We review a recent statistical equilibrium model of self-organization in a generic class of focusing, nonintegrable nonlinear Schrodinger (NLS) equations. Such equations provide natural prototypes for nonlinear dispersive wave turbulence. The primary result is that the statistically preferred state for such a system is a macroscopic solitary wave coupled with fine-scale turbulent fluctuations. The coherent solitary wave is a minimizer the Hamiltonian for a fixed particle number (or L~ 2 norm squared). The predictions of the statistical model are compared with direct numerical simulations of the NLS equation, and it is demonstrated that the model describes the long-time average behavior of solutions remarkably well. In particular, the statistical theory accurately captures both the coherent structure and the spectrum of the solution of the NLS system in the long-time state. Finally, we discuss the dynamics for continuum systems.
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机译：我们在一般类聚焦，不可抗性非线性Schrodinger（NLS）方程中审查了最近的自组织统计均衡模型。这样的等式为非线性色散波湍流提供自然原型。主要结果是这种系统的统计上优选的状态是耦合具有微级湍流波动的宏观孤立波。相干孤立的波是固定粒子数（或L〜2 NUM Squared）的最小化器。将统计模型的预测与NLS方程的直接数值模拟进行比较，并且证明该模型描述了解决方案的长时间平均行为。特别地，统计理论在长时间状态下精确地捕获NLS系统的溶液的相干结构和频谱。最后，我们讨论了连续体系的动态。
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