Parametric Lie Group Actions on Global Generalised Solutions of Nonlinear PDEs: Including a Solution to Hilbert’s Fifth Problem(452 Mathematics and Its Applications)

This book presents global actions of arbitrary Lie groups on large classes of generalised functions by using a novel parametric approach. This new method extends and completes earlier results of the author and collaborators, in which global Lie group actions on generalised functions were only defined in the case of projectable or fibre-preserving Lie group actions. The parametric method opens the possibility of dealing with vastly larger classes of Lie semigroup actions which still transform solutions into solutions. These Lie semigroups can contain arbitrary noninvertible smooth mappings. Thus, they cannot be subsemigroups of Lie groups. Audience: This volume is addressed to graduate students and researchers involved in solving linear and nonlinear partial differential equations, and in particular, in dealing with the Lie group symmetries of their classical or generalised solutions.

Table of Contents:
1. Introduction.- 2. Actions on functions, difficulties.- 3. Parametric representation of functions.- 4. Action on parametric representations.- 5. Parametric functions as solutions.- 6. Rarefaction waves and Riemann solvers of the nonlinear shock wave equation.- 7. Arbitrary nonlinear Lie group actions on generalised functions.- 8. Nonprojectable Lie group symmetries of rarefaction waves and Riemann solvers.- 9. General parametric approach to Lie symmetries.- 10. Projectable Lie group actions and Hilbert’s fifth problem.- 11. Nonprojectable Lie group actions and an answer to Hilbert’s fifth problem.- 12. Singularities and the nowhere dense algebras of generalised functions.- 13. Lie semigroup actions and semisymmetries.

Review :
E.E. Rosinger Parametric Lie Group Actions on Global Generalised Solutions of Nonlinear PDEs Including a Solution to Hilbert’s Fifth Problem "This book presents a novel approach to Lie group actions on ordinary and generalized functions, based on parametric representation. This allows a global definition of arbitrary nonlinear Lie group actions on functions, including generalized functions. The parametric approach also makes possible a global definition for Lie semigroup actions. It is shown that the usual Lie group symmetries of classical solutions of smooth nonlinear PDEs will remain Lie group symmetries of generalized solutions of such equations." —MATHEMATICAL REVIEWS