This is an updated, revised and extended translation from the Russian original work Symmetry Analysis and Exact Solutians of Equations of Nonlinear Mathematical Physics, Nauk:a, Kiev, © 1989 AU Rights Reserved

We generalize classical Yang-Mills theory by extending nonlinear constitutive equations for Maxwell fields to non-Abelian gauge groups. Such theories may or may not be Lagrangian. We obtain conditions on the constitutive equations specifying the Lagrangian case, of which recently discussed non-Abelian BornInfeld theories are particular examples. Some models in our class possess nontrivial Galilean (c → ∞) limits; we determine when such limits exist and obtain them explicitly.

Recent experimental results on slow light heighten interest in nonlinear Maxwell theories. We obtain Galilei covariant equations for electromagnetism by allowing special nonlinearities in the constitutive equations only, keeping Maxwell's equations unchanged. Combining these with linear or nonlinear Schrödinger equations, e.g. as proposed by Doebner and Goldin, yields a Galilean quantum electrodynamics.MSC: 78A25, 78A97, 81B05, 81G10.

We introduce new classes of Schrödinger equations with time-dependent potentials which are transformable to the free particle equation through non-local transformations. These non-local transformations arise when considering the potential systems of the Schrödinger equation. Explicit formulae are given for the potentials and the corresponding solutions related to the solutions of the free particle equation.

We extend and solve the classical Kolmogorov problem of finding general classes of Kolmogorov equations that can be transformed to the backward heat equation. These new classes include Kolmogorov equations with time-independent and time-dependent coefficients. Our main idea is to include nonlocal transformations. We describe a step-by-step algorithm for determining such transformations. We also show how all previously known results arise as particular cases in this wider framework.