Lax Pairs, Riemann-Hilbert Transforms and Affine Algebras for Hidden Symmetries in Certain Nonlinear Field Theories

Abstract

Two recently proposed approaches, the H-transformation method and the Riemann-Hilbert transform method, to the group structure of infinite-parameter hidden symmetries in certain nonlinear field theories are briefly reviewed, and the relationship between them and their respective advantages are discussed. Once the Lax pair or linearization system in question is known, these methods provide simple and systematic procedures for both finding explicit expressions of hidden symmetry transformations and deriving the infinitely many commutation relations among them. The hidden symmetry algebras (usually of the affine type) for, e.g., 2-D chiral models, 4-D self-dual Yang-Mills equations and 2-D reduced gravity have been obtained or extended by applying these methods.