aerodynamics, Building 12, P.O. Box 395, Pretoria 0001, South Africa.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Aug;80(2 Pt 2):026609. doi: 10.1103/PhysRevE.80.026609. Epub 2009 Aug 31.
The amplitude of oscillations of the freely wobbling kink in the varphi(4) theory decays due to the emission of second-harmonic radiation. We study the compensation of these radiation losses (as well as additional dissipative losses) by the resonant driving of the kink. We consider both direct and parametric driving at a range of resonance frequencies. In each case, we derive the amplitude equations which describe the evolution of the amplitude of the wobbling and the kink's velocity. These equations predict multistability and hysteretic transitions in the wobbling amplitude for each driving frequency--the conclusion verified by numerical simulations of the full partial differential equation. We show that the strongest parametric resonance occurs when the driving frequency equals the natural wobbling frequency and not double that value. For direct driving, the strongest resonance is at half the natural frequency, but there is also a weaker resonance when the driving frequency equals the natural wobbling frequency itself. We show that this resonance is accompanied by the translational motion of the kink.
在φ(4)理论中，自由摆动的纽结振荡幅度会因二次谐波辐射的发射而衰减。我们研究了通过对纽结进行共振驱动来补偿这些辐射损失（以及额外的耗散损失）。我们考虑了在一系列共振频率下的直接驱动和参数驱动。在每种情况下，我们都推导出了描述摆动幅度和纽结速度演化的幅度方程。这些方程预测了每个驱动频率下摆动幅度的多稳定性和滞后转变——这一结论通过完整偏微分方程的数值模拟得到了验证。我们表明，当驱动频率等于自然摆动频率而非其两倍时，会出现最强的参数共振。对于直接驱动，最强共振发生在自然频率的一半处，但当驱动频率等于自然摆动频率本身时也会有较弱的共振。我们表明，这种共振伴随着纽结的平移运动。
