孤子局域共振激发的一种物理模型

A new physical model for the localized resonant emission of solitons

作为非线性领域的重要内容,孤子自从被发现以来就得到了广泛的关注。虽然目前关于孤子的传播及其相互作用性质等方面的研究已经相当成熟,但是对于孤子的产生问题仍然没有系统的认识。孤子产生问题的深入研究,不仅对于理解自然界及物理系统中的非线性现象十分重要,对孤子现象的工程应用也极为重要。基于这些原因,一些孤子激发方法的研究不断被提出。在非线性阵列带隙中的能量超透射现象激发孤子理论的基础上,局域共振孤子激发的方法被提出。该方法通过在半无限长β-FPU链中加入缺陷来引入局域共振机制,不仅有效降低了孤子激发的临界驱动振幅,还实现了孤子激发的可控操作。然而,虽然理论分析与数值模拟都证明了这一方法的可行性,但关于该方法的相关实验验证还是空白。为了进一步将该方法推向实际实验,提出了一种简易可行的实验方案,并做了深入的理论分析与大量的数值模拟。该方案给出一种物理模型,该模型以线性弹簧为基本元件,利用结构的几何非线性来构建β-FPU链,并以耦合摆阵列来实现。通过大量数值模拟证明了该方法的可行,同时研究了孤子释放周期与驱动频率、振幅的关系,结果与前述理论一致。此外,还通过数值模拟研究了孤子的激发周期与缺陷处阻尼大小的关系,以及单元质量不均匀性对孤子激发与传播的影响,为进一步实验提供依据。

As one of the major field of nonlinear science, solitons have gotten lots of attention academically since they are discovered. However, although theories on their properties have been sufficient and mature, study on creation of solitons is still incomplete. Thorough research on creation of solitons is not only fundamental to understanding solitonic phenomena occurring in nature and physical systems, but also important to engineering applications of solitons. For those reasons, plenty of researches on approaches to solitons emission have been proposed continually. For an improvement upon the methods based on the energy supra-transmission in band gaps of nordinear lattices, the so-caUed resonant emission method is proposed. According to this method, mechanism of local resonation is applied by introducing impurity in a semi-infinite β-FPU chain. By doing this, the new method not only reduces drive amplitude that is needed to induce soliton emission dramatically, but also make solitons emission manageable. Although theoretical analysis and numerical calculation have proved the method＇s feasibility, experimental verification is not proposed. To push the theory further forward to experiment, we proved a feasible experimental scheme with simple physical model, and operated thorough theoretical analysis and numerical investigation. The new model is consist of simple linear springs, and realizes the β-FPU chain by designing the geometrical nonlinearity of the structure intelligently. The numerical investigations in our work have not only confirmed the feasibility of the scheme, but also have studied the special relationship between the duration of soliton emission and the driving amplitude and frequency. The results are consistent with the conclusion from the fore-mentioned paper. In addition, to provide reference for further experiments, we studied the special relationship between the duration of soliton emission and the damping on impurity union, as well as the influence of unions＇ inhomogeneity on the emission and propagation of solitons.