Group theory and nonlinear phenomena in physics

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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.