Jordan R, Josserand C
Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Road, Worcester, Massachusetts 01609-2280, USA.
Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics. 2000 Feb;61(2):1527-39. doi: 10.1103/physreve.61.1527.
We present a statistical equilibrium model of self-organization in a class of focusing, nonintegrable nonlinear Schrodinger (NLS) equations. The theory predicts that the asymptotic-time behavior of the NLS system is characterized by the formation and persistence of a large-scale coherent solitary wave, which minimizes the Hamiltonian given the conserved particle number (L2-norm squared), coupled with small-scale random fluctuations, or radiation. The fluctuations account for the difference between the conserved value of the Hamiltonian and the Hamiltonian of the coherent state. The predictions of the statistical theory are tested against the results of direct numerical simulations of NLS, and excellent qualitative and quantitative agreement is demonstrated. In addition, a careful inspection of the numerical simulations reveals interesting features of the transitory dynamics leading up to the long-time statistical equilibrium state starting from a given initial condition. As time increases, the system investigates smaller and smaller scales, and it appears that at a given intermediate time after the coalescense of the soliton structures has ended, the system is nearly in statistical equilibrium over the modes that it has investigated up to that time.
我们提出了一类聚焦的、不可积非线性薛定谔（NLS）方程中的自组织统计平衡模型。该理论预测，NLS系统的渐近时间行为的特征是形成并持续存在一个大规模的相干孤立波，在给定守恒粒子数（L2范数平方）的情况下，该孤立波使哈密顿量最小化，同时伴有小规模的随机涨落或辐射。这些涨落解释了哈密顿量的守恒值与相干态哈密顿量之间的差异。将统计理论的预测与NLS直接数值模拟的结果进行了对比，结果显示出了极佳的定性和定量一致性。此外，对数值模拟的仔细检查揭示了从给定初始条件开始直至达到长期统计平衡态的瞬态动力学的有趣特征。随着时间的增加，系统研究的尺度越来越小，并且似乎在孤子结构合并结束后的某个给定中间时刻，系统在其截至该时刻所研究的模式上几乎处于统计平衡状态。
