Group theory and nonlinear phenomena in physics

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1.Continuous Symmetries of Discrete Equations. The objective of this research program is to turn Lie group theory into an effective tool for solving difference equations, as it already is for differential ones. The program has two aspects. In the first one, the difference equations and lattices are given a priori, i.e. come from the description of a discrete physical system and the aim is to find their symmetry group and use it to solve the difference system analytically. The second aspect concerns continuous phenomena described by differential equations with a nontrivial symmetry group. Here the aim is to discretize the equation while preserving its symmetries and then to use the obtained difference system to obtain improved numerical solutions of the original equation. During the coming five years my students and I will concentrate on this program for partial differential equations and multidimensional difference ones. In particular we will study discrete relativistic and nonrelativistic quantum systems on lattices. For linear systems we will use umbral calculus to impose Lorentz or Galilei invariance, for nonlinear one we will construct symmetry adapted (and transforming) lattices. An interesting byproduct of this research is the use of umbral calculus to produce a class of orthogonal polynomials particularly well suited to describe flattened optical beams in laser physics. 2. Superintegrable Systems in Classical and Quantum Mechanics. These are systems with more integrals of motion than degrees of freedom (like the Kepler-Coulomb system). All bounded trajectories in such systems in classical mechanics are periodic. In quantum mechanics they have degenerate energy levels and are exactly solvable. We have recently discovered an infinite family of such systems in a plane and will investigate its properties. I mention that the Hohmann Transfer, a fundamental procedure for the positioning of satellites and orbital maneuvering of interplanetary spacecraft is based on the superintegrability of the Kepler system.