Group theory and nonlinear phenomena in physics

我计划集中在两个相关的领域，即经典和量子理论中的超可积性和离散方程的连续对称性。它们与两者预期的影响之间的联系是，我打算在连续和离散的时空中发现和研究新的超可积和精确可解的经典和量子系统。在离散的情况下，这样的系统是由差分方程组而不是微分方程组控制的。基本的工具是李代数理论及其推广。*1.超积分。*超可积系统是哈密顿系统，它允许的运动积分比它们的自由度多。最著名的，也是唯一旋转不变的是谐振子和开普勒-库仑系统。出于几个原因，它们具有物理上的兴趣。它们是完全可以解决的。运动的积分形成了有趣的非阿贝尔代数。在经典力学中，极大超可积系统的所有有界轨迹都是闭合的，并且运动是周期的。在量子力学中，它们表现出能级的“偶然”简并。直到最近，超可积系统还被认为是极其罕见的。最近，我们证明了极大超可积系统的无限族的存在，其运动积分是动量的任意阶多项式。具有非交换积分代数的超可积系统是无限维孤子系统的有限维模拟。*在接下来的5年里，我计划系统地研究具有运动积分的超可积标量系统，它们是动量的N阶多项式，并研究它们与量子力学中的超对称性和Painleve超越理论的联系。我未来的工作还将包括具有磁场的系统和具有非零自旋的粒子。*2.离散方程的合成。*这是一个通用而雄心勃勃的程序，其目标是将李群理论转变为一种有效的工具，用于求解描述离散的线性和非线性物理现象的方程。该计划有两个方面。一种是解析的，即利用李理论得到这些离散方程的精确解析解。另一种是与几何积分理论相交叉的数值方法。这里的思想是使用对称适应的格子来离散微分方程，同时保持它们的整个李点对称群，或者至少整个对称群的一个大的物理子群。我们证明了对于常微分方程组，保持Lie点对称性改善了数值计算，特别是接近解的奇异性。在接下来的5年里，我计划从这个角度集中学习偏微分方程式。程序的一部分是发展关于晶格的量子理论，同时保持洛伦兹、伽利莱或共形不变性。**

I plan to concentrate on two related areas namely Superintegrability in Classical and Quantum Theories and Continuous Symmetries of Discrete Equations.The connection between them and the expected impact of both is that I intend to find and study new superintegrable and exactly solvable classical and quantum systems in both continuous and discrete space-time. In the discrete case such systems are are governed by difference equations, rather than differential ones. The essential tool is Lie algebra theory and its generalizations.***1. SUPERINTEGRABILITY.***Superintegrable systems are Hamiltonian systems that allow more integrals of motion than they have degrees of freedom. The best known ones and only rotationally invariant ones are the harmonic oscillator and the Kepler-Coulomb system. They are of physical interest for several reasons. They are exactly solvable . The integrals of motion form interesting non-Abelian algebras. In classical mechanics all bounded trajectories in maximally superintegrable systems are closed and the motion is periodic. In quantum mechanics they exhibit "accidental" degeneracy of energy levels. Until recently superintegrable systems were considered to be extremely rare. Recently we have shown that infinite families of maximally superintegrable systems exist with integrals of motion that are polynomials of arbitrary order in the momenta. Superintegrable systems with their non-Abelian algebras of integrals are finite-dimensional analogs of infinite dimensional soliton systems.***In the next 5 years I plan to systematically study superintegrable scalar systems with integrals of motion that are polynomials of order N in the momenta and to investigate their connection with supersymmetry in quantum mechanics and with the theory of Painleve transcendents. My future work will also include systems with magnetic fields and particles with nonzero spin.*** ***2.SYMMETRIES OF DISCRETE EQUATIONS.***This is a general and ambitious program the aim of which is to turn Lie group theory into an efficient tool for solving equations describing discrete linear and non-linear physical phenomena. The program has two aspects. One is analytic, namely to use Lie theory to obtain exact analytic solutions of these discrete equations. The other is numeric and intersects with the theory of geometric integration. Here the idea is to use symmetry adapted lattices to discretize differential equations while preserving their entire Lie point symmetry groups, or at least a large physical subgroup of the entire symmetry group. We have shown that for ordinary differential equations preserving Lie point symmetries improves the numerics, specially close to singularities of solutions. In the next 5 years I plan to concentrate on partial differential equations from this point of view. Part of the program is to develop quantum theory on lattices while preserving Lorentz, Galilei, or conformal invariance. **