Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom.
Proc Natl Acad Sci U S A. 2013 Feb 26;110(9):3237-41. doi: 10.1073/pnas.1215325110. Epub 2013 Feb 11.
In this work, we examine the important theoretical question of whether dispersion relations can arise from purely nonlinear interactions among waves that possess no linear dispersive characteristics. Using two prototypical examples of nondispersive waves, we demonstrate how nonlinear interactions can indeed give rise to effective dispersive-wave-like characteristics in thermal equilibrium. Physically, these example systems correspond to the strong nonlinear coupling limit in the theory of wave turbulence. We derive the form of the corresponding dispersion relation, which describes the effective dispersive structures, using the generalized Langevin equations obtained in the Zwanzig-Mori projection framework. We confirm the validity of this effective dispersion relation in our numerical study using the wavenumber-frequency spectral analysis. Our work may provide insight into an important connection between highly nonlinear turbulent wave systems, possibly with no discernible dispersive properties, and the dispersive nature of the corresponding renormalized waves.
在这项工作中，我们研究了一个重要的理论问题，即色散关系是否可以仅由具有无线性弥散特征的波之间的纯非线性相互作用产生。我们使用两个非弥散波的典型例子，展示了非线性相互作用如何在热平衡中确实会产生类似有效弥散波的特征。从物理上讲，这些示例系统对应于波湍流理论中的强非线性耦合极限。我们使用 Zwanzig-Mori 投影框架中得到的广义 Langevin 方程，推导出相应的色散关系的形式，该关系描述了有效弥散结构。我们通过波数-频率谱分析的数值研究验证了该有效色散关系的有效性。我们的工作可能为高度非线性的湍流波系统之间的一个重要联系提供了启示，这些系统可能没有明显的弥散特性，以及相应的正则化波的弥散性质。
