The multi-dimensional six-wave interaction system is derived in the context of nonlinear optics. Starting from Maxwell's equations, a reduced system of equations governing the dynamics of the electric and polarization fields are obtained. Using a space-time multi-scale asymptotic expansion, a hierarchy of coupled equations describing the spatio-temporal evolution of the perturbed electric and polarization fields are derived. The leading order equation admits a six-wave ansatz satisfying a triad resonance condition. By removing secular terms at next order, a first order in space and time quadratically nonlinear coupled six-wave interaction system is obtained. This resulting system is tied to its integrable counterpart which was postulated by Ablowitz and Haberman in the 1970s. A reduction to a space-time shifted nonlocal three-wave system is presented. The resulting system is solved using the inverse scattering transform, which employs nonlocal symmetries between the associated eigenfunctions and scattering data; soliton solutions are then obtained. Finally, an infinite set of conservation laws for the six-wave system is derived; one is shown to be connected to its Hamiltonian structure.
From nothing to something II: nonlinear systems via consistent correlated bang
中国古代圣人老子说，一切都来自于\emph{\bf \em "nothing"}。\rm在第一个字母（中国。Phys. Lett. 30（2013）080202），已经通过简单原理（Dao）从“无”获得了无限多个离散可积系统。第二章引入了一个新的概念--相容关联爆炸，得到了一些可积的非线性动力学系统，如连续非线性薛定谔方程（NLS），（势）Korteweg de Vries（KdV）方程，（势）Kadomtsev-Petviashvili（KP）方程和sine-Gordon（sG）方程。这些非线性系统通过适当的“道”、移宇称、电荷共轭、延迟时间反转、移交换、移宇称旋转等方法从无到有。
Special solutions to five autonomous integrable partial difference equations via the third and sixth Painlevé equations and the Garnier system in two variables
Institute of Engineering, Tokyo University of Agriculture and Technology（东京农工大学工学研究院）
Integrable Twelve-Component Nonlinear Dynamical System on a Quasi-One-Dimensional Lattice
Bogolyubov Institute for Theoretical Physics, The National Academy of Sciences of Ukraine（乌克兰国家科学院波格留博夫理论物理研究所）；Subbotin Institute of Geophysics, The National Academy of Sciences of Ukraine（乌克兰国家科学院苏博京地球物理研究所）
Minimal nonintegrable models with three-site interactions
School of Physics, Northwest University（西北大学物理学院）；Institute of Modern Physics, Northwest University（西北大学现代物理研究所）；Shaanxi Key Laboratory for Theoretical Physics Frontiers（陕西理论物理前沿重点实验室）；Peng Huanwu Center for Fundamental Theory（彭桓武基础理论中心）；Fundamental Discipline Research Center for Quantum Science and Technology of Shaanxi Province（陕西省量子科学与技术基础学科研究中心）
Long-time asymptotics of the integrable defocusing Wadati-Konno-Ichikawa equation with a finite-genus algebro-geometric background
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics and Information Science, Zhongyuan University of Technology（中原工学院数学与信息科学学院）
On integrability of Hirota-Kimura type discretizations. Experimental study of the discrete Clebsch system
R. Hirota和K.木村发现可积离散的欧拉和拉格朗日顶部，给予双有理映射。他们的方法是对W。Kahan定理，适用于任何对相变量具有二次依赖性的向量场。根据T.因此，Hirota-Kimura型离散化可以被考虑用于经典力学的许多可积系统。由于一个显着的和不太好理解的机制，这样的离散化似乎继承了所有代数完全可积系统的可积性。我们介绍了一个严格的研究这种离散化的可积性的实验方法。应用这种方法Hirota-Kimura型离散化的Clebsch系统导致发现四个功能独立的运动积分的离散时间系统，这原来是复杂得多的积分的连续时间系统。进一步，我们证明了离散时间Clebsch系统的每个轨道位于六维相空间中的四个二次曲面的交点上。类似的结果保持Hirota-Kimura型离散化的所有通勤流的Clebsch系统，以及$so(4)$欧拉顶。
