Non-integrability of measure preserving maps via Lie symmetries

阻尼Hénon-Heiles系统的Painlevé分析、Prelle-Singer方法、对称性与可积性

Painlevé Analysis, Prelle-Singer Approach, Symmetries and Integrability of Damped Hénon-Heiles System

本研究针对修正阻尼Hénon-Heiles系统，通过扩展Painlevé分析确定其在三种参数约束下具备Painlevé性质，用Prelle-Singer方法构造运动积分、推导非点对称，证明其可积性并给出两类精确解。

Symmetry and integrability for stochastic differential equations

We discuss the interrelations between symmetry of an Ito stochastic differential equations (or systems thereof) and its integrability, extending in party results by R. Kozlov [J. Phys. A ${\bf 43}$ (2010) \& ${\bf 44}$ (2011)]. Together with integrability, we also consider the relations between symmetries and reducibility of a system of SDEs to a lower dimensional one. We consider both "deterministic" symmetries and "random" ones, in the sense introduced recently by Gaeta and Spadaro [J.Math. Phys. ${\bf 58}$ (2017)].

非线性波的机器学习：用于计算机辅助发现方程、对称性、守恒律和可积性的数据驱动方法

Machine Learning of Nonlinear Waves: Data-Driven Methods for Computer-Assisted Discovery of Equations, Symmetries, Conservation Laws, and Integrability

本文探讨了机器学习与非线性波研究结合的最新进展，介绍了深度学习、数据驱动方程发现等方法在发现守恒律和可积性中的应用，强调了这些方法对非线性波研究工具的补充作用。

On integrability of the Kontsevich non-abelian ODE system

We consider systems of ODEs with the right hand side being Laurent polynomials in several non-commutative unknowns. In particular, these unknowns could be matrices of arbitrary size. An important example of such a system was proposed by M. Kontsevich. We prove the integrability of the Kontsevich system by finding a Lax pair, corresponding first integrals and commuting flows. We also provide a pre-Hamiltonian operator which maps gradients of integrals for the Kontsevich system to symmetries.

Laplacian Growth 的保可积性正则化

Integrability-preserving regularizations of Laplacian Growth

针对 LG 与 DLA 长时间解标度性质缺乏恰当理论的问题，本文提出构建保可积性正则化 LG 的思路，并阐明其与 DLA 的联系。

Measure-preserving symmetries and reversibilities of ordinary differential systems

We prove that measure-preserving symmetries of an $n$-dimensional differential system preserve its divergence and the divergence derivatives along the solutions. Also, we prove that measure-preserving reversibilities preserve odd-order divergence derivatives along the solutions, and that even-order derivatives are multiplied by $-1$. We apply such results to find all the area-preserving symmetries and reversibilities of planar Lotka-Volterra and Liénard systems.